.~~/~ Dr. Karen L. Sauer, Dissertation Director pi,. ~cgu:= Dr. Maria Dworzecka, Committee Member

Transcription

1 SPIN-DAMPING IN AN ULTRA-SENSITIVE TUNABLE RF ATOMIC MAGNETOMETER by Orang Alem A Dissertation Submitted to the Graduate Faculty of George Mason University in Partial FuLfillment of the Requirements for the Degree of Doctor of Philosophy Physics Committee:.~~/~ Dr. Karen L. Sauer, Dissertation Director pi,. ~cgu:= Dr. Maria Dworzecka, Committee Member Dr. Mingzhen Tian, Committee Member ~~ Dr. John Schreifels, Committee Member Dr Mike Summers, Department Chair Dr. Timothy Born, Associate Dean for Academic and Student Affairs, College of Science Dr. Vikas Chandhoke, Dean, College of Science Date: ---l7-1-1.c!...-19l.j--/-i...2"~ _ Summer Semester 2011 George Mason University Fairfax, VA

2 Spin-Damping in an Ultra-Sensitive Tunable RF Atomic Magnetometer A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at George Mason University By Orang Alem Master of Science, Applied and Engineering Physics George Mason University, January 2007 Bachelor of Science, Physics George Mason University, January 2001 Bachelor of Science, Mathematics George Mason University, January 2001 Director: Dr. Karen L. Sauer, Professor Department of Physics and Astronomy Summer Semester 2011 George Mason University Fairfax, VA

3 Copyright c 2011 by Orang Alem All Rights Reserved ii

4 Dedication I dedicate this work to the memory of my father, Ali Alem, and to the strongest person I know, my mother, Kianoush Alem. My realization and appreciation for everything she has done and sacrificed runs deep. She has supported me unconditionally in all of my endeavors. I owe everything that I am and have accomplished to them. iii

5 Acknowledgments First and foremost I would like to thank my advisor, Dr. Karen L. Sauer. She pushed me when I needed to be pushed, guided me when I was lost, and gave me room to grow. I could not have asked for a better mentor, both as a teacher and as a scientist. Without her this work would not have been possible. Next, I d like to thank Dr. Mike Romalis for his collaboration. My gratitude is also extended to the faculty and staff of the physics department at George Mason University who supported my education through the years. I would like to thank my lab mates, Dave Prescott and Michael Malone, for their technical, moral, and humorous support. Thanks to Laura Cawthorn and Phillip Naudus for their contributions to this project. Thanks to my committee members, Dr. Maria Dworzecha, Dr. Mingzhen Tian, and Dr. John Schreifels for their time and feedback. Much gratitude to my editing team, Megan Van Dyke and Keyana Corliss for many hours of editing and proofreading of this manuscript and to Farhang Alem for the great pictures. My appreciation goes out to my friends for their understanding during the last few months and a special thanks to my family for their love and support. Particular thanks to Megan for making the tough moments easier. This work was supported in part by NSF Grant and iv

6 Table of Contents Page List of Tables vii List of Figures viii Abstract x 1 Introduction Atomic Magnetometer with Spin-Damping - Theoretical Approach Tunable RF Atomic Magnetometer Tuning and the Quadratic Zeeman Effect Optical Pumping Perturbation by an RF Field Magnetometer Response Detection by Faraday Rotation Sensitivity Spin-damping Feedback with Finite-Gain Amplifiers Frequency-Dependent Feedback of the Magnetometer Noise with and without Damping Measuring Signal and Noise with Quadrature Detection Atomic Magnetometer - Physical Realization and Noise Limitations The K Cell Vacuum System Filling Process Pressure Broadening K Atomic Density The Oven Coil Assembly Sample Holder and the Excitation Coils The Shielding Pump and Probe Lasers v

7 3.7 Balanced Polarimeter Spectrometer Experimental Noise Contribution Magnetometer Sensitivity Optimization and Tuning of the Magnetometer Magnetometer Sensitivity and Linewidth Results Spin-Damping in an RF atomic magnetometer Sensitivity and Bandwidth under Constant Spin-Damping Measuring the Damping Factor Signal Suppression Noise Suppression The Limit of Noise Suppression Gain in Magnetometer Bandwidth Ringing and Spin-Damping Ringing in an Atomic Magnetometer Damped Ringing in Time Domain Sensitivity Versus Acquisition Window SNR with Ringing and Spin-Damping NQR Measurements Conclusion Bibliography vi

8 List of Tables Table Page 3.1 Volumes of vacuum system sections Magnetic field strength of the field coils vii

9 List of Figures Figure Page 1.1 Diagram of an RF atomic magnetometer Spin-damping schematic Energy level diagram for K Plot of Breit-Rabi equation for K Zeeman transition frequencies Schematic of feedback mechanism Predicted SNR and bandwidth as a function of damping factor Magnetometer schematic K cells Vacuum system Optical linewidth of K Optical depth as a function of temperature Calculated K atomic density Magnetometer linewidth as a function of pump power Transmitted light as a function of temperature Oven schematic Coil assembly schematic Smallest measured magnetometer linewidth Offset coils Sample holder schematic Excitation coil tuning circuit Measured excitation coil field amplitude Zemax simulation of pump beam expansion Pump and probe intensity profile after 1st expansion Pump beam intensity profile after Keplerian expansion Pump and probe laser noise measurements Polarimeter circuit Measured photodiode responsitivity viii

10 3.22 Dynamic range of the Tecmag spectrometer Measured Tecmag spectrometer sensitivity Beam alignment diagram Magnetometer signal and linewidth as a function of pump power Magnetometer response to on-resonance signal in the time domain Magnetometer response to an RF signal as a function of frequency Magnetometer noise spectra Difference in measured shot noise Optimal magnetometer sensitivity Suppressed magnetometer signal response for various damping factors Signal suppression Suppressed magnetometer noise for various damping factors Filtered noise on extreme spectra wings Noise suppression Quadratic fit of the suppressed noise power density for data set Quadratic fit of the suppressed noise power density for data set Magnetometer SNR bandwidth with spin-damping Ringing and signal with and without spin-damping Ringing with and without spin-damping Signal amplitude as a function of window size Noise amplitude as a function of window size Sensitivity as a function of window size Magnetometer SNR in the presence of ringing and spin-damping Measured NQR signal with a short dead time and spin-damping NQR calibration curves with spin-damping ix

11 Abstract SPIN-DAMPING IN AN ULTRA-SENSITIVE TUNABLE RF ATOMIC MAGNETOME- TER Orang Alem, PhD George Mason University, 2011 Dissertation Director: Dr. Karen L. Sauer Optically pumped radio frequency (RF) atomic magnetometers have been shown to have an improved sensitivity over standard tuned coils for frequencies less than 50 MHz, making these RF magnetometers attractive for the detection of nuclear quadrupole resonance (NQR) and low-field nuclear magnetic resonance (NMR) signals. In an atomic magnetometer a linearly polarized probe beam measures, through Faraday rotation, the transverse atomic magnetization induced by the resonant RF signal. The resonance, or Larmor, frequency of the magnetometer is easily tuned with a small magnetic field. We construct an atomic magnetometer based on a potassium vapor cell with a measured sensitivity of 0.22 ± 0.02 ft/ Hz and a detection bandwidth of 334 ± 11 Hz when tuned to 423 khz, giving a Q of over While high Q magnetometers are sensitive to weak magnetic fields, they are also sensitive to magnetic transients, such as those associated with the excitation pulses needed for magnetic resonance. The ringing created by such transients can obscure the signal of interest. This is particularly detrimental for magnetic resonance signals that decay faster than the transverse relaxation rate. We demonstrate that by feeding back part of the optical signal to orthogonal electromagnetic coils, this unwanted ringing can be quickly damped

12 out leaving the magnetometer ready for the detection of the signal. This negative feedback as applied to the K spins is called spin-damping and can be used to relax the K atoms faster by more than two orders of magnitude. Using spin-damping we reduce the dead-time before data acquisition from 0.8 to 0.2 ms in the detection of NQR signals from ammonium nitrate, one of the so-called fertilizer bomb explosives with an NQR frequency of 423 khz. Furthermore, we discover that spin-damping can not only be used to suppress signals, it can also be used to suppress noise. We show that it can be used to suppress both environmental noise and more fundamental quantum noise, such as photon shot and spin projection noise. The latter suppression opens the door for its use in quantum control to initialize spin systems before measurements. For phase-sensitive detection, as is often used in magnetic resonance, spin-damping suppresses signal and noise in such a way as to increase the sensitivity bandwidth of the magnetometer. We prove, both theoretically and experimentally, that it is possible to increase the magnetometer detection bandwidth by a factor of 3 or more without significant loss in sensitivity. Such gains in bandwidth for a high Q magnetometer translate into savings in detection time, particularly for substance detection where the frequency of the signal of interest may not be completely known.

13 Chapter 1: Introduction Magnetic fields are a fundamental part of nature. The detection of magnetic fields is of great value as a tool for both the study and understanding of the fundamentals of nature and for use in day-to-day application in many different fields. Magnetometers take advantage of different physical principles such as Faraday induction [1], Hall effect [2], the Josephson effect and Meissner effect in superconducting quantum interference device (SQUID) [3], the Zeeman effect in optically-pumped atomic magnetometers [4], and others. A comprehensive list of magnetic sensor technologies employing these effects can be found in [5]. Induction coil detection, also known as pickup coils, is a mature and highly robust technique that has seen use in a wide range of applications for decades. The fundamental limitation of induction coil magnetometery is Johnson noise, particularly at low frequencies where this becomes of consequence as its sensitivity scales with frequency [6]. One way to compensate for this loss at low frequency is the use of super conducting coils [7]. However, the use of superconductivity requires cryogens. The very sensitive SQUID magnetometers also use cryogens. The versatility of SQUID magnetometers to measure both static and radio-frequency (RF) fields with high sensitivity (on the order of T/ Hz) has resulted in the wide spread use in applications such as low-frequency detection in biomagnetism, geomagnetism, and metrology [8] and high frequency nuclear magnetic resonance (NMR) signal detection [9]. However, the cryogenic operation requirements can make practical application in many situations very difficult and costly. Higher temperature SQUID magnetometers, operating at the temperature of liquid nitrogen (77 K), have sensitivities that suffer in comparison to the ultrasensitive liquid He magnetometers [10]. The need for magnetic detectors with better sensitivity has prompted the emergence 1

14 of ultrasensitive atomic magnetometers in the last decade [11 13]. Atomic magnetometers whose sensitivity is rather flat with frequency [6] provide a new level of precision for the detection of both static and low frequency magnetic fields with potential sensitivities limited by quantum fluctuations, including spin-projection noise as governed by the Heisenberg uncertainty principle [14, 15]. Recent advancements in atomic magnetometers have yielded sensitivities that equal or better SQUID magnetometers. An ultrasensitive atomic magnetometer operating in the spin-exchange relaxation-free (SERF) regime [16] with a sensitivity of 0.16 ft/ Hz (at 30 Hz) has been demonstrated [13] for low frequencies, while RF atomic magnetometers have reached sensitivities of 0.24 ft/ Hz (at 423 khz) [17]. They have already found many uses from magnetoencephalography [18, 19] to electromagnetic geophysical exploration [20] and the detection of nuclear magnetic resonance signals [21,22]. Small chip-scale atomic magnetometers have been shown to match the performance of SQUID sensors without requiring superconducting magnets or cryogenics [23] [24]. One particular area of interest in recent years has been the use of sensitive RF atomic magnetometers for the detection of explosive materials using NQR spectroscopy [25]. While NMR techniques are capable of identifying certain materials with a high degree of accuracy, the requirement of a large static magnetic field renders this technology impractical for use in certain applications. NQR spectroscopy is similar to NMR except it does not require a strong externally applied static magnetic field for the necessary energy splitting of the nuclei [26]. Instead, the resonance signal arises from the energy level splitting caused by the interaction of quadrupole nuclei (spin > 1/2) with the electric field gradient arising from the surrounding charge distribution in the crystal [27]. The highly coupled interaction of the quadrupole nuclei with the unique electric field gradient of each substance provides an NQR frequency signature that is different for each substance. As a result, NQR spectroscopy becomes a very attractive technique for the detection of materials that contains nuclei with spin > 1/2, such as many explosives and narcotics. The improved sensitivity of atomic magnetometers for NQR detection translates into more efficient detection [17], where a 2

