The Pennsylvania State University The Graduate School Eberly College of Science THE RAPID CONTROL OF INTERACTIONS IN A TWO-COMPONENT FERMI GAS

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1 The Pennsylvania State University The Graduate School Eberly College of Science THE RAPID CONTROL OF INTERACTIONS IN A TWO-COMPONENT FERMI GAS A Dissertation in The Department of Physics by Ronald William Donald Stites c 2012 Ronald William Donald Stites Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2013

2 The dissertation of Ronald William Donald Stites was reviewed and approved by the following: Kenneth M. O Hara Associate Professor of Physics Dissertation Advisor, Chair of Committee Milton W. Cole Distinguished Professor of Physics Kurt Gibble Professor of Physics David S. Weiss Professor of Physics John V. Badding Professor of Chemistry Richard W. Robinett Professor of Physics Associate Department Head, Director of Graduate Studies Signatures are on file in the Graduate School.

3 Abstract In this dissertation, we describe a variety of experiments having application to ultra-cold atomic gases. While the majority of the experimental results focus on the development of a novel laser source for cooling and manipulating a gas of fermionic 6 Li atoms, we also report on a preliminary investigation of rapidly controlling interactions in a two-component Fermi gas. One of the primary tools for our ultra-cold atomic physics experiments is 671 nm laser light nearly resonant with the D 1 and D 2 spectroscopic lines of ultracold fermionic 6 Li atoms. Traditionally, this light is generated using dye lasers or tapered amplifier systems. Here we describe a diode pumped solid state ring laser system utilizing a Nd:YVO 4 gain crystal. Nd:YVO 4 has a 4 F 3/2 4 I 13/2 emission line at 1342 nm. This wavelength is double the 671 nm needed for our experiments. As a part of this investigation, we also measured the Verdet constant of undoped Y 3 Al 5 O 12 in the near infrared for constructing a Faraday rotator used to drive unidirectional operation of our ring laser. As an alternative method to achieve unidirectional, single-frequency operation of the laser, we developed a novel scheme of self-injection locking where a small portion of the output beam is coupled back into the cavity to break the symmetry. This technique is useful for high-power, single-frequency operation of a ring laser because lossy elements needed for frequency selection and unidirectional operation of the laser can be removed from the internal cavity. In addition to our laser experiments, we also drive Raman transitions between different magnetic hyperfine states within 6 Li atoms. For atoms in the two lowest hyperfine states, there exists a broad Feshbach resonance at Gauss whereby the s-wave scattering length diverges, resulting in strong interactions between the two species. By using two phase locked lasers to drive a transition from a strongly interacting state to a weakly interacting state, we can rapidly control the interaction strength of a two component Fermi gas. iii

4 Table of Contents List of Figures List of Tables Acknowledgments vii xvi xvii Chapter 1 Introduction Interactions in an Atomic Gas Probing Strongly Correlated Systems The Rapid Control of Interactions A Self-Injected 1342 nm Ring Laser Outline Chapter 2 Fermi Gases Properties of 6 Li s-wave Scattering Theory Scattering Resonances Interactions in a 1-5 Mixture Chapter 3 The Interaction of Atoms with Light The Two Level Atom The AC Stark Effect Magnetic Dipole Transitions Raman Transitions iv

5 Chapter 4 The Verdet Constant of Y 3 Al 5 O Verdet Constant Theory Y 3 Al 5 O 12 Properties Measurement Setup Cylindrical Magnet Verdet Measurement Conclusion Chapter 5 Constructing a 1342 nm Ring Laser Cavity Layout Reflectors Nd:YVO 4 Crystal Pump Laser Cooling the Crystal Additional Components Cavity Modeling Conclusion Chapter nm Ring Laser Operation Measuring the Nd:YVO 4 Gain The Free Running Laser Modeling Power Output Unidirectionality with a Faraday Rotator Unidirectionality with Self Injection Frequency Tuning Intracavity Tuning External Cavity Tuning Conclusion Chapter 7 Preparing a Two-Component Fermi Gas Ultra-High Vacuum System The Oven Region The Zeeman Slower The Experimental Region Vacuum Pumps Laser System Overview v

6 7.3 Magnetic Field Coils Magneto-Optical Trapping Optical Dipole Traps Atomic Imaging Chapter 8 The Rapid Control of Interactions Initial Preparation Phase Locking of Raman Lasers Preliminary Investigations of a Non-Interacting Gas The Next Steps Chapter 9 Conclusions and Outlook Conclusions Outlook Appendix A Mathematica Code for Calculating Scattering Lengths 137 Bibliography 152 vi

7 List of Figures 2.1 The hyperfine energy level structure for 6 Li showing the 2 2 S ground state and the two 2 2 P excited states The magnetic field dependence of the energy for the F = 1/2 and F = 3/2 ground states of 6 Li. As the applied field becomes stronger, the atoms enter the Paschen-Back regime, whereby the atom becomes electron spin polarized and the three way splitting arises due to difference in the nuclear spin of the atoms A Feshbach resonance occurs when the least bound state of the close channel molecular potential is tuned via a magnetic field to be resonant with the collision energy of the particles in the entrance channel The s-wave scattering length as a function of applied magnetic field for atoms in the two lowest hyperfine ground states of 6 Li [1]. Note the broad Feshbach resonance located at G. By arbitrarily tuning the applied magnetic field around this resonance, we can change the strength of the two-body interaction from being infinitely repulsive to infinitely attractive The result of our coupled channel calculation for s-wave scattering between atoms in states 1 and 2. To simplify the calculation, we model the singlet and triplet molecular potentials as finite square well potentials. Despite this oversimplication, the calculation reproduces the broad Feshbach resonance located at G which can be observed as a divergence in the s-wave scattering length. The location and width of this resonance is in good agreement with a coupled channel calculation which uses accurate singlet and triplet molecular potentials. The good agreement gives us confidence in our coupled channels calculations using a simple model for the potentials vii

8 2.6 Our simplified coupled-channels model for scattering of atoms in states 1 and 2 also predict the presence of a narrow Feshbach resonance. While the position of this resonance is slightly higher in magnetic field than what has been experimentally observed, this discrepancy could easily be explained by the different shape of our simplified potentials The predicted s-wave scattering lengths as a function of magnetic field for a 1-5 mixture. With no Feshbach resonance, the s-wave scattering length increases to a value of approximately -3 Bohr at a magnetic field corresponding to gauss. In this way, interactions of a two-state 1-2 mixture at this field could be quickly suppressed by a simple transfer from state Energy level diagram for a simplified atomic system. A two level atom with ground state g and excited state e interacts with an oscillating electric field having a frequency of ω. This beam is detuned by an amount δ from the transition frequency ω 0 corresponding to the energy difference between the two atomic states The shift of the energy levels of a two level atomic system due to the presence of an oscillating electromagnetic field. This AC Stark effect results in a resonance frequency shift to higher frequencies when the detuning of the beam is negative and lower frequencies when the detuning is positive The energy level diagram for a two photon Raman transition. Laser L1 drive the atom from the ground state to a virtually excited intermediate state i. Laser L2 then drives the atom from state i back to the desired excited state. It is important to note that the population in state i is virtual, and losses due to spontaneous emission are not allowed. By employing this two photon technique, one can drive state changing transitions where they would otherwise be forbidden, such as when Δl = The transmission spectrum of undoped YAG from 200 to 6500 nm. Note the broad transmission extending from the visible well into the mid-infrared [2] viii

9 4.2 Light from an external cavity diode laser (ECDL) passes through a half-wave plate (HWP) and polarizer (P) pair. A double Fresnel rhomb (FR) is used to adjust the polarization of the light incident on a cylindrical magnet (M). The light is then split by a polarizing beam splitter (PBS) before falling on two photodetectors (DET) which record the power of each beam (a) Physical layout and dimensions of the right hollow cylindrical magnet housing the undoped YAG crystal. The crystal, of length L c =18.0mm is located inside the magnet having length L =19.0 mm and an outer diameter 2b =22.2 mm. The inner bore of the magnet has a diameter 2a =6.3 mm. (b) The measured magnetic field of the magnet at various distances from the end facets. The dashed line represents a fit to the measured data for the magnetization M Verdet constant of undoped YAG in the near infrared. Each data point represents the average of one hundred measurements. The inset provides a closer look at the 1300 nm to 1350 nm range. Error bars are indicative of the standard error of the mean. The dashed line is a fit to the data as per equation 4.3, demonstrating the dispersive nature of the Verdet constant Energy level diagram of Nd:YVO 4. The crystal has a 4 I 9/2 4 F 5/2 pump absorption line at 808 nm and two emission lines ( 4 F 3/2 4 I 13/2 and 4 F 3/2 4 I 11/2 ) at 1342 nm and 1064 nm respectively Artistic 3D rendering of the Nd:YVO 4 laser cavity Experimental setup for the laser cavity. A Nd:YVO 4 crystal located inside of a bow-tie ring cavity is double end pumped by two 25 Watt diode arrays. The cavity consists of a fully reflecting mirror (M1), an output coupler (M2), and two dichroic mirrors (M3 and M4) which reflect light at 1342 nm while transmitting light at 808 and 1064 nm. With nothing to break the symmetry of the system, the laser will lase bidirectionally, resulting in the gain being shared by the clockwise and counterclockwise modes Machine drawing for homemade mirror mount for the dichroic mirror M3 in our 1342 nm laser cavity Machine drawing for homemade mirror mount for the dichroic mirror M4 in our 1342 nm laser cavity ix

10 5.6 The crystal structure of an yttrium ortho-vanadate. Yttrium orthovanadate crystalizes as a zircon tetragonal (tetragonal bipyramidal) structure, leading to a natural birefringence along its a and c axes. Shown centered is the yttrium ion surrounded by its vanadium and oxygen neighbors. This figure is adapted from [3] The absorption spectrum of a 0.27% doped Nd:YVO 4 crystal in the region of the 808 nm band, reproduced from [4]. This plot shows a peak absorption coefficient of approximately 9.4 cm 1 at nm Copper mount used to hold the Nd:YVO 4 crystal inside the laser cavity. One end of the mount is connected to a thermo-electric cooler used to extract heat from the crystal (see section 5.5. The other end of the mount holds the crystal in a narrow finger-like region, allowing the crystal to be located in the cavity without obstructing the beam path Copper cover used to sandwich the gain crystal against the copper mount Adapter for connecting the sma fiber from the DUO-FAP laser to the homemade lens mount for the 2:1 pump imaging system Homemade lens mount used to house the imaging optics for focusing the 808 nm pump laser light onto the Nd:YVO 4 gain crystal. Two lenses, of focal lengths 60 and 30 mm are located so as to provide a 2:1 imaging system, focusing light from the 800 μm diameter pump laser fibers to a diameter of 400 μm on the gain crystal itself This home built mount is used to hold the lens mount shown in figure By doing so, it allows the lenses used for imaging the pump laser light onto the gain crystal to be precisely positioned via a translation stage connected to this mount by the adapter plate shown in figure Machine drawings for an adapter connecting the mount in figure 5.12 to the stainless steel gothc-arch xyz translation stage Cartoon showing the interface between the copper thermal reservoir and the water cooled mount [5]. Two heat sinks are located in very close proximity to one another so as to enable heat transfer without making a physical connection. This enables heat to be transferred across the interface without coupling any vibration from the water cooled mount into the copper thermal reservoir (and subsequently, the laser gain crystal) x

11 5.15 ABCD matrices used for a paraxial resonator analysis of our ring laser cavity. Included are matrices for propagation of length d in a uniform medium with index of refraction n as well as a thin lens of focal length f The waist of the gaussian beam for one round trip of propagation inside the ring laser cavity. The zero position reference is taken to be the center of the gain crystal Experimental setup for measuring the unsaturated small signal gain of the Nd:YVO 4 crystal when pumped by 2 25 Watt pump beams. The transmitted power of an extended cavity diode laser (ECDL) whose wavelength is tunable from 1335 to 1350 was measured by a photodetector (DET) after having passed through the crystal. Not shown in the setup are the lenses used to mode match the waist of the probe laser with the pump lenses as they co-propagate through the crystal The unsaturated gain profile of our Nd:YVO 4 crystal from 1335 nm to 1350 nm when pumped by 2 25 Watt pump beams. The gain profile is quite broad, spanning several nanometers, and peaks at approximately 1342 nm. Also of note is a broad excited state absorption band spanning from 1336 nm to 1341 nm The gain profile of our Nd:YVO 4 crystal from 1341 to 1345 nm. The profile has a peak value of 1.87 at a corresponding wavelength of nm. A dashed line has been added as a guide to the eye (see text) Experimental setup for measuring the angular dependence of the unsaturated small signal gain of the Nd:YVO 4 crystal. The transmitted power of an extended cavity diode laser (ECDL) whose wavelength is set at 1342 nm was measured by a photodetector (DET) after having passed through the crystal. A half-wave plate (HWP) is used to rotate the polarization of light prior to transmission through the crystal Unsaturated small signal gain as a function of polarization angle, measured with respect to vertical. The dashed line represents a sinusoidal fit to the data (see text) xi

12 6.6 Measurement of the power stability for the free running laser cavity. For most of the time, the power is split evenly between the two directions of operation, indicating bidirectionality. However, during some instances, the laser operates in unidirectional mode, whereby the recorded power is either twice as high or zero, depending on the direction Measurement of the frequency characteristics of our free running ring laser. The graph shows an output of a 300 MHz Fabry-Perot interferometer. The presence of four distinct peaks is indicative of the fact that the output laser frequency is multi-longitudinal mode in structure Results of a beam profile measurement made by the Mode Master, manufactured by Coherent, Inc. The output of our free running laser is coupled into the Mode Master, which measures values for the beam radius as the beam propagates over a given distance. From these measurements, several spatial properties of the laser beam can be determined and reported Power output from the laser cavity when driven unidirectionally using an intracavity Faraday Rotator for seven different output couplers. The dashed line represents a theoretical fit to the data used to determine the unsaturated small signal gain and intracavity scattering losses for our home built ring laser Experimental layout for our novel scheme of self injection. A small portion of the output light from our laser is picked off from the main beam using a half-wave plate (HWP) and a polarizing beam splitter (PBS). This light then travels through a Faraday rotator (FR) and another half-wave plate to change its polarization back to vertical. The weak beam is then injected back into the cavity, where it causes stimulated emission in the gain crystal, breaking the symmetry of the ring laser and driving unidirectional operation. Also included in this loop are three lenses (L1, L2, L3) used to shape the injected beam for mode matching (see text) Measurement of the power stability for the self-injected laser cavity. Unlike the free running laser, the unidirectional operation of the self-injected laser prevents mode competition in the gain crystal resulting in a drastic reduction of intensity noise on the output of the laser beam xii

13 6.12 The longitudinal mode structure of the self-injected laser as measured by a 300 MHz Fabry-Perot interferometer. Unlike the case of the free running laser cavity, the lack of mode competition in the gain crystal enables single frequency operation The spectral output of a heterodyne measurement of the linewidth of our self-injected laser. The laser output is beat with the output of a tunable ECDL and sent to a spectrum analyzer. From the full width at half maximum of this beat note measurement, it is shown that the linewidth of our self-injected laser is no larger than 150 khz The output power of our self-injected laser as a function of output coupling, δ 1, for seven different output couplers. The dashed line represents a fit to our data with the intracavity reflectivity being the only free parameter. Compared to the free running laser, we observe a 4.5% increase in output power for the optimum output coupler The periodic transmission measurement of a thin silicon etalon as a function of wavelength. The transmission was measured from 1330 nm to 1350 nm using a tunable extended cavity diode laser at normal incidence to the etalon. The dashed line represents a fit to the data of equation 6.18, indicating an etalon thickness of 34.2 μm The output power of our laser as a function of wavelength when tuned via a 34.2 μm intracavity silicon etalon. The dashed line represents a fit to the data when considering the convolution of the etalon transmission, the small signal gain of the laser crystal, and the increase in intracavity scattering losses due to the insertion of the etalon (see text) The output power of the laser cavity as a function of wavelength when tuned via a 250 μm silicon etalon located in the external loop region of our self-injected laser. By merely rotating the etalon, we were able to observe a tuning range that exceeded 38.9 GHz for our laser setup xiii

14 7.1 The experimental layout for our UHV system used to cool, trap, and manipulate a gas of fermionic 6 Li atoms. The apparatus is divided into three regions, namely an oven used to provide a source of hot atoms, a Zeeman slower used to reduce the temperature of the atoms, and an experimental region where the cool atoms are then trapped and cooled further for use in our experiments. Also shown are a variety of vacuum pumps used to maintain the low pressure required for our experiments. This figure has been adapted from reference [6] Layout of the Zeeman slower used in our apparatus [6]. Three electromagnet coils are wired in series to produce a spatially dependent magnetic field that shifts the energy levels of the atoms to be continuously on resonance with a laser beam. The absorption of photons from this counter propagating laser beam causes a momentum transfer to the atoms, reducing their velocities and subsequently cooling the atoms as they enter the experimental chamber Lateral cross section view of the experimental region of our apparatus. From this view, one can see the location of the MOT coils, the Feshbach coils, and the rf coils used to drive magnetic dipole transitions in our trapped atoms. This figure has been adapted from reference [6] a) The basic layout for the magneto-optical trapping of 6 Li. Three pair of red detuned orthogonal laser beams overlap the zero magnetic field location between two magnetic coils in anti-helmholtz configuration. By choosing the proper polarizations, the combination of magnetic and optical fields provide a restorative cooling force on the atoms located in this spatially overlapped region. b) The energy level diagram showing the cooling and repumping laser transitions for our MOT (solid lines). The dashed lines show the available channels for spontaneous emission, indicating the necessity of the repumping laser Cartoon layout of the three lasers that overlap to form the optical dipole force trap for our experiments. Two 100 watt 1064 laser beams intersect at a relative angle of 11 with each beam having awaistof30μ. A third 1070 nm 100 watt laser overlaps the other two, forming a deep trap used to capture atoms from our MOT. Also shown is the side view of one of the anti-helmholtz coils used to provide the spherical quadrupole trap for our MOT xiv

15 7.6 Absorption image of a cloud of 6 Li atoms trapped in a) two 1064 nm beams forming a crossed dipole trap b) a single 1070 nm beam c) a combination of the two 1064 nm beams and the 1070 nm beam Diagram showing the components making up the phase lock feedback loop. A small amount of power from each of two diode lasers are combined on a high bandwidth photo detector. The beat note signal of the two beams is then split where it takes one of two paths. On the first path, the signal is amplified before being fed into an optical phase-lock loop circuit where the beat note is compared to a reference oscillator and a corrective signal is fed back to the piezo and the FET of the laser diode. The signal is also amplified on the other path before being mixed with another reference oscillator, filtered, and coupled to the bias-t connector of the same laser. In this way, we can lock both the frequency and phase of the two diode lasers to arbitrary values The beat spectrum of our two phase locked lasers locked at 1.6 GHz. The central peak of 0.35 dbm corresponds to the power in the carrier while the total power from the occupied bandwidth measurement (2.2 dbm) can be used to determine the carrier power fraction, and thus the mean-square phase error of our lock Measurement of the fractional population of atoms in state 2 as a function of frequency for a magnetic dipole transition. The frequency of the applied microwave rf field was scanned over a range of ±30 khz relative to the central transition frequency of GHz for two different pulse duration times of 100 ms and 200 ms Rabi oscillations demonstrating the transfer of atoms from state 2 to state 5 using a two-photon Raman transition. The decay in coherence is attributed to the small size of the Raman beams causing a spatially dependent Rabi frequency for atoms located in different parts of the trap xv

16 List of Tables 2.1 g-factors and Hyperfine constants for the 2 2 S and 2 2 P electronic states of 6 Li Measured reflectivities of available output couplers for our homemade 1342 nm laser. Measurements were made at normal incidence and at 15 for vertically polarized light. The δ value of each output coupler is another way of representing its reflectivity (also at 15 ). For a further explanation, see the text xvi

17 Acknowledgments This Dissertation would not have been possible without the endless support of a number of people who I am honored to consider my mentors, colleagues, and friends. It has been said that it takes a village to raise a child. In the same vein, I would argue that it also takes a village to complete a Ph.D. First and foremost I would like to thank Ken O Hara for the opportunity to conduct research in his lab. His positive attitude, keen insight, fervent devotion, and incessant creativity has served as a constant inspiration, not only to press on through seemingly impossible obstacles, but to also dream big with lofty goals and aspirations. I could not have asked for a better advisor. During my tenure in the O Hara lab, I have also had the opportunity to rub elbows with some of the brightest up and coming researchers in the field. The hard work of Johnny Huckans, Jason Williams, Eric Hazlett, and Yi Zhang have greatly contributed not only to the success of our research lab, but to my own personal success as well. I would also like to thank Kurt Gibble and Dave Weiss for their vast knowledge and endless patience, both during formal course instruction and during our weekly AMO journal club meetings. Likewise, I would also like to extend thanks to the other members of my dissertation committee, Milton Cole and John Badding, whose willingness to serve in this capacity is truly humbling. Next I would like to thank my parents, Ron and Connie Stites. Without their support I could not have achieved success. The completion of this dissertation should serve as a great example of what can be accomplished through the positive encouragement and ceaseless dedication of family. Finally, I would like to thank my new family, starting with my beautiful wife, Jenna. Her unconditional love and devotion has truly blessed me. Whenever I m having a bad day and need someone to listen, I can take comfort in knowing that she has been and will always continue to be there for me. I only hope that in some small way I can eventually repay her the endless debt of gratitude that I owe. Last, I would like to thank my daughter, Hannah. Even though I have only known her xvii

18 for a few short months, she has already transformed me teaching me new lessons every day about love, patience, and overcoming adversity. There s no affliction in the world that one of her smiles cannot cure. xviii

19 Chapter 1 Introduction Ever since Bohr first published his research on the structure of the atom in 1913 [7], atomic physics, along side the development of quantum mechanics, has been one of the most studied and fruitful fields of modern physics. However, while the field had been dominated for many decades by traditional spectroscopy-type experiments, it has been over the course of the last 30 years that the atomic physics community has experienced a revolution in research direction, experimental techniques, and fundamental results. Ushering in this new era of research has been the development of new techniques for the cooling, trapping, and manipulation of ultra-cold neutral atomic gasses [8]. One of the reasons that physicists have been so interested in cooling and trapping neutral atomic gases is because these atomic systems provide an ideal test bed for the simulation of few and many body quantum systems. This is due to the fact that many of their experimental parameters (such as atomic density, temperature, and scattering length) can easily be tuned by the application of optical and magnetic fields. Additionally, for the temperature scales involved in these experiments, typically spanning six orders of magnitude from a few hundred nano-kelvin to a few milli-kelvin, the theoretical treatment of interactions in these systems can be greatly simplified. In this way, fundamental questions regarding the nature of collisions, chemical reactions, and thermodynamics can be investigated using these experimental systems. After it s initial conception in 1975, the cooling and trapping of ultra-cold gases using lasers began to achieve success in the early 1980s by Chu at Bell Laboratories

20 2 in New Jersey [9]. In these experiments, nearly resonant counter propagating laser light was used to reduce the velocity (and hence the temperature) of sodium atoms. Though not trapped, it was demonstrated that atoms experiencing this three dimensional viscous force were cooled to approximately the Doppler limit of 240 μk. This limit represents the minimum temperature obtainable using this optical molasses technique [10]. Also in 1985, Phillips, Metcalf, and colleagues at the National Bureau of Standards (now NIST) first demonstrated the use of field gradients from a spherical quadrupole magnetic trap to confine these cold atoms [11]. Repeating the experiments of Chu, they found that the atoms in the trap had been cooled to a temperature of 40 μk, a factor of six below the theoretical cooling limit. To explain cooling below this Doppler limit, the idea of polarization gradient cooling was independently and simultaneously published by Dallibard and Cohen-Tannoudji [12] as well as Unger, et al. [13]. For the development of methods to cool and trap atoms with laser light, Chu, Phillips, and Cohen-Tannoudji would later share the Nobel Prize in Physics in Once it had been demonstrated that atoms could be cooled even lower than the theoretically predicted limit, additional interest was spawned in further cooling the gas. For an ensemble of ultra-cold atoms, one way of parameterizing the gas is by it s phase space density ρ = nλ 3 db, where n is the density of the gas and λ db = 2π 2 / (mk B T ) is the thermal de Broglie wavelength for atoms of mass m and temperature T. When the phase space density equals unity, the de Broglie wavelength is comparable to the interparticle spacing. It had been predicted that a system of identical particles with integer spin (bosons) would undergo a phase transition if the phase space density is further increased to whereby the ground state energy level becomes macroscopically occupied. This so-called Bose- Einstein Condensate (BEC) was first predicted by Einstein in 1925 [14]. To achieve a BEC, the temperature of the atomic gas had to be cooled even further. It wasn t until several years later in 1995 that this hurdle was surpassed using the idea of forced evaporative cooling [15]. In forced evaporative cooling, atoms were magnetically trapped in a quadrupole trap in the presence of an rf magnetic field. This rf field was resonant for atoms with the highest energy and would cause a spin flip transition to a magnetically untrapped state. By selectively removing the most energetic atoms from the trap and allowing for rethermalization,