15 factor of 10 improvement in sensitivity corresponds to a factor of 100 improvement in detection time, or requiring a tenth of the amount of substance for a similar sized signal. The main limitation of NQR spectroscopy for practical application is the inherently low signal-to-noise ratio (SNR). The strength of the resonant signal is proportional to the population difference between the energy levels, or hν/kt, where hν is the transition energy between the energy states and kt is the Boltzmann thermal energy. Typical NQR frequencies for most common explosives, such as TNT or ammonium nitrate (a common fertilizer used in the Oklahoma city bombing), are in the MHz range [28]. In contrast, the nuclear magnetic dipole interaction with a strong external field, say 1 Tesla for example, produces an NMR resonant frequency of 43 MHz for hydrogen atoms. As a result, NQR signals are approximately one to two orders of magnitude weaker than a typical NMR signal. The high sensitivity of an RF atomic magnetometer makes it an ideal detector for the weak NQR signals at low frequencies [17]. A simple functional diagram of an atomic magnetometer is given in Fig An RF atomic magnetometer relies on the atomic Zeeman transitions of an optically pumped alkali vapor for the detection of an external RF magnetic field. A vapor cell containing an alkali metal, in an applied static field B 0, will have a ground state Zeeman energy splitting with a Larmor frequency ω L = γ B 0, where γ is the gyromagnetic ratio of the alkali atom. Optical pumping of the alkali vapor by circularly polarized resonant photons [29] can induce nearly 100% polarization of the atomic spins into a single spin state forming a net magnetization vector parallel to B 0. An orthogonal RF magnetic field B RF, with frequency equal to the Larmor frequency, will induce Zeeman transitions within the alkali population resulting in the tipping of the net magnetization vector into the transverse plane. The magnitude of the transverse component of the net alkali magnetization vector in the rotating frame is proportional to the amplitude of B RF. The transverse magnetization can be detected through the Faraday rotation, ϕ x, of an off-resonance linearly polarized light passing through the alkali vapor. The rotation of the polarization is detected by a balanced polarimeter and the output of the magnetometer is a voltage signal that is proportional to the magnitude 3

16 φ x (t) B RF B 0 Pump Pump y x z Probe K Cell M y M x Figure 1.1: Diagram of a tunable RF atomic magnetometer. A static magnetic field B 0 (blue) induces Zeeman splitting with transition frequency ω L = γ K B 0. Two circularly polarized pump beams (pink) optically pump the K spins into a single Zeeman state, with near 100% polarization, creating a net K magnetization aligned with B 0. An RF magnetic field (orange), with the same frequency as the K Larmor frequency, induces magnetization M x along ˆx, which is detected by the Faraday rotation of the linearly polarized probe beam (green). of the detected RF magnetic field. In resonance detection such as NQR, a sample placed close to the vapor cell is excited with an RF magnetic pulse at the same frequency as the tuned magnetometer. The magnitude of these excitation pulses is typically 12 orders of magnitude greater than the NQR signal. To protect the spins, an offset field is applied during the excitation pulse to detune the K spins away from the frequency of the excitation field [17]. Without the offset field the excitation pulse would destroy the K polarization, which would take tens of milliseconds to return and would render the magnetometer useless during that time. However, the application of the offset field, if done too quickly, can cause ringing in the K spins. Any rapid amplitude fluctuations of a magnetic field that do not allow sufficient time for the 4

17 adiabatic transition of the atomic spins with the field will induce ringing. Through the decay of this ringing the magnetometer experiences a dead time during which it is unable to detect any weak RF fields. This is particularly detrimental to the NQR detection of certain explosives like TNT, where the resonance signal has a characteristic decay time of less than one millisecond [25]. Such a signal would be obscured by the ringing noise. In atomic magnetometers, ringing can be quickly eliminated through a spin-damping mechanism. The sensitivity of atomic magnetometers is ultimately limited by quantum fluctuations, where the Heisenberg uncertainty principle sets a limit on the projection of the atomic spins. There has been much interest, particularly in magnetometery and quantum information science, in the manipulation of quantum states through feedback in order to reach and surpass this standard quantum limit through spin-squeezing [30 32]. The approach taken by Geremia, Stockton, and Mabuchi in [32], takes part of the optical signal from a spin polarized ensemble of cold atoms and uses it to continuously apply quantum feedback in order to change the variance in one of the transverse spin components, or, stated differently, to reduce the uncertainty in the desired spin-state. Although they were unsuccessful in reproducing their results due to the inherent difficulty with a small population of cold atoms in zero field [33], their idea is intriguing. We follow their basic approach but with a hot atomic vapor in a small magnetic field. We demonstrate that in this system it is possible to suppress noise below the spin-projection limit. Simply stated, spin-damping provides quantum control over the atomic spins. Using continuous feedback spin-damping makes it possible to suppress fundamental quantum noise, such as photon shot noise and spin-projection noise, in a closed loop system. Spin-damping can also be used to quickly initialize the ensemble quantum state of a system. For example, it can quickly re-initialize the spin polarization of the population after perturbation by an external field. Spin-damping is a technique that takes part of the optical signal from the magnetometer and uses it to apply an anti-parallel magnetic field produced through electromagnetic coils to the K atoms, as shown in Fig The gain of the damping field, which is proportional to 5

18 Probe B 0 B damp K Pump Balanced polarimeter Attenuator /Switch B perturb Electromagnetic coils Photodiode φ shifter _ Spectrometer 10 db Coupler Figure 1.2: Schematic of the spin-damping mechanism. A perturbing magnetic field B perturb sets the optically pumped K atoms precessing around B 0. The resulting transverse magnetization rotates the polarization of the probe beam, which is detected and converted to an electrical signal by a balanced polarimeter. Part of this electric signal is phase corrected and fed back through electromagnetic coils to produce a damping field (B damp ) that is anti-parallel to B perturb, resulting in an active damping of the K spin perturbation. the magnetometer signal, can be adjusted to control the degree of signal suppression with the negative feedback. By suppressing the output of the magnetometer, it is possible to increase the detection bandwidth of the magnetometer without any degradation of sensitivity. This technique is particularly useful for detection using excitation pulses since the ringing induced by the pulses can be quickly suppressed. This work will present a theoretical and experimental description of spin-damping in an ultrasensitive tunable RF atomic magnetometer. Chapter 2 will present a theoretical description of a tunable RF atomic magnetometer and spin-damping by describing the effects 6

19 of negative feedback on signal and noise and the magnetometer detection bandwidth. Chapter 3 will describe the physical realization of an atomic magnetometer with discussion on design characteristics and experimental and environmental noise sources. The experimental results of magnetometer sensitivity and linewidth measurements is given in chapter 4 and in chapter 5, we present spin-damping results which will show: the suppression of noise to below the fundamental quantum limits of photon shot noise and spin-projection noise, the gain in the magnetometer bandwidth without loss in sensitivity by applying spin-damping during signal acquisition, and the use of spin-damping to quickly damp ringing and regain magnetometer sensitivity. 7

20 Chapter 2: Atomic Magnetometer with Spin-Damping - Theoretical Approach 2.1 Tunable RF Atomic Magnetometer The performance of an RF atomic magnetometer is evaluated by two factors: sensitivity and linewidth. Sensitivity is the measure of magnetometer noise given in units of Tesla per square root of Hertz. Linewidth is defined as the full width at half max (FWHM) frequency response of the magnetometer given in Hz. The response of the magnetometer is a transfer function H mag that relates the magnetometer input, an RF magnetic field B RF, to the output, the voltage signal from the balanced polarimeter V out, or H mag V out B RF. (2.1) The magnetometer response is measured in units of V/T at the output of the balanced polarimeter, or as rad/t for the rotated angle of the linearly polarized probe beam. The sensitivity of the magnetometer is limited by environmental noise (thermal Johnson, experimental and external field, and optical noise) and quantum noise (photon shot, light shift, and spin-projection noise). In this chapter, we will present a theoretical description of a tunable RF atomic magnetometer with spin-damping. This description is presented in two parts. The first section develops the framework for a tunable RF atomic magnetometer and the second introduces spin-damping. The description of the atomic magnetometer is formulated in three parts: the tuned optically pumped steady state, perturbation of the atomic spins by an RF magnetic field that is to be detected, and detection of the perturbed spins by the Faraday rotation 8

21 of a linearly polarized probe light by balanced polarimetry. We conclude the first part by describing the three quantum noise limits of an atomic magnetometer. The second part introduces spin-damping by a theoretical formulation of magnetometer response under negative feedback, the suppression of environmental and quantum noise, and the broadening of the magnetometer signal-to-noise ratio (SNR) bandwidth. This section concludes with a description of noise measurement with quadrature detection Tuning and the Quadratic Zeeman Effect Our atomic magnetometer uses a potassium (K) vapor cell. The two most abundant naturally occurring isotopes of potassium are 39 K at 93.26% and 41 K at 6.73% [34]. Hence, all atomic properties related to potassium will be that of the 39 K isotope. When potassium atoms are placed in a weak external magnetic field B 0, along ẑ, the hyperfine structure (HFS) of the ground and excited states will undergo Zeeman splitting, as depicted in Fig The external magnetic field is considered weak if the difference in energy between the Zeeman sublevels is much smaller than the HFS transition energy. The K HFS transition frequency v HFS is MHz [35]. In contrast, for magnetic field strengths of less than 1 G, the Zeeman transition frequencies are less than 1 MHz. In a weak external field the hyperfine energy levels split into 2F + 1 magnetic sublevels, represented by the quantum numbers m F, where F is the maximum quantum number for the atomic angular momentum given by F = I + L + S, the summation of nuclear spin, electron orbital, and electron spin angular momentum. For K in the ground state, where L = 0, the electron spin S = 1/2, and the nuclear spin angular momentum I = 3/2, the possible values are F = 1 and F = 2. The magnetic sublevels m F are the set of possible quantized values of the angular momentum F z along ẑ, where m F = F,.., 0,.., F. Detection of an RF signal requires the absorption of RF magnetic energy by the K atoms. The K Zeeman transition frequency, or the Larmor frequency, is tuned to the frequency of 9

22 2 P 3/2 2 P 1/2 Fine Splitting Hyperfine Splitting F 2 ν HFS 1 Zeeman Splitting m F D nm D nm Induced optical transitions by left circularly polarized D 1 light 2 S 1/2 2 ν HFS I = 3/ Figure 2.1: The hyperfine and Zeeman energy splitting of K in a weak magnetic field showing the allowed energy transitions from the ground state to the first excited state when pumped with D 1 left circularly polarized light. the RF field though the magnitude of B 0 by ω L = γ K B 0, (2.2) where γ K = 2π 700 khz/g is the gyromagnetic ratio for 39 K [14]. It is convenient if all transition frequencies between any two adjacent Zeeman sublevels are equal. This holds true for small magnetic field strengths, but as the external magnetic field strengths increase, the transition energy between different Zeeman sublevels is no longer linear in B 0. In general, it is difficult to calculate the transition energies between the sublevels in the non-linear regime. However, for states where either I or J are less than or equal to 1/2, the 10

23 energy level splitting has a quadratic characteristic, referred to as the Quadratic Zeeman Effect. In such intermediate magnetic fields, the energy for each m F sublevel is calculated with the Breit-Rabi Equation [35], E mf = hv HFS 2(2I + 1) g Iµ n B 0 m F ± hv HFS (1 + 4m F x 2 (2I + 1) + x2 ) 1/2, (2.3) where h is the Plank constant, g I is the nuclear g-factor, and µ n is the nuclear magneton. The dimensionless parameter x is defined as (g J + g I m e m p ) µ BB 0 hv HFS, where m e is the rest mass of the electron, m p is the rest mass of the proton, g J is the Lande g-factor, and µ B is the Bohr magneton. The plus sign is used for energy states of the zero-field hyperfine level F = I +1/2 and the minus sign for F = I 1/2. The plot of the Breit-Rabi equation for 39 K is given in Fig The non-degenerate transition frequencies between all Zeeman sublevels as a function of B 0 around 0.6 G is shown in Fig This field strength corresponds to the frequency of 423 khz, one of the characteristic nuclear quadrupole resonance frequencies of ammonium nitrate. Also in Fig. 2.3, we see that at 0.6 G the transition frequencies can differ by as much as 2 khz. For a 100% population in a stretched state, F = 2, m F = ±2, this width of course does not come in to play. For polarization less than 100% however, this spread in frequencies would broaden the observed Zeeman transition Optical Pumping Optical pumping uses resonant photons to create a large population imbalance or polarization in the ground or excited Zeeman energy states [36]. When subject to a laser light with resonant wavelength of D 1 = nm (for 39 K) [34], the K atoms are excited, or pumped, from the 2 S 1/2 ground state to the 2 P 1/2 first excited state. With the beam aligned in the direction of B 0, and depending on the polarization of the incident photons, conservation of angular momentum requires that the absorption transition must obey the selection rules F = 0, ±1 and m F = ±1 [35]. Furthermore, for circularly polarized light the allowed 11

24 Frequency (MHz) F=2 m F =2 F=2 m F =1 F=2 m F =0 F=2 m F = 1 F=2 m = 2 F F=1 m = 1 F F=1 m F =0 F=1 m F = Magnetic Field (Gauss) Figure 2.2: The plot of the Breit-Rabi equation for potassium. transitions are reduced to the HFS transitions of F = 0, ±1 with Zeeman transition of m F = +1 for left circularly polarized light, or m F = 1 for right circularly polarized light. Figure 2.1 shows all possible transitions for a left circularly polarized pump beam. Once pumped to the first excited state, the inevitable transitions back to the ground state sub-levels will occur with some fixed non-zero transition probability for each of the individual ground state sub-levels. However, as stated above, due to selection rules for left circularly polarized light, no K atom in the m F = 2 ground state sub-level will get optically pumped. Since there is a non-zero probability of transition from the pumped state to the m F = 2 sublevel of the ground state, each iteration of optical absorption and emission leaves the atomic population with an excess population in this sublevel. As a result, with sufficient photon intensity, a few iterations of optical pumping produces a population that is nearly all in the m F = 2 spin state. It is important to note here that, the presence of 12

25 Zeeman Transition Frequency (khz) F=1, 1 to 0 F=1, 0 to 1 F=2, 2 to 1 F=2, 1 to 0 F=2, 0 to 1 F=2, 1 to Magnetic Field (Gauss) Figure 2.3: The Zeeman transition frequencies between all ground magnetic sublevels m F of potassium near 0.6 G. a quenching gas is necessary to ensure that the excited atoms do not radiate a resonant photon isotropically through the cell during de-excitation and cause the depolarization of other polarized spins by absorption of this photon [37]. Using the alkali electron spin, we show that the atomic spin polarization P z increases with increasing optical pumping rate R OP. The alkali electron spin polarization in the ẑ direction is given by P (e) z = 2 S z = s z R OP R OP + R SD, (2.4) where S z is the mean electron spin, s z is the degree of circular polarization of the pump beam, and R SD is the rate of electron spin destruction. The relationship between the 13

26 electron spin and the atomic spin is given by 1 + ϵ(i, P (e) z ) = F z S z, (2.5) with ϵ(i, P (e) z polarization where P (e) z ) = (5 + P z (e)2 )/(1 + P z (e)2 ) for I = 3/2 [38]. With nearly complete spin 1, ϵ(i, P z (e) ) 3, and S z F z /4, we find P z (e) = F z = P z, (2.6) 2 where P z is the polarization of the total atomic angular momentum in ẑ. For a left circularly polarized light, P z = 1 corresponds to the entire population of K atoms all occupying the m F = 2 state Perturbation by an RF Field The energy of an alkali atom, in the presence of a uniform static magnetic field B 0 in ẑ and an RF magnetic field B RF in ŷ, can be calculated using the Hamiltonian H = H 0 +H HFS +H RF, where H 0 = P 2 2m e Ze2 4πε 0 r + e2 4πε 0 N 1 i=1 ψ(ri ) 2 r i r d3 r i + e 2 8πε 0 m 2 ec 2 (S L) (2.7) r3 is the zero-field Hamiltonian describing the central electrostatic field of the atom, the repulsion between the electrons, and the spin-orbit interaction. Here, Z = 19 is the number of protons for potassium, e is elementary charge, ε 0 is the permittivity of free space, and c is the speed of light. The hyperfine structure Hamiltonian in the presence of an external 14

27 magnetic field is given by H HFS = A J (I J) + B J 2I(2I 1)J(2J 1) [3(I J)2 + 3 (I J) 2 I(I + 1)J(J + 1)] + g J µ B (J B 0 ) g Z µ n (I B 0 ), (2.8) where the first term is the hyperfine structure describing the interaction between the nuclear magnetic moment, µ I = g I µ n I, and the magnetic field at the nucleus produced by the valence electron and A J is the magnetic hyperfine structure constant determined experimentally. The second term vanishes since the interaction constant B J = 0 for potassium in the ground state, where J = 1/2. This is due to the spherical symmetry of the charge distribution from the single valence electron, which is in an s-shell orbital. The last two terms are the contributions from the interaction of the external field with the electronic and nuclear magnetic moments of the potassium atom. In Eqn. 2.8, g J is the Landé g-factor, and g I is the nuclear g-factor. The Hamiltonian for an interacting resonant RF radiation field, B RF = ŷb 1 cos ω L t is H RF = µ F B RF = g F µ B F y B 1 cos ω L t, (2.9) where µ F = g F µ B F is the total atomic magnetic moment and ω L is the angular frequency of the RF field. Here, g F is the effective g-value given by [35] g F = g J ( F (F + 1) + J(J + 1) I(I + 1) ) 2F (F + 1) m e F (F + 1) + I(I + 1) J(J + 1) g I ( ). (2.10) m p 2F (F + 1) Expressing F y in terms of the ladder operators, F + = F x + if y and F = F x if y, and 15

28 substituting the complex notation for cos ω L t, we get H RF = g F µ B B 1 (F + F )(e iωlt + e iωlt ), (2.11) 4i The general solution for the time-dependent Schrödinger equation ψ(r, t) i = H ψ(r, t), (2.12) t describes the evolution of the state of a potassium atom with Hamiltonian H = H 0 + H 1, where H 0 = H 0 + H HFS represents the dominant Hamiltonian and the RF radiation Hamiltonian, H 1 = H RF, represents the perturbing Hamiltonian. The above is simplified if we assume a wavefunction of the form ψ(r, t) = a F,mF (t) F, m F e ief,mf t/. (2.13) F,m F The spatial wavefunctions, F, m F, are the eigenstates of the time-independent energy equation H 0 F, m F = E F,mF F, m F. (2.14) Inserting ψ(r, t) into Eqn and simplifying, we get i ȧ F,mF (t) F, m F e ief,mf t/ = a F,mF (t)h 1 F, m F e ief,mf t/. (2.15) F,m F F,m F Substituting Eqn for H 1 in the above equation and equating terms for each of the 16

29 F, m F states, we get the following simplified expressions for the ȧ F,mF (t), 2gF µ B B 1 ȧ 1,1 (t) = a 1,0 (e 2iωLt + 1), (2.16) 4 ȧ 1,0 (t) = ȧ 1, 1 (t) = 2gF µ B B 1 [a 1,1 (e 2iωLt + 1) a 1, 1 (e 2iωLt + 1)], 4 2gF µ B B 1 a 1,0 (e 2iωLt + 1), 4 ȧ 2,2 (t) = g F µ B B 1 2a 2,1 (e 2iωLt + 1), 4 ȧ 2,1 (t) = g F µ B B 1 [2a 2,2 (e 2iωLt + 1) 6a 2,0 (e 2iωLt + 1)], 4 ȧ 2,0 (t) = g F µ B B 1 [ 6a 2,1 (e 2iωLt + 1) 6a 2, 1 (e 2iωLt + 1)], 4 ȧ 2, 1 (t) = g F µ B B 1 [ 6a 2,0 (e 2iωLt + 1) 2a 2, 2 (e 2iωLt + 1)], 4 ȧ 2, 2 (t) = g F µ B B 1 2a 2, 1 (e 2iωLt + 1). 4 The exponential terms in Eqn have minimal contribution to the Hamiltonian, therefore 17

30 by the secular approximation we get 2gF µ B B 1 ȧ 1,1 (t) = a 1,0, (2.17) 4 ȧ 1,0 (t) = ȧ 1, 1 (t) = 2gF µ B B 1 [a 1,1 a 1, 1 ], 4 2gF µ B B 1 a 1,0, 4 ȧ 2,2 (t) = g F µ B B 1 2a 2,1, 4 ȧ 2,1 (t) = g F µ B B 1 [2a 2,2 6a 2,0 ], 4 ȧ 2,0 (t) = g F µ B B 1 [ 6a 2,1 6a 2, 1 ], 4 ȧ 2, 1 (t) = g F µ B B 1 [ 6a 2,0 2a 2, 2 ], 4 ȧ 2, 2 (t) = g F µ B B 1 2a 2, 1. 4 The potassium atoms being nearly completely polarized to the m F = 2 state implies the probability amplitude a 2,2 (t) is much greater than the other amplitudes in Eqn. 2.17, reducing the equations of interest to ȧ 2,2 (t) = g F µ B B 1 a 2,1, (2.18) 2 ȧ 2,1 (t) = g F µ B B 1 a 2,2, (2.19) 2 18

31 which have solutions a 2,2 (t) = a 2,2 (0) cos ( g F µ B B 1 2 a 2,1 (t) = a 2,1 (0) cos ( g F µ B B 1 2 t) a 2,1 (0) sin ( g F µ B B 1 t), (2.20) 2 t) + a 2,2 (0) sin ( g F µ B B 1 t). (2.21) Magnetometer Response We now need to calculate the expectation value of the total angular momentum vector F in the ˆx and ŷ. For a K atom with F x µ x and F y µ y, the approximate expectation values in both transverse directions, only keeping terms with amplitude a 2,2 (t), is given by µ x = g F µ B ψ F + + F ψ, 2 g F µ B (a 2,2(t)a 2,1 (t)e iω Lt + a 2,1(t)a 2,2 (t)e iω Lt ). (2.22) µ y = g F µ B ψ F + F ψ, 2i g F µ B (a i 2,2(t)a 2,1 (t)e iωlt a 2,1(t)a 2,2 (t)e iωlt ). (2.23) In general, the probability amplitudes can be expressed as a F,mF (0) = a F,mF (0) e iϕ and a F,m F (0) = a F,mF (0) e iϕ. For a large ensemble of K atoms, the phase factors e iϕ are isotropically distributed and so the net phase of the population is zero. Expansion of Eqns and 2.23 using Eqns and 2.21, followed by simplification and consideration of the isotropic distribution of the phase factors, gives µ x = g F µ B ( a 2,1 (0) 2 a 2,2 (0) 2 ) sin (g F µ B B 1 t) cos (ω L t), (2.24) 19