21 3 the temperature of the trapped gas was lowered until BEC was observed [16, 17, 18]. It was for the observation of BEC that Ketterle, Wieman, and Cornell shared the Nobel Prize in Physics in Today, the record for the coldest matter in the universe at 500 pk is held by an ultracold gas that had been cooled using these same techniques [19]. For a system of identical particles with half-integer spin (fermions), the experimental story line is fairly similar. In 1957, Bardeen, Cooper, and Shrieffer published a series of papers attempting to explain superconductivity as a microscopic effect concerning the bose condensation of pairs of fermions [20, 21]. Using the same techniques of cooling, trapping, and forced evaporative cooling using a magnetic trap, evidence for the first observation of a non-interacting degenerate Fermi gas (DFG) of the fermionic isotope 40 K was demonstrated by Jin s group in 1999 [22]. 1.1 Interactions in an Atomic Gas As mentioned above, the first experimental realization of a DFG in 1999 by Jin was for 40 K atoms evaporatively cooled in a magnetic trap. In this experiment, it was noted that when the gas was released from it s magnetic trapping potential and allowed to expand before imaging, the time-of-flight absorption image of the cloud demonstrated isotropic ballistic expansion. This ballistic expansion is the characteristic signature of a weakly-interacting or non-interacting atomic gas. For a strongly interacting atomic gas, however, this isotropic expansion is not expected. As the mean free path becomes shorter than the size of the cloud, the atoms no longer behave ballistically, but rather hydrodynamically, resulting in multiple collisions as the cloud expands. This anisotropic or hydrodynamic expansion was first theorized for a BEC [23], but was later expanded to include the superfluid nature of a strongly interacting Fermi gas [24, 25] One tool that atomic physicists have at their disposal for tuning interaction strengths in ultra-cold atomic samples is the application of a DC magnetic field in the vicinity of a Feshbach resonance [26] where the s-wave scattering length diverges. At this resonance for fermions, there exists a universality that connects the unitary Fermi gas to an ideal Fermi gas [27, 28]. The tuning of the s-wave

22 4 scattering lengths were first experimentally observed in an atomic gas by Ketterle in 1998 [29] as demonstrated by an increase in the two-body loss for the system near this resonance. For fermionic 6 Li, there was also predicted to be a broad Feshbach resonance located in the vicinity of 834 Gauss [30]. In 2002, O Hara et al. performed an experiment on 6 Li trapped in a conservative optical potential, rather than a magnetic one [31]. This optical potential had the benefit of trapping non-magnetically trapable states of 6 Li. By releasing the cloud of atoms from the optical trap in the vicinity of a Feshbach resonance, this strongly interacting Fermi gas expanded anisotropically in the transverse direction of the cigar shaped optical trap while remaining nearly stationary in the axial direction [32]. In contrast to ballistic expansion where the column density of the absorption image measurement evolves as 1/t 2, it was found that for anisotropic expansion, the density decreases only as 1/t. To explain this observed anisotropy, an expansion on the theory of superfluid hydrodynamics was employed [24]. It should be noted that since this first experiment demonstrating the anisotropic expansion of a highly interactive 6 Li Fermi gas, the same hydrodynamic expansion has been observed in a 40 K degenerate Fermi gas [33] as well as for a rotating gas [34, 35]. 1.2 Probing Strongly Correlated Systems Highly correlated atomic systems, such as strongly interacting Bose-Einstein Condensates and Degenerate Fermi Gases, represent a class of materials where a variety of novel many-body phenomenon can be explored. Such phenomenon include Mottinsulator states, superfluidity, anti-ferromagnetic ordering, frustrated spin systems, the BEC/BCS crossover, and a variety of other exotic states. To investigate these systems, however, several innovative techniques have been developed to probe the atomic systems. One such technique invokes the idea of Bragg diffraction. Bragg diffraction in an atomic gas is based on similar principles of Bragg diffraction in a crystal system. For an n th order Bragg diffraction, two laser beams can be used as an optical standing wave to stimulate a 2n photon Raman process where n photons are absorbed by one of the beams and then emitted into the other. Conservation of energy requires (np recoil ) 2 /2M = n δ n where P recoil =2 ksin(θ/2) is

23 5 the recoil momentum, k =2π/λ, λ is the wavelength of the light, M is the atomic mass, and δ n is the frequency detuning for the two lasers. When this condition is met, some of the atoms of the BEC or DFG will diffract from the standing wave and leave the cloud with a momentum n δ n. The first application of Bragg diffraction to a BEC was by Phillips group in 1999 [36]. In this experiment, Bragg diffraction was used as a tool to coherently split a BEC into two components. Additional use of Bragg diffraction were employed shortly thereafter to measure the excitations spectrum ω(k) aswellasthe static structure factor S(k) for these systems [37, 38]. Additionally, it was demonstrated that Bragg scattering could be used to excite phonons in a BEC [39]. Up to this point, all the experiments utilizing Bragg spectroscopy had been performed on a weakly interacting gas. In 2008, Vale s group in Australia investigated the crossover from BEC to BCS in a fermionic gas of 6 Li near a Feshbach resonance using Bragg spectroscopy [40]. In this experiment, it was demonstrated that using Bragg spectroscopy as a measurement tool in a strongly interacting gas was non-trivial. As atoms are diffracted they quickly undergo collisions distorting the diffracted cloud and obfuscating the measurement. Another tool that has been employed to measure the structure of the highly correlated atomic systems is the use of spatial quantum noise interferometry [41]. For this technique, the atoms are released from their trapped state and allowed to expand before an absorption image is taken of the sample. From this image, spatial correlations are computed to gather information about the initial structure of the system, such as pair correlations for fermions in momentum space [42]. While this has been proven to be quite an effective tool for weakly interacting atomic samples, again this technique has drawbacks in strongly interacting systems. As the gas is released, collisions quickly destroy the coherence of the cloud and the information gathered using these spatial correlations is lost. It should be noted that other experiments have been conducted to measure the excitation spectrum of these correlated systems using additional techniques. One example is the use of rf photoemission spectroscopy to measure the excitation spectrum of a degenerate fermi gas (e.g. see ref. [43]). Additionally, in the group of Cornell, photon correlation measurements of the laser beams themselves have recently been demonstrated for Bragg spectroscopy [44].

24 6 1.3 The Rapid Control of Interactions With regard to the complicating difficulties that a strongly interacting Fermi gas imposes on the measurements of Bragg diffraction and spatial correlations in a highly correlated atomic system, in this dissertation we investigate a method of rapidly controlling atomic interaction strengths by quickly changing the s-wave scattering length for a two-component Fermi gas. To do so, we first prepare an atomic sample of 6 Li atoms in it s two lowest magnetic hyperfine levels 1 and 2. The s-wave scattering length for atoms in these two states can be widely tuned via an applied magnetic field in the vicinity of a broad Feshbach resonance located at 834 Gauss. By pulsing on two co-propagating laser beams, a two photon Raman transition drives the entire population of atoms in the highly interacting state 2 to a weakly interacting state 5. The time scale of this Raman transition is on the order of a few microseconds much faster than the time scales required to drive the same transition utilizing an rf field via a magnetic dipole transition. In this way, the s-wave scattering length can be rapidly reduced by several orders of magnitude over the course of a few microseconds. To demonstrate this reduction in interaction strength, we will measure the time of flight expansion of the two-component cloud of atoms as they are released from their trapping potential as a function of time. As mentioned above, for atoms in states 1 and 2 near the Feshbach resonance, the s-wave scattering length is larger than the average interparticle spacing and the sample is in the so called hydrodynamic regime. In this regime, the cloud will expand anisotropically at a rate of 1/t. This expansion will then be compared to the expansion of atoms in states 1 and 5 at the same magnetic field. Atoms first prepared in states 1 and 2 will be release from the trap. Immediately upon their release, the atoms in state 2 are transferred to state 5 using the two-photon Raman pulse. The time of flight expansion of this cloud is again measured and shown that the cloud now expands ballistically at a rate of 1/t 2. This technique of rapidly controlling the interaction strength of a two-component gas has direct application to studies of Bragg spectroscopy and spatial correlations as outlined above.

25 7 1.4 A Self-Injected 1342 nm Ring Laser In addition to our studies of rapidly controlling interactions in a two-component Fermi gas, a large portion of this dissertation will also be dedicated to describing the construction of a novel laser source for use in 6 Li atomic experiments. Traditionally, the primary source of the 671 nm laser light used for 6 Li spectroscopy was generated using dye lasers. While demonstrating a great deal of versatility in the wavelengths of light able to be produced using these lasers, dye lasers also have several drawbacks. Since the gain medium of these lasers is a liquid jet stream, small bubbles in the stream, pressure fluctuations, and even fluctuations in the stream path can lead to large, noticeable inconsistencies in the repeatability of our ultra-cold atomic gas experiments. In particular, it was noted that the shot to shot fluctuations in the number of atoms varied by as much as 50% when using a dye laser in our experiments [6]. Recently, the development of semiconductor tapered amplifier systems at 671 nm have improved on these instabilities, reducing our shot to shot fluctuations to less than 10%. However, these tapered amplifiers do not come without problems of their own. First and foremost, the available power from one of these chips is currently limited to 500 mw. After spatially filtering the beam and sending it through a pair of optical isolators to protect the chip from back scattering, the amount of usable power is already reduced to 300 mw, requiring several amplifier systems for our experiments. In addition, these tapered amplifiers also have a finite lifetime, determined experimentally to be on the order of two years. Over time, the power output from these chips declines even further, making the successful completion of experiments very challenging. It was to this end that we developed a solid state laser system utilizing a Nd:YVO 4 gain crystal. The 4 F 3/2 4 I 13/2 transition in Nd:YVO 4 has a wavelength of 1342 nm, double that of the 671 nm light needed for our 6 Li experiments. Thus, the 671 nm light we require can be generated by frequency doubling the 1342 nm light with non-linear optics. Also, in constructing this laser, we have developed a novel scheme of self-injection locking whereby a small portion of the output power of the ring laser cavity is re-injected to drive unidirectionality and single frequency operation. Also during the course of these experiments, we

26 8 have measured the Verdet constant of undoped YAG in the near infrared. Doing so enabled us to construct a home made Faraday rotator for inclusion in our laser cavity to compare the output of our laser using this self-injection technique to a more traditional method for driving unidirectionality. In the end, we have demonstrated power outputs on the order of 3 Watts at 1342 nm. Even for a modest efficiency of 50%, frequency doubling this light should provide us with a high powered stable alternative to dye lasers and tapered amplifiers for application to our 6 Li experiments. Additionally, this power may also enable us to create a deep optical lattice for Raman cooling of atoms captured directly from a MOT, similar to the technique described in references [45] and [46]. 1.5 Outline This dissertation is laid out as follows. Chapter 2 will provide the basic background information regarding the energy level structure of 6 Li and how this structure is changed by the presence of an applied magnetic field. In addition, a review of s-wave scattering theory and scattering resonances as they apply to 6 Li will be provided. Finally, we will report on a simplified calculation for determining the scattering length of atoms located in states 1 and 2 as well as in states 1 and 5. Chapter 3 will begin with a discussion of atomic transitions while approximating the atom as a two level system. A description of light shifts due to the ac Stark effect will follow then some background on magnetic dipole transitions. Finally, a brief discussion of two photon Raman transitions will be presented. Chapter 4 describes our studies of the Verdet constant of undoped YAG in the near infrared. By measuring the magnetic field of a right hollow cylindrical magnet with an axial bore hole, we can determine the total applied magnetic field to an undoped YAG rod located along its hollow axis. By measuring the rotation of the plane of polarization for a probe laser beam sent through this magnet housing, we can extract values of the Verdet constant. We report measurements of the Verdet constant for wavelengths ranging from 1300 to 1350 nm as well as at 1064 nm. From these measurements of the dispersive nature of the Verdet constat, we can extract other properties of the YAG crystal, such as its electron band gap energy,

27 9 and compare these values to previous measurements. Chapters 5 and 6 are devoted to describing the construction and testing of our 1342 nm self-injected ring laser cavity. First, the details pertinent to the design, layout, and physical properties of the ring laser cavity will be described followed by an extensive investigation into various measurements, including the gain profile, the power as a function of output coupler, the transverse mode profile, the longitudinal mode profile, and the spectral linewidth of the beam. In addition, we introduce the novel idea of self-injection. We then compare our measurements for the self-injected laser to those using more traditional methods to drive unidirectional operation, such as the inclusion of a home built Faraday rotator utilizing the Verdet effect measured in chapter 4. Finally, we demonstrate frequency tuning of this laser by using an etalon located both inside the laser cavity as well as in the external self-injection loop region. In Chapter 7, we describe our experimental system for the creation of a twocomponent Fermi gas. Included in this discussion is an in depth look at our ultra-high vacuum system used to conduct these investigations. Additional details pertaining to the tapered amplifier laser systems used to generate the 671 nm laser light necessary for these experiments will be provided as well as information about the magnetic field coils used for both trapping our atoms in a magnetooptical trap as well as manipulating the s-wave scattering length via a Feshbach resonance. Finally, a brief discussion of using absorption imaging to measure various parameters of our atomic sample will be presented. In Chapter 8, we will report measurements of the rapid control of interactions in a two-component Fermi gas. We will begin by describing the process of phase locking the output of two lasers for use as Raman beams to drive transitions within our atoms. We also characterize the degree to which these lasers are locked by looking at the phase noise from the beat note signal of these two beams. Additionally, we will report on our investigations driving 2 5 state transitions using microwave fields and two-photon Raman pulses for a non-interacting Fermi gas. We will also provide information about the progress made in rapidly controlling these interactions near the broad Feshbach resonance in 6 Li. Next, Chapter 9 will summarize all of the research reported in this dissertation as well as provide direction for future experiments in our lab group. Finally,

28 10 Appendix A will provide the Mathematica code that we used to calculate the s-wave scattering lengths for atoms in different magnetic hyperfine states.

29 Chapter 2 Fermi Gases For our experiments with ultra cold Fermi gasses, we are primarily concerned with two-state mixtures of 6 Li in the two lowest energy hyperfine electronic spin states. Because we are dealing with atoms whose temperature scales are below the centrifugal barrier required for higher partial wave contributions to scattering, the interactions between two fermions can be best described by s-wave collisions. While the Pauli exclusion principle prevents identical fermions in the same spin state from interacting, having atoms in two different spin states allows for these collisions to occur. These s-wave collisions can be parameterized by a single value the s-wave scattering length. For certain two-state mixtures, the s-wave scattering length can be tuned via a broad Feshbach resonance that allows us to tune the interaction strength of the gas from strongly attractive to strongly repulsive by simply adjusting an applied magnetic field. In this way, we can prepare a twostate mixture of atoms with arbitrarily strong interactions. On the other hand, certain two-state mixtures have small s-wave scattering lengths and interact only weakly. By rapidly changing the internal state for one of our trapped states to a third non-interacting state, we can rapidly reduce the interaction strength by several orders of magnitude. This chapter describes the properties of fermionic 6 Li atoms used in our experiment. Section 2.1 will describe the atomic hyperfine structure of 6 Li and the Zeeman shift of the energy level of those states in the presence of an applied magnetic field. Section 2.2 will introduce the idea of s-wave scattering for two-atom collisions in our gas while section 2.3 describes how we can tune this s-wave scatter-

30 12 ing length using Feshbach resonances. Finally, section 2.4 will describe our method for estimating the s-wave scattering lengths for other internal spin states. 2.1 Properties of 6 Li The fermionic 6 Li isotope studied in our ultra-cold atomic physics lab shares similar properties with all other alkali metal atoms in that each of these atoms has a single unpaired valence electron, greatly simplifying the energy level structure of the atom. Since the majority of our studies will involve the ground 2 2 S and excited 2 2 P electron states, this section will look at the fine and hyperfine splitting of these states in the presence of a magnetic field. The fine structure of the atomic energy levels originates from the spin-orbit coupling of the valence electron and the electric field of the nucleus. The angular momentum J of the atom is written as the sum of the spin angular momentum of the electron S and the angular momentum L. For 6 Li, the ground state has values ofs=1/2andl=0,leading to a total angular momentum J = 1/2. For the excited states, L = 1, yielding two J values (1/2 and 3/2) corresponding to the fine structure splitting of the D line into the D1 and D2 spectroscopic lines. Additional splitting of these lines emerges when one considers the interaction of the total angular momentum J with the angular momentum of the nucleus I. 6 Li has a nuclear angular momentum value I = 1, causing hyperfine splitting in terms of the total atomic angular momentum F, where F = J + I. The energy level diagram showing the fine and hyperfine structure of 6 Li is shown in figure 2.1. The values for the ground and excited energy levels were reported in reference [47]. Additional information about the atomic structure of lithium can be found in reference [48]. For the majority of our experiments, the energy levels of the atoms will be shifted due to the Zeeman interaction with a magnetic field. To calculate the effects of the Zeeman interaction on the energy level structure, we need to first examine the total Hamiltonian for this system. At small magnetic fields, the Zeeman shift of the energy levels is no longer small compared to the hyperfine splitting for the 2 2 S ground state and 2 2 P excited states of lithium. Because of this, F is no longer a good quantum number and the magnetic and hyperfine interactions must be

31 P 3/2 F = 1/2 F = 3/2 F = 5/2 4.4 MHz GHz D 2 = nm F = 3/2 2 2 P 1/2 F = 1/ MHz D 1 = nm F = 3/2 2 2 S 1/2 F = 1/ MHz Figure 2.1. The hyperfine energy level structure for 6 Li showing the 2 2 S ground state and the two 2 2 P excited states. looked at in the J m J,I m I basis. The total interaction Hamiltonian is given by [49] H int = H hf + H ZE (2.1) where H hf is the hyperfine interaction Hamiltonian given by H hf = A j I J + B J[3(I J) 2 +3/2(I J) I(I +1)J(J + 1)] 2I(2I 1)J(2J 1) (2.2) and Zeeman interaction energy Hamiltonian, H ZE is H ZE = μ B (g J J + g I I) B (2.3) where A J and B J are the magnetic dipole and electric quadrupole hyperfine constants for an atom in state J and μ B is the Bohr magneton. Values for these parameters as well as the relevant g-factors for the 2 2 S ground state and 2 2 Pex-

32 14 Property Symbol Value g J (2 2 S 1/2 ) Total Electronic g-factor g J (2 2 P 1/2 ) g J (2 2 P 3/2 ) Nuclear Spin g-factor g I A 2 2 S 1/2 /h Magnetic Dipole Constant (MHz) A 2 2 P 1/2 /h A 2 2 P 3/2 /h Electric Quadrupole Constant (MHz) B 2 2 P 3/2 /h Table 2.1. g-factors and Hyperfine constants for the 2 2 Sand2 2 P electronic states of 6 Li. cited states of 6 Li are given in table 2.1. For atoms in the ground state, the angular momentum quantum number L =0. As a consequence of this, the angular wavefunction is spherically symmetric and the electric quadrupole constant is zero, allowing the Hamiltonian to be analytically solved through diagonalization into six eigenstates labeled 1 through 6 from least to most energetic [50]. These product states are 1 =sinθ + 1/2 0 cos Θ + 1/2 1 2 =sinθ 1/2 1 cos Θ 1/2 0 3 = 1/2 1 4 =cosθ 1/2 1 +sinθ 1/2 0 5 =cosθ + 1/2 0 +sinθ + 1/2 1 6 = 1/2 1 (2.4) where the coefficients are given by sin Θ ± = cos Θ ± = Z ± = 1 1+(Z± + R ± ) 2 /2 1 sin 2 Θ ± μb ( g J (2 2 S 1/2 )+g I ) ± 1 A 2 2 S 1/2 2 R ± = (z ± ) (2.5)

33 15 4. μ Energy (Joules) 2. μ μ μ Magnetic Field (Gauss) Figure 2.2. The magnetic field dependence of the energy for the F = 1/2 and F = 3/2 ground states of 6 Li. As the applied field becomes stronger, the atoms enter the Paschen-Back regime, whereby the atom becomes electron spin polarized and the three way splitting arises due to difference in the nuclear spin of the atoms. The eigenenergies of the ground state levels 1 through 6 as a function of magnetic field are shown in figure 2.2. As we apply a magnetic field, the degeneracy of the F = 1/2 and F = 3/2 states are quickly lifted and the six states become resolved. For field values where the energy μ B B is greater than the magnetic dipole constant A 2 2 S 1/2, the atoms enter the Paschen-Back reginme whereby the product states become good approximations of the eigenstates of the system. In this regime, the six states become electron spin polarized with 1 through 3 representing the electron spin down (-1/2) states and 1 through 3 representing the electron spin up (+1/2) states. The three states in each group thus correspond to different nuclear spins of the atom. 2.2 s-wave Scattering Theory In this section, we take a theoretical look at the scattering of two neutral atoms interacting via a short ranged molecular potential. The general problem of scattering has been covered in many quantum mechanics textbooks (e.g. [51, 52]) as well as in previous dissertations within our lab group [6, 53]. The treatment here will primarily follow that of reference [6]. In the center of mass frame of the two colliding particles, the problem reduces

34 16 to the scattering of a single particle off the spherically symmetric potential V(r). For large distances, we can approximate V(r) by the van der Waals potential for neutral atoms having aj=1/2ground state. This van der Waals potential falls off approximately as V (r) C 6 (2.6) r 6 where C 6 is the van der Waals coefficient. At shorter distances, as the atoms begin to approach one another, they begin to experience a strong repulsion as their electrons clouds begin to interact with one another, ultimately becoming infinitely repulsive as r 0. The potential well created by these two interacting neutral particles can support many bound states (a diatomic molecule) and can be approximated, for states near the potential minimum, by a Morse potential[54]. For low energy collisions, the characteristic length of the interaction potential is defined by the van der Waals length scale ( ) 1/4 2MC6 l vdw = (2.7) where is Planck s constant divided by 2π and M is the reduced mass of the two colliding particles. For ultra-cold atoms with high polarizabilities, such as 6 Li, l vdw can be much larger than the size of the atom on the order of several nanometers. One way the effect of a scattering event on two particles undergoing a collision can be interpreted is as a phase shift on the atomic wave function of the particle. To understand this, we first represent an incident particle as a plane wave traveling in the +ẑ direction with momentum k. After scattering, the wave function of the scattered particle in the asymptotic limit will consist of a plane wave plus a spherical wave, and can be represented as 2 Ψ k = e ikz + f(θ, φ) eikr r (2.8) where the effect of the scattering potential V(r) is contained within the scattering amplitude f(θ,φ). From this scattering amplitude, the differential scattering cross

35 17 section can be determined by dσ dω = f(θ, φ) 2. (2.9) Since the scattering potential V(r) is a central potential, we can express the scattering process in terms of an expansion of partial waves. Furthermore, since the potential has no angular dependence, depending only on r, the scattering amplitude is only a function of θ, where θ is the angle between the incident +ẑ direction and the direction of the outgoing wave. As a result, we can recast the scattering amplitude as a series expansion of Legendre polynomials f(θ) = l=0 (2l +1) e2iδ l 1 2ik P l(cos θ). (2.10) where l represents the value for the orbital angular momentum of the partial waves in our expansion and δ l is the phase shift associated with that partial wave function. For ultra-cold 6 Li atoms, the long range centrifugal barrier associated with the van der Waals scattering potential, of the form 2 l(l +1)/(2mr 2 ), has an associated temperature of 6.5 mk [55]. As the temperature of the 6 Li atoms in our experiment will almost always be lower than this value, we only need to consider s-wave (l =0) collision terms in our theoretical model. With this assumption, equation 2.10 becomes f = e iδ sin(δ 0 0). (2.11) k and the total cross section σ simplifies to σ = dω f 2 = dω sinδ 0 k 2 =4π sin2 δ 0 k 2. (2.12) In the zero energy limit, s-wave collisions are characterized by an s-wave scattering length a. It can be shown that tan(δ 0 ) -ka as k 0 [56]. Therefore, for the low-energy atoms in our experiment, we can define the scattering length a as tanδ 0 (k) a lim. (2.13) k 0 k

36 18 Using this identity in equation 2.11 for the scattering amplitude yields lim f = lim sinδ 0 (k) k 0 k 0 k and the total cross section for the collision becomes = a (2.14) σ =4πa 2. (2.15) The physical interpretation of the s-wave scattering length a is simply the distance between the center of the scattering potential and the location on the r-axis where the asymptotic wave function crosses zero. It is important to note that the sign of the s-wave scattering length relates important information about the nature of the two-particle interaction. For attractive potentials, the scattering length is negative, whereas for repulsive potentials the scattering length will be positive. 2.3 Scattering Resonances In section 2.2 we showed how the s-wave scattering length can be used to describe the interaction between two scattering particles whose temperatures are colder than the centrifugal barrier required to freeze out higher partial scattering waves. Now, we want to take a look at manipulating the value of that s-wave scattering length by utilizing Feshbach resonances to enhance the value of that scattering length. The enhancement of the two-body scattering length was first studied in the context of nuclear physics of H. Feshbach [57] and then atomic physics by U. Fano [58]. The basic premise for this resonance can be seen in figure 2.3. Two atoms approach each other in the triplet molecular potential (entrance channel) that has a non-zero magnetic moment and can be tuned in a magnetic field. The singlet state (closed channel) has a zero magnetic moment and is at a higher energy that exceeds the available kinetic energy of the atoms. The atoms are energetically forbidden from making the transition to the closed channel. Note that the singlet state molecular potential becomes energetically accessible for small internuclear

37 19 Closed Channel (Singlet) Incident Energy Δμ B 0 Entrance Channel (Triplet) 0 Figure 2.3. A Feshbach resonance occurs when the least bound state of the close channel molecular potential is tuned via a magnetic field to be resonant with the collision energy of the particles in the entrance channel. separations. However, because the two scattering states have different magnetic moments (Δμ), the relative energy of the entrance channel with respect to the closed channel can be tuned by applying a DC magnetic field. As the least bound molecular state is tuned such that the energy of the molecular state is just above the energy of the entrance channel, the s-wave scattering length becomes very large and negative, diverging when the two states are resonant [59]. Similarly, as the bound state of the closed channel is just below the energy of the entrance channel, the s-wave scattering length becomes very large and positive, again diverging when the two states are resonant. In this way, the scattering length of the two particles can be tuned by simply applying a DC magnetic field near one of these scattering resonances (see figure 2.4). The variation of the s-wave scattering length, a, as a function of magnetic field B can be described by ( a(b) =a bg 1 Δ ) (2.16) B B 0 where a bg is the background scattering length for a magnetic field far from the resonance field B 0 and Δ describes the width of the resonance [60]. It is important to note that Δ can have both positive or negative values.