32 µ y = g F µ B ( a 2,1 (0) 2 + a 2,2 (0) 2 ) sin (g F µ B B 1 t) sin (ω L t). (2.25) With the probe beam along the ˆx, µ x is the component of interest influencing the polarization of the probe beam. In the frame of the angular momentum vector F which, as can be seen from Eqns and 2.25, is rotating in the x y plane with Larmor frequency ω L, we get the expression µ x = g F µ B ( a 2,1 (0) 2 a 2,2 (0) 2 ) sin ( g F µ B B 1 t). (2.26) At very low RF fields, where g F µ B B 1 / 1, we get µ x = g F µ B ( a 2,1 (0) 2 a 2,2 (0) 2 )( g F µ B B 1 t). (2.27) The Bloch equation, given in [27], with gyromagnetic ratio γ K = g F µ B /, introduces an additional term to the rate of change of µ x, which takes into account the transverse relaxation time T 2, given by d µ x dt = g F µ B γ K B 1 ( a 2,1 (0) 2 a 2,2 (0) 2 ) µ x T 2. (2.28) The transverse relaxation time T 2 is the characteristic time for an exponential decay of the net transverse polarization of the potassium population, and will be presented following this derivation. The solution to the first order differential equation, Eqn. 2.28, with initial condition of F x = F y = 0 at time t = 0, is µ x = g F µ B γ K B 1 ( a 2,1 (0) 2 a 2,2 (0) 2 )T 2 (1 e t/t 2 ). (2.29) 20

33 With P z 1, where a 2,2 1 and a 2,1 0, a 2,1 (0) 2 a 2,2 (0) 2 1, we get µ x = g F µ B γ K B 1 T 2 (1 e t/t 2 ). (2.30) For the K atoms, the expectation value of the ˆx component of the net atomic angular momentum, in the non-rotating frame, for t, is F x = γ K B 1 T 2 cos (ω L t), (2.31) and so, we can define the transverse polarization P x as P x F x F z = 1 2 γ K B 1 T 2. (2.32) An expression for T 2, for a regime where the Zeeman transition frequencies are nondegenerate and R OP R SD, is given in [38] as T 1 2 = R OP 4 + R [ SER SD RSE + 4iωL 2 R /πν ] HF S R OP 5R SE + 8iωL 2 /πν, (2.33) HF S where R OP is the potassium optical pumping rate [39], R SD is the potassium spin destruction rate [40,41], and R SE is the spin-exchange rate [42]. R[] corresponds to the real part of the [ ] RSE +4iωL expression where, for ω L 2π 1MHz, R 2 /πν HF S 5R SE +8iωL 2 /πν 1 HF S Detection by Faraday Rotation The transverse polarization will cause an optical or Faraday rotation of the linear polarization of a beam of light going through the potassium cell parallel to P x. The angle of 21

34 rotation of the plane of polarization is directly proportional to P x and is given by [29, 43] ϕ x = 1 2 l xr e cfn K D(ν pr )P x P z, (2.34) where l x is the length of the cell traversed by the probe beam, r e is the classic electron radius, f 1/3 is the oscillator strength of the D 1 transition [44], and D(ν pr ) = (ν pr ν D1 ) (ν pr ν D1 ) 2 + (Γ P /2) 2 (2.35) is the dispersion profile, where ν D1 is the frequency of the D 1 transition and Γ P is the FWHM of the Lorentzian distribution. We measure the rotation angle ϕ x using a balanced polarimeter, which will be described in more detail in the following chapter. Equations 2.32 and 2.34 show that the rotation angle ϕ x is directly proportional to the transverse relaxation time T Sensitivity As mentioned earlier, there are three factors limiting the sensitivity of our magnetometer. The first source is spin-projection noise. The uncertainty principle for non-commuting operators F x and F y gives an uncertainty relation δf x δf y F z /2, where in general, δf x = δf y (non-squeezed state). For N atoms the uncertainty in ˆx is then δf x = F z /2N. As derived in more detail later, if the field is close to zero ( ω L 1 T 2 ), the autocorrelation function is δf x e t/t 2. As measured for some time t, the uncertainty for the transverse polarization is δp x = δf x t /F z, where δf x t = δf x (2T2 /t). This leads to an expression for spin projection noise [14], in units of T/ Hz, of δb SPN = 1 8. (2.36) γ K F z n K V T 2 22

35 For a field such that ω L 1 T 2 we show in section b that this overestimates the spin-projection noise by 2. Due to the discrete nature of light particles, there exists a lower bound for the sensitivity of the magnetometer due to photon shot noise. The variance in the angle ϕ x due to shot noise is given by [14] δϕ x = 1 2Φpr η, (2.37) which is measured as rad/ Hz. Here, Φ pr is the photon flux of the probe beam at the photodiode. The photodiode quantum efficiency η is the ratio between the number of electrons generated to the number of photons hitting the photodiode, given by η = I/e P/hν, (2.38) where I is the current from the photodiode, P is the power of the incident beam, and ν is the frequency of the incident probe beam. In terms of magnetic sensitivity, the shot noise can be expressed, using Eqns. 2.32, 2.34 and 2.37, as δb PSN = δϕ x ϕ x B RF = 4 l x r e cfn K D(ν pr )P z γ K T 2 2Φpr η. (2.39) There is also light shift noise from the quantum fluctuations in the polarization of the probe beam arising from the ac-stark shift. One can think of this light shift as an induced magnetic field which causes fluctuations in the atomic spins proportional to polarization fluctuations of the probe beam. These fluctuations result in magnetic noise given by [14] δb LSN = r ecfd(ν pr ) 2Φ pr, (2.40) 8γ K A 23

36 where A is the probe beam cross-section. The total magnetic noise for the magnetometer, with the probe beam sufficiently detuned from resonance so that D(ν pr ) 1/(ν pr ν D1 ), is given by δb = R prod 8 + γ K nk V T 2 32 R pr ODT2 2η, (2.41) where OD = σ 0 n K l x is the optical depth of the probe beam at resonance wavelength and σ 0 is the resonance photon absorption cross-section. The optical pumping rate by the probe beam R pr is given by [39] R pr = σ ν pr A Φ 0e σ νprn K l z /2, (2.42) where Φ 0 is the photon flux of the probe beam incident upon the cell, n K is the atomic number density of K in the cell, and σ νpr is the photon absorption cross-section at the probe wavelength, given in general by ( 1 σ ν = r e c 3 L D 1 (ν) + 2 ) 3 L D 2 (ν), (2.43) where r e is the electron radius, c is the speed of light, and the 1/3 and 2/3 are the oscillator strengths for the D 1 and D 2 transitions of potassium, respectively [44]. L D1 (ν) and L D2 (ν) are Lorentzian distribution functions of the form L D (ν) = Γ op /2 (ν ν D ) 2 + (Γ op /2) 2, (2.44) centered on the D 1 and D 2 frequencies, respectively, and the Γ op is the optical absorption linewidth. Note that the incident photon flux Φ 0 and the flux at the photodiodes Φ pr are 24

37 related by Φ pr = Φ 0 e σ νpr n Kl x. To find the theoretical minimum sensitivity, limited by spin-projection noise, we minimize Eqn as a function of x, where x = R pr ODT 2. The minimum value for of δb for a given cell volume is found at x = 16 η, which corresponds to the expression R pr σ νpr n K l x T 2 = 16 η. In addition to the above mentioned fundamental quantum noise, other environmental noise contributions δb env may also limit the sensitivity of an atomic magnetometer. In the operating range of our magnetometer the noise contributions are δb env > δb PSN δb SPN > δb LSN. Environmental noise is independent of the magnetometer parameters as it scales with the magnetometer response and therefore cannot be reduced through magnetometer optimization. The largest quantum noise contribution is photon shot noise, Eqn. 2.39, which can be minimized by maximizing T 2, the K polarization, and qe q/2, where q = σ νpr n K l x. In the following chapter, we will discuss possible environmental and experimental noise sources and their contribution to the overall magnetometer sensitivity. In the next section, we will present spin-damping and a more rigorous development of the three fundamental and environmental magnetic noise sources. 2.2 Spin-damping Feedback with Finite-Gain Amplifiers Spin-damping is reminiscent of a finite gain amplifier under negative feedback. If the amplifier has a gain of α and the feedback subtracts a fraction β of the amplifier output, the input back into the amplifier is V in βv out and the final output is [45] V out = α 1 + αβ V in (2.45) or a closed loop gain of G = α 1+αβ as compared to the open loop gain α. The loop gain is the product of α and β; the corresponding loop gain for the magnetometer we will call the 25

38 Balanced polarimeter Amplifier φ x Optical signal V out = cp x Electrical signal B 1 + _ B out P Magnetometer x P x = γ K T 2 B out / 2 B f b = β V out Orthogonal coils Switch Figure 2.4: Schematic of feedback with the magnetometer under steady-state conditions. As in section and 2.1.4, the transverse polarization P x of the K spins, created by a resonant B 1, is optically detected, ϕ x, by the magnetometer. The optical signal is converted to a voltage signal V out by the polarimeter which is then amplified. Part of the amplified signal is fed back through orthogonal coils which produce the damping field B fb. Feedback is applied in such a way that the net magnetic field is B out = B 1 B fb. Therefore the loop gain, or damping factor, is (cβ γ K T 2 /2). damping-factor, DF. Another quantity of interest which will appear often in the equations governing damping in a magnetometer is 1 + αβ, or in our case 1 + DF, which is called the return difference. As shown in Fig. 2.4, for the magnetometer the input signal is the magnetic field to be sensed and the output is the tipped K magnetization created by the magnetic field. The final output of the magnetometer, as recorded by a spectrometer, is directly proportional to the tipped magnetization. In the next section, we describe the feedback mechanism in the case of the magnetometer and calculate the frequency dependence of this feedback Frequency-Dependent Feedback of the Magnetometer The Bloch equations for optical pumping [35] are used to define the frequency dependency of α, or equivalently, the evolution of the potassium s net mean spin F = 1 N N i=1 F in the presence of an alternating magnetic field. The magnetometer is sensitive to nearly 26

39 resonant alternating magnetic fields, B RF = B 1 cos(ωt + θ)ˆr, (2.46) whose direction ˆr = ˆx cos η +ŷ sin η is perpendicular to the dominant static magnetic field. Due to the strong optical pumping along the direction of the applied magnetic field B 0 ẑ, and the relatively weak strength of B 1, F z is treated as constant which is close to its fully pumped value of F. In this limit, the Bloch equations can be reduced to dp dt = iγ K B RF (ˆx + iŷ) + iω L P P T 2, (2.47) where the Larmor frequency ω L = γ K B 0 = γ K B 0, T 2 is the transverse relaxation time in the presence of the pumping light, and P F x +i F y F z. Converting to a frame rotating at ω close to ω L, Eqn becomes ˆP t = iγ K B RF (ˆx + iŷ) e iωt i(ω ω L ) ˆP ˆP T 2, (2.48) where ˆP Fx +i Fy F z = e iωt F x +i F y F z. The primed coordinate system corresponds to the rotating frame and the unprimed to the laboratory frame. At this point we invoke the secular approximation, particularly to the B RF term which contains highly oscillatory terms, γ K B RF (ˆx + iŷ) e iωt e i(ωt+θ) + e i(ωt θ) = γ K B 1 (cos η + i sin η) e iωt (2.49) 2 γ KB 1 e iη+iθ = ω 1 e iϕ, (2.50) 2 27

40 where the Rabi frequency is ω 1 γ K B 1 /2 and ϕ η + θ. Therefore, Eqn can be written as ˆP t = iω 1 e iϕ i(ω ω L ) ˆP ˆP T 2, (2.51) which has the solution ( ) ˆP = iω 1e iϕ T 2 1+i(ω ω L )T 2 1 e i(ω ωl)t e t T 2 + ˆP 0e i(ω ωl)t e t T 2, (2.52) where ˆP 0 = F x (0)+i F y (0) F z is the transverse polarization of ˆP at time t = 0. This expression is equivalent to Eqn derived via wavefunctions, instead of the Bloch equations as used here, in the limits assumed for Eqn. 2.30, namely, ω = ω L, ˆP 0 = 0, ϕ = π 2, and F z = 2. For t the steady-state solution is ˆP ss = iω 1e iϕ T 2 1+i(ω ω L )T 2. (2.53) The linearly polarized light of the probe is rotated in proportion to the K magnetization along its direction of propagation. The rotation is measured using a balanced polarimeter and is read out as voltage from a differential amplifier. This signal is sent through a pre-amplifier before being recorded by the spectrometer, using quadrature detection. The voltage is directly proportional to the magnetization along the direction of the probe beam V out = c F [ ] x F z = cr Fx + i F y [ = cr ˆP e iωt], (2.54) F z or in phasor notation Ṽ out = c ˆP, (2.55) 28

41 where c is a constant of proportionality converting the atomic polarization to the output voltage. The output of the pre-amplifier is also coupled to the feedback circuit and sent to a set of saddle coils, which produce a field perpendicular to the pump direction. Therefore, the feedback field [ B fb = B fb cos(ωt + θ )ˆr = R β Ṽ out e iωt] ˆr, (2.56) where β can be a complex constant whose phase can be set using a phase shifter, and ˆr = ˆx cos η + ŷ sin η is determined by the direction of the saddle coils. Equivalently to Eqn B fb e iθ = β Ṽ out. (2.57) The feedback field makes the contribution to the Bloch equations in the rotating frame through γ K B fb (ˆx + iŷ) e iωt e i(ωt+θ ) + e i(ωt iθ ) ( = γ K B fb cos η + i sin η ) e iωt (2.58) 2 = γ K 2 B fbe iθ e iη = γ K 2 eiη β Ṽ out (2.59) ( γk ) 2 eiη β c ˆP = iβ ˆP, (2.60) where in the second to last line Eqn has been used, and in the last line we define the ( ) constant β = i γk2 e iη β c. Notice how the direction defined by η, and/or the complex constant β, can be used to set the phase of the feedback. 29

42 With the inclusion of the damping field, Eqn becomes ˆP t ( ) = iω 1 e iϕ i(ω ω L ) ˆP ˆP 1 + β. (2.61) T 2 Defining 1 T 2 = 1 T 2 + β, the solution to Eqn is the same as Eqn. 2.52, with T 2 replacing T 2 ˆP = iω 1 e iϕ T i(ω ω L )T 2 ) (1 e i(ω ωl)t e t T 2 + ˆP 0 e i(ω ωl)t e t T 2. (2.62) If β is positive and real the net effect of the feedback is to reduce ˆP (t ) as well as broaden the magnetometer s bandwidth; this is called spin-damping since the spinpolarization is damped. Note that if β is chosen to be 1 T 2, we have a singular solution at ω = ω L. In the case when ω = ω L and t = ˆP ss = iω 1 e iϕ T i(ω ω L )T 2 (2.63) = i ω 1e iϕ 1 + βt 2 T 2. (2.64) So the effect of the field on the magnetometer has been effectively reduced by 1 + βt 2. Comparing to Eqn. 2.45, T 2 acts as the gain α in the finite-gain amplifier and we define the dimensionless damping factor as DF = βt 2, so the return difference is 1 + βt 2. Note that the phase of the transverse polarization, and hence the phase of the signal does not change if β is real; this is one way to determine the correct phase of the feedback signal to use for damping. 30

43 2.2.3 Noise with and without Damping In order to understand the average frequency content of our noise, we take the mean square value of the Fourier transform of the noise signal F{S(t)} or equivalently what is known as the periodogram P Tacq (ω) 1 T acq F{S(t)} 2, (2.65) where T acq is the duration in time of the acquisition. We can relate the periodogram to the autocorrelation function R N (τ) = S(t)S(t + τ) through [46]: 1 Tacq ( F{S(t)} 2 = 1 τ ) R S (τ)e iωτ dτ, (2.66) T acq T acq T acq where the angle brackets indicate an ensemble average. In the limit that T acq is much larger than the characteristic decay time of R(τ) with τ, the integral approaches what is known as the two-sided power spectral density function P (2) (ω) R S(τ)e iωτ dτ [47]. Noise at frequency ω is added in at several places of the magnetometer - environmental magnetic noise, light shift noise, spin-projection noise, photon shot noise, and noise added through the amplification stage. The first three represent white noise which is colored through the detection by the magnetometer. The last two are white noise contributions under normal detection by the magnetometer, but which become colored under the presence of feedback. a. Environmental Noise and Light Shift Noise Working in a frame rotating at ω L, the Bloch equations are ˆP (t) t = iγ K δb rot (t) ˆP (t) T 2, (2.67) 31

44 where δb rot (δb x + iδb y )e iωlt is the magnetic noise in the rotating frame, either from the environment or from light shift noise masquerading as magnetic noise. Assuming the noise is white and isotropic in the laboratory, δb rot will also be white. While Eqn relates the transverse polarization in the rotating frame to the magnetic noise, what we measure is V out = cr ˆP e iω Lt [ ]. The relationship between the power spectral density of V out and ˆP can be found by first examining the autocorrelation function corresponding to each: R Vout (τ) = c2 F z 2 E [ F x (0) F x (τ) cos ω L τ F x (0) F y (τ) sin ω L τ ], (2.68) where E [] corresponds to the expectation value of an ensemble of groups of spin, ie, F represents the mean spin over the cell, while E [ F ] represents the mean spin over the cell as averaged over separate experiments. Within the frame rotating at the Larmor frequency there is not expected to be any correlation between F x and F y, so that R Vout (τ) = c2 F z 2 E [ F x (0) F x (τ)] cos ω Lτ. (2.69) For the complex transverse polarization, the autocorrelation function is R P (τ) = 1 F z 2 E[ F x (0) F x (τ) + i F x (0) F y (τ) i F y (0) F x (τ) + F y (0) F y (τ)]. (2.70) Again, we expect the cross-terms to vanish and further E [ F x (0) F x (τ)] = E [ F y (0) F y (τ) ], as there is no preferred direction in the rotating frame. Therefore, we can equate R Vout (τ) = c2 2 R P (τ) cos ω L τ. (2.71) 32

45 The corresponding power spectral density is P (2) V out (ω) = c2 2 dτe iωτ R P (τ) cos ω L τ (2.72) = c2 4 [ ] P p (2) (ω ω L ) + P p (2) (ω + ω L ). (2.73) We find the transfer function h(ω) = ˆp (ω) δb rot(ω), by taking the Fourier transform of Eqn. 2.67: iωˆp (ω) = iγ K δb rot (ω) ˆp (ω) T 2 (2.74) h(ω) = ˆp (ω) δb rot (ω) = iγ K (iω + 1 T 2 ). (2.75) The lower case corresponds to the Fourier transform of the same function in upper case, for example ˆp (ω) = F{ ˆP (t)}. The ratio of spectral density of the output P (2) p (ω) to that of the input s Brot is simply h(ω) 2. Therefore, given that δb rot (ω) has flat spectral density, the spectral density of the output is P p (2) (ω) = h(ω) 2 P (2) B env = γ2 K T ω 2 T2 2 P (2) B env, (2.76) where P (2) B env is the spectral density of the environmental magnetic noise. Therefore, in the limit of long T acq, the noise observed in periodogram is P T (ω) = P (2) V out (ω) = c2 4 P(2) B env [ γ 2 K T (ω ω L ) 2 T 2 2 γk 2 + T (ω + ω L ) 2 T2 2 ]. (2.77) The first term in [] dominates when ω is close to ω L, and we can ignore the second term in 33

46 the limit that (ω + ω L ) 2 T If the acquisition time T acq is not long, we can express the periodogram P T (ω) = Tacq T acq Tacq T acq ( 1 τ ) F 1 {s out (ω)}e iωτ dτ (2.78) T acq ( 1 τ ) e τ /T 2 e iωτ dτ, (2.79) T acq which corresponds to the convolution of sinc 2 (ω) with the Lorentzian in Eqn Adding in the feedback to Eqn ˆP t = iγ K δb rot (t) + iγ K B fb (ˆx + iŷ) e iω Lt ˆP T 2 (2.80) = iγ K δb rot (t) + iγ K B fb (t)e iη e iω Lt ˆP T 2, (2.81) where B fb = B fb (t)ˆr. To find the transfer function between the output in voltage and the input in the form of the environmental noise, we begin by taking the Fourier transform of Eqn. 2.81: iωˆp (ω) = iγ K δb rot (ω) + iγ K e iη b fb (ω + ω L ) ˆp (ω) T 2. (2.82) The Fourier transform of the feedback field is proportional to the Fourier transform of the output voltage V out b fb (ω) = β (ω)v out (ω), (2.83) where β (ω) is determined by the feedback circuitry, as seen in Eqn In turn, the 34

47 Fourier transform of the output voltage is related to the Fourier transform of ˆP (t) by v out (ω) = cf{r{ ˆP (t)e iω Lt }} (2.84) [ = c F{ F x (t) F z e iω Lt + e iω Lt 2 + i F y (t) F z e iω Lt e iω Lt 2 ] } (2.85) [ fx (ω ω L ) = c + f x (ω + ω L) + i f y (ω ω L) i f y (ω + ω ] L). (2.86) 2 F z 2 F z 2 F z 2 F z Therefore, combining Eqn and Eqn. 2.86, to find B fb in terms of ˆp, [ b fb (ω + ω L ) = cβ fx (ω) (ω + ω L ) 2 F z + f x (ω + 2ω L) + i f y (ω) 2 F z 2 F z if y (ω + 2ω ] L) 2 F z [ cβ fx (ω) (ω L ) 2 F z + if y (ω) ] 2 F z (2.87) = cβ (ω L ) ˆp (ω), (2.88) 2 where in the second line we have invoked the secular approximation, i.e., the power spectrum of the transverse polarization in the rotating frame must be dominated by frequencies close to ω = 0, as well as assumed that the transfer function β (ω) is fairly constant over the frequency range of interest and can be approximated by β (ω L ). Therefore, the Eqn can be simplified to ) ( ) iωˆp (ω) = iγ K δb rot (ω) (β + 1T2 1 ˆp (ω) = iγ K δb rot (ω) T 2 ˆp (ω), (2.89) where all the constants are swept into β = i γ Kcβ (ω L ) 2 e iη. Equation 2.89 has the same solution as in Eqn. 2.77, except T 2 is replaced with T 2. 35