38 Scattering Length (a 0 ) Magnetic Field (Gauss) 2000 Figure 2.4. The s-wave scattering length as a function of applied magnetic field for atoms in the two lowest hyperfine ground states of 6 Li [1]. Note the broad Feshbach resonance located at G. By arbitrarily tuning the applied magnetic field around this resonance, we can change the strength of the two-body interaction from being infinitely repulsive to infinitely attractive. As discussed in section 2.1, the application of a magnetic field to a 6 Li atom splits the F=1/2 and F=3/2 ground states into six energy levels labeled from lowest to highest energy. At sufficiently high magnetic fields, greater than 100 Gauss, the atoms enter the Paschen-Back regime, whereby they become electron spin polarized with states 1-3 having electron spin down and 4-6 having electron spin up. For the experiments outlined in this dissertation, we want to tune s- wave interaction strength for atoms in the lowest two spin states. Because of the atomic properties of 6 Li, the background triplet s-wave scattering length has an anomalously high value of a bg = a 0 where a 0 0.5Å is the Bohr radius [61]. Figure 2.4 shows the variation of the scattering length as a function of applied field from a coupled channels calculation based on a precise measurement of the interaction parameters [1]. As you can see, for states 1-2 there exists a very broad Feshbach resonance located at a magnetic field B 0 of G having a width Δ of -300 G [26]. It is this Feshbach resonance that we will use to tune the interaction strength of our two-component Fermi gas.

39 Interactions in a 1-5 Mixture While the s-wave scattering length for a two-component 1-5 mixture has not been explicitly measured or calculated, these scattering lengths becomes important in our experiments as a means to rapidly switch off interactions in a twocomponent Fermi gas. Therefore, we would like to have an estimate of the scale of the interactions. To do this, we perform a full coupled channels calculation of s-wave scattering modeling the singlet and triplet scattering potentials as finite square-well potentials. The model, with the Mathematica code explicitly reproduced in appendix A, approximates the singlet and triplet potentials for two body interaction as square well potentials. The depths and widths of the square well potentials are chosen to reproduce the known scattering lengths a s and a t for the singlet and triplet potentials as well as the binding energy of the most weakly bound molecular state of the singlet potential. The known values for the background scattering lengths, a s and a t, are 45.5 Bohr and Bohr [61]. There are three parameters we adjust in this model: the widths of both potentials (R) and the depths (V t and V s ) of the triplet and singlet potentials respectively. These parameters are chosen to reproduce the singlet and triplet scattering lengths and the binding energy of the ν =38 th vibrational bound state of the singlet molecular potential. In order to more accurately match the observed location of the Feshbach resonances, we can adjust the width of the square wells. The model works as follows. We calculate the s-wave scattering phase shift (and thereby the scattering length) by solving the coupled channels Schrodinger equation for all s-wave scattering channels that can be coupled by the spherically symmetric molecular potentials (i.e. those channels with the same z-projection of the total spin angular momentum). We will consider s-wave scattering of atoms in states 1 and 5. The z- component of the total spin angular momentum in this case is m F = +1. When the atoms are asymptotically separated, the two-body spin wave function for the incident wave is {1, 5}, an anti-symmetrized combination of states 1 and 5 defined above. In principle, the molecular potentials could couple this incident pair of atoms in state {1, 5} to any channel with a two-body spin wave function

40 22 having m F = +1. The two possibilities are {2, 6} and {4, 6}. However, since these channels are energetically closed for the energies and fields of interest, the asymptotic two-body spatial wave function in these channels decay exponentially. Thus, the asymptotic two body wave function (for r )is u(r) = [ e ikr + S 1,5 e ikr] {1, 5} + S2,6 e κ 2,6r {2, 6} + S4,6 e κ 4,6r {4, 6} where S 1,5, S 2,6, and S 4,6 are the amplitudes for the scattered waves in each respective channel. The two-body spin wave functions {1, 5}, {2, 6}, and {4, 6} are relevant when the atoms are asymptotically separated as they are eigenstates of the hyperfine and Zeeman interaction Hamiltonians. However, when the atoms are in close proximity such that the singlet and triplet potentials dominate over the hyperfine and Zeeman interactions, the relevant two-body spin wave functions should instead be expressed in terms of the total electron spin S s 1 + s 2 and the total nuclear spin I = i 1 + i 2. Again, the two-body spin wave functions must be antisymmetric under exchange of all particle labels. Also, the two-body spin wave functions of interest must have the same z-projection of the total spin angular momentum as the incoming spin state (i.e. m F = 1 for our example). The relevant two-body spin wave functions in terms of electron spin singlet and electron spin triplet states are (written in the S, m S ; I, m I basis): 0, 0; 2, 1, 1, 0; 1, 1, 1, 1; 1, 0. In the simplified coupled channels model we consider, we assume that the singlet/triplet spin states diagonalize the interaction Hamiltonian for r<r(i.e. the singlet and triplet molecular potentials dominate) and the spin states {1, 5}, {2, 6}, {4, 6} diagonalize the interaction Hamiltonian for r>rwhere we assume the molecular interactions are zero. The relative spatial wavefunction u(r) and its derivative must be continuous at r = R. If we model the singlet and triplet molecular potentials as square wells of radius R, the value of u and u at r = R are given by u(r) =A S sin [k(r a s )] and u (R) =A S k cos [k(r a s )]

41 23 for particles in the spin singlet state and u(r) =A T sin [k(r a t )] and u (R) =A T k cos [k(r a t )] for particles in a spin triplet state. Here, a s and a t are the singlet and triplet scattering lengths. Putting all of this together, continuity of u and u at r = R requires A S sin [k(r a s )] 0, 0; 2, 1 + A T,0 sin [k(r a t )] 1, 0; 1, 1 +A T,1 sin [k(r a s )] 1, 1; 1, 0 = ( e ikr + S 1,5 e ikr) {1, 5} + S2,6 e κ 2,6R {2, 6} +S 4,6 e κ 4,6R {4, 6} and A S k cos [k(r a s )] 0, 0; 2, 1 + A T,0 k cos [k(r a t )] 1, 0; 1, 1 +A T,1 k cos [k(r a s )] 1, 1; 1, 0 = ( ike ikr + iks 1,5 e ikr) {1, 5} κ2,6 S 2,6 e κ 2,6R {2, 6} κ 4,6 S 4,6 e κ 4,6R {4, 6}. If we project out the states 0, 0; 2, 1, 1, 0; 1, 1, and 1, 1; 1, 0 in each of the two equations above we obtain six equations which allow us to determine the six unknowns: A S, A T,0, A T,1, S 1,5, S 2,6, and S 4,6 in the limit k 0. Knowing the scattering amplitude S 1,5 for the {1, 5} channel allows us to determine the s- wave scattering phase shift and thereby the s-wave scattering length for {1, 5} collisions. Other than the six unknown quantities A S, A T,0, A T,1, S 1,5, S 2,6, and S 4,6, all of the other variables in the equations above can be determined. The singlet and triplet scattering lengths for 6 Li are known to be a s = a 0 and a t = 2140 a 0. We choose the radius R of the square well potential to give the correct binding energy for the most weakly bound vibrational state of the singlet potential. This bound state in the singlet potential gives rise to the Feshbach resonances in 6 Li.

42 Scattering Length (a 0 ) x Magnetic Field (Gauss) 860 Figure 2.5. The result of our coupled channel calculation for s-wave scattering between atoms in states 1 and 2. To simplify the calculation, we model the singlet and triplet molecular potentials as finite square well potentials. Despite this oversimplication, the calculation reproduces the broad Feshbach resonance located at G which can be observed as a divergence in the s-wave scattering length. The location and width of this resonance is in good agreement with a coupled channel calculation which uses accurate singlet and triplet molecular potentials. The good agreement gives us confidence in our coupled channels calculations using a simple model for the potentials. The decay constants κ 2,6 and κ 4,6 are determined from the energies of the twobody spin states {2, 6} and {4, 6} relative to the energy of the two-body spin state for the entrance channel ( {1, 5} ). When projecting out the states 0, 0; 2, 1, 1, 0; 1, 1, and 1, 1; 1, 0 in the equations above, we compute Clebsch- Gordan coefficients such as 0, 0; 2, 1 {1, 5}. Before looking at the s-wave scattering length as a function of magnetic field for a 1-5 mixture, we first want to verify the accuracy of this numerical approximation for the well known values of a 1-2 interaction. By adjusting R to a value of Bohr, we can see in figure 2.5 that the calculation predicts a broad Feshbach resonance centered at a magnetic field of gauss. Remarkably, in addition to the broad Feshbach resonance, the calculation also predicts the existence of a narrow Feshbach resonance at gauss (see figure 2.6). While the location of this resonance is at a slightly higher magnetic field than what has been

43 Scattering Length (a 0 ) Magnetic Field (Gauss) Figure 2.6. Our simplified coupled-channels model for scattering of atoms in states 1 and 2 also predict the presence of a narrow Feshbach resonance. While the position of this resonance is slightly higher in magnetic field than what has been experimentally observed, this discrepancy could easily be explained by the different shape of our simplified potentials. experimentally observed ( G) [53], again, it s predicted presence provides confidence in our simplified model. Looking now at the model for the s-wave scattering length for atoms in a 1-5 mixture, the results (using the same parameters as those used to determine the 1-2 scattering lengths above) are shown in figure 2.7. For values near zero magnetic field, the scattering length is large and negative, dominated by the anomalously large background scattering length for the triplet potential. As the applied magnetic field increases, however, the scattering length increases as well, approaching zero. For values around the gauss Feshbach resonance seen for a 1-2 mixture, the 1-5 scattering length is around -3 Bohr. Therefore, if we were to create a 1-2 mixture of atoms at a magnetic field near the broad Feshbach resonance where the scattering length is very large (hundreds of thousands of Bohr), we could quickly turn off interactions between the two atoms by changing the spin state from 2 5, thus reducing the s-wave scattering rate by several orders of magnitude.

44 26 0 Scattering Length (a 0 ) Magnetic Field (Gauss) Figure 2.7. The predicted s-wave scattering lengths as a function of magnetic field for a 1-5 mixture. With no Feshbach resonance, the s-wave scattering length increases to a value of approximately -3 Bohr at a magnetic field corresponding to gauss. In this way, interactions of a two-state 1-2 mixture at this field could be quickly suppressed by a simple transfer from state 2 5

45 Chapter 3 The Interaction of Atoms with Light In this chapter, we consider the effect of electromagnetic radiation on two and three-level atoms in several different physical contexts relevant to this dissertation work. We will discuss coherent population transfer and the AC Stark shift in a two level atom and population transfer via a two-photon Raman transition in a three-level atom. The Stark shift is relevant for trapping neutral atoms with laser light and the magnetic dipole and Raman transitions are relevant for transferring atoms between different hyperfine ground states in 6 Li. The approach taken will be a semi-classical one, whereby the radiation will be treated using a classical electric field while the atom itself will be treated quantum mechanically. While the derivations of the atom-light interactions here will predominantly follow that of reference [10], the topic itself has been covered many times, with reference [62] in particular providing a more in depth discussion on the topic. We will begin our discussion in section 3.1 where we derive from first principles equations describing the coherent evolution of the atom between two internal quantum states. To do so, we make use of the approximation that the radiation driving the transition within the atom is far detuned from all other energy levels in the multi-level atomic system. In this way, we can simplify the problem by assuming that the atom itself can be approximated as a two level system. As the presence of electromagnetic radiation can perturb the energy levels of the atomic system, in section 3.2 we take a theoretical look at the effect of this AC Stark shift, or light shift, on the energy levels of the atom. The AC Stark shift from a focussed laser beam can be useful for providing an essentially conservative trapping poten-

46 28 tial for neutral atoms provided the laser beam is detuned far from resonance. In section 3.3 we describe the selection rules and dipole matrix elements for using rf and microwaves to drive magnetic dipole transitions between these states. Finally, in section 3.4 we look at the coherent evolution of atomic population between two states using two coherent lasers whose energy difference is equal to that of the difference between the two energy levels of the atom. This two photon Raman transition will utilize an intermediate third level that will become virtually populated as the transition occurs. These Raman transitions will become critical to our experiments involving the rapid control of interactions in a two-component Fermi gas. 3.1 The Two Level Atom As mentioned above, in this section we will derive from first principles the coherent evolution of the internal quantum state of an atom as it interacts with electromagnetic radiation. While the internal structure of the atom can be quite complicated, we begin our discussion with a simplification. Instead of concerning ourselves with every possible energy level within the atomic system, we will instead reduce the atom to a two level system consisting of a ground state g and an excited state e as shown in figure 3.1. While these two energy levels can, in principle, be any two energy levels within the atomic system where transitions are allowed, we will primarily be concerned with transitions between different hyperfine Zeeman sub-levels of our 6 Li gas, for example between 1 and 2 (see figure 2.2). We begin the derivation from the time dependent Schrodinger equation i Ψ t = HΨ (3.1) where the Hamiltonian, H, consists of two components H = H 0 + H I (t). (3.2) The eigenfunctions and eigenvalues of the time independent Schrodinger equation describe the unperturbed energy levels and wave functions for the atom as discussed

47 29 e δ ω ω 0 g Figure 3.1. Energy level diagram for a simplified atomic system. A two level atom with ground state g and excited state e interacts with an oscillating electric field having a frequency of ω. This beam is detuned by an amount δ from the transition frequency ω 0 corresponding to the energy difference between the two atomic states. in section 2.1. The time dependent interaction Hamiltonian, on the other hand, will describe the interaction of coherent radiation with an oscillating electric field which perturbs the energy levels. To solve the time dependent part of the Schrodinger equation, we assume a solution of the form Ψ n (r,t)=ψ n (r)e ient/ (3.3) for a wave function with energy E n, where we have separated out the time independent wave function ψ n. For our two level system, the spatial wave functions satisfy H 0 ψ g (r) =E g ψ g (r) H 0 ψ e (r) =E e ψ e (r). (3.4) It is important to note that these solutions do not satisfy the time dependent Schrodinger equation as written in equations 3.1 and 3.2. The complete solution to the full Hamiltonian can be expressed as Ψ(r,t)=c g (t)ψ g (r)e ie gt/ + c e (t)ψ e (r)e ie et/ (3.5) where c g (t) and c ge (t) are the probability amplitudes of the atom being in the

48 30 ground and excited states respectively. Rewriting equation 3.5 in Dirac notation, where c g (t) is shortened to c g and ω g =E g /, yields Ψ(r,t)=c g g e iω gt + c e e e iω et. (3.6) For the system to be properly normalized, the two time dependent probability amplitudes c g and c e must obey the relation c g 2 + c e 2 =1. (3.7) We now look at the effect of an oscillating electric field E = E 0 cos(ωt) asa perturbation described by the interaction part of the Hamiltonian H I = er E 0 cos (ωt). (3.8) This interaction Hamiltonian corresponds to the energy of an electric dipole -er in the presence of an electric field. Here, r is the position of the electron with respect to the center of mass of the atom. The effect of this interaction will be to mix the two states with energies E g and E e. In our atomic system, the wavelength of the radiation will always be much larger than the size of the atom. That is to say, λ a 0 where a 0 is the bohr radius. Because of this, we can make the approximation that the amplitude of the electric field is nearly uniform over the atomic wave function. This approximation is known as the dipole approximation. By substituting equation 3.6 into the time dependent Schrodinger equation 3.1, we find two coupled differential equations ic g = Ωcos (ωt) e iω0t c e (3.9) and ic e =Ω cos (ωt) e iω0t c g (3.10) where we have defined ω 0 =(E e E g )/ and the Rabi frequency Ω is defined by Ω= g er E 0 e = e φ g(r)r E 0 φ e (r)d 3 r. (3.11)

49 31 We can take the amplitude E 0 outside the integral in equation Therefore, for linearly polarized light along the x-axis, we obtain Ω= ex ge E 0 2 (3.12) where X ge = g x e. (3.13) We now turn our attention back to solving the coupled system of differential equations 3.9 and These two equations can be rewritten as [ ic g = c e e i(ω ω 0 )t + e ] i(ω+ω Ω 0)t 2 (3.14) and [ ic e = c g e i(ω+ω 0 )t + e ] i(ω ω Ω 0)t 2. (3.15) It is at this point that we can make another approximation. For our experiments, the frequency of the radiation is close to the energy level resonance of the atoms. Because of this, the magnitude of the detuning is small such that ω 0 ω ω 0. Therefore, in each of these two equations, the term containing (ω + ω 0 )t oscillates much faster than the term (ω ω 0 )t and subsequently much faster than any interaction time in the system. As a result, we can make the rotating wave approximation, whereby each term in the differential equations containing this summation term goes to zero. Taking this approximation, our two differential equations can now be written as ic g = c e e iδt Ω (3.16) 2 and iδt Ω ic e = c g e (3.17) 2 where we have defined δ =(ω ω 0 ). Combining these two equations yields a second order differential equation of the form d 2 c e dt 2 + iδ dc 2 e dt + Ω 2 c e = 0 (3.18)

50 32 For the initial condition that at time t = 0 the entire population is in the ground state, the solution of this differential equation gives the time dependent probability of the atoms being in the excited state as c e (t) 2 = Ω2 sin2 Ω 2 ( ) Ω t where we have defined the generalized Rabi frequency Ω as 2 (3.19) Ω 2 =Ω 2 + δ 2. (3.20) On resonance, when δ = 0 and Ω = Ω, the population evolves as ( ) Ωt c e (t) 2 =sin 2. (3.21) 2 The interpretation of this result is relatively straightforward. In the presence of an oscillating electric field, ignoring spontaneous emission, the atomic population will coherently oscillate back and forth between the ground and excited states in a sinusoidal fashion. When the electric field is on resonance, such that the detuning of the laser light is zero with respect to the difference in energy levels, it becomes possible to have complete transfer of the atomic population from the ground to excited state and back. Additionally, this treatment introduces the idea of a π pulse. On resonance, if the product of the Rabi frequency and pulse length is an odd-integer multiple of π then the entire population will be transferred from the ground to the excited state. Similarly, if the product of the Rabi frequency and pulse length is an even-integer multiple of π then the population will remain in it s initial state. In this way, it becomes possible to transfer any arbitrary amount of population from one state to the other simply by selecting an appropriate pulse length for the coherent radiation beam. 3.2 The AC Stark Effect In addition to changing the state populations, the presence of an oscillating electromagnetic field also causes a shift in the energy level eigenvalues of the atom

51 33 [8]. This shift in energy levels is known as the AC Stark effect. To gain theoretical insight into this effect, we begin by redefining the values for the probability amplitudes using new variables c g = c g e iδt/2 (3.22) and c e = c e e iδt/2. (3.23) Differentiating 3.22 with respect to time yields c g = c g e iδt/2 iδ 2 c ge iδt/2. (3.24) Now, if we multiply this equation by i and make use of equations 3.16, 3.17, 3.22, and 3.23, we find the equations reduced a pair of of first order differential equations, namely i c g = 1 2 (δ c g +Ω c e ) (3.25) and i c e = 1 2 (Ω c g δ c e ). (3.26) To solve this system of first order differential equations, we can rewrite them in in matrix form as ( ) ( )( ) i d δ Ω cg cg 2 2 =. (3.27) dt c e c e Ω 2 δ 2 This matrix of equations will have solutions of the form ( ) ( ) cg a = e iλt (3.28) c e b where λ is an eigenvalue of the system. To solve for the eigenvalues, we need to set the determinant of the matrix equal to zero, namely δ 2 λ Ω ( ) 2 ( ) 2 2 δ Ω Ω δ λ = λ2 =0. (3.29)

52 34 δ Light Shift Figure 3.2. The shift of the energy levels of a two level atomic system due to the presence of an oscillating electromagnetic field. This AC Stark effect results in a resonance frequency shift to higher frequencies when the detuning of the beam is negative and lower frequencies when the detuning is positive. From this equation, one can easily see that the eigenvalues for the system are λ = ±(δ 2 +Ω 2 ) 1/2 /2. In the absence of the perturbing radiation, that is, when Ω = 0, the unperturbed eigenvalues are simply λ = ±δ/2, which corresponds to two energy levels spaced by an energy δ as shown in figure 3.2. These two states correspond to a ground state having energy E g and an excited state having energy E g + ω, corresponding to the ground state plus a photon of the radiation field. This picture of the atom plus a photon is referred to as a dressed atom [62]. For the case when the the frequency detuning of the laser light is quite large with respect to the Rabi frequency, the eigenvalues become ( ) δ λ ± 2 + Ω2. (3.30) 4δ That is to say, in the presence of the perturbing radiation, the energy levels experience a light shift of Δω light = Ω2 4δ. (3.31) It is important to note that this light shift equation is valid for both positive and negative values of detunings, as this effect will be utilized in our experiments to trap atoms at the intersection of two far red-detuned laser beams.

53 Magnetic Dipole Transitions The experiments described in this dissertation primarily make use of the hyperfine states within the ground 2 S 1/2 manifold of 6 Li (see figure 2.2). Because each of these six states all lie within the same l = 0 S-level, transitions between them do not change the orbital angular momentum of the atom s electrons. Therefore, in order to conserve angular momentum, transitions between these levels cannot be driven by an electric dipole transition, as described in section 3.1. To avoid violating these selection rules and still conserve angular momentum, we instead employ the use of magnetic dipole transitions. Because magnetic dipole transitions only change the magnetic quantum number of the system, they are extremely useful for driving transitions between atomic energy levels when Δl =ΔL = 0. The transition matrix element between hyperfine levels is given by μ eg e μ B g (3.32) where μ is the sum of the magnetic dipole moment of the electron and the nucleus, defined as μ = g S μ B S + g I μ B I (3.33) where g S and g I are the gyromagnetic ratios of the electron and nucleus, μ B is the Bohr magneton, and S and I are the electron and nuclear spin. Also B = B 0 cos ωt is the oscillating magnetic field. For our 6 Li system in a magnetic field, transitions between states 1, 2, and 3 or 4, 5, and 6 can be driven using rf fields. However, transitions requiring an electron spin flip (say from state 2 to 5 ) require a higher energy and microwave fields must be employed. Finally, it s of interest to note that the size of the magnetic dipole matrix element is much smaller than that of an electric dipole matrix element. This is due to the fact that the ratio of the magnetic field to the electric field, B / E, equals 1/c for an electromagnetic wave. Therefore, magnetic dipole transitions will be much slower than electric dipole transitions for allowed transitions within an atom.