48 b. Spin Projection Noise In the absence of magnetic noise, the solution to the Bloch equation in the Larmor rotating frame for a group of N atoms becomes ˆP = ˆP 0 e t T 2. (2.90) Assuming optical pumping keeps F z constant, we can rewrite this as F x (t) + i F y (t) = { F x (0) + i F y (0)}e t T 2 e iω Lt (2.91) F x (t) = { F x (0) cos ω L t F y (0) sin ω L t}e t T 2 (2.92) F y (t) = { F y (0) cos ω L t + F x (0) sin ω L t}e t T 2 (2.93) The expectation values of a time-independent operator A S, can be written either in the Schrodinger representation, denoted by S, or the Heisenberg representations, denoted by H, [48] A (t) = ψ(t) A S ψ(t) (2.94) = ψ H A H (t) ψ H. (2.95) The two representations are related by ψ H = ψ S (0) (2.96) A H (t) = U (t, 0)A S U(t, 0), (2.97) where U(t, 0) is the evolution operator. Note that the operators in both representation are equal when t = 0, since U(0, 0) = 1 36

49 Therefore, using Eqns. 2.92, 2.94, and 2.95, F x (t) = ψ S (0) F H,x (t) ψ S (0) =( ψ S (0) F H,x (0) ψ S (0) cos ω L t ψ S (0) F H,y (0) ψ S (0) sin ω L t)e t T 2 = ψ S (0) (F H,x (0) cos ω L t F H,y (0) sin ω L t) e t T 2 ψ S (0), (2.98) which implies F H,x (t) = (F H,x (0) cos ω L t F H,y (0) sin ω L t) e t T 2. (2.99) Following ref. [49], we can use this Heisenberg operator to define the symmetrized spin-spin correlation function R Fx (τ) = 1 2 F H,x(τ)F H,x (0) + F H,x (0)F H,x (τ) = ψ S (0) F H,x (0) 2 cos ω L τ 1 2 (F H,y(0)F H,x (0) + F H,x (0)F H,y (0)) sin ω L τ ψ S (0) e τ T 2 = ψ S (0) F 2 x cos ω L τ 1 2 (F yf x + F x F y ) sin ω L τ ψ S (0) e τ T 2. (2.100) For the case of fully polarized atoms in the state i m F,i = ±F, then substituting in F x = 1 N N i=1 F x,i and F y = 1 N N i=1 F y,i, the correlation function becomes R Fx = 2 2N {F (F + 1) F z 2 } cos ω L τe τ T 2 = 2 τ 2N F cos ω Lτe T 2. (2.101) For atoms that are randomly polarized in the x y directions and partially polarized in the third direction, the correlation function would be the same as above, but with F replaced 37

50 with F +(F 2 F 2 z ). Therefore, the net effect of having a sample only partially polarized is to increase the spin projection noise. When, as is this case with K, there are two hyperfine levels for the ground state with F = I ± 1/2, one can assume a common spin-temperature [50,51] between the levels, and calculate F F F 2 z to replace F in Eqn , to find the spin-projection noise. In the opposite limit of an unpolarized sample, we can compare our correlation function to ref. [49], who treats quantum mechanically the case of F = 1/2 and an unpolarized and undamped single spin. Under these conditions, F F F 2 z = 1/2 and our derived correlation function becomes 2 4 cos ω Lτe τ T 2, which corresponds to their two-sided power spectrum P (2) F x (ω) = e iωτ R Fx (τ)dτ, (2.102) or ( P (2) F x (ω) = 2 4 T T 2 2 (ω ω L) 2 + T T 2 2 (ω + ω L) 2 ). (2.103) Returning to the damped and fully polarized spin-f case, the corresponding one-sided power spectrum is [47] P (1) F x (ω) = 2 0 [ e iωτ + e iωτ ] R Fx (τ)dτ (2.104) ( = 2 2 2N F T T 2 2(ω ω L) 2 + T T 2 2 (ω + ω L) 2 ). (2.105) Using the steady-state value of the transverse polarizations, Eqn. 2.63, the power spectrum expressed in magnetic field is ( P (1) B (ω) = 4 NF γk 2 T T (ω ω L) T 2 38 ) 2 (ω + ω L) 2.. (2.106)

51 Note that this result is equivalent to that given by Ref. [14] only in the case that ω L = 0; this reflects the absence of cos ω L τ in the autocorrelation function of Ref. [14]. For ω close to ω L, and ω L T 2 1, the noise power is smaller by 2 than when ω L = 0. Note that for a sample not fully polarized 1 F would be replaced with F F F 2 z F z 2 in Eqn c. Photon shot noise In the frame rotating with the Larmor frequency, again in the absence of magnetic noise but with feedback, the Bloch equation is ˆP t = iγ K B fb (ˆx + iŷ) e iωt ˆP T 2 (2.107) or, using Eqns & 2.83, iωˆp (ω) = iβ (ω)γ K e iη v out (ω + ω L ) ˆp (ω) T 2. (2.108) Here V out is a combination of the signal from ˆP and photon shot noise V psn (t), or through Fourier transform, and Eqn. 2.86, v out (ω) = [ fx (ω ω L ) v psn (ω) + c + f x (ω + ω L) + i f y (ω ω L) i f y (ω + ω ] L), 2 F z 2 F z 2 F z 2 F z (2.109) which is equivalent to v out (ω + ω L ) v psn (ω + ω L ) c 2 p (ω). (2.110) 39

52 Substituting Eqn into Eqn , we find iω[v out (ω + ω L ) v psn (ω + ω L )] = i c 2 β (ω)γ K e iη d(ω)v out (ω + ω L ) v out(ω + ω L ) v psn (ω + ω L ) T 2 (2.111) v out (ω)[i(ω ω L ) + β + 1 T 2 ] = v psn (ω)[i(ω ω L ) + 1 T 2 ] (2.112) or a transfer function of h(w) 2 = 2 v out v psn = (ω ω L ) T2 2 (ω ω L ) (2.113) T 2 2 Therefore, P (1) V out = T 2 2 T (ω ω L ) 2 T (ω ω L ) 2 T 2 2 P (1) V psn, (2.114) where P (1) V psn is the spectral density of the photon shot noise in volts squared per hertz. d. Total Noise The total one-sided power spectral density of the output voltage is then, using Eqn , Eqns and 2.54 with F z = 2, and Eqn. 2.77, P (1) V out = T 2 2 T (ω ω L ) 2 T (ω ω L ) 2 T 2 2 ( P (1) V psn + c2 2N T (ω ω L ) 2 T 2 2 ) + c 2 γ 2 K T 2 2 /4 1 + (ω ω L ) 2 T 2 2 P (1) B env (2.115) 40

53 where P (1) V psn and P (1) B env are also taken as one-sided power spectral densities. Again both of which are assumed to be white. Combining like terms in Eqn , the total noise becomes V out = A2 n + (ω ω L ) 2 T (ω ω L ) 2 T 2 P (1) V psn, (2.116) P (1) 2 where the power spectral density on resonance A 2 is A 2 n = T 2 c 2 T 2 T 2 2N + T 2 2 T 2 2 ( c 2 ) 4 T 2 2 γkp 2 (1) B env + P (1) V psn. (2.117) e. Bandwidth of the SNR For quadrature detection in which the phase of the signal is known, the absorptive signal can be separated from the dispersive signal, permitting a 2 increase in SNR over measuring the magnitude of a resonant signal [52]. As explained in more detail in the next section, Sec , the real part of the Fourier transform of a quadrature detected signal gives a SNR of V out, where the phase of the spectrometer has been set to match the absorptive P (1) Va /T acq component V a of the signal V out. Using Eqns and 2.63, the absorptive part of V out is V a = c γ K B 1 T 2 /2 1 + (ω ω L ) 2 T 2. (2.118) 2 Using this and the power spectral density of Eqn , the magnetometer SNR becomes SNR = ( c 2 4 T 2 2 γ2 K P(1) B env + P (1) V psn + c2 T 2 cγ K B 1 Tacq T 2 /2 2NR + P(1) V ext R 2 + (ω ω L ) 2 T 2 2 P(1) V psn ) (1 + (ω ωl ) 2 T 2 2 R2), (2.119) 41

54 where R = T 2 T 2 = (DF + 1) 1, so that the value of R is between one, corresponding to no damping, and 0, corresponding to infinite damping. We have also included in the above expression the noise that is added external to the spin-damping loop P (1) V ext recorded by the spectrometer. as it is being In general, the bandwidth of the signal to noise ratio depends on the relative contribution to the total noise. We now look at different limiting cases. In the limit that photon shot noise dominates, Eqn reduces to SNR = SNR 0, (2.120) 1 + (ω ωl ) 2 T2 2(R2 + 1) + (ω ω L ) 4 T2 4 R2 where SNR 0 is the SNR at ω = ω L, and does not change with damping. The linewidth of this SNR increases from ω HW HM = 1 T 2 with no damping to ω HW HM = damping, with most of the increase occurring for damping factors under 10. In the more fundamental limit, in which spin-projection noise dominates, 3 T 2, for infinite SNR = SNR 0 R, (2.121) 1 + (ω ωl ) 2 T2 2 R2 where SNR 0 is the undamped SNR at ω = ω L. It is apparent that on resonance the SNR decreases by R = 1 1+DF as damping increases, but also broadens at the same time ω HW HM = 3 RT 2. In the limit that environmental noise or light shift noise or really any noise source that continually drives the K atoms into having a non-zero transverse polarization, the dominate noise expression simplifies to SNR = SNR 0. (2.122) 1 + (ω ωl ) 2 T2 2 R2 42

55 The HWHM linewidth of this expression is ω HW HM = 3 T 2 R, (2.123) so that the width continues to broaden while the SNR at ω = ω L, SNR 0, remains the same for all values of damping. In principle, the photon shot noise could be reduced to negligible amounts by the use of a higher probe power. This would also help to eliminate the external contribution of noise, as the signal would be higher. Under these conditions the value of R for which the bandwidth expansion comes at a significant cost to SNR is 1 1 R 4 T 2 2γ2 K P(1) B env T 2 2N. (2.124) If, as is typical, the spin-projection noise is 10 times smaller than the environmental noise, or a factor of 100 in power, significant loss of signal to noise corresponds to a damping factor on the order of 100 or greater. In chapter 5, we study two different cases with varying amount of environmental noise; see the theoretical prediction based on the two data sets in Fig The one with less environmental noise (data set 1) has the ratio of environmental noise to photon shot noise to spin-projection noise to extraneous out of loop noise(2.3:1:0.29:0.03). From this figure, for damping factor (DF = 1 R 1) of 20 the SNR linewidth increases by a factor of 3.4 with SNR SNR 0 = 86%. Data set 2, with more environmental noise, has the ratio (3.2:1:0.28:0.03) and a SNR linewidth increase of 3.9 at DF = 20 with SNR SNR 0 = 93%. As seen in Fig. 2.5, the magnetometer SNR linewidth increases linearly up to about a damping factor of 100, at which point it increases more rapidly. The sensitivity however, remains relatively constant for damping factors of up to 20. Therefore, for damping factors of 20 or less, the magnetometer gains in bandwidth with 43

56 15 SNR bandwidth (normalized) DF Data 1 Data 2 SNR Data 1 Data DF Figure 2.5: Predicted SNR for two data sets showing the expansion of magnetometer SNR bandwidth as damping factor increases (top graph). For damping factors of up to 20, we see a linear increase in the magnetometer bandwidth while magnetometer sensitivity remains relatively constant (bottom graph). minimal loss of sensitivity Measuring Signal and Noise with Quadrature Detection The input to the spectrometer S(t) with quadrature detection is real-valued. The signal is split into two components and is mixed with two reference signals at frequency ω L that are 90 phase shifted with respect to each other. The output of one channel is labeled X(t) and that of the other one Y (t), which we can represent as a complex number Z(t), Z(t) = X(t) + iy (t) = A(cos ω L t + i sin ω L t)s(t). (2.125) 44