54 36 i Δ L2 L1 e δ g Figure 3.3. The energy level diagram for a two photon Raman transition. Laser L1 drive the atom from the ground state to a virtually excited intermediate state i. Laser L2 then drives the atom from state i back to the desired excited state. It is important to note that the population in state i is virtual, and losses due to spontaneous emission are not allowed. By employing this two photon technique, one can drive state changing transitions where they would otherwise be forbidden, such as when Δl = Raman Transitions In section 3.3 it was noted that electric dipole transitions were forbidden for the hyperfine state changing transitions in the electronic ground state that we are primarily interested in for our experiments. Here we examine another possibility for making these transitions, namely a two photon Raman transition. Two photon Raman transitions use two laser beams of frequency ω L1 and ω L2 to drive transitions from state g to state e by way of an intermediate state i as demonstrated in figure 3.3. While an intermediate level i is used to help aid in the transition, it s important to note that at no time is there any real excitation into this virtual level. The mathematics behind the two photon Raman process is very similar to that derived in section 3.1 with the following differences. Since we are now dealing with a two-component electric field of the form E = E L1 cos (ω L1 t)+e L2 cos (ω L2 t), the detuning δ of the transition is now defined as δ =(ω L1 ω L2 ) (ω e ω g ). (3.34)

55 37 Also of importance is the detuning of laser L1 with respect to the intermediate level i. This detuning Δ is defined as Δ=ω g + ω L1 ω i (3.35) as shown in figure 3.3. Also similar to the two level treatment, we can define an effective Rabi frequency for the two photon transition. It should come as no surprise that the effective Rabi frequency Ω eff has contributing components from the electric fields of each of the two lasers driving the transition, namely Ω eff = Ω eiω ig 2Δ = e er E L2 i i er E L1 g. (3.36) 2 (ω i ω g ω L1 ) In this way, it can be shown that the excited state population also oscillates in time as it did for electric dipole transitions, except now using Ω eff instead of Ω. On resonance, this means that the excited state population evolves via the functional form ( ) c e (t) 2 =sin 2 Ωeff t. (3.37) 2 It s also important to note that the evolution from state g to state e is coherent in nature and the idea of a π pulse driving all of the population from state g to state e is extended to this treatment as well. In fact, the only requirement for using two laser frequencies to drive the transition is that the two Raman beams must be phase coherent with respect to one another. This can be accomplished experimentally by using the same laser and an acousto-optic modulator to generate the two beams for difference frequencies in the rf, or by phase locking two different lasers when the difference frequency is larger. Finally, because this scheme utilizes electric dipole transitions instead of magnetic dipole transitions, the speed with which one can make state changing transitions is greatly increased. In this way, we can rapidly change the internal state of the atomic sample, and thus rapidly changing the interaction strength when the s-wave scattering length has been tuned near a Feshbach resonance.

56 Chapter 4 The Verdet Constant of Y 3 Al 5 O 12 The rotation of the polarization of light by a medium in the presence of a magnetic field was first observed by Faraday in 1845 while examining the effect of magnetic fields on bulk glasses. Using these experiments as a basis for his own investigation, Verdet conducted a series of experiments to systematically quantify the dependence of the rotation angle on the various parameters involved. As a result of these pioneering works on the interplay of light and magnetic fields, the Faraday effect has since been utilized to construct numerous types of optical devices. At the heart of each of these devices is a crystal characterized by the Verdet constant a measure of polarization rotation of transmitted light for an applied magnetic field over the length of the crystal. Here we describe a measurement of the Verdet constant of undoped Y 3 Al 5 O 12 (YAG) in the near infra-red. While previous measurements have been made for the Verdet constant of undoped YAG in the visible spectrum [63, 64], our measurements represent, to the best of our knowledge, the first to be made for light at longer wavelengths. Our motivation for this experiment stemmed from our desire to construct a home built Faraday rotator which, when inserted into a 1342 nm ring laser cavity, drives unidirectional operation of the laser. For ring laser cavity configurations, two stable modes of operation coexist, corresponding to the bidirectional propagation of traveling waves inside the cavity. This bidirectional operation is not always desirable as the gain from the lasing medium is shared, dividing the total available power among the two directions. To force unidirectional operation (and subsequently yield nearly twice the power output for a given beam) one can in-

57 39 sert an optical isolator into the cavity. However, for some laser wavelengths and intracavity intensities commercial isolators do not exist. One solution is to use an optically active crystal combined with a half wave plate to form a homemade Faraday rotator. For the proper orientation, this system rotates the polarization of circulating light for only one direction of propagation inside the ring laser cavity. As gain media and intracavity losses can be polarization dependent [65], this rotation is sufficient to break the symmetry of the system and drive unidirectional laser operation. This chapter highlights these measurements of the Verdet constant for undoped YAG as well as associated parameters that can be extracted from these measurements. Section 4.1 frames the theoretical groundwork for the phenomenon responsible for this effect. Section 4.2 provides background information concerning the bulk properties of YAG crystals. The experimental setup for measuring the Verdet constant is laid out in section 4.3 with section 4.4 describing our measurements of the magnetic field of a hollow right cylindrical magnet. Finally, the results of the measurements are presented in section 4.5, followed by a concluding section Verdet Constant Theory As mentioned above, when linearly polarized light is incident on an optically active medium in the presence of a magnetic field, the polarization will rotate. This rotation by an angle dφ can be described per the equation dφ = VB z L c (4.1) where B z is the total applied magnetic field along the propagation axis of a crystal medium of length L c, and V is the Verdet constant of the material. The origin of this rotation stems from the fact that the applied magnetic field causes a Zeeman splitting of the involved energy levels within the crystal. Linearly polarized light can be decomposed into components of both right and left circular polarization. Because of the energy level splitting associated with the application of a magnetic field, each of these components will experience a slightly different index of refraction within the material. As a result of this difference in index of refraction, the

58 40 phase relationship between these two components will change as they propagate through the crystal, corresponding to a rotation of the polarization vector. The Verdet constant for a single oscillator inside of an optically active medium is given by nv = K (4.2) (ω0 2 ω 2 ) 2 where n is the index of refraction of the material, ω is the frequency of the photon and ω 0 is the resonant frequency of the oscillator [66]. It is important to note that the Verdet constant is dispersive in nature (dependent on the wavelength of the incident photon) not only for ω, but for the index of refraction n as well. Summing up the states of all allowed direct transitions in the conduction and valence bands [67] yields the relation nv = K x ω 2 [[ ] x 1+x 4 x [ 2 1 x 1+x] ] (4.3) where we define x E/E g with E = ω being the photon energy, E g is the band gap energy of the crystal, and K is a dimensionless parameter unique to each material. It should be noted that for energies larger than the band gap, the index of refraction becomes imaginary and the material becomes opaque. Therefore, by simply measuring the rotation angle dφ as a function of wavelength for a crystal of a given length in a known magnetic field, we can determine E g and K and extrapolate the rotation for any desired wavelength, applied field, and crystal length. 4.2 Y 3 Al 5 O 12 Properties As the ultimate goal of this investigation was to construct a Faraday rotator for intracavity insertion into a 1342 nm ring laser, we needed to identify and characterize a material with sufficiently large optical activity to provide enough rotation when constrained by our cavity size and available magnetic fields. One prime candidate for such a device is undoped Y 3 Al 5 O 12 (YAG). First grown commercially for scientific use in 1962 [68], YAG has been widely used within the optics community

59 41 Figure 4.1. The transmission spectrum of undoped YAG from 200 to 6500 nm. Note the broad transmission extending from the visible well into the mid-infrared [2]. as a host material for various types of rare-earth doped laser gain media, including erbium, ytterbium, thulium, and, most commonly, neodymium [69]. Undoped YAG, however, is also useful, as some of those same properties that make it an ideal candidate for active laser gain crystals also make it an ideal candidate for passive devices as well. One of the characteristics that makes YAG an ideal material is its broadband spectral transparency. The transmission spectrum of undoped YAG is shown in figure 4.1 [2]. From the graph, one can see that YAG has excellent transmission over the range of 0.2 μm to5.5μm, covering the near infra-red spectrum that we are interested in. In addition, YAG has a high thermal conductivity of 12.9 W m 1 K 1 and a low dn/dt value of K 1 at 1064 nm [70]. When subjected to the high powers circulating inside of a ring laser cavity, these values enable the crystal to efficiently dissipate any absorbed power with minimal modulation of the refractive index. As a result, thermal lensing effects are greatly reduced in these crystals [71]. The crystal structure of YAG is body-centered cubic, having one lattice point in the center of the cubic unit cell in addition to the eight corner points. Such a lattice structure exhibits symmetry along the three crystal axes in real space. It is

60 42 because of this symmetry that undesirable adverse effects, such as birefringence, are absent within the crystal. Therefore, as linearly polarized light propagates through the crystal, in the absence of an applied magnetic field, there is no phase delay in the transverse electric and magnetic field polarizations. The undoped YAG single crystal rods used in this experiment were grown via the Czochralski process by United Crystals. A total of six crystals with a diameter of 5 mm and a length L c = 18 mm were produced. The crystals were cut along the 111 axis and the two ends of the rods were polished for high optical purity. For three of these crystals, including the one used for the measurements reported here, a broadband anti-reflective coating centered at 1342 nm was applied by United Crystals to the end facets to reduce reflective scattering losses when inserted into the ring laser cavity. 4.3 Measurement Setup The setup used in this experiment is similar to that of reference [72] and is shown in figure 4.2. As mentioned before, to measure the Verdet constant of undoped YAG, we must first determine the optical activity of the crystal. To do so, we measure the rotation of the polarization of light as it propagates through the crystal in the presence of a known magnetic field One of the sources of laser light used in this experiment is derived from a DL100 External Cavity Diode Laser (ECDL) manufactured by Toptica Photonics. For an ECDL in the Littrow configuration, the output of the laser diode is collimated with a short focal length aspheric lens and is incident on a diffraction grating. The first order of diffraction from this grating is coupled back through the collimation lens and into the laser diode cavity while the zeroth order continues on for use. By rotating the angle of the diffraction grating, the wavelength of the laser can be tuned from approximately 1300 nm to 1350 nm for this laser. The other source of laser light used in this experiment is derived from a commercial Nd:YAG non-planar ring oscillator (NPRO) laser. This laser was originally manufactured by Lightwave Electronics (part number 126N ), now a subsidiary of JDS Uniphase. This single mode laser outputs 500 mw of light at 1064 nm has a linewidth of less than 5 khz (5 ms integration time).

61 43 DET DET PBS M P FR HWP ECDL Figure 4.2. Light from an external cavity diode laser (ECDL) passes through a halfwave plate (HWP) and polarizer (P) pair. A double Fresnel rhomb (FR) is used to adjust the polarization of the light incident on a cylindrical magnet (M). The light is then split by a polarizing beam splitter (PBS) before falling on two photodetectors (DET) which record the power of each beam. To make the measurement, radiation from either source was coupled into the APC end of an APC/FC polished panda style polarization maintaining fiber purchased from Thorlabs (part number PM980-XP). This fiber has a lower wavelength cutoff of 920 ± 50 nm, well below the desired 1064 and 1342 nm wavelengths that we are interested in. The fiber has a numerical aperture of 0.12 and a mode field diameter of 6.6 ± 0.7 μm at a wavelength of 980 nm. An APC/FC fiber was used to prevent power fluctuations in the transmission of the fiber due to etalon effects. Additionally, by coupling into the APC end, the laser source (and summarily, the test wavelength) could be changed without affecting the steering of the beam after the light is launched out of the fiber. After the light is launched from the output of the fiber, it first passes through a half-wave plate and a Glan-Taylor polarizer oriented at a fixed 45 with respect to vertical. By rotating the half wave plate, we ensure a constant 300 μw of power for each measurement of the polarization rotation. Next, a double Fresnel rhomb, acting as another half wave plate, is used in conjunction with a polarizing beam splitter to split the laser light into two beams of equal power. Unlike a standard zero or multi-order wave plate, the phase delay of the Fresnel rhomb is wavelength insensitive over several hundreds of nanometers, making it an ideal

62 44 device when taking measurements over a wide range of frequencies. To measure the optical activity of the YAG crystal, a magnet with uniform magnetization along its cylindrical axis and a hole bored down its center containing the crystal is inserted between the Fresnel rhomb and the polarizing beam splitter. The end facets of the crystal have a broadband anti-reflective coating to minimize reflective losses of the incident laser light. After the beam splitter, the beams are then incident upon a pair of balanced photodetectors and the power of each is simultaneously recorded. These power measurements are then used to determine the optical activity of the crystal. Linearly polarized light incident at 45 on a polarizing beam splitter relative to its s and p axes, in the absence of any rotation, will split into two beams of equal power. That is to say, I 1 = I 2 = I 0 /2 where I 1 and I 2 are the intensity of each beam as measured on detectors 1 and 2 respectively and I 0 is the total intensity incident on the beam splitter. For a small rotation of the polarization of the beam, a change in intensity equal to I 1 = I o 2 (1 2dφ),I 2 = I o (1+2dφ) (4.4) 2 will occur [73]. As a result, the difference in the voltage output of the two detectors, proportional to the difference of the incident intensities, assuming identical transimpedance gains, is V 1 V 2 2I o dφ. (4.5) Therefore, by measuring the voltages of each detector, the small rotation angle (optical activity) can be determined by dφ 1 V 1 V 2. (4.6) 2 V 1 + V 2 The balanced photodetectors used in this experiment are a commercial device also purchased at Thorlabs (model number PDB-150C). This switchable gain detector is made from InGaAs and has a wavelength response ranging from 800 to 1700 nm. The active area of each detector is 0.3 mm in diameter and has a max response of 1 Amp/Watt. Each detector also has a fast monitor output that is proportional to the power incident on the photodetector faces. It is this fast mon-

63 45 itor output that we record to measure the optical activity (and thus the Verdet constant) of the undoped YAG. To record the output from the fast monitors, we employ the use of a U6 Data Acquisition device manufactured by the LabJack Corporation. This data acquisition card contains a 16 bit analog to digital converter and interfaces with a computer via a USB connection. To record the data, a simple LabVIEW program was written to sample the analog outputs of the fast monitors 100 times at a rate of 10 Hz. The average and standard deviations of these measurements are then used to calculate the rotation angle of the polarization vector, as per equation Cylindrical Magnet As mentioned before, the magnetic field applied to the undoped YAG crystal is generated by a right hollow cylindrical magnet of uniform axial magnetization. The magnet used for this experiment is a rare earth N50 neodymium magnet. As diagrammed in figure 4.3a, the magnet is of length L =19.0 mm and has an outer diameter 2b =22.2 mm. The inner bore of the magnet has a diameter 2a =6.3 mm. To determine the field along the cylindrical axis of the magnet, we begin our analysis with Maxwell s equations, notably B=0 and H=0, and continue via the analysis of [74]. Using the relation B=H+4π M, it can be readily shown that 2 Φ=4π M (4.7) where Φ represents the magnetic scalar potential. For a right-cylindrical permanent magnet with uniform magnetization along its axis, M vanishes everywhere except on the pole faces. Solving equation 4.7 using a spherical coordinate system, the potential at each pole face obeys Laplace s equation 2 Φ = 0 and can be represented as Φ(r, θ) = inf l=0 [ Al r l + B l r ( l+1)] P l (cos θ) (4.8) where P l (cos θ) is a Legendre polynomial of order l. The coefficients A l and B l are determined by the axial boundary conditions of equation 4.8, which is determined

64 46 Position (meters) a L L c 2b (a) Position (meters) (b) Magnetic Field (Tesla) Position (meters) Figure 4.3. (a) Physical layout and dimensions of the right hollow cylindrical magnet housing the undoped YAG crystal. The crystal, of length L c =18.0mm is located inside the magnet having length L =19.0 mm and an outer diameter 2b =22.2 mm. The inner bore of the magnet has a diameter 2a =6.3 mm. (b) The measured magnetic field of the magnet at various distances from the end facets. The dashed line represents a fit to the measured data for the magnetization M. by using the Green s function method to be of the form [ Φ(r, θ) =2πM r2 + R 2 r] (4.9) where R is the radius of the pole face. Accounting for the contribution of the field due to the superposition of each of the four pole faces (two ends of radius a with two holes of radius b), we find the total magnetic scalar potential along the

65 47 symmetry axis to be Φ(z,0) = 2πM + ( a 2 + z L ) 2 ( + b z L ) 2 2 ( a 2 + z + L ) 2 ( b z + L ) 2 (4.10) 2 Finally, taking the gradient of the magnetic scalar potential yields the magnetic field along the symmetry axis of the magnet z B z =2πM L z L a 2 +(z L2 )2 b 2 +(z L 2 )2 z + L z + L (4.11) a 2 +(z + L2 )2 b 2 +(z + L 2 )2 To determine the magnetization of the magnet, the magnetic field was measured using a standard gaussmeter at several locations along its symmetry axis extending out from the magnet faces. The measured magnetic field is shown in figure 4.3b. We then fit these measurements with equation 4.11, shown by the dashed line in figure 4.3b. From this fit, we determine the magnet has a magnetization M=0.125 T for our hollow cylindrical magnet. Given the value for the magnetization, it now becomes possible to determine the total applied magnetic field along the symmetry axis of the magnet over the length of the undoped YAG crystal. Assuming the crystal is centrally located within the bore of the magnet, we can integrate equation 4.11 over the length of the crystal. Doing so yields a total applied magnetic field of T m applied along the length of the crystal. 4.5 Verdet Measurement With the measurements of the rotation angle, dφ, of the polarization vector as a function of wavelength and knowledge of the integrated applied magnetic field, it

66 48 Figure 4.4. Verdet constant of undoped YAG in the near infrared. Each data point represents the average of one hundred measurements. The inset provides a closer look at the 1300 nm to 1350 nm range. Error bars are indicative of the standard error of the mean. The dashed line is a fit to the data as per equation 4.3, demonstrating the dispersive nature of the Verdet constant. now becomes possible, via equation 4.1 to determine the Verdet constant of the undoped YAG crystal as a function of wavelength. Figure 4.4 shows the Verdet constant as a function of wavelength from 1300 to 1350 nm as well as at 1064 nm. Each data point represents the average of one hundred measurements with error bars indicating the standard error of the mean. The Verdet constant from 1300 nm to 1350 nm was found to be of the order 1.38 rad/t m. Additionally, the optical activity of the YAG crystal at 1064 nm corresponds to a Verdet constant of 2.12 rad/t m. As mentioned in section 4.1, the Verdet constant is dispersive in wavelength. The dashed line in figure 4.4 is a fit of equation 4.3. From this fit, we find that our undoped YAG crystal has an energy band gap value of 8.1 ev ± 0.3 ev and a K- parameter of 596 rad/t m ± 52 rad/t m, consistent with previous measurements made with visible light[64].

67 Conclusion The optical activity of undoped YAG was measured via a pair of balanced photodetectors and the field of a hollow cylindrical magnet containing the crystal was determined. From these two results, we report, for the first time, values for the Verdet constant of undoped YAG in the near-infrared. By fitting this data with an analytical coupled oscillator theory, we can extract values for the energy band gap and K-parameter for the undoped YAG crystal and compare them with previous measurements. With this knowledge, it now becomes possible to construct an appropriate Faraday rotator for insertion into a high power ring laser cavity to drive unidirectional operation.

68 Chapter 5 Constructing a 1342 nm Ring Laser As extensively discussed in chapter 2, many of the experimental investigations undertaken in our atomic physics laboratory revolve around the cooling, trapping, and manipulation of fermionic 6 Li. As one would expect, one of the vital tools needed to conduct these experiments is a source of coherent laser radiation whose wavelength is resonant with the atomic sample. For 6 Li, the D 1 and D 2 spectroscopic lines lie at a wavelength of 671 nm. Therefore, for experiments involving ultra-cold 6 Li gases, we require several laser sources near 671 nm with sufficiently high power and good spectral qualities. For many ultra-cold atomic gas experiments, with species such as Rubidium and Cesium, a tunable Ti:Saph laser is sufficient to generate the required wavelengths of light. However, in the case of 6 Li, the emission spectrum of Ti:Saph lasers are such that the gain is not high enough at 671 nm to be a viable source of radiation. Historically, dye lasers have been used to generate the required wavelength. In fact, the first experiments conducted in our research group were carried out with a Coherent-899 dye laser pumped by a frequency doubled Nd:YAG solid state laser. While these lasers are very versatile, producing over 1 Watt of power and having a tuning range of several hundred nanometers, the spectral linewidth properties of the emitted light are limited to a few MHz at best. Only by implementing several aftermarket improvements, such as dye dampeners and external cavity locking schemes, has the linewidth of these lasers been reduced. Additionally, power stability issues and intensity fluctuations lead to shot-to-shot variation in the number of atoms trapped in subsequent experimental runs.

69 51 A more recent solution to this problem arrived with the development of semiconductor chip based tapered amplifier systems. These amplifier chips, when seeded with tens of milliwatts of power from a seed laser, can produce 500 mw of 671 laser light. Since these systems are diode based, the spectral output of the beam is a few hundred khz. However, while the laser used to seed these amplifiers has a TEM 00 transverse mode profile, often the output of the tapered amplifiers is a mix of TEM 00 and higher order modes. Because of this, the coupling efficiency of light from a tapered amplifier into a fiber optic waveguide is reduced to 50% or less, limiting the available power for the experiment. While two or more laser-amplifier systems could be used in parallel to generate the required power, a different solution to this problem may be prudent. Here we present another solution for generating light at 671 nm namely, a solid state laser system. Solid state lasers are much more stable in both frequency and power than dye lasers and the transverse mode profile of the output laser beams can be spatially manipulated to be a TEM 00 mode. Power outputs of multiple Watts have been achieved in solid state systems, making them an ideal candidate for the construction of a 671 nm laser source. The only problem with a solid state laser solution is the fact that there are currently no known gain crystals that have a 671 nm lasing transition. However, Nd:YVO 4, whose energy level diagram can be seen in figure 5.1, may present a solution to that problem. Nd:YVO 4 as a lasing medium is a four level laser system, with a 4 I 9/2 4 F 5/2 pump absorption line at 808 nm and two emission lines ( 4 F 3/2 4 I 13/2 and 4 F 3/2 4 I 11/2 ) at 1342 nm and 1064 nm respectively. We have constructed a double end pumped solid state ring laser using this Nd:YVO 4 crystal to produce over 3 Watts of power at 1342 nm. An artistic rendering of this cavity can be seen in figure 5.2. Having demonstrated this power at 1342 nm, we should then be able to frequency double the laser output by locking this laser to a frequency doubling cavity containing a non-linear optical crystal. While we have not demonstrated a very high conversion efficiency as of yet, with a well-optimized doubling cavity we could expect to have a 50% conversion efficiency, producing over 1.5 Watts of power at 671 nm. During the construction of this laser we also developed a novel technique of self-injection [75]. At 1342 nm, high power optical isolators are not commercially available for insertion into the cavity. Using this novel technique, a small amount

70 52 4 F 5/2 4 F 3/2 808 nm 880 nm 1064 nm 1342 nm 4 I 15/2 4 I 13/2 4 I 11/2 4 I 9/2 Figure 5.1. Energy level diagram of Nd:YVO 4. The crystal has a 4 I 9/2 4 F 5/2 pump absorption line at 808 nm and two emission lines ( 4 F 3/2 4 I 13/2 and 4 F 3/2 4 I 11/2 ) at 1342 nm and 1064 nm respectively. of the output power of the laser is mode matched back into the cavity to drive unidirectional operation. Doing so allows us to remove nearly all the intracavity elements of our laser. By reducing the number of elements located inside the cavity, not only will the laser have a higher output power (due to a reduction in intracavity scattering losses), but the cavity length can be reduced resulting in a larger free spectral range. For our laser system having high gain, this additional power due to the reduction of scattering loss is not significant. However, this technique is broadly applicable to all ring lasers. Additionally, we demonstrate tunability of the system via an etalon, both within the cavity itself as well as in the path of the light injected back into the laser. Section 5.1 will detail the overall design and layout of the ring laser cavity. Section 5.2 will discuss the dichroic mirrors used in pumping our laser crystal as

71 53 Figure 5.2. Artistic 3D rendering of the Nd:YVO 4 laser cavity. well as detailing measurements of the reflectivity of several output couplers used throughout these investigations. Section 5.3 will provide a discussion about the physical properties of the Nd:YVO 4 gain crystal used in our setup. Section 5.4 will discuss the methods employed to pump the Nd:YVO 4 crystal while section 5.5 will discuss our setup employed to cool the crystal. Section 5.6 will detail additional components that could be inserted into the laser cavity for various reasons. Finally, section 5.7 will present a paraxial resonator analysis of our cavity, followed by a concluding section Cavity Layout Since the goal of this project was to construct a single frequency laser that can be implemented in an ultra-cold atomic physics experiment, cavity linewidth and frequency noise considerations played an important role in the design and construction of the laser system. To this end, it was decided that the laser cavity itself would be mounted within an enclosure that had been machined out of a single block of aluminum. In addition, a top plate aluminum lid was also employed

72 54 M1 Direction M2 Pump Pump M3 Nd:YVO4 M4 Figure 5.3. Experimental setup for the laser cavity. A Nd:YVO 4 crystal located inside of a bow-tie ring cavity is double end pumped by two 25 Watt diode arrays. The cavity consists of a fully reflecting mirror (M1), an output coupler (M2), and two dichroic mirrors (M3 and M4) which reflect light at 1342 nm while transmitting light at 808 and 1064 nm. With nothing to break the symmetry of the system, the laser will lase bidirectionally, resulting in the gain being shared by the clockwise and counterclockwise modes with through holes that enabled the mirrors of the laser cavity to be anchored, not only to the heavy aluminum base, but to the lid as well. This increased the mirror stability and vastly improved frequency noise while reducing mode hopping in the output beam. The laser cavity layout is shown in figure 5.3. Four reflectors with infinite radii of curvature form a folded bow-tie ring cavity around the gain medium. Since we ultimately require single mode operation, a ring cavity configuration was chosen over a standing wave cavity to prevent gain competition (and thus multimode operation) due to spatial hole burning in the gain medium[76, 77]. The tradeoff in using a ring cavity geometry is that ring lasers support bidirectional lasing operation, with the gain being shared between the two symmetric modes of oscillation, namely clockwise and counterclockwise traveling waves in the laser cavity. Unless an effort is made to break the symmetry of the laser oscillation, there will be two output beams emitted from the output coupler and the total power of the laser will be shared between the two. In addition to choosing a ring geometry to promote single frequency operation, we also wanted to minimize the total round trip path length of the cavity as well.

73 55 For a ring laser, the free spectral range (FSR) of a cavity is given by FSR = c L (5.1) where L represents the round trip path length of the cavity and c is the speed of light. For our bow-tie cavity, L = meters, corresponding to a FSR of 1.40 GHz. This means that the stable modes of oscillation for this laser will be spaced, in frequency, by 1.40 GHz. For the 1342 nm wavelength that we are interested in, this FSR corresponds to a Δλ value of nm. Therefore, if the gain profile is sharply peaked, or can be made so by the use of an etalon, the laser will be forced to operate in single mode. 5.2 Reflectors Here we describe the four reflectors that make up the bow-tie ring cavity (mirrors M1 M4 in figure 5.3). Mirror M1 is a highly reflective (taken to have a reflectivity R=1) with infinite radius of curvature. This mirror has a broadband dielectric coating covering the 1342 nm transition that we are primarily interested in. Mirrors M3 and M4 are dichroic mirrors with infinite radii of curvature and were manufactured by Advanced Thin Films as well. Each of these mirrors has a dielectric coating that either transmits or reflects light depending on the wavelength. Because these mirrors need to allow pump laser light to transmit through to the crystal gain medium, they have an anti-reflective coating centered at 808 nm. Additionally, because we do not want the cavity to lase on the 4 F 3/2 4 I 11/2 transition of Nd:YVO 4, they also have an anti-reflective coating centered at 1064 nm as well. Finally, because we are interested in the 4 F 3/2 4 I 13/ nm transition, the mirrors have a highly reflective coating at this wavelength. Because of space limitations and cavity design, these two mirrors needed to be mounted as close to the gain crystal as possible while still maintaining an unobstructed path length. To accomplish this, a pair of custom mirror mounts were manufactured by the Engineering Machine Shop at Penn State. Although these mirrors mounts do not allow for the adjustments that a traditional kinematic

74 56 Label 0 R@15 δ 70% % Laser # # % % Table 5.1. Measured reflectivities of available output couplers for our homemade 1342 nm laser. Measurements were made at normal incidence and at 15 for vertically polarized light. The δ value of each output coupler is another way of representing its reflectivity (also at 15 ). For a further explanation, see the text. mirror mount would, they were designed to have slight adjustments in their position via set screws when mounted in the laser cavity. Machine drawings for these two mirror mounts can be seen in figures 5.4 and 5.5. To complete the cavity, a partially reflecting mirror M2 is used as an output coupler for the laser. Seven different output couplers (all flat with infinite radii of curvature) were available and used in the experiments that follow. Each of these output couplers were manufactured by CVI Laser. To experimentally determine the reflectivity, R, of each output coupler, the transmission of each output coupler was measured using vertically polarized light from an external cavity diode laser (Toptica DL100) at a wavelength of 1342 nm. By measuring the incident and transmitted power of this test laser for each output coupler, the transmission, T, for each is determined. Using the simple relation R+T =1, we thus determine the reflectivity. These measurements were made for both normal incidence (θ=0 )as well as at an angle of 15. The normal incidence measurement was used to verify the quoted reflectivity of the output coupler while the 15 measurement was used to determine the reflectivity under normal experimental conditions. The results of these measurements are summarized in table 5.1. Also reported in table 5.1 is the reflection coefficient of each output coupler using δ notation for mirror reflectivities, where R = exp[ δ] [78].

75 TAP THRU 1 HOLE /8 DRILL THRU C'BORE FOR #4 SOC. HD. (7/32 C'BORE, 3/16 DP.) 2 HOLES /2 C'BORE WITH 3/8 C'BORE OFFSET BY TITLE DICHROIC MIRROR MOUNT I 1 ALUMINUM SIZE CAGE CODE DWG NO REV SCALE 1:1 CONTACT SHEET O'HARA Figure 5.4. Machine drawing for homemade mirror mount for the dichroic mirror M3 in our 1342 nm laser cavity.

76 58 D C B A /2 C BORE WITH 3/8 C BORE OFFSET BY /8 DRILL THRU C BORE FOR #4 SOC. HD. (7/32 C BORE, 3/16 DP.) 2 HOLES A A TAP THRU 1 HOLE SECTION A-A TITLE DICHROIC MIRROR MOUNT II MATERIAL QTY. 1 ALUMINUM SIZE CAGE CODE DWG NO A4 SCALE 1: REV CONTACT O HARA SHEET D C B A Figure 5.5. Machine drawing for homemade mirror mount for the dichroic mirror M4 in our 1342 nm laser cavity.

77 Nd:YVO 4 Crystal The gain crystal used to construct our 1342 nm solid state laser was a 0.27% neodymium doped yttrium ortho-vanadate (YVO 4 ) crystal. First recognized as an important laser material in 1966 [79], yttrium ortho-vanadate is an excellent host material for laser crystal construction due to its physical and optical properties. The structure of the YVO 4 crystal is zircon tetragonal (tetragonal bipyramidal) [3], and shown in figure 5.6. The lattice parameters of the crystal are a = b = Å and c = Å [80]. Because of this internal structure, YVO 4 is naturally birefringent. The Sellmeier equations [81] are n 2 e = λ λ2 n 2 o = λ λ2 (5.2) where λ is the wavelength in units of μm. Because of this natural birefringence there will be a polarization dependence to the laser gain [65], a fact that we will exploit to drive unidirectionality in our ring laser. Also, this polarized output avoids undesirable thermally induced birefringence in our laser. One potential drawback to YVO 4 crystals are their low thermal conductivity. Parallel to the c-axis, the thermal conductivity coefficient is 5.23 W m 1 K 1 while perpendicular to the c-axis, the coefficient is 5.10 W m 1 K 1 [80]. These values are about one third those of YAG and prevent good heat dissipation within the crystal. As a result, power scaling may be limited in these crystals due to thermal damage. Also, this lack of dissipation leads to thermal lensing within the crystal [71, 82]. As shown in the energy level diagram of figure 5.1, Nd:YVO 4 has a 4 I 9/2 4 F 5/2 pump absorption line at 808 nm. Nd3+ ions in YVO 4 have a large absorption cross section at this wavelength, with a peak value of about cm 2 [83] Figure 5.7 shows the absorption spectrum of a 0.27% doped Nd:YVO 4 crystal in the region of the 808 nm band, reproduced from [4]. At this concentration, this plot shows a peak absorption coefficient of approximately 9.4 cm 1 at nm. As for stimulated emission from the Nd:YVO 4 gain crystal, the 4 F 3/2 4 I 13/2

78 60 Figure 5.6. The crystal structure of an yttrium ortho-vanadate. Yttrium orthovanadate crystalizes as a zircon tetragonal (tetragonal bipyramidal) structure, leading to a natural birefringence along its a and c axes. Shown centered is the yttrium ion surrounded by its vanadium and oxygen neighbors. This figure is adapted from [3]. transition at 1342 nm, though not as strong as the 1064 nm 4 F 3/2 4 I 11/2 line, still has a stimulated emission cross section value of 6± cm 2 [84]. Coupled with a high fluorescent lifetime of 110 μs, such a crystal seems well suited for the construction of a 1342 nm solid state laser. The Nd:YVO 4 crystals used to construct the laser described in this chapter were manufactured by Coretech Crystal, a division of Shanda Luneng Information Technology Co., Ltd. in Shandong China. The crystals were rectangular prisms in shape, 10 mm in length with ends measuring 3 mm by 3 mm square. The crystals were cut along their a-axis, with the c-axis being rotated by 45 degrees relative to the flat 3 x 10 mm faces. The crystal is mounted such that the c-axis is vertical. Because of the natural birefringence and polarization dependence in the ortho-vanadate crystal, this particular rotation and orientation of the crystal establishes the polarization axis of the laser to be in the vertical direction. For the desired laser output wavelength of 1342 nm, the crystals were doped with neodymium. It has been shown that the pump power fracture limit of Nd:YVO 4 is inversely proportional to the doping concentration [85]. Because of

79 61 Figure 5.7. The absorption spectrum of a 0.27% doped Nd:YVO 4 crystal in the region of the 808 nm band, reproduced from [4]. This plot shows a peak absorption coefficient of approximately 9.4 cm 1 at nm. this, for higher power operation with pump sources at the tens of Watts level, as is our case, the optimum concentration for a Nd:YVO 4 crystal is in the range of 0.25% to 0.4% at. [86]. It is for this reason that we chose to use a doping concentration value of 0.27% at. The end facets of the gain crystal were also coated with anti-reflection coatings at 808 nm, 1064 nm, and 1342 nm. The 808 nm coating prevents reflective losses of the pump light incident upon the crystal, enabling more power to be absorbed and thus higher output laser powers. The 1064 and 1342 nm coatings are included to prevent reflections of spontaneously emitted light at the crystal-air interface. The YVO 4 crystal has an index of refraction of 2.18, leading to Fresnel reflections of approximately 13.7% at the surface. If no measures are taken to prevent these reflections, the crystal itself can act as a standing wave cavity and begin to lase, reducing the excited state population of the gain available for the larger ring cavity. The crystal is mounted in a copper mount shown in figure 5.8 and held in place by a cover (figure 5.9). Between the crystal and the mount is a layer of indium foil that provides a compressible barrier to prevent stress fractures of the crystal while providing a good thermal contact between the mount and the crystal. The copper mount used to hold the crystal is also part of the cooling system described in section 5.5.

80 * / TAP 1/4 DEEP 2 PLACES * #41 DRILL 1/4 DP TAP THRU 4 PLACES TAP 3/8 DP. 4 PLACES TITLE GAIN MEDIUM MOUNT MATERIAL QTY COPPER CONTACT 1:1 O HARA SCALE PHONE Figure 5.8. Copper mount used to hold the Nd:YVO 4 crystal inside the laser cavity. One end of the mount is connected to a thermo-electric cooler used to extract heat from the crystal (see section 5.5. The other end of the mount holds the crystal in a narrow finger-like region, allowing the crystal to be located in the cavity without obstructing the beam path.

81 63 SECTION A-A (SCALE 2:1) A A #48 (0.076) DRILL THRU C BORE FOR #1 SOC. HD. (5/32 C BORE, DEEP) 2 HOLES * * / * TITLE GAIN CRYSTAL COVER MATERIAL COPPER QTY. 1 SIZE CAGE CODE DWG NO REV A4 SCALE 1:1 CONTACT O HARA SHEET Figure 5.9. Copper cover used to sandwich the gain crystal against the copper mount.

82 Pump Laser The pump light used to excite the gain crystal is derived from a DUO-FAP laser system. This laser system is a commercially available product from Coherent and can produce 60 total Watts of power (two 30 Watt beams) at a wavelength of 808 nm. This wavelength is resonant with the 4 I 9/2 4 F 5/2 absorption line for Nd:YVO 4. The spectral width of the laser light is quoted as being less than 3 nm, with an intensity noise of less than 1% rms. Inside the DUO-FAP laser, light from two laser diode arrays are coupled into two 800 μm diameter multi-mode fibers protected by an armored jacket. These fibers are then used to steer the beam to double end pump the crystal as shown in figure 5.3. To launch the light from the fibers and onto the crystal, two coupled lens mounts are used for each fiber. The first mount, shown in figure 5.10, is used as an adapter and spacer, providing an sma connection to the armored jacket fiber as well as a connection to the homemade lens mount shown in figure Inside this homemade lens mount are two lenses of focal length 60 mm and 30 mm respectively. By locating the lenses such that the first is 60 mm from the end of the fiber sma connector and the second is 30 mm from the center of the gain crystal, a 2:1 imaging system is used to image the pump light from the DUO-FAP laser onto the crystal gain medium. Because of the lenses used in this imaging system, the beam spot size on the crystal is approximately 400 μm. To properly align the pump light on the center of the gain crystal, each mount, shown in figure 5.10, was mounted to another homemade laser mount, shown in figure This second mount then connects to an adapter plate, shown in figure 5.13, which is connected to a stainless steel gothic-arch translation stage (model 9061-XYZ) from New Focus. These translation stages have a 25 by 25 mm square platform that the adapter plate mounts on to. These translation stages allow for 6.25 mm of travel in each of the x, y, and z directions. Used in combination with Model 9354 actuators (also from New Focus), this allows for precision alignment of the pump laser beams onto the gain crystal for maximum output laser power.

83 65 M dia SMA Connector Figure Adapter for connecting the sma fiber from the DUO-FAP laser to the homemade lens mount for the 2:1 pump imaging system.

84 66 A A SECTION A-A 0.787" (20 MM) OUTSIDE DIAMETER M20x1.25 THREAD 1.008" DIA " - 40 TAP TITLE PUMP LENS MOUNT MATERIAL ALUMINUM QTY. 2 SIZE CAGE CODE DWG NO REV A4 SCALE 1:1 CONTACT O HARA SHEET Figure Homemade lens mount used to house the imaging optics for focusing the 808 nm pump laser light onto the Nd:YVO 4 gain crystal. Two lenses, of focal lengths 60 and 30 mm are located so as to provide a 2:1 imaging system, focusing light from the 800 μm diameter pump laser fibers to a diameter of 400 μm on the gain crystal itself.

85 *SLIP FIT NO PLAY WITH STAINLESS STEEL TUBE PROVIDED * TAP THRU 6 HOLES TITLE OPTICAL DIODE MOUNT MATERIAL ALUMINUM QTY. SIZE CAGE CODE DWG NO REV A4 SCALE 1:1 CONTACT O HARA SHEET Figure This home built mount is used to hold the lens mount shown in figure By doing so, it allows the lenses used for imaging the pump laser light onto the gain crystal to be precisely positioned via a translation stage connected to this mount by the adapter plate shown in figure 5.13.

86 /32 DRILL THRU 5/16 C BORE 0.1 DEEP 4 HOLES TAP THRU 4 HOLES TITLE QTY. ADAPTER PLATE 1 MATERIAL ALUMINUM SIZE CAGE CODE DWG NO REV A4 SCALE 1:1 CONTACT O HARA SHEET Figure Machine drawings for an adapter connecting the mount in figure 5.12 to the stainless steel gothc-arch xyz translation stage.

87 Cooling the Crystal Because Nd:YVO 4 is a four level laser with a 4 I 9/2 4 F 5/2 absorption line at 808 nm and a 4 F 3/2 4 I 13/2 emission line at 1342 nm, there exists an energy difference between the excited pump state and the upper level of the lasing transition (72 nm) as well as a difference between the lower level of the lasing transition and the ground state of the system (427 nm). These so called quantum defects have an adverse affect on laser operation as these wavelength mismatches are converted to heat via phonon excitation [87]. It should be noted that the term quantum defect is ambiguous and is unrelated to Rydberg atom physics. Additionally, while the YVO 4 crystal is highly transmissive at 1342 nm, its transmission is not perfect, leading to additional heating of the crystal as well. Therefore, because of the power levels involved with the construction of this laser, actively cooling the laser crystal is necessary. As mentioned in section 5.3, the crystal is located at the end of a copper finger that serves as a mount to hold its position in the laser cavity. In addition to this function, the mount also plays a role in actively cooling the crystal, as the other end of the mount is connected to a larger copper block via a thermo-electric cooler (TEC) that acts as a thermal mass. The TEC used in this experiment is a UT8,12,F2,2525,TA,W6, UltraTEC manufactured by Melcor Corporation, now a division of the Laird Group. This TEC is 2.5 cm square and has the capacity to absorb 69 Watts of power at its cool face. For our laser setup, we operate the TEC at a constant 4 amps of current. Simply cooling the laser crystal via a TEC to the copper block reservoir would not be an effective solution, however, unless heat in that reservoir can be also be extracted through other means. This extraction occurs by way of an innovative setup [5] involving two metal heat sinks and a water cooled mount, depicted in figure As can be seen in the picture, the heat sink on the copper reservoir is located in very close proximity to the heat sink connected to another copper mount, this one cooled by a circulating water system. The spaces in between the nearly touching heat sinks are filled with standard ceramic heat sink paste that acts as a medium for the transfer of heat without creating a solid physical connection. The reason such lengths are taken to isolate the two heat sinks from

88 70 Copper Thermal Reservoir Water Cooled Mount Figure Cartoon showing the interface between the copper thermal reservoir and the water cooled mount [5]. Two heat sinks are located in very close proximity to one another so as to enable heat transfer without making a physical connection. This enables heat to be transferred across the interface without coupling any vibration from the water cooled mount into the copper thermal reservoir (and subsequently, the laser gain crystal). direct physical contact is because the flow of water through the copper mount can cause vibration in the cooling system mounts, which can then be transferred to the crystal. These crystal vibrations have been observed to be strong enough to cause frequency noise on the output of the laser beam. Finally, the water cooled copper block is cooled by a rack mounted solid state thermoelectric thermal control unit manufactured by ThermoTech. This thermal control unit circulates an 80/20 mixture of distilled water and corrosion inhibited Glycol at a temperature of -5 C and is itself cooled by the chilled water line of the building. In all, this cooling system enables to maintain the crystal at a temperature of approximately 18 C when operating at full pump power capacity. 5.6 Additional Components The only other permanent optical element located inside the cavity besides the Nd:YVO 4 crystal is a 710 μm diameter spatial pinhole filter located halfway between mirror M1 and output coupler M2. At this location in the resonator cavity, the circulating beam has a waist of 148 μm and an infinite radius of curvature (see section 5.7). In the absence of this filter, the laser output was observed to be a mixture of Hermite-Gaussian modes, as determined by the transverse mode measurement described in section 6.2. This spatial filter ensures a high quality

89 71 transverse mode structure to the output laser beam by increasing the losses for those higher order modes. 5.7 Cavity Modeling To determine the full characteristics of the beam propagating inside of the ring laser cavity, as well as the thermal lensing of our Nd:YVO 4 crystal due to the two pump beams, we perform a paraxial resonator analysis (ABCD analysis) for the elements contained within the ring laser cavity assuming a lowest order Hermitegaussian mode [88]. For the laser cavity layout shown in figure 5.3, the two main components of the laser cavity that need to be modeled in this analysis are the contributions of free propagation through a uniform medium as well as that of thin lenses. The corresponding ABCD matrices for each of these two elements are detailed in figure For this analysis, we determined the ABCD matrix corresponding to the elements inside of our laser cavity for one complete round trip. Choosing the center of our Nd:YVO 4 crystal as our reference plane, a round trip in the cavity consists of the following elements: 1) 5 mm of free space propagation in a medium with an index of refraction of 2.18, 2) a thermal lens of unknown focal length, 3) 192 mm of free space propagation in air (index of refraction of 1.0), 4) A second thermal lens of unknown focal length, and 5) another 5 mm of free space propagation in a medium with an index of refraction of From this round trip ABCD matrix, we can define reference values for the radius of curvature, R 0, and spot size, w 0 such that and w 0 = R 0 = 2B D A (5.3) Aλ 1 (5.4) π 1 m 2 where m A+D 2 and λ is the wavelength of the light. From these values, we can then create a complex q-parameter for our system q 0 =( 1 R 0 i λ πw 0 ) 1. (5.5)

90 72 Uniform medium Index n and length d d 1 0 d/n 1 Thin lens Focal length f f 1-1/f 0 1 Figure ABCD matrices used for a paraxial resonator analysis of our ring laser cavity. Included are matrices for propagation of length d in a uniform medium with index of refraction n as well as a thin lens of focal length f. From the complex q-parameter for the reference plane, we can then determine q(z) for any arbitrary position inside the cavity q(z) = A(z)q 0 + B(z) C(z)q 0 + D(z) (5.6) where A(z) is the A component of the ABCD matrix for an arbitrary location z from the reference plane (similarly for B,C, and D). Finally, with this q(z), the waist and radius of curvature can be determine for that arbitrary position inside the cavity λ w(z) = 1 Im( ) (5.7) q(z)π and 1 R(z) = Re( 1 ) (5.8) q(z) To experimentally determine the unknown values of the effective focal lengths due to thermal lensing at each end of the gain crystal, we perform a measurement of the beam size and mode properties, as discussed in section 6.2. From these measurements, for a location halfway between mirror M1 and output coupler M2, the beam was determined to have a waist of 148 μm and infinite radius of curvature. Therefore, using this measured value in our complex ABCD calculation above, we find that the only solution that would yield this waist would be for the thermal lenses at the ends of the Nd:YVO 4 to have an effective focal length of cm. With this knowledge, it now becomes possible to determine the waist and radius

91 Waist (meters) Position (meters) Figure The waist of the gaussian beam for one round trip of propagation inside the ring laser cavity. The zero position reference is taken to be the center of the gain crystal. of curvature of the beam at any arbitrary location inside of our ring laser cavity. Figure 5.16 shows the waist of the gaussian beam as it propagates through one round trip of the cavity, beginning at our reference point located at the center of the Nd:YVO 4 crystal. From this analysis, the waist of the beam at two critical locations can be determined. First, as was directly measured, at a location halfway between mirror M1 and output coupler M2, the waist is 148 μm. Second, as the beam propagates through the active gain crystal, its waist has a value of 316 μm, an important value in determining the expected power out of the laser cavity. 5.8 Conclusion In this chapter, we have described the layout and construction of a four mirror bow-tie ring laser cavity for application to experiments involving the cooling and trapping of ultra-cold fermionic 6 Li gasses. A 10 mm long a-cut 0.27% at. doped Nd:YVO 4 crystal with its c-axis rotated to the vertical position was used as the gain medium for this laser. The crystal was double end pumped by two fiber coupled 25 Watt diode arrays resonant with the 808 nm 4 I 9/2 4 F 5/2 absorption line. The reflectors used to build the cavity were dielectrically coated so as to be transmissive

92 74 at both 808 and 1064 nm while being highly reflective at 1342 nm. These coatings suppressed lasing in the cavity of the 4 F 3/2 4 I 11/2 emission line at 1064 nm while enabling the laser to operate on the 4 F 3/2 4 I 13/2 emission line at 1342 nm. An extensive multi-step cooling system was employed to maintain a reasonable operating temperature of the lasing crystal in order to prevent thermal damage to the crystal. This system also reduced frequency noise resulting from vibrational coupling of the water cooling system to the crystal. Also, a theoretical analysis was done on the cavity to determine the beam properties within the laser cavity, information that will become pertinent when conducting a theoretical analysis of the expected power out of the cavity. In chapter 6 we will continue our study of this 1342 nm ring laser, measuring its output characteristics while operating in both free running and unidirectional mode. To drive unidirectionality, two methods will be employed, involving the use of a home-built Faraday rotator element as well as a novel scheme of selfinjection. Various quantitative parameters of these two methods will be compared. Also, studies involving the tuning of the laser using both intra-cavity and external cavity etalons will be performed. Preliminary results pertaining to frequency doubling the 1342 nm light to 671 nm will also be reported.