57 Before passing to an analog to digital converter, a low pass filter gets rid of the unwanted high frequency components, set usually to cut off frequencies higher than 1 2 t, where 1/ t is the sampling rate of the converter. For instance, for a signal equal to B cos ωt, where ω is close to ω L. The signal after mixing and filtering becomes Z(t) = X(t) + iy (t) = AB 2 e i(ω ω L)t. (2.126) We take the discrete Fourier transform of this expression to find the real signal is equal to B A 2 T acq t acquisition. as long as (ω ω L ) is an integral multiple of 2π/T acq, where T acq is the time of Looking instead at the Fourier transform of the signal s(ω), we find x(ω) = A 2 {s(ω ω L) + s(ω + ω L )}, (2.127) y(ω) = i A 2 {s(ω ω L) s(ω + ω L )}, (2.128) z(ω) = As(ω ω L ). (2.129) The power spectral density of z(ω) is therefore the same as that of s(ω) multiplied by A 2 and shifted by the reference frequency ω L. In frequency space, the effect of the low-pass filter is ideally to apply a unit step function to z(ω) from a frequency of 1 2T acq to 1 2T acq. Alternately, one can think of this as applying a band-pass filter to s(ω), represented by multiplying by a unit step-function from ν 0 1 2T acq to ν T acq. The filtering done on our system gives a spectrum that decreases but does not go to zero at the extreme edges, and therefore implies that some noise may be folded back in. This may be the source of the external noise observed in our experiment; this noise would be particularly evident for higher damping factors. The one-sided spectral density (positive frequencies only) can be expressed in terms of 45

58 the periodogram P Tacq (ω) 1 T acq s(ω) 2, (2.130) as P (1) V out = lim 2E [ P Tacq (ω) ]. (2.131) T acq Therefore, the root means square noise of the real or imaginary part, assuming these are equivalent, of the discrete spectrum of Z(t) becomes P (1) V out 1 A T acq 2 T acq t, (2.132) in the limit of long T acq. Therefore the SNR for this measurement is SNR = B. (2.133) P (1) 1 V out T acq For a SNR of 1 then B min = P (1) V out 1 T acq (2.134) or sensitivity of P (1) V out with a bandwidth of 1 T acq. 46

59 Chapter 3: Atomic Magnetometer - Physical Realization and Noise Limitations In this chapter the physical realization of an atomic magnetometer is presented and the design and construction of key elements are described. Our magnetometer is operated in a regime where the dominant fundamental noise is photon shot noise. In addition to fundamental noise, the sensitivity of our magnetometer is limited by environmental noise. Environmental noise includes: thermal Johnson noise from any conductors close to the K cell, laser noise, vibrational noise, grounding noise through the field coils, electronic noise, and external electromagnetic noise. The characteristics and noise contributions of each element are theoretically or experimentally determined and discussed. A schematic of our magnetometer is illustrated in Fig The K Cell At the heart of the magnetometer is the K cell. A few of our cells are pictured in Fig These cells are not available commercially and are designed, made, and filled by us with help from a professional glass blower. During the course of this research a number of cells were prepared and used, and two main processes controlling the functioning life span of the K cells were identified; the K diffusion into the glass walls and the accumulation of K film on the optical surfaces, which will be discussed below. The cell preparation starts with commercially available stock rectangular glass tubing that is formed into a cell by a professional glass blower [53]. Our cells are made of Pyrex glass (code 7740), which we have chosen for its ubiquitousness. It is readily available in stock rectangular tubes, which is necessary because it allows for the K vapor to be uniformly pumped across the entire volume of cell. The measured alkali vapor density in glass cells, 47

60 λ = D1 line of potassium λ off-d1 resonance Pump Laser Probe Laser z y x V out 1. Differential amplifier 2. Photodiodes A & B 3. Polarizing beam splitter 4. Mu metal shields 5. Faraday shield 6. Magnetic coil assembly 7. Tapered laser power amplifier 8. Potassium cell 9. Oven 10. Plano-convex lens 11. Aspheric lens 12. λ/4 wave plate 13. Mirror 14. Glen-Laser linear polarizer 15. λ/2 wave plate 16. Polarizing beam splitter 17. Variable beam expander 18. Fixed 7x beam expander Figure 3.1: Schematic of out atomic magnetometer. The pump beam is amplified in power (7), expanded by a variable beam expander (17), split into two beams (16), circularly polarized (12), and expanded again by a Keplerian beam-shaping telescope (10,11), before pumping the potassium cell (8), which is heated by the oven (9). The probe beam is similarly amplified in power (7) and expanded, by two beam expanders (17,18), before passing through a λ/2 plate (15) and Glan-Laser linear polarizer (14). It is expanded again by a Keplerian beam-shaping telescope, passes through the K vapor (8), and the change in the angle of linear polarization is detected by a balanced polarimeter (1,2,3). like Pyrex, has generally been found to vary and is typically less than the expected density [54]. The result of the cell preparation process is a glass cell containing solid potassium metal (ideally located in the stem of the cell), along with helium and nitrogen gas, and minimal impurities. To reach the desired K vapor density only a few milligrams of K metal is required in each cell. However, as will be discussed shortly, the quantity of K metal is not trivial and can impact the lifespan of the cell as well as the operation of the magnetometer. The addition of the two gases is also critical to the operation of the magnetometer. The helium atoms act as a buffer between the K atoms and the cell walls, 48

61 New Cell K metal a Stem c b d K film Cell # 4 Cell # 2 < 1 month K diffusion Insufficient K supply complete absorption Cell # 1 > 18 months Figure 3.2: Picture of some K cells. (a) A new cell with K metal in the stem and bottom surface. (b) A cell with extensive K film on the optical surfaces. (c) A cell with very little K supply that lasted for less than a month before all the K diffused in to the glass. (d) A cell with greater initial supply of K that lasted for over 18 months and had the best measured sensitivities. In the last two cells no visible droplets of K remain. greatly reducing the rate of K collisions with the cell walls which significantly reduces the K spin destruction rate [55]. N 2 acts as a quenching gas providing the optically excited K atoms a means of transferring energy kinetically with the diatomic nitrogen molecule. Without a quenching gas the optically pumped excited atoms will fluoresce isotropically, causing significant depolarization of the optically pumped K atoms [37]. The total number of vaporized K atoms, N, in the cell is a limiting factor for magnetometer sensitivity, see Eqn [14]. Increasing N requires either increasing the operating 49

62 temperature T of the magnetometer or increasing the cell volume V. The operating temperature is limited by the permeation rate of helium through the glass [56, 57]; it increases exponentially as a function of temperature. For this reason, the maximum operating temperature of the magnetometer is limited to approximately 180 C. Other types of glass, such as aluminum silicate, which can operate at much higher temperatures with helium, are not readily available in the desired shape and dimensions and require custom fabrication, which can introduces other difficulties and expense. Therefore, increasing N to improve magnetometer sensitivity requires increasing the volume of the K cell. The dimensions of our cells are chosen to reach an acceptable balance between optimal sensitivity and experimental constraints. The dimensions are l x = 6 cm l y = 4 cm l z = 4 cm. The longer dimension l xˆx helps to increase the path length of the probe beam resulting in a larger signal, see Eqn The first process that affects the functional lifespan of the cell is the diffusion of K atoms into the glass walls. Over time, K atoms will diffuse into the silica network of the Pyrex glass causing a darkening of the cell walls as shown in Fig The rate of diffusion increases exponentially with temperature [58, 59]. Similar darkening of Pyrex glass by sodium vapor has been observed by Laux and Schulz [59]. The consequence of this darkening is the attenuation of the transmitted intensity of the laser light. Depending on the initial quantity of K metal in the cell, the color of the walls can range from a light yellow to a deep amber, as shown in the bottom pictures in Fig Typical clear glass attenuates transmitted light by about 4% per surface (or 39% per cell) [60]. A well-seasoned cell with a high degree of diffusion (dark amber) is measured to attenuate the transmitted power through the cell by up to an additional 6%. The loss of light intensity inside the cell can be partially compensated for by increasing the power of the incident pump and probe lasers. However, any loss of light on the detection side, meaning the attenuation of informationcarrying probe light as it exits the cell, translates into a lower effective quantum efficiency for the photodiode detectors, which, from Eqn. 2.39, correlates to an increase in the photon shot noise. This diffusion of alkali atoms can be reduced or avoided if necessary by using 50

63 other types of glass with a lower portion of silica or by coating the glass surfaces with other materials, as has been done for sodium cells with calcium fluoride [59] and for rubidium cells with paraffin [61]. The more significant consequence of diffusion is the reduction in the supply of K metal in the cell. If not sufficient in quantity initially, the K supply can be completely absorbed into the glass walls leaving none to produce the necessary vapor density. Such was the fate of two of our cells. Cell 1 (bottom right in Fig. 3.2) lasted many months before the entire supply of potassium completely diffused into the walls and turned the cell into a deep amber color. Cell 2 (bottom left in Fig. 3.2), with a small initial supply of K metal, turned a light yellow color and lasted only a few weeks. The second process that effects the lifespan of the cell is the accumulation of K film on the inside surfaces of the optical walls. K film is far more detrimental to the operation of the magnetometer than diffusion, as it typically reduces the transmitted light intensity by an additional 40 70% and more importantly, can be a significant source of Johnson noise as will be described later in this section. When the cell is heated the K atoms, through the process of vaporization and re-condensation, will eventually migrate and accumulate on the coolest regions of the cell walls. The rate of this process depends on the net cell temperature and the degree of temperature inhomogeneity across the surfaces. The limitations of an oven which needs to accommodate four optical surfaces, as will be discussed in section 3.2, where the optical surfaces cannot be heated directly, causes an experimental challenge for maintaining a uniform temperature gradient across the cell Vacuum System The setup for producing the K cells consists of a vacuum section coupled to a filling station, capable of receiving a glass string with various types of attached cells as shown in Fig Evacuation of air and impurities from the cells and the string to sufficient levels requires a system capable of ultra high vacuum (UHV). We use a Drytel 1025 dry (oil free) dual pump vacuum system, capable of reaching a base pressure of torr, and an Extorr 51

64 Vol. 6 Vol. 4 Baratron Pressure Gauge Extorr Vol. 5 Vol. 3 Purifier Vol. 1 N 2 He Flow Control Valve Known Volume Vol. 2 Vacuum Pump Cells Retort Tube Figure 3.3: Schematic of the vacuum system and filling station with an attached glass string containing three Pyrex cells. XT100M vacuum measurement probe with a residual gas analyzer (RGA) and pressure sensors. The XT100M measures pressure via two separate gauges. The Pirani sensor is rated to measure from 1 atm down to torr. However, its accuracy depends critically upon its orientation within the system and is therefore not used for precision measurements. The ion gauge is capable of pressures ranging from 10 2 torr down to 10 9 torr. The quadruple RGA is capable of measuring 1 to 100 atomic mass units (amu). For accurate pressure measurements above 1 torr, necessary during the gas filling stage, we use a Baratron (MKS Instruments 870B) with a single ended ultra-clean absolute pressure transducer capable of measuring from 0.5 torr to approximately 5000 torr with an accuracy of 1%. The supply of nitrogen and helium gas are research grade or % pure. The tanks are connected to the vacuum system through two sets of pressure regulating valves. To further purify these gases a Gatekeeper 35K gas purifier is used. A bypass system around the Gatekeeper is installed in order to protect the purifier during system evacuation and cell preparation. 52