93 Chapter nm Ring Laser Operation With construction of our 1342 nm solid state ring laser cavity complete, it now becomes necessary to measure the output characteristics of our laser for use in experiments involving ultra-cold 6 Li gases. One important general limitation to ring laser geometries is that these lasers support bidirectional operation, leading to instabilities[89]. To combat this, several schemes have been employed to drive unidirectional operation, such as the inclusion of intracavity optical isolators, the use of acousto-optic modulators[90], polarization dependent output couplers[91], and non linear crystals for sum-frequency mixing[92]. The use of such implements, however, typically involves the insertion of additional elements into the laser cavity, increasing the amount of intracavity scattering loss and reducing the overall power output of the laser. In this chapter, we report a novel technique of selfinjection locking for forced unidirectional laser operation. Because this technique does not involve any additional cavity elements, we can achieve stable unidirectional operation with higher output powers than what can otherwise be achieved using traditional methods. We begin in section 6.1, where measurements of the small signal gain of the Nd:YVO 4 crystal are reported as a function of wavelength. Also measured is the angular dependence of that small signal gain at 1342 nm, a consequence of the natural birefringence in the crystal. Section 6.2 presents our measurements of the free running output of the laser without making any attempts to either frequency tune or drive unidirectional operation of the field within the cavity. The longitudinal mode structure of the laser light is examined using a Fabry-

94 76 Perot interferometer and the transverse mode structure is measured with an M 2 value reported. Also, fluctuations in intensity of the laser output are qualitatively reported. Section 6.3 provides a theoretical framework for modeling the expected output power of the laser given values for the unsaturated small signal gain as well as cavity scattering losses. Next, in section 6.4, we report values for the laser output when driven unidirectionally by including a home made Faraday rotator inside the laser cavity. In section 6.5 we introduce a novel scheme of self-injection to drive unidirectional operation and compare this method to that of section 6.5. Section 6.6 reports results of using etalons to tune the frequency of our lasers and section 6.7 will conclude the chapter. 6.1 Measuring the Nd:YVO 4 Gain In this section we report measurements of the unsaturated small signal gain for a single pass as both a function of wavelength and also as a function of polarization angle. For the wavelength dependent gain measurement, the unsaturated small signal gain of the Nd:YVO 4 crystal was measured when pumped by the 2 25 Watt pump beams. The basic layout for this measurement is shown in figure 6.1. To facilitate this measurement, the output coupler M2 of the laser cavity was removed and an extended cavity diode laser (ECDL) whose wavelength is tunable from 1335 to 1350 nm was used. Not shown in figure 6.1 but included in the setup for this measurement were several lenses used to ensure high spatial mode overlap of the probe laser with the pump beams inside the crystal. Also, as the gain is polarization dependent, the electric field of the probe beam was rotated so as to be aligned with the vertical c-axis of the crystal. The results of our small signal gain measurement are reported in figure 6.2. From the figure, it can be noted that the gain bandwidth spans several nanometers. Also significant is a negative gain or absorption feature from approximately 1336 to 1341 nm. This feature is the result of excited state absorption of the crystal and has been previously reported [93]. While this feature is interesting to note, it s presence does not prevent laser operation at the desired 1342 nm wavelength in the Nd:YVO 4 crystal. More important to our discussion of the 1342 nm laser behavior, however, is

95 77 M1 DET ECDL Pump Pump M3 Nd:YVO4 M4 Figure 6.1. Experimental setup for measuring the unsaturated small signal gain of the Nd:YVO 4 crystal when pumped by 2 25 Watt pump beams. The transmitted power of an extended cavity diode laser (ECDL) whose wavelength is tunable from 1335 to 1350 was measured by a photodetector (DET) after having passed through the crystal. Not shown in the setup are the lenses used to mode match the waist of the probe laser with the pump lenses as they co-propagate through the crystal. the small signal gain peak located at approximately 1342 nm. To get a better and more insightful understanding of this peak, a zoomed in view of the gain profile from 1341 nm to 1345 nm is shown in figure 6.3. As a guide to the eye, we have fit this gain profile over this region to a triple gaussian summed function of the form Gain(λ) =y + Ae (λ λ 1 ) 2 w 1 + Be (λ λ 2 )2 w 2 + Ce (λ λ 3 )2 w 3 (6.1) where λ 1,2,3 represent the center wavelengths of the three gaussians used for the fit and w 1,2,3 represents the corresponding widths. For the measured values of our gain profile, A(B, C) = ( , ), λ 1 (λ 2,λ 3 ) = (1342.2, ), w 1 (w 2,w 3 )= ( , ), and y = Also, from this data, we observe that the gain profile peaks at approximately nm with a value of This gain value of 1.87 corresponds to a small signal unsaturated gain coefficient, α m0, of where α = ln(gain) (6.2) In addition to using values of the small signal gain to determine the expected power output of the laser, we also have an interest in driving unidirectional operation of the laser within the ring cavity. One way to do this would be to exploit

96 78 Figure 6.2. The unsaturated gain profile of our Nd:YVO 4 crystal from 1335 nm to 1350 nm when pumped by 2 25 Watt pump beams. The gain profile is quite broad, spanning several nanometers, and peaks at approximately 1342 nm. Also of note is a broad excited state absorption band spanning from 1336 nm to 1341 nm. the polarization dependence of the gain medium. By inserting a Faraday rotator coupled with a properly oriented half-wave plate into the laser cavity, light traveling in one direction around the cavity would experience a slight rotation in its polarization while light from the other direction would not experience any rotation at all. This small rotation in polarization, coupled with the fact that the Nd:YVO 4 crystal is birefringent, should be enough to break the symmetry of the cavity and force unidirectional operation. We also wanted to measure how how the small signal gain of the laser changes as a function polarization. To measure this polarization dependence, we use a setup similar to the previous gain measurement, portrayed in figure 6.4. Like the previous measurement, an ECDL is mode matched to the gain crystal and incident upon a photodetector. However, in this case, a half-wave plate is also used before the gain crystal to rotate the angle of the polarization of light before the gain crystal. Also, for this measurement, the wavelength of the ECDL was set to nm. Figure 6.5 shows the angular dependence of the unsaturated small signal gain

97 79 Figure 6.3. The gain profile of our Nd:YVO 4 crystal from 1341 to 1345 nm. The profile has a peak value of 1.87 at a corresponding wavelength of nm. A dashed line has been added as a guide to the eye (see text). for our laser crystal from 0 to 90 degrees. As expected, the gain has a peak value when the polarization of the probe laser light is parallel to the vertically rotated c-axis of the crystal at 0 degrees. As expected, for this particular polarization, the gain has a measured peak value of 1.84, consistent with the measurements of gain as a function of wavelength for this value as reported above. To further characterize this angular dependence, we fit the measured data points in figure 6.5 to an offset cosinusoidal function of the form Gain = y 0 + Acos(fθ). (6.3) From this fit, we find values of for A, for y 0, and for f. Therefore, when the polarization is aligned vertically with the c-axis of the gain crystal, the fitted gain has a value of 1.85 and when the polarization is perpendicular to the c-axis, the gain has a value of This fit information about the angular dependence of the gain will become useful when examining the effects of inserting a Faraday rotator into the laser cavity to drive unidirectionality, as described in section 6.4.

98 80 M1 DET ECDL HWP Pump Pump M3 Nd:YVO4 M4 Figure 6.4. Experimental setup for measuring the angular dependence of the unsaturated small signal gain of the Nd:YVO 4 crystal. The transmitted power of an extended cavity diode laser (ECDL) whose wavelength is set at 1342 nm was measured by a photodetector (DET) after having passed through the crystal. A half-wave plate (HWP) is used to rotate the polarization of light prior to transmission through the crystal. Figure 6.5. Unsaturated small signal gain as a function of polarization angle, measured with respect to vertical. The dashed line represents a sinusoidal fit to the data (see text). 6.2 The Free Running Laser As expected, in the absence of any attempts to drive unidirectional operation of the laser cavity, the 1342 nm ring laser that we have constructed operates in bidirectional mode the light circulates in both the clockwise and counterclockwise directions. This can be easily demonstrated by looking at the light emitted from

99 81 the output coupler M2 (see figure 5.3). For light traveling in the clockwise direction, laser radiation will be emitted from the output coupler traveling in the plane parallel to the gain crystal. For light traveling in the counter-clockwise direction, laser radiation will be emitted from the output coupler traveling at an angle of 30 degrees with respect to the clockwise direction output. For the output coupler with a reflectivity of (labeled 70% in table 5.1), 1.62 Watts of laser power were measured in each of two output beams corresponding to the two directions of bidirectional operation of the laser cavity. To measure the power, a PM30 thermopile power detector manufactured by Coherent, Inc. was used. This air convection cooled power meter can measure laser beam powers up to 30 Watts. Because it is a thermopile detector, there is a 2 second response time with each measurement. Because of this, fluctuations in the intensity of the laser light faster that 0.5 Hz will not be discernable with this detector To qualitatively measure high frequency intensity noise in this free running laser beam, a small portion (on the order of a few tens of mw) of the output of one beam was picked off and measured with a model DET10C biased InGaAs Photodetector from Thorlabs. This photodetector has an active area of only 0.8 mm 2. Because of this small active area and the fact that it is a photodiode rather than a thermopile detector, the response time is much faster, having a rise time of only 10 ns. Figure 6.6 shows the intensity of the output beam for a scan of 10 ms. From the plot, one can get a sense of the competition between the two directions of operation. For the majority of the time, the power is split evenly between the two directions, indicated by approximately 20 mw of power. However, at some times, the power the power measured on the beam is nearly twice this value or zero, indicating that during brief periods the laser is operating unidirectionally. Next, we wanted to investigate the frequency mode structure of our free running laser. To measure the mode structure, the laser output was coupled into a Fabry-Perot interferometer. The Fabry-Perot used in this measurement was also manufactured by Coherent, Inc. and has a free spectral range of 300 MHz. The output scan of this Fabry-Perot measurement is recorded in figure 6.7. From the figure, one can see that the frequency output of the free running laser is operating in multi-longitudinal mode. In fact, for this particular measurement, four distinct frequencies are discernable. It is important to note that for our desired applica-

100 82 Figure 6.6. Measurement of the power stability for the free running laser cavity. For most of the time, the power is split evenly between the two directions of operation, indicating bidirectionality. However, during some instances, the laser operates in unidirectional mode, whereby the recorded power is either twice as high or zero, depending on the direction. tion of such a laser to experiments involving 6 Li spectroscopy, single frequency operation is crucial. Finally, in addition to the longitudinal mode structure of the laser, we also wanted to measure the transverse mode structure of the beam as well. To do this, we used a device called a Mode Master, also manufactured by Coherent, Inc. This device samples the beam radius at a variety of locations in order to determine the divergence of the beam. From this information, a variety of beam parameters can be computed and reported. The output of the Mode Master scan for this free running laser is reported in figure 6.8. One important value measured and reported by the Mode Master is the M 2 value of our laser. Also known as the beam quality factor, the M 2 value of the laser refers to the ratio of the beam parameter product (the product of the beams divergence with its waist) to that of an ideal gaussian beam. For lasers operating in a pure TEM 00 mode, the M 2 value of the measurement, by definition, would be one. As lasers can operate in higher transverse Hermite-Gaussian modes, this number can be much larger as the beam diverges faster for these higher modes.

101 83 Figure 6.7. Measurement of the frequency characteristics of our free running ring laser. The graph shows an output of a 300 MHz Fabry-Perot interferometer. The presence of four distinct peaks is indicative of the fact that the output laser frequency is multilongitudinal mode in structure. From the output of the Mode Master, we note that the averaged radial M 2 value for our laser is 1.04, indicative of the fact that, for all intents and purposes, the laser is operating in the TEM 00 mode. This is not surprising as we have included a pinhole within the laser cavity to prevent lasing in all higher order transverse modes (see section 5.6). Another important measurement reported by the Mode Master is the waist of the laser beam. From this measurement, it is reported that the beam has a waist of μm. From this value, we can then calculate the value for the waist of the beam inside the ring laser cavity. Not shown in figure 5.3, we use a 17.5 cm focal length lens located 17.5 cm from the midpoint between mirror M1 and output coupler M2 in order to collimate the laser beam. By knowing that the beam has a mode radius of μm at the location of this collimating lens, we can use ABCD matrices to propagate the beam backwards in order to determine that the beam must have a waist of 147 μm at the halfway position between mirror M1 and output coupler M2. The value for the beam waist at this point is critical for determining the properties of the propagating beams inside the cavity, as well as

102 84 Figure 6.8. Results of a beam profile measurement made by the Mode Master, manufactured by Coherent, Inc. The output of our free running laser is coupled into the Mode Master, which measures values for the beam radius as the beam propagates over a given distance. From these measurements, several spatial properties of the laser beam can be determined and reported. for determining the unknown effective focal lengths for thermal lensing inside the gain medium (see section 5.7). It is also interesting to note, from the output of the Mode Master measurement, that there exists a slight astigmatism to our laser, on the order of 9%. This slight astigmatism is not surprising, considering the effects of thermal lensing in our gain crystal. Because the crystal is birefringent, thermal lensing caused by the pump beams will cause different distortions to the laser wavefront along different directions, contributing to an astigmatic beam [94]. 6.3 Modeling Power Output It now becomes necessary to develop a theoretical description of the expected output power of our laser cavity, taking into consideration the gain of the laser crystal, the intracavity reflectivity, and the reflectivity of the output coupler. By then measuring the output power of the laser for a variety of output coupler reflectivities,

103 85 this theory will enable us to fit the data and perform an independent verification of our unsaturated small signal laser gain as well as determine the optimum output coupler for the greatest efficiency of our laser cavity. The following derivation follows that of reference [78], however with slight modifications for a ring laser cavity instead of a standing wave setup. Also, for this derivation, it is assumed that the laser is already operating unidirectionally, as will be the case for the two measurements of laser power vs. output coupler that follows. For a laser cavity operating in continuous wave steady state conditions, the round trip gain for the signal intensity inside the laser cavity must equal the losses due to intracavity scattering and output coupling. If we assume a traveling ring laser cavity with homogeneously saturable gain medium, the growth the traveling wave I(z) inside the cavity will be given by the equation di(z) dz =[α m α 0 ]I(z) (6.4) where α 0 is the cavity reflective loss coefficient and α m is the saturated gain coefficient. For a homogeneously saturable gain medium, the saturated gain coefficient as a function of position along the axis of the laser is α m = α m0 1+ I(z) I sat (6.5) where α m0 is the unsaturated small signal gain coefficient and I sat is the saturation intensity of the transition. As a reminder, the small signal gain coefficient is related to the small signal gain of the crystal via the relation Gain = e α m0. (6.6) Also, the saturation intensity, I sat, of the transition is defined as the intensity that reduces the small signal absorption coefficient down to half its small signal value and is given by I sat = ω (6.7) στ eff where is Planck s constant divided by 2π, ω is the angular frequency of the

104 86 transition, σ is the stimulated emission cross section for the transition and τ eff is the fluorescent lifetime of the transition. For the 4 F 3/2 4 I 13/2 transition at 1342 nm, σ is reported to have a value of 6± cm 2 and τ eff is 110 μs [84]. This yields, per equation 6.7, a saturation intensity I sat = Watts/m 2. If we now make the assumption that the output coupler reflectivity R 1 is close to unity such that the intensity I(z) remains fairly constant during the length of the cavity, we can then make the approximation that I(z) I circ (6.8) where I circ is the unidirectional circulating intensity inside the laser cavity that is independent of cavity position. With this approximation, the saturated gain coefficient also becomes independent of position within the cavity and can be written as. α m = α m0 (6.9) 1+ I circ I sat As mentioned above, for the laser to be operating in steady state, the round trip gain for a single pass must equal the round trip losses for a single pass as well. In other words α m = α m0 = α 1+ I circ 0 + ln( 1 ) δ 0 + δ 1 (6.10) I sat R 1 where e δ 0,1 are the cavity reflectivity (due to scattering losses) and output coupler reflectivity respectively. Solving equation 6.10 for I circ we find α m0 I circ =[ 1] I sat. (6.11) δ 0 + δ 1 For a lightly coupled laser cavity, the output coupler transmissions are given by T 1 =1 R 1 δ 1. Therefore, the intensity output from the laser would thus be δ 1 I circ or α m0 I circ = δ 1 [ 1] I sat. (6.12) δ 0 + δ 1 Finally, as intensity is defined as power per unit area, the power output of the laser cavity can thus be determined by α m0 Power = δ 1 [ 1] I sat πw0 2 (6.13) δ 0 + δ 1

105 87 where w0 2 is the waist of the circulating laser beam inside the gain crystal. As mentioned above, this derivation for the power output from our laser cavity was for one dimensional transport through a homogeneously saturable gain medium. For our laser system here, this is not the case and equation 6.9 must be modified appropriately [95]. For situations where the beam converges or diverges in the gain medium, a one dimensional plane wave treatment is insufficient to describe the incremental gain and higher order terms must be considered to account for this radial radiation transport. A more exact treatment of this additional consideration would be α m = α m0 [1 + γω2 Icirc 1+ I circ I sat 4 + I sat ] (6.14) 1+ I circ I sat where γ = 2.88 (6.15) ρ 2 is the quadratic term of the zero-order bessel function used to describe the gain profile of radius ρ. The second term of 6.14 becomes 0.72 ω 2 /ρ 2. In many single mode oscillators, ω 2 /ρ 2 is about 0.5. The last term of 6.14 describes the part of the radial radiation transport that comes from the change of the saturated gain profile by the gaussian intensity distribution. For optimized laser systems, I circ /I sat >> 1 causing the third term to go to one. Therefore, with these corrections, the actual saturated gain parameter becomes α m = 1.64α m0 1+ I circ I sat (6.16) where the 1.64 prefix is the result of radial transport of radiation in the crystal. Using this value, we can thus rewrite equation 6.13 to reflect this modification, making the equation for the expected power output of the laser as a function of output coupler reflectivity Power = δ 1 [ 1.64α m0 δ 0 + δ 1 1] I sat πw 2 0. (6.17)

106 Unidirectionality with a Faraday Rotator Traditional techniques to drive unidirectional operation of a ring laser cavity typically involve some form of intracavity element that provides directionally dependent loss to break the symmetry of the system. One such device, a Faraday rotator, rotates the plane of polarization of the electric field vector in a directionally independent way. This device can be combined with a half-wave plate, oriented such that the fast axis will rotate the plane of polarization back to its original orientation for one direction while causing further rotation for the other direction of transmission. Doing so provides directionally dependent rotation resulting in directionally dependent loss for our gain crystal. Unfortunately, at the wavelength and power levels of our 1342 nm laser, such a device does not exist commercially. Therefore, we have constructed our own unidirectional element consisting of a home built Faraday rotator and a multi-order half-wave plate for insertion into our laser cavity In section 4.4, the applied magnetic field on a 10 mm long undoped YAG crystal from a right hollow cylindrical magnet of uniform magnetization was measured to be T m along the transmission axis of the crystal contained within the bore of the magnet. In section 4.5 it was found that the Verdet constant of undoped YAG was 1.38 rad/t m. Therefore, our home built Faraday rotator, when inserted into the ring laser cavity will not rotate the plane of polarization for the clockwise direction, but will rotate the plane of polarization for the counterclockwise direction by radians (2.53 degrees). Since our gain crystal is polarization dependent, the small signal gain experienced by this rotated polarization is determined to be less than that for the unrotated polarization as per equation 6.3. While this difference in gain may seem trivial, it is indeed enough to drive unidirectional operation of our laser. Using this intracavity rotation element, we have measured the output power of the unidirectional laser for the seven different output couplers listed in table 5.1. The results of this measurement are shown in figure 6.9. For a reflectivity, R 1, of 0.804, it was found that the laser had a peak in output power of 3.10 Watts. Also included in this plot is a theoretical fit of equation 6.17 to our data. From this fit we can extract values for the unsaturated small signal gain coefficient α m0

107 Output Power (W) Output Coupler (d 1 ) Figure 6.9. Power output from the laser cavity when driven unidirectionally using an intracavity Faraday Rotator for seven different output couplers. The dashed line represents a theoretical fit to the data used to determine the unsaturated small signal gain and intracavity scattering losses for our home built ring laser. and the cavity reflectivity value R 0. For this measurement, we find α m0 to be 0.637, corresponding to a single pass gain of 1.89 and R 0 to be 87.3%. It is very interesting to note that this value of 1.89 for the unsaturated small signal gain is very close to the independently measured value of 1.87 reported in section 6.1 using a probe laser beam. 6.5 Unidirectionality with Self Injection It has been anecdotally reported that in 1865 Christiaan Huygens noticed that the pendulums of two clocks in his room invariably locked into synchronization when they were hung close enough to one another yet remained free running when moved farther apart [96]. While this coupling of the harmonic oscillators of the pendulums was determined to result from vibrational coupling within the wall they were hung on, the phenomenon of coupled oscillators still persists through a variety of systems, including electric circuits [97, 98] and optical cavities [99, 100]. This idea of injection locking, whereby radiation from a master laser is injected

108 90 into the cavity of a slave laser, controlling its spectral properties, has been often used to control the output of high power lasers, on the order of several Watts, using only a weak (tens of milliwatts) master laser beam [101]. In fact, one report demonstrated injection locking of a 13 Watt cw Nd:YAG ring laser using only a 40 mw laser diode [102]. It is from these studies of using a weak injected laser beam to control the output characteristics of a more powerful laser cavity that the novel idea of self injection was derived. Diagramed in figure 6.10, the basic idea of self injection is as follows [75]. A small amount of output laser light from one output beam (one direction of intracavity circulation) is picked off using the combination of a halfwave plate and a polarizing beam splitter. This low power beam then propagates through the external self injected loop region, where it passes through a Faraday rotator, acting as a unidirectional light diode, allowing light to only travel in the forward direction. The weak probe beam is then aligned such that the path of its propagation is aligned perfectly counter to that of the other output direction of the laser (corresponding to the opposite direction of circulation within the cavity). In this way, the weak beam is injected back into the laser cavity mode. Also, another half wave plate is used to change the polarization of injected beam back to vertical. The addition of this beam to the cavity mode causes stimulated emission in the gain crystal in the same direction from which the weak beam is sourced, breaking the symmetry of the ring laser cavity, and driving unidirectional operation of our laser. In addition to the half-waveplates, the polarizing beam splitter, and the Faraday rotator, three lenses are also included in the external loop in order to mode match the weak injected beam to the laser cavity. The first lens, L1, has a focal length of 17.5 cm and is located 17.5 cm from the midpoint of mirror M1 and output coupler M2 of the cavity. This lens also acts as a collimation lens for the laser for the high powered beam that does not get reflected by the polarizing beam splitter. Also in the loop are two lenses (L2 and L3) having focal lengths of 100 cm and -10 cm respectively that act as a telescope. From an ABCD calculation of the beam parameters, at the output coupler M2, the output beam has a waist of μm and a radius of curvature of meters while the injected beam has a waist of μm and a radius of curvature of meters. While not perfect, these

109 91 FR M HWP M1 Direction M2 L3 L2 HWP L1 PBS Pump Pump M3 Nd:YVO4 M4 Figure Experimental layout for our novel scheme of self injection. A small portion of the output light from our laser is picked off from the main beam using a half-wave plate (HWP) and a polarizing beam splitter (PBS). This light then travels through a Faraday rotator (FR) and another half-wave plate to change its polarization back to vertical. The weak beam is then injected back into the cavity, where it causes stimulated emission in the gain crystal, breaking the symmetry of the ring laser and driving unidirectional operation. Also included in this loop are three lenses (L1, L2, L3) used to shape the injected beam for mode matching (see text). two beams are well coupled and the majority of the power injected back into the laser cavity will be sufficiently mode matched to drive unidirectional operation. For all the measurements presented in this dissertation used to quantify and characterize the operation of our self-injected laser, a constant 2% of the output power of the primary beam is picked off by the polarizing beam splitter to be injected back into the laser cavity. When this self-injected beam is properly aligned into the laser cavity, the output power of the laser was measured to be 3.24 Watts for the output coupler with a reflectivity of This power is twice what was measured for a single output of the free running laser that was reported in section 6.2, indicating unidirectional operation. Also, as expected, the high power beam corresponding to the wrong direction of circulation within the laser cavity that emits from the rejection port of the Faraday rotator in the external loop disappears when the weak beam is injected back into the cavity, further evidence of unidirectional operation. As in section 6.2, we again want to qualitatively measure high frequency intensity noise of this self-injected laser. A small portion of the output beam was again picked off and measured on a fast photodetector. Figure 6.11 shows the intensity

110 92 Figure Measurement of the power stability for the self-injected laser cavity. Unlike the free running laser, the unidirectional operation of the self-injected laser prevents mode competition in the gain crystal resulting in a drastic reduction of intensity noise on the output of the laser beam. fluctuations of the output beam for a scan of 10 ms. Comparing this plot to figure 6.6, one sees that the intensity noise of the self-injected laser is much quieter than that of the free running laser. This result is not surprising, given the now lack of mode competition between the two directions of oscillation within the laser cavity. In addition to looking at intensity noise fluctuations on the output of this selfinjected laser setup, we also wanted to look at the longitudinal mode structure corresponding to the frequency components of the output light. To do so, we again used a 300 MHz Fabry-Perot interferometer, as we did in section 6.2. The output of the interferometer is reproduced in figure From the figure, one can see that in addition to reducing the intensity noise of the output beam, the lack of mode competition inside the laser crystal also causes the laser to operate at a single frequency, a very desirable and necessary quality for application to atomic physics experiments. In addition to verifying single mode operation, we also wanted to to check the spectral linewidth of the output laser beam. To do so, we perform a heterodyne measurement whereby the output of our self-injected laser is combined with light

111 93 Figure The longitudinal mode structure of the self-injected laser as measured by a 300 MHz Fabry-Perot interferometer. Unlike the case of the free running laser cavity, the lack of mode competition in the gain crystal enables single frequency operation from a tunable ECDL using a 2 2 single mode broadband fiber coupler manufactured by Thorlabs (part number FC APC). The ECDL is detuned by 735 MHz with respect to the ring laser and one of the outputs from the fiber coupler is sent onto a fast photodetector where the difference frequency of the two lasers will create a beat note. This beat note signal is then sent from the photodetector into a spectrum analyzer and is reproduced in figure The full width half maximum of the signal measured by the spectrum analyzer was found to be 150 khz, indicating that the spectral linewidth of the self-injected laser is less than or equal to 150 khz. As the reported value for the linewidth of the ECDL ranges from 100 khz to 1 MHz, it is impossible to draw a conclusion as to which laser provides the dominant contribution to our linewidth measurement. Regardless, 150 khz should be sufficient for our application to lithium experiments. Finally, we wanted to measure the power output of the self-injected laser as a function of output coupler, as we did in section 6.4. As before, seven different output couplers were used and the total output power of our laser was recorded for each using a thermopile detector. Figure 6.14 shows the result of this measurement. As before, the output coupler having a reflectivity of resulted in

112 94 Figure The spectral output of a heterodyne measurement of the linewidth of our self-injected laser. The laser output is beat with the output of a tunable ECDL and sent to a spectrum analyzer. From the full width at half maximum of this beat note measurement, it is shown that the linewidth of our self-injected laser is no larger than 150 khz. the highest output power of 3.24 Watts. This corresponds to a 4.5% increase in total power for the optimum output coupler. We also fit the data in figure 6.14 with equation 6.17 and included the fit as a dashed line in figure As the pump powers and alignments for the Faraday rotator laser measurement and the self-injected laser were the same, the unsaturated small signal gain coefficient, α m0, was held to be a constant 0.637, leaving the intracavity reflectivity R 0 to be the only free parameter. From this fit, R 0 was determined to be 88.8%. This 1.5% increase in intracavity reflectivity is not surprising, given that, for the self-injected setup, the Faraday rotating element was removed from the laser cavity, reducing the intracavity scattering losses. Therefore, by implementing this novel scheme of injection locking, the output power of a ring laser can be increased due to the reduction of intracavity elements. As there is nothing unique about the bidirectional operation of our diode pumped solid state ring laser, we would anticipate this self-injection technique to be broadly applicable to any ring laser cavity.

113 95 Figure The output power of our self-injected laser as a function of output coupling, δ 1, for seven different output couplers. The dashed line represents a fit to our data with the intracavity reflectivity being the only free parameter. Compared to the free running laser, we observe a 4.5% increase in output power for the optimum output coupler. 6.6 Frequency Tuning If we were only concerned with creating a high powered, single frequency, narrow linewidth laser our task would be complete at this point. However, as is the case with most laser applications in atomic physics, the frequency of the laser must be tunable as well if we are to use such a laser for ultra-cold lithium experiments. From our measurements of the gain profile in section 6.1, the gain peaks and the free running laser operates at a wavelength of approximately nm. If we were to frequency double this light, the resulting wavelength of nm would still be red detuned from the nm D 2 line of 6 Li. Therefore, we must tune the wavelength of the solid state ring laser to a value of nm if we wish to use the frequency doubled light. To frequency tune the spectral output of our ring laser, we will employ the use of various optical etalons. When inserted into the laser cavity, the etalon acts as an additional optical resonator with its transmission periodically varying as a function of frequency. On resonance, the reflections of the two surfaces destruc-

114 96 tively interfere and cancel each other out, resulting in full transmission through the etalon. At the anti-resonances, however, these reflections do not entirely cancel each other out and there will be some reflected loss from the etalon. In this way, the gain profile of the laser cavity can be affected. Also, as the etalon is rotated, the resonance frequencies of the etalon are slightly changed. In this way, the frequency of the laser can be tuned by merely tilting the etalon. For a small tilt angle θ, the transmission, T, of an etalon is given by T = (1 R) 2 1+R 2 2Rcos[ 4πnd λ θ2 (1 + )] n 2 (6.18) where R is the reflectivity of the etalon, n is the index of refraction, and d is the etalon thickness. For our laser, we will use silicon etalons to tune the frequency. The index of refraction of silicon at 1342 nm is 3.5, resulting in a reflectivity of In the subsections that follow, we describe our efforts to frequency tune our laser using both an intracavity etalon as well as an etalon located in the external loop for our self-injected laser Intracavity Tuning To tune the laser using an intracavity etalon, a 32 μm silicon etalon was purchased from Light Machinery for use. Because of the uncertainty in the etalon thickness as quoted by Light Machinery (±2 μm), the first task was to measure the actual etalon thickness. To do so, the etalon was oriented normal to a wavelength tunable extended cavity diode laser (ECDL). The transmission of the etalon as a function of wavelength was measured from 1330 to 1350 nm and is reproduced in figure Also included in this figure is a fit to the etalon transmission using equation 6.18 for an angle θ = 0 and a reflectivity R = From this fit, we find that the etalon has an actual thickness of 34.2 μm. Knowledge of this thickness is important in determining the expected power and wavelength output of the laser cavity when tuned via an intracavity etalon. To tune the wavelength of our self-injected laser, the 34.2 μm etalon was inserted into the bow tie ring cavity on a rotation mount that allowed for the angular adjustment of the etalon. Also, for this measurement, the output coupler of the

115 Transmission Wavelength (nm) Figure The periodic transmission measurement of a thin silicon etalon as a function of wavelength. The transmission was measured from 1330 nm to 1350 nm using a tunable extended cavity diode laser at normal incidence to the etalon. The dashed line represents a fit to the data of equation 6.18, indicating an etalon thickness of 34.2 μm. laser cavity was changed from one with a reflectivity of to one with a reflectivity of While this change resulted in a reduction in power for the output of the laser cavity, it was determined that an output coupler with a higher reflectivity (lower loss) was necessary in order to tune the laser over a wider range. This can be understood when considering the fact that in order for the cavity to lase, the product of the gain and the cavity losses must be greater than one. Since the gain of the laser is non-uniform (see figure 6.3), the reflective losses in the cavity must be reduced as the wavelength is tuned away from the peak in the gain profile. The power output of the laser as a function of wavelength via intracavity etalon tuning is shown in figure The wavelength of the laser light was measured using an Agilent 86120b wave meter. As the etalon was rotated within the laser cavity, 13 different stable modes were observed. From this measurement, it can be seen that using an intracavity etalon, we are able to tune the laser over a broad range of wavelengths from nm to nm, covering the desired nm value for the wavelength doubled D 2 line of 6 Li. This range corresponds to a tunability of over 200 GHz.

116 98 Figure The output power of our laser as a function of wavelength when tuned via a 34.2 μm intracavity silicon etalon. The dashed line represents a fit to the data when considering the convolution of the etalon transmission, the small signal gain of the laser crystal, and the increase in intracavity scattering losses due to the insertion of the etalon (see text). It can be also be noted that in figure 6.16 the power output of the laser cavity is smaller than that reported in figure 6.14 for a self-injected laser having an output coupler reflectivity of To understand this, we convolved the transmission of the etalon (equation 6.18) with the measured small signal gain (equation 6.3 in order to determine the gain profile of the laser when an intracavity etalon is inserted in the cavity. Using this convolution as the new small signal gain value, we fit the data in figure 6.16 with equation 6.17, where the only free parameter is the intracavity scattering loss parameter, δ 0. From a least squares fit to the data, it was found that the reflectivity of the cavity R 0 was 86.5%, a value lower than the 88.8% measured before. This discrepancy indicates that the insertion of the 34.2 μm silicon etalon into the laser cavity results in an additional reflective losses of 3.6%.

117 Power (Watts) Wavelength (nm) Figure The output power of the laser cavity as a function of wavelength when tuned via a 250 μm silicon etalon located in the external loop region of our self-injected laser. By merely rotating the etalon, we were able to observe a tuning range that exceeded 38.9 GHz for our laser setup External Cavity Tuning Finally, as a proof of principle experiment, we wanted to demonstrate the tunability of our laser system using an etalon located in the external cavity loop of our selfinjected laser. To do this, we again use a silicon etalon, however this time, the etalon used has a reported thickness of 250 μm. Also for this measurement, the output coupler was changed back to having a reflectivity of and the polarizing beam splitter was changed to reflect 30% of the light from the main beam and into the output coupler. This was done so as to increase the amount of power being coupled back into the laser cavity via the external loop. The output power as a function of wavelength measured by tuning the angle of the external cavity etalon is shown in figure As can be noted, 14 stable modes of operation were observed by rotating the etalon in the external cavity spanning a range from nm to nm, corresponding to a tuning range of 38.9 GHz. It is also interesting to note that the data seem to be paired, with the pairs being separated in wavelength by nm. We believe this to be the result of a mode hop of the external loop region. The output coupler,

118 100 beam splitter, and external cavity mirror make up a loop whose path length was designed to be incommensurate with a multiple of the intracavity path length of the ring laser cavity. In this way, the loop region can be thought of as an etalon whereby small changes in the path length could cause mode hops in the output frequency of the laser cavity. To test this hypothesis, the path length of the external loop region was reduced to be an even multiple of four times the path length of the ring laser cavity. The polarizing beam splitter was located on a translation stage that could be driven by a piezo-electric transducer (PZT) to finely control its position. In this way, it was observed that when the path length was as near as possible to an integer multiple of the cavity length, the wavelength of the laser could be greatly tuned by merely applying a voltage on the PZT resulting in a small external cavity length change. Using this technique, a range from nm to nm was observed, covering the desired nm doubled wavelength for 6 Li. Because the cavity length of the external loop was extremely sensitive to motion at this even multiple, it was impossible to record the power output of the laser as a function of wavelength and further investigation of this phenomenon may be warranted. 6.7 Conclusion In this chapter, we have described the operation of our home built diode pumped solid state ring laser system operating at a wavelength of 1342 nm. We began by measuring the unsaturated small signal gain of our laser as well as the angular dependence of the measurement (stemming from the birefringent nature of the gain crystal). We ve also investigated the intensity noise and frequency characteristics of both the free running laser as well as that for unidirectional operation. We ve performed an ABCD matrix calculation to determine the beam properties within the laser cavity and have used this information to theoretically calculate the expected power output as a function of output coupler, which we have also measured. Additionally, we constructed a home built Faraday rotator to drive unidirectional operation of the laser as well as developed a novel technique of self injection laser control as well. Finally, by using etalons both inside the cavity and in the external loop of this self injected region we have demonstrated tunability of our

119 101 laser. The next step necessary for implementing this laser system in ultra-cold atomic physics experiments involves the efficient frequency doubling the laser light. While we have observed a modest 10% efficiency using a single pass through periodically poled lithium niobate (PPLN), preliminary investigations of using PPLN and lithium triborate as non-linear crystals in a four mirror frequency doubling ring cavity have been met with very limited success. While some red light at 671 nm has been observed (on the order of a few μw), the use of these crystals within a cavity have been limited due to the fact that as we attempt to actively lock the cavity length on resonance with the input 1342 nm light, power buildup in the nonlinear crystals cause the crystals to heat and expand, changing the optical path length and preventing the cavity from locking. Regardless, by overcoming these technical difficulties, the use of the 1342 nm ring laser as a high power source of 671 nm light will be a welcome tool in the ultra-cold 6 Li community.

120 Chapter 7 Preparing a Two-Component Fermi Gas In order to prepare a two-component gas of ultra-cold 6 Li atoms for our studies of the rapid suppression of interactions in a fermi gas, many different experimental subsystems must be constructed and operated in parallel. In this chapter we will outline the basic components for each of these subsystems. As this dissertation expands upon the research of previous graduate students in the lab of Dr. Ken O Hara, many of the subsystems that follow have been previously described in great detail. However, for completeness, it is prudent to include an overview for each of these systems. For additional information and more in-depth discussion, the reader is directed to references [6] and [103]. The basic outline of this chapter is as follows. In section 7.1, the various components comprising the ultra-high vacuum system required to isolate our experiment from the surrounding environment are described. This isolation helps ensure that the trapped atoms will be long lived in the various magnetic and optical traps due to the reduction in background gas collisions commensurate with the vacuum. In section 7.2 we discuss the laser systems used to cool and trap 6 Li atoms from a hot atomic vapor. Section 7.3 describes the various coils of wire used to generate the magnetic fields and magnetic field gradients for the manipulation of the hyperfine energy level structure of the 6 Li atoms. In section 7.4, we describe the preliminary trapping of atoms in a Magneto-Optical Trap (MOT), where the temperature of the atoms is reduced by over six orders of magnitude.

121 103 Once the atoms have been sufficiently cooled and trapped in a MOT, section 7.5 describes the process of loading the atoms into an optical dipole trap. Finally, in section 7.6 we will describe the setup for imaging the cloud of atoms using the technique of absorption imaging. 7.1 Ultra-High Vacuum System The basic layout of our experimental apparatus is shown in figure 7.1. This apparatus is composed of three primary regions, namely a lithium oven used to produce a hot and effusive source of atomic lithium gas, a slower region whereby a narrow collimated beam of lithium atoms are cooled in preparation to be trapped, and finally an experimental chamber where the atoms are further cooled and trapped using a variety of magnetic fields in preparation for further experimentation. To maintain the ultra-high vacuum required for our experiments, a series of ion pumps, titanium sublimation pumps, and non-evaporable getters are constantly used to keep the pressure in the apparatus low. Finally, it should be noted that a gate valve is also included in the system to isolate the the oven region from the zeeman slower and experimental region for when we need to open the oven to atmosphere and replenish the lithium source The Oven Region The oven used as a source of 6 Li atoms for our experiments primarily consists of a 1-1/3 conflat flange half nipple containing approximately 2 grams of solid lithium. This nipple is located inside a two inch diameter form fitting aluminum mount that provides excellent thermal contact with the half nipple ensuring an even distribution of heat. The aluminum mount itself is wrapped with band heaters that heat the solid lithium to a temperature of 435 C, well above the C melting temperature for lithium. This results in a vapor pressure of Torr in the oven itself [48]. At the top of the oven, the half nipple is connected via a 90 right angle elbow to a blank conflat flange with a 0.25 inch diameter whole drilled in the center. This hole, approximately 0.15 inches deep, acts as a nozzle for the hot lithium atoms as

122 ) Ion Pump 4) Gate Valve 2) Ti-Sub Pump 5) Zeeman Slower 3) Oven Region 6) Experimental Chamber Figure 7.1. The experimental layout for our UHV system used to cool, trap, and manipulate a gas of fermionic 6 Li atoms. The apparatus is divided into three regions, namely an oven used to provide a source of hot atoms, a Zeeman slower used to reduce the temperature of the atoms, and an experimental region where the cool atoms are then trapped and cooled further for use in our experiments. Also shown are a variety of vacuum pumps used to maintain the low pressure required for our experiments. This figure has been adapted from reference [6]. they are emitted from the oven [104]. Because lithium can be very reactive with copper, a nickel gasket is used at this nozzle in order to prevent the breakdown of the gasket and the formation of lithium salts. Additionally, to prevent lithium from condensing at this point, additional band heaters are used at the nozzle location to raise its temperature to 435 C. This ensures that the lithium atoms will condense back into the oven region, which determines the vapor pressure of the atomic gas. The number density of the atoms inside the oven chamber region can be calculated as [48] n 0 = P k B T = cm 3. (7.1) Using this value for the number density, it then becomes possible to determine the atomic flux Φ as well as the diffusion rate of the atoms through the nozzle Ṅ [105]

123 105 8k B T πm Φ= n 0 v 4 = cm 2 s 1 Ṅ =ΦA s = s 1 (7.2) where v = = 1610 m/s is the velocity of the atoms in the vapor and A s is the area of the nozzle. Also included in the oven region is a UHV compatible shutter manufactured by Uniblitz. This shutter is aligned so as to be in the path of the atomic beam. By activating this shutter, it becomes possible to block the atomic beam and prevent the atoms from propagating down to the experimental region of our apparatus. This function is important, as it prevents undesirable collisions between the atomic beam and the trapped samples of ultra-cold atoms, limiting their lifetime in the trap The Zeeman Slower To preliminarily cool the atoms emitted from the nozzle of the hot oven region, we employ the use of a counter propagating resonant laser beam. The basic principle behind this cooling scheme is as follows. As an atom absorbs a photon from the counter propagating beam, it experiences a change in momentum equal to the momentum of the absorbed photon. This excited atom will then undergo a spontaneous emission event whereby the photon will then be re-emitted in a random direction. After the absorption and emission of many photons, the momentum kicks associated with the spontaneously emitted photon will cancel each other out, leaving a net change in momentum along the direction of the cooling beam corresponding to the initial directionally dependent absorption of the photon. The maximum force associated with the absorption of a photon can be calculated as F max = kγ (7.3) 2 where k is the momentum of the absorbed photon and Γ is the spontaneous emission rate of the atomic transition (the factor of 2 originating from the equal population of the ground and excited states). For the 2 S 1/2 to 2 P 3/2 cycling transi-

124 106 tion used here for our 6 Li atoms, this corresponds to a large maximum acceleration of m/s 2. One problem that must be overcome in decelerating atoms using laser light is the Doppler shift of the light due to the motion of the atoms. As the atoms absorb photons and begin to slow down, the frequency of the light seen by the atoms is shifted until the beam is no longer resonant with the cycling transition. Because of this, only a small velocity class of atoms can absorb photons at any given time. One way to overcome this limitation is through the implementation of a Zeeman slower [10]. A Zeeman slower is a device that provides a spatially dependent magnetic field along the direction of atomic motion. As the atoms travel down the slower region scattering photons and reducing their velocity, the magnetic field experienced by those atoms change as well, causing a spatially varying shift of their energy levels keeping the atoms in resonance with the laser light. The Zeeman slower that we use in our experiment can be seen in figure 7.2. Three separate coils of wire are connected in series with one another to provide a spatially dependent magnetic field. The first two spatially inhomogeneous coils are wrapped around a long section of vacuum tubing while the third coil is located at the end of the experimental apparatus and is used to cancel the magnetic field inside the experimental chamber at the position of the magneto-optical trap. The wires used in these coils are hollow, enabling them to be water cooled. The magnetic field as a function of location with respect to the apparatus is shown in the subset of figure 7.2. The laser light used to drive the cycling transition originates from the spectroscopy laser setup (see section 7.2). Approximately 35 mw of power is coupled into a polarization maintaining fiber and launched down the slower axis of the apparatus. The frequency of the light is red detuned by 814 MHz from the D 2 line of 6 Li and is right circularly polarized via a λ/4 wave plate prior to the chamber. The beam has a diameter of approximately 15 mm at the entrance window of the experimental chamber and is focused on the nozzle in the oven region. At the zero crossing inside the slower region, the light is on resonance with the moving atoms. The intensity of the beam is approximately four times the saturation intensity of the transition and the velocity of the atoms at this location is approximately 540 m/s. To prevent the atoms from decaying into the 2 S 1/2 F =1/2 dark state, an

125 107 Section 3 Section 2 Section 1 λ/4 λ/2 λ/2 EOM Slower Field Profile 1000 B (Gauss) Slower Launch -800 Figure 7.2. Layout of the Zeeman slower used in our apparatus [6]. Three electromagnet coils are wired in series to produce a spatially dependent magnetic field that shifts the energy levels of the atoms to be continuously on resonance with a laser beam. The absorption of photons from this counter propagating laser beam causes a momentum transfer to the atoms, reducing their velocities and subsequently cooling the atoms as they enter the experimental chamber. electro-optic modulator is used to provide sidebands at ± 228 MHz for the slower light. The blue detuned sideband repumps the atoms back into the F=3/2 state where they can again absorb the carrier photons and continue the cooling process. As mentioned above, the coils are design to cancel out the magnetic field inside the experimental chamber after the Zeeman slower. In this region the atoms have an average velocity of 30 m/s. This velocity is small enough to be captured via a Magneto-Optical trap as described in section The Experimental Region At the end of the slower region of our apparatus, an 8 inch multi-conflat spherical octagon chamber acting as our experimental region is side mounted by one of its 8 2-3/4 conflat flanges. The benefit of using this type of a chamber, as opposed to a glass cell, is that the 7 remaining side ports as well as the two larger ports can be sealed with optical windows that have been anti-reflective coated for both

126 nm light as well as 1064 nm light. These windows provide excellent optical access for the variety of lasers required for the additional cooling and trapping of the pre-cooled atoms in this region. Additionally, the two larger windows are also recessed, providing nearly 90 of optical access to the center of our experimental region. The recess also allows for the close proximity of additional coils of wire used to generate the electric field gradients of our MOT as well as the bias magnetic field of our Feshbach magnets (see section 7.3). Inside the experimental region, two rf coils are also mounted to the chamber using groove grabbers. These rf coils are used to drive magnetic dipole transitions between different magnetic sub levels of our 6 Li atoms. By locating these coils within the chamber, the power required to drive the transition is greatly reduced, enabling efficient transfer Vacuum Pumps In order to maintain ultra-high vacuum (UHV) inside the apparatus, several different techniques for pumping out the system are employed. While roughing pumps and turbo pumps are used to initially reduce the pressure inside the chamber from atmosphere down to Torr, these pumps are quite noisy, causing vibrations in the system. Furthermore, these pumps are unnecessary once we engage additional pumps for continuous use in the system. The three main types of pumps that are used continuously in the system are ion pumps, titanium sublimation pumps, and non-evaporable getters. The principle behind the operation of ion pumps is that background gasses become ionized and accelerated via a strong dc electric field toward a chemically active cathode [106]. For the ion pumps used in our setup, this titanium cathode is specially designed for removing large amounts of noble gases and hydrogen from the system, operating at pressures ranging from 10 2 to Torr. Titanium sublimation pumps are used to supplement the pumping of the ion pumps in our system. Since titanium is a highly reactive metal, we can coat the inside walls of our experimental chamber with a thin layer of titanium to aid in the removal of unwanted atoms and molecules in our system. To coat the walls, a high current (on the order of 50 Amps) is sent through a titanium filament located

127 109 within the vacuum chamber. This current causes the sublimation of a thin layer of titanium on the walls. This thin layer can pump the chamber at a rate of 10 l s 1 cm 2. Over time, the titanium layer becomes saturated and the walls must be coated with a new layer of titanium to maintain pumping efficiency. Non-evaporable getters work based on similar principles to titanium sublimation pumps in that they absorb highly reactive molecules from the vacuum chamber. These getters are composed of a TiZrV alloy deposited on an amagnetic strip that can be cut to the appropriate size for use in our vacuum system. These strips can pump at rates of 0.1 l s 1 cm 2, and, much like the titanium, they must be reactivated from time to time via heating as they surface becomes saturated. Using a combination of ion pumps, titanium sublimation pumps, and nonevaporable getters, the day to day operating pressure of our apparatus is approximately Torr in our oven region and Torr in our experimental region. Differential pumping between the two regions is made possible by a four inch long, quarter inch diameter copper tube that separates these two regions. This tube restricts the conductance between the two sections of our apparatus and allows the oven region to operate at a higher pressure at its full operating temperature without increasing the undesirable background gas collisions inside the experimental region. 7.2 Laser System Overview In this section, we describe the laser systems that produce the 671 nm light used in our experiments for cooling, trapping, manipulating, and probing an ultra-cold 6 Li fermi gas. It is intended that this section will provide a brief overview of the laser systems used for these application without overwhelming the reader with the numerous details pertaining to the two setups. For a more thorough discussion of the individual components in each of these systems, please see the excellent description in reference [6]. Each of the two systems used to generate the required power to complete the desired tasks originates from a tapered amplifier that is seeded by a grating stabilized, tunable, extended cavity diode laser (ECDL) manufactured by Toptica. The tapered amplifiers work by taking the relatively low injected power of the seed

128 110 laser (on the order of 10 mw) and amplifying that power to approximately 500 mw. The first system described, referred to as the spectroscopy laser, generates the power necessary to operate the Zeeman slower and iodine lock while the second system, the experimental laser, produces the power for the cooling and repumping lasers of the MOT as well as the imaging laser for our absorption imaging setup described in section 7.6. In order to produce light at the proper frequency for cooling and trapping 6 Li, the frequency of the amplified laser light from the spectroscopy laser system is locked using sub-doppler saturated absorption spectroscopy of molecular iodine [107]. In particular, the ECDL laser was frequency locked to the a 1 0 hyperfine singlet of the R(142)5-6 rovibronic transition of the B0 + u X0 + g electronic system in 1 27I 2 [108]. This singlet was useful as it provides a strong enough line strength for high resolution feedback to lock the spectroscopy laser. Additionally, this rovibration transition is red-detuned from the 2 S 1/2 F =3/2 hyperfine transition by only 931 MHz, making it an ideal frequency reference for our laser system. The remainder of the light from the spectroscopy laser system is then summarily coupled into a polarization maintaining fiber for use in the Zeeman slower as described in section 7.1. As for the experimental laser system, prior to amplification, a small portion of the output of the ECDL is picked off and coupled into another polarization maintaining fiber where it is used for imaging the cloud of atoms as described in section 7.6. The remainder of the seed laser light is then amplified to 500 mw of power using another tapered amplifier setup. After amplification, a small portion of the beam is picked off and is then combined with a small portion of the output of the spectroscopy laser onto a high speed photodiode for beat-note locking the experimental laser to the spectroscopy laser, as described in reference [109]. In this way, the experimental laser is continuously phase-frequency locked to the spectroscopy laser (which is in turn frequency locked to the iodine reference) and can be tuned between over a range of 50 MHz to -2.1 GHz from the D 2 line of lithium. The remainder of the amplified light is then split into two paths for use as the cooling and repumping beams for our MOT as described in section 7.4.

129 111 Feshbach Coils MOT Coils RF Coils Zeeman Slower Recessed Viewports MOT Coil Housing Figure 7.3. Lateral cross section view of the experimental region of our apparatus. From this view, one can see the location of the MOT coils, the Feshbach coils, and the rf coils used to drive magnetic dipole transitions in our trapped atoms. This figure has been adapted from reference [6]. 7.3 Magnetic Field Coils In addition to the various optical laser systems necessary to facilitate the cooling and trapping of our ultra-cold atomic gases, a variety of magnetic field coils are necessary as well. In this section we will describe three sets of magnetic field coils used in our apparatus, namely a pair of anti-helmholtz coils used to create a magnetic field gradient for the MOT, a pair of Helmholtz coils used to provide the DC magnetic field to tune the s-wave scattering length and interaction strength of our two-component Fermi gas near a Feshbach resonance and a pair of rf coils located inside the experimental region of the vacuum chamber that will be used to drive magnetic dipole transitions between various hyperfine sub-levels of the atoms. The location of each of these coils can easily be discerned from the lateral cross section of the experimental region of our apparatus as shown in Figure 7.3. We begin our discussion by taking a closer look at the MOT coils. These coils are actually repurposed coils from previous experiments involving three-component fermi gasses [6]. In these experiments, the coils were used in Helmholtz configuration to apply a dc magnetic field to boost the old Feshbach coils. Now, these coils

130 112 are wired in anti-helmholtz configuration and provide a magnetic field gradient for the MOT. The coils themselves consist of ten turns of copper wire that fit within the recessed window housing, as seen in figure 7.3. The are located within a water cooled housing that keeps their temperature below the 200 C limit for our window vacuum seals. They have a radius of 3, a separation of 2.6, and are wired in series. Under normal operating conditions, approximately 25 amps of current from a Sorenson power supply flow through the coils, providing an on axis magnetic gradient of 30 gauss/cm for our MOT. The Feshbach coils used to provide a dc bias field on our trapped atoms consist of two independent coils of 16 turn shielded copper wire. This wire has a 3/16 3/16 cross sectional area with a 1/16 1/16 hollow core that is used to flow water through for the removal of heat. The coils are located on the outside of the 8 viewports of our experimental system (see figure 7.3) and are separated by 3.75, approximately satisfying the Helmholtz condition. Each coil is also contained within a plexiglass mount that prevents shifting of the coils as the large currents are quickly turned off and on. It should be noted that great lengths have been taken to control and stabilize the current that flows through each of the Feshbach coils in our system. While an excellent and thorough description of the control electronics has been described in the appendix of reference [103], a brief summary is provided here. The current for each coil is supplied by two current supplies (Agilent 6690A) wired in parallel for master/slave operation. Together, these supplies are capable of providing up to 880 amps of current for each coil, well in excess of that needed to generate the Gauss for the broad Feshbach resonance as described in section 2.3. The output current of the two supplies are continuously measured via a Danfysik 867 current transducer. The current then passes through an array of MOSFETs wired in parallel before continuing on to the Feshbach coils. By applying a gate voltage to the MOSFET arrays, we can set the current flowing through the coils in our system. To apply the appropriate gate voltage for the desired magnetic field, the following feedback loop is employed. First, a commanded current from the timing computer is converted from an analog signal to a digital signal using a field programable gate array (FPGA). This is done so that the signal, as it travels from

131 113 the timing computer across the room to the control electronics, does not pick up any noise on the analog signal. Once across the room, the signal is then sent into another FPGA that is used as a PID control circuit. The digital output of this FPGA is then converted to an analog signal again via a digital to analog converter which then both feeds back to the second FPGA and controls an analog control circuit. The analog control circuit is used to control the gate voltage of the MOS- FET bank which controls the output of the current supplies. As mentioned above, the current flow is measured via the Danfysik transducers, which then feeds back to the analog controller. Though seemingly complicated, this setup enables us to have a magnetic field stability on the order of 250 μg an impressive feat for an 800 Gauss field. The final set of coils used in our experiment are a pair of rf coils located within the vacuum of the experimental region. These coils consist of a single loop of a copper strip measuring 0.25 tall and 0.01 thick. As seen in figure 7.3, these loops are shaped into race-track shaped structures approximately 3.5 long and 1 wide. The two loops are separated by 1 and centered on the location of the atomic gas. In this way, the rf current applied to the loops act as a nearly uniform oscillating magnetic field in the ŷ direction. These loops are connected to bnc feed-throughs in the walls of our UHV vacuum system. The signal applied to the rf coils originates from either a home build direct digital synthesizer circuit or an Agilent waveform generator before being amplified and sent to the coils. In this way, the frequency of the rf signal can be tuned to be on resonance with the atoms so as to drive transitions between different magnetic hyperfine levels of the atoms via magnetic dipole transitions. 7.4 Magneto-Optical Trapping Even though the velocities of the atoms exiting the Zeeman slower region have been reduced to approximately 30 m/s, this is still too hot of a temperature for the atoms to be directly loaded into a crossed dipole trap for our experiments. Because of this, we first load the atoms into a magneto-optical trap (MOT) to provide additional cooling. The MOT is a technique that combines a spherical quadrupole trap with three

132 114 a)b) F = 1/2 2 2 P 3/2 F = 3/2 4.4 MHz σ + F = 5/2 σ + σ Cooling F = 3/2 Repumping 2 2 P 1/2 D 2 = nm F = 1/ MHz σ + σ σ F = 3/2 2 2 S 1/ MHz F = 1/2 Figure 7.4. a) The basic layout for the magneto-optical trapping of 6 Li. Three pair of red detuned orthogonal laser beams overlap the zero magnetic field location between two magnetic coils in anti-helmholtz configuration. By choosing the proper polarizations, the combination of magnetic and optical fields provide a restorative cooling force on the atoms located in this spatially overlapped region. b) The energy level diagram showing the cooling and repumping laser transitions for our MOT (solid lines). The dashed lines show the available channels for spontaneous emission, indicating the necessity of the repumping laser. pairs of counter propagating σ + and σ polarized laser beams as seen in figure 7.4a. These three pairs of counter propagating beams are orthogonally oriented and overlap each other in the center of the experimental chamber at the midpoint between the MOT coils. Each beam is red detuned from the 2 S 1/2 F =3/2 2 P 3/2 D 2 line of 6 Li. This detuning is chosen such that as an atom travels across the region of space where these beams overlap, there will be a component of the motion of the atom that travels counter to at least one of the six laser beams. For this direction, cooling light is doppler shifted into resonance and the atoms will receive a momentum kick in the opposite direction, similar to the principle behind the Zeeman slower. This type of cooling has been dubbed optical molasses because of the viscous nature of the cooling beams [10][9]. Because the absorption of a photon from the MOT cooling beams causes a small amount of heating as well, there is a lower limit to the temperature that can be obtained using this technique, known as the doppler limit T D. Assuming that the intensity of the cooling beams I is much less than the saturation intensity, I sat,

133 115 the minimum temperature achievable by a MOT is given by [8] k B T = Γ 4 1+ ( 2δ Γ 2δ Γ ) 2 (7.4) where Γ is the natural linewidth of the transition and δ =(ω ω 0 ) is the detuning of the laser. From the equation, one can see that the temperature T D is at a minimum when δ =Γ/2, yielding a lower limit of 142 μk for 6 Li using this technique. While these six laser beams can provide additional cooling for the atoms, a spherical quadrupole magnetic field gradient is required to provide a restorative confining force for the atoms [10, 8]. The polarization of the six beams and the orientation of the anti-helmholtz coils are chosen such that as an atom is displaced from the zero field location between the anti-helmoltz coils, it experiences a net magnetic field that will cause a Zeeman shift on its energy levels. This shift brings the detuned laser light from the cooling beams onto resonance with the atom, which then begins to scatter photons and experience a restoring force pointing in the direction of the zero field location. In this way, the atoms are both cooled and trapped using a combination of magnetic fields and optical beams, giving rise to the name magneto-optical trap. As mentioned previously, the 2 S 1/2 F =3/2 2 P 3/2 transition is not a closed transition in 6 Li. Atoms that have been excited to the 2P 3/2 state have a finite probability of spontaneous decay back down to the 2 S 1/2 F =1/2 state, which is a dark state to the cooling laser light. If no corrective action were to be taken to plug this leak, the atoms would quickly be pumped to this lower ground state and no cooling could effectively take place. To compensate for this, additional beams resonant with the 2 S 1/2 F =1/2 2 P 3/2 transition are used to repump the atoms back into resonance with the cooling laser light, as shown in figure 7.4b. For our experiment, the cooling and repumping light used for the MOT are derived from the experimental laser. Light of the two appropriate frequencies are coupled into a 2 6 fiber optic coupler that mixes the two input frequencies and the six beams used for the MOT. This results in 10 mw of power for the cooling laser and 2 mw of power for the repumping laser in each of the six beams used in the MOT. Also, in our experiments, we use a two stage cooling procedure to optimize the cooling of our trap, as described in reference [6]. In this way, we are

134 116 able to produce a cloud of 10 8 atoms having a temperature of 200 μk with a peak density of atoms/cm 3 for loading into a optical dipole trap for additional cooling and experimentation. 7.5 Optical Dipole Traps As mentioned in section 3.2, the presence of an oscillating electric field can shift the energy levels of an atom via the AC Stark effect. In many atomic physics experiments, this light shift for an atom in the ground state can be exploited to create an additional trapping potential for the atoms because of their polarizability. The basic premise for this optical dipole trap is as follows. The dipole force associated with the gradient of the electric field is given by [10] F dipole = δ 2 Ω Ω δ 2 +Ω 2 /2+Γ 2 /4 z (7.5) where δ = ω ω 0 is the detuning of the light from resonance, Ω is the Rabi frequency of the light, and Γ is the linewidth of the transition. When the detuning δ Γ and for an intensity such that δ Ω, the dipole force equals the derivative of the light shift F dipole ( ) Ω 2. (7.6) z 4δ Therefore, the light shift for an atom in its ground state acts as a potential in which the atoms can be trapped. This treatment also can be expanded to three dimensions as F dipole = U dipole (7.7) where U dipole Ω2 4δ = Γ 8 Γ δ I (7.8) I sat where we have defined [10] I 2Ω2 I sat Γ. (7.9) 2 When δ is negative, the light shift provides an attractive trapping potential whereby the atoms are forced to the region of highest intensity. In this way, a focused laser beam can be used to create this optical dipole force trap.

135 117 Y X 1070 nm 1064 nm 1064 nm Figure 7.5. Cartoon layout of the three lasers that overlap to form the optical dipole force trap for our experiments. Two 100 watt 1064 laser beams intersect at a relative angle of 11 with each beam having a waist of 30μ. A third 1070 nm 100 watt laser overlaps the other two, forming a deep trap used to capture atoms from our MOT. Also shown is the side view of one of the anti-helmholtz coils used to provide the spherical quadrupole trap for our MOT. It should also be noted that the presence of this trapping potential could also cause atoms to become ejected from the trap due to the spontaneous emission of photons from the light. The photon scattering rate [10] is equal to the spontaneous emission rate times the excited state population, given by R scatt = Γ 2 In the limit that the detuning δ Γ, R scatt becomes Ω/2 δ 2 +Ω 2 /2+Γ 2 /4. (7.10) R scatt Γ 8 Γ 2 I. (7.11) δ 2 I sat Comparing this equation to the dipole potential, we see that the scattering rate goes as I/δ 2 whereas the trapping potential goes as I/δ. Therefore, in order to provide an effective trapping potential and minimize the effects due to spontaneous emission, the wavelength of the trap laser must operate at a sufficiently large detuning.

136 118 Figure 7.6. Absorption image of a cloud of 6 Li atoms trapped in a) two 1064 nm beams forming a crossed dipole trap b) a single 1070 nm beam c) a combination of the two 1064 nm beams and the 1070 nm beam. For our experiments, three intersecting laser beams are used to create our optical dipole trap as shown in figure 7.5. Two of the laser beams are derived from a 1064 nm fiber laser while the third beam, along the x-direction, is derived from a 1070 nm fiber laser. Because these wavelengths are far detuned from the 671 nm D 1 and D 2 lines of 6 Li, the effects of spontaneous emission on atoms trapped in these potentials are minimal. As a result, the intensity gradient of each of these beams must be very high in order to provide a practical trapping potential. For our experiment, both the 1064 and 1070 nm laser beams are approximately 100 Watts in power in each beam and have been focused down to a waist of 30 μm. This creates a trapping potential that is 400 μk deep and able to trap the 200 μk atoms from our magneto-optical trap. Additionally, sideband excitations of the trap by misaligned Raman beams indicate that this crossed dipole potential has trap frequencies ω =2π (7.2, 3.4, 11.2) khz. In figure 7.6, we show false color absorption images of our atomic cloud trapped in either the crossed 1064 nm laser beams, the 1070 nm laser beam, or a combination of all three. 7.6 Atomic Imaging As seen in the false color image in figure 7.6, experimental information about our cloud of trapped atoms is measured and recorded via a process known as absorption imaging. The basic premise behind absorption imaging is this. The trapped atoms are exposed to a laser beam of resonant light that is incident upon a CCD camera. If there are atoms in the trap, they will absorb photons from the reso-

137 119 nant light and scatter them in a random direction, leaving a shadow on the CCD camera corresponding to the location of the atoms. From this absorption image, information about the total atom number, number of atoms in each spin state, temperature, density, and quantum degeneracy can all be determined, making it the primary source of data collection in our experiments. For a probe beam with initial intensity I 0 propagating through a cloud of atoms, the intensity will be attenuated by the atoms according to I = I 0 e D (7.12) where D is the optical depth of the cloud. For a probe beam that is resonant with a single transition having saturation intensity I sat, the optical depth is given by D = D I I sat + 4Δ2 Γ 2 (7.13) where Δ is the detuning of the probe beam from the atomic resonance with natural linewidth Γ and D 0 is given by D 0 = σ 0 ñ (7.14) where σ 0 is the polarization averaged resonant scattering cross section and ñ is the column density, corresponding to the spatial density integrated over the ẑ direction. For our experiments, all images taken of the cloud of atoms occur in the so called high field limit. In this limit, the atoms are imaged in the presence of a magnetic field causing a Zeeman splitting of the hyperfine energy levels, as described in section 2.1. In this way, the probe light used to image the cloud can be resonant with only one transition, enabling us to determine the individual populations of the different magnetic hyperfine states. The probe beam is derived from the experimental laser system (see section 7.2) and is σ polarized, propagating along the positive ẑ quantization axis of our system. This beam is tuned to be on resonance with the ± 1, ± 1, 0 transition in the Jm J Im I basis. This transition is a closed cycling transition and does not need repumping light, as would be the case for zero magnetic field imaging. The actual experimental procedure for taking an absorption image of the cloud

138 120 of atoms is as follows. First, a short 20 μs resonant imaging pulse I abs is pulsed on the atoms and the resulting shadow is recorded by the CCD camera. This pulse is sufficiently strong so as to rapidly heat all of the atoms out of the trap due to spontaneous emission. Next, a reference image I ref is taken using an identical pulse ms after the first. Finally, a third background image I bkgrd is taken several hundred milliseconds later. The transmission of the imaging light as detected by the camera can then be calculated as T (x, y) = I abs(x, y) I bkgrd (x, y) I res (x, y) I bkgrd (x, y). (7.15) From this transmission, the density distribution n(x, y) can then be extracted as n(x, y) = 1 ln [T (x, y)] (7.16) σ 0 M 2 where M is the magnification of the lens system in front of the CCD camera. This density distribution is the primary measurement of our system and is useful for extracting numerous experimental parameters as mentioned above. For additional information about the imaging process, see references [6] and [103].

139 Chapter 8 The Rapid Control of Interactions In this chapter, we present the development of methods to control interactions in a two-component Fermi gas. As mentioned before, for atoms in the two lowest hyperfine magnetic spin states of fermionic 6 Li ( 1 and 2 ), there exists a very broad Feshbach resonance in the vicinity of where the s-wave scattering length diverges. In addition, a much more narrow Feshbach resonance having a width of 0.1 Gauss located at Gauss has also been experimentally verified [53]. Near this narrow resonance at a field of Gauss the scattering length crosses zero, allowing us to investigate a non-interacting gas. We will first develop methods to transfer atoms between states 2 and 5 near this zero-crossing of the scattering length and then study Raman transfer near the broad Feshbach resonance. To rapidly control interactions in this gas, we will use two phase locked lasers to quickly drive Raman transitions between different magnetic hyperfine levels. As calculated in section 2.4, for values around the gauss Feshbach resonance seen for a 1-2 mixture, the 1-5 s-wave scattering length is around -3 Bohr. Therefore, by creating a 1-2 mixture of atoms at a magnetic field near the broad Feshbach resonance where the scattering length is very large, we quickly control the strength of interactions between the two atoms by driving a transition from 2 5, reducing the s-wave scattering rate by several orders of magnitude. This chapter is divided up as follows. Section 8.1 will describe the basic procedure for preparing an initial two-state Fermi gas starting with loading atoms from a MOT. Next, we will discuss the various techniques employed to phase lock two

140 122 lasers in section 8.2. In section 8.3, we demonstrate the rapid transition from state 2 to state 5 for a two-component mixture at the Gauss zero-crossing. Finally, in section 8.4, we will discuss the next steps for our experiment. 8.1 Initial Preparation To initially prepare a two-component Fermi gas for our experiments, we begin by loading a magneto optical trap (MOT) with 6 Li atoms that have been pre-cooled using a Zeeman slower as described in section 7.1. Atoms are then loaded into the MOT for approximately 1 second. At this point, the detuning and intensity of the cooling and repumping laser light are changed so as to be closer to resonance and less intense. This reduces the temperature to approximately 300 μk cold enough to then be loaded into the 1064 nm crossed dipole trap. Once the atoms are loaded into the crossed dipole trap, the repumping light is turned off and the atoms are optically pumped from the 2 S 1/2 F = 3/2 state into the 2 S 1/2 F = 1/2 ground state. After optical pumping, the atoms are in an incoherent mixture of states 1 and 2. Next, a magnetic field of approximately 572 Gauss is applied to the atoms by the Feshbach coils. This causes the atoms in states 1 and 2 to undergo collisions with a positive s-wave scattering length. It is at this point the atoms are further cooled via forced evaporative cooling by reducing the intensity of the 1064 nm crossed dipole trapping lasers. During the evaporative cooling process (and with a final pulse shortly thereafter) we want to prepare the atoms in a mixture of states 1 and 2. Todoso, we first transfer the atoms in state 2 to state 3 using rapid adiabatic passage. Light from the imaging laser is then used to clear the atoms from state 3 and another rapid adiabatic passage sequence transfers the remaining atoms from state 3 to state 2. For all these experiments, all the atoms are effectively removed by the first clearing pulse, leaving now only atoms in state 1. At the end of evaporation, an additional 1070 nm optical dipole trap is turned on, providing an additional tight confinement for the atoms along the ŷ direction. Finally, a π/2 pulse is applied via the rf coils located inside the experimental region, creating a balanced mixture of atoms in states 1 and 2. It is at this point that the applied magnetic field can be changed to the desired field of interest and further

141 123 experimentation can be undergone. 8.2 Phase Locking of Raman Lasers To drive a Raman transition between two different hyperfine levels in an atom, the two lasers must be phase coherent. The simplest way to generate two phase coherent beams is to use one laser and an acousto-optic modulator (AOM). By driving the AOM with an rf-signal whose frequency corresponds to the frequency of the hyperfine transition, light from the 0th and 1st order of diffraction off the AOM crystal can be used. This technique works well for transitions in 6 Li where the spin of the electron does not need to be flipped during the transition. That is, for transitions between states 1, 2, and 3, it is experimentally convenient to drive the AOM with the 80 MHz signal necessary for these transitions. For transitions involving an electron spin flip, such as 2 to 5, the process is not as simple. As discussed in section 2.1, the difference in energy between states 2 and 5 scale as twice the bohr magneton (2.8 MHz per Gauss) for applied magnetic fields. Therefore, calculating the difference in energy between the two states, we find that at the Gauss 1-2 zero crossing, the corresponding frequency for a 2-5 transition is approximately 1.5 GHz. Near the broad Feshbach resonance at Gauss, the separation is even larger on the order of 2.35 GHz. Because of these microwave frequencies required for the transition, it is challenging to use a single laser source and an AOM to generate the two laser frequencies as the efficiencies at these frequencies are quite low. Additionally, we desire to broadly scan the frequency difference to investigate BEC/BCS crossover physics, but can only generate a difference frequency equal to the resonance of the AOM. Therefore, we must instead phase lock two lasers with the appropriate difference frequencies. The method used to phase lock two lasers is outlined in figure 8.1. This technique is modeled after that used in reference [109]. The output of two DL100 ECDL from Toptica Photonics operating near 671 nm are incident upon two polarizing beam splitters where a small portion of each beam is picked off (approximately 1 mw of power in each). The two strong beams are combined and coupled into a polarization maintaining fiber where they are routed to the main experiment

142 124 DL100 To Experiment Bias-T FET Piezo DL100 RF Signal Generator DET OPLL +15 db Amp +15 db Amp Split +15 db Amp +15 db Amp Phase Advance Filter Low Pass 80 MHz RF IF LO Mix RF Signal Generator Figure 8.1. Diagram showing the components making up the phase lock feedback loop. A small amount of power from each of two diode lasers are combined on a high bandwidth photo detector. The beat note signal of the two beams is then split where it takes one of two paths. On the first path, the signal is amplified before being fed into an optical phase-lock loop circuit where the beat note is compared to a reference oscillator and a corrective signal is fed back to the piezo and the FET of the laser diode. The signal is also amplified on the other path before being mixed with another reference oscillator, filtered, and coupled to the bias-t connector of the same laser. In this way, we can lock both the frequency and phase of the two diode lasers to arbitrary values. for use in driving Raman transitions. The weak beams are also combined and launched into a fiber, where they are incident upon a high speed photo-detector (ν-focus model 1580). This photo-detector is fast enough (12 GHz bandwidth) to measure the beat note frequency between the two diode lasers. The output of this detector is then split using an rf-splitter from Mini-Circuits. For slow feedback to the lasers, one of the outputs from the splitter is amplified by two +15 db low noise amplifiers (Mini-Circuits model ZX E-S+) and connected to the input of an optical phase lock loop (OPLL) circuit, as described in reference [109]. The OPLL circuit used in this experiment primarily consists of a digital phasefrequency-discriminator chip (Analog Devices ADF4107). This chip measures the beat note signal and digitally divides the frequency by a factor of 16. The frequency and phase of this divided signal is then compared to a reference signal generated by an 1 GHz MXG Waveform Generator by Agilent Technologies (model N5181A). This waveform generator allows the user to program the desired difference fre-

143 125 quency between the two lasers for the OPLL circuit to lock. For small differences between the beat and reference signals, the phase-frequency-discriminator chip produces a feedback current proportional to this difference. This feedback current is then either integrated for feedback to the piezo input of one of the DL100 lasers (with a bandwidth of approximately 50 khz), or amplified for direct feedback to the laser diode current via a faster FET input (with a bandwidth of approximately 5 MHz limited by the gain bandwidth product of the electronics). In addition to using the OPLL circuit to lock the laser using a piezo and an FET, we also apply feedback to the faster bias-t input of the diode laser. The other output of the rf splitter mentioned above is again amplified by two +15 db low noise amplifiers before being mixed with an rf signal generated by another Agilent N5181A. This Agilent generates frequencies up to 3 GHz and is set such that the frequency is exactly 16 times that used for the OPLL circuit. Additionally, the time reference for each of these two generators are synched to a 10 MHz GPS signal, ensuring a consistent phase difference between their outputs. This phase difference can then be adjusted by the user on the 3 GHz Agilent waveform generator. The output of the mixer is then sent through an 80 MHz low pass filter followed by a phase advance filter [110] before being coupled into the bias-t input of the same laser. In this way, we can now provide feedback on the locked laser with a bandwidth of up to 1 GHz for phase locking our two diode lasers. To characterize how well our lasers are phase locked, we can look at the output of the beat note signal on a spectrum analyzer (Hewlett Packard HP8595E). From this signal, we can measure the carrier power fraction by looking at the ratio between the power in the carrier signal to the total power in the beat note signal. This carrier power fraction is related to the mean-square phase error Δφ 2 by the relation [111] exp ( Δφ 2 ) = P carrier P (ν)dν (8.1) where P carrier is the power in the carrier and P (ν) is the total power in the beat spectrum. Figure 8.2 shows the beat spectrum of our laser phase locked to an arbitrary 1.6 GHz value. From the figure, the total power in the beat spectrum was measured to be 2.2 dbm with 0.35 dbm power in the carrier. This corresponds

144 126 Figure 8.2. The beat spectrum of our two phase locked lasers locked at 1.6 GHz. The central peak of 0.35 dbm corresponds to the power in the carrier while the total power from the occupied bandwidth measurement (2.2 dbm) can be used to determine the carrier power fraction, and thus the mean-square phase error of our lock. to a mean-square phase error of 0.42 rad 2.