65 The cells start the preparation stage attached to a glass string, a cylindrical tube of approximately 70 cm long that is fused to the vacuum system through a glass to stainless steel flange. At the other end of the string is a small bubble and a retort tube used to hold the K supply that will eventually be used to fill the cells. The cells are attached to the mid-section of the string by a stem (glass tubing) with a narrow waist in the middle that makes it easier to detach the cells from the string once they are filled. In evacuating and subsequently filling the cells with the intended quantity of N 2 and He gas, it is useful to measure the volume of the different sections to understand the rate of evacuation and filling. For convenience, the system is divided into five sections using isolation valves, as shown in Fig In addition, a reference chamber of known volume, in the form of a 151 ± 7.5 cm 3 stainless steel cylindrical capsule, is temporarily attached to the system. The entire system, with the exception of section 1 (V 1 ), is evacuated to below 10 5 torr. Once the desired vacuum is reached, all the valves are closed and the sections are isolated from each other. The first step is to release the air in V 1, which was isolated from the vacuum system, into the adjacent section V 2 and measure the air pressure, P 2, of the combined volumes (V 1 + V 2 ). Next, V 3 is opened to the air in V 1 and V 2 and the air pressure P 3 is recorded. This procedure continues sequentially until all sections are open to the system, including the cylindrical capsule of known volume. Starting with a fixed quantity of air in the system at constant temperature, the ideal gas law gives the following 53

66 relations P 2 (V 1 + V 2 ) = P 1 V 1 (3.1) P 3 (V 1 + V 2 + V 3 ) = P 1 V 1 P 4 (V 1 + V 2 + V 3 + V 4 ) = P 1 V 1 P 5 (V 1 + V 2 + V 3 + V 4 + V 5 ) = P 1 V 1 P 6 (V 1 + V 2 + V 3 + V 4 + V 5 + V 6 ) = P 1 V 1 P 7 (V 1 + V 2 + V 3 + V 4 + V 5 + V cm 3 ) = P 1 V 1. where P 1 = 1 atm. Solving the above set of equations yields the volume of each section, given in table 3.1, with a total system volume of 795 ± 32 cm 3. Before the filling process begins, the cylindrical capsule is removed from the vacuum system to minimize any unnecessary sources of impurities. Table 3.1: Volumes of vacuum system sections Section Volume (cm 3 ) V ± 23 V ± 21 V 3 62 ± 4 V ± 0.6 V ± 1.2 V ± 1.2 The first step in filling the cells is to break open the pre-scored ampule containing approximately one gram of 99.95% pure K, place it inside the retort tube, and seal the retort opening by melting it closed with a torch. Prior to filling the cells with K and the two gases, it is critical to remove as much of the contaminants as possible from the string, most notably oxygen and water vapor, from the inside surfaces of the glass. Contaminants can react with the potassium and reduce the available supply. Furthermore, resulting byproducts, such 54

67 as potassium oxide, are strongly paramagnetic and can decrease the polarization of the K spins through the introduction of strong magnetic field gradients in the cell [62]. The purification process begins with evacuating the cells to UHV, generally < torr. The RGA helps detect impurities in the system by monitoring the partial pressures of atoms or molecules up to 100 amu. When the total pressure in the system has reached UHV the glass string is wrapped with heating tape, except for the bubble and retort tube containing the K ampule, and is baked for a few days at approximately C, well above the operating temperature of 180 C. The heat vaporizes contaminants adhered to the surfaces, providing a greater level of purification. This is evident by the increase in total system pressure when the glass is initially heated. The retort tube is spared this heating so as not to vaporize the K metal and have it condense at some unintended place such as inside the vacuum system. Many of the procedures described in the remainder of this sub-section are described by Hunter Middleton in [63]. Once the pressure returns to UHV, the string outside the heating tape is baked with a flame; a relatively cool, yellow flame from a glass blowing torch is passed over the outside surfaces of the glass string in a slow and methodical fashion. Flame baking will vaporize more of the remaining contaminants, particularly in regions that are not effectively heated by the heating tape. A hot (oxygen rich) blue flame may seem to be more effective at removing contaminants, but with the glass under high vacuum, over heating can soften the glass sufficiently to have it collapse onto itself. Also, for the same reason, it is very important not to hold the cooler flame in one region for too long. This procedure is repeated and monitored by the RGA until no sudden spikes in the mass analyzer are observed and the pressure stays at UHV. As before, the retort tube containing the K capsule is omitted from flame baking to protect the K metal from vaporizing. Hence, the retort section is considered contaminated and will be removed from the string prior to filling of the cells. But first, the K is chased from the retort tube to the small bubble on the string adjacent to the retort. This procedure requires some time and patience. When a surface containing K is heated, the vaporized potassium will recondense onto any cooler 55

68 surfaces in the vicinity. Therefore, it is critical to uniformly heat all the glass surfaces that are off-limits to the K and only leave cool the intended target where the K is to be moved. This chasing also serves to distill the K and leave behind any potassium oxide that was formed when the metal was exposed to air prior to evacuation in the retort tube. With sufficient K chased into the bubble, the connection to the retort tube is heated with a hot flame and the retort is pulled off the string. Finally, the heating tape is removed from the string and the cells are ready to be filled Filling Process The filling process begins by chasing some milligrams of K metal into each cell. If the amount of K is too small, sufficient atomic vapor density cannot be reached, and if too much, it could contribute to K film buildup on the optical surfaces. More importantly, the K film and any excess K metal droplets in the cell at normal operating temperature, can be a source of magnetic noise from the thermal motion of the electrons in the metal. Using the derivations by Lee and Romalis in [64], the noise contribution from both the excess metal droplets and the K film can be calculated. For a K droplet with a radius equal to the skin depth of t = 0.2 mm at 423 khz, the thermal Johnson noise at the center of the cell, a distance of a = 2 cm, is given by δb droplet = 4 µ 0 kt σt 5 15π a 3, (3.2) where µ 0 is the vacuum permeability constant, k is Boltzmann s constant, T is the cell temperature, and σ is K conductivity. At 180 C, the calculated noise from a single droplet is ft/ Hz, which is far below the 0.1 ft/ Hz photon shot noise of our magnetometer. However, the thermal noise from a single thin film of K with radius r = 3 mm, given by δb film = 1 ( µ 0 kt σt 8π a a 2 /r 2 ), (3.3) 56

69 can generate magnetic noise of 0.30 ft/ Hz. This noise is a factor of three greater than the photon shot noise limited sensitivity of the magnetometer, which makes K film a potential source of significant environmental noise and particularly detrimental to the sensitivity of the magnetometer. While filling the cells, it is important to get the K down the stem and through the opening while also being careful not to get any of the metal on the optical surfaces. If the stem gets blocked by the chased K, it will prevent the filling of the cell with nitrogen and helium gases. In the event of unwanted K on the optical surfaces it can be carefully chased off with a cool flame while making sure not to heat the surfaces too much as the K will react with the Pyrex and turn the glass yellow. Once enough K is chased into each cell the valve on the vacuum pump is closed and the filling of the gases can start. For sufficient atomic density of the buffer and quenching gases, the finished cells must contain approximately torr of nitrogen gas and 650 torr of helium gas. Keeping the combined gas pressure in the cell below 1 atm permits us to pull-off the cells with a torch without the use of cryogens. The entire system up to the gas tanks is evacuated, with the purifier only briefly evacuated due to its nature. Volumes 5 and 6 of Fig. 3.3 are then closed off. A bolus of nitrogen gas fills volume 6 and the valves are opened to let the nitrogen gas flow through the purifier. The flow control valve serves to limit the flow of the gas through the purifier. This procedure is repeated until a pressure of 60 torr is indicated with the Baratron pressure transducer. This approach can be more efficient by taking into account the relative size of volume 6 to volumes 1, 2, and 3, see table 3.1, and the regulator on the gas tank. When the intended pressure is reached and the nitrogen valves are fully closed, the process is repeated for helium until a pressure of 700 torr is reached. Now with sufficient amounts of K, nitrogen, and helium in each cell, using a hot flame, the cells are separated from the string at the waist of each stem. In addition to the K cells, a similar procedure, minus the filling, is used to prepare four longer evacuated glass cells; two sets of two with sizes matching the dimensions of the two different optical surfaces of the K cell. These cells will act as oven windows by sitting flush 57

70 against the four optical surfaces of the K cell and protecting the light from the turbulent hot air molecules in the vicinity of the oven, which can add magnetometer noise through the pump and probe lasers [17] Pressure Broadening The addition of nitrogen and helium gases to the cell, while critical for achieving a high degree of spin polarization, cause a perturbation of the K energy states through the interaction forces between the K atoms and the gases, which broadens the K spectral linewidth [35]. The width of the line is proportional to the atomic density of the perturbing gases and is referred to as pressure broadening. The pressure broadened linewidth of K per unit density for both helium and nitrogen gas are experimentally measured by Lwin and Mc- Cartan in [65] at 410 ± 10 K, with γ He = 0.49 ± 0.01 nhz/(atom/cm 3 ) and γ N2 = 0.78 ± 0.01 nhz/(atom/cm 3 ). For cell 1, with partial gas pressures of P He = 8.62 ± Pa and P N2 = 8.13 ± Pa at room temperature, the atomic density of the two gases in the cell are n He = 2.13 ± atom/cm 3 and n N2 = 2.01 ± atom/cm 3. These values predict a total pressure broadening (FWHM), taking into account the T 0.39 scaling factor and the temperature of our cell at C (or 381 K), of Γ OP = (γ He n He ) + (γ N2 n N2 ) = 11.8 ± 0.3 GHz. (3.4) This measurement was made at 108 C instead of our normal operating conditions at 180 C because the sample is optically thick at the latter temperature, see Fig The experimentally measured K optical linewidth of 12.1 ± 0.3 GHz, Fig. 3.4, is in agreement with the predicted value. The peak absorption of incident light occurs at a wavelength of nm corresponding to the D 1 line for K atoms [34]. At typical operating temperature of 180 C, Γ OP = 12.6 Ghz for the cell of Fig The transmitted intensity of a beam of 58

71 ln(transmitted Power) (arb. units) Peak at ± THz ± nm FWHM = 12.1 ± 0.3 GHz Measured Power Lorentzian Fit Cell Temperature = ± 0.3 C Frequency (THz) Figure 3.4: The measured linearly polarized light transmission through the potassium cell at Earth s field, fitted to a Lorentzian function of the form AΓ 2 OP 4(ν ν c) 2 + Γ 2 OP, where A is peak amplitude, Γ OP = F W HM, and ν c is the peak wavelength. The width of 12.1 ± 0.3 GHz is in agreement with the calculated value of 11.8 ± 0.3 GHz. Maximum absorption occurs at wavelength of nm corresponding to the D 1 line of K. Note, gaps in data are due to mode-hopping of the laser. light passing through a vapor cell, given by [35], is I = I 0 e n Kσ ν l x, (3.5) or in terms of light transmittance ( ) I ln = n K σ ν l x, (3.6) I 0 where I 0 is the incident intensity, l x is the path length of light through the K vapor, and σ ν is the K photon absorption cross-section, given by Eqns and 2.44, with a F W HM = Γ OP. 59

72 1.0 Normalized Transmitted Power Cell Temperature ( C) Figure 3.5: The transmitted power of linearly-polarized resonant light at zero field is shown as a function of cell temperature. As temperature increases, the number density increases greatly and the cell is optically thick at our normal operating conditions ( 180 C) K Atomic Density The atomic density of the K vapor inside the cell, n K, is directly dependent on cell temperature T, as given by the ideal gas law n K = P V /kt, (3.7) where k is Boltzmann constant. The K vapor pressure P V is given by P V = 10 ( /T ), (3.8) 60

73 for potassium in the solid phase, where T < 63.4 C (melting point of potassium), and P V = 10 ( /T ), (3.9) for potassium in the liquid phase [66]. In Fig. 3.6, the plot of Eqn. 3.7 shows the exponential dependence of n K on cell temperature. At 180 C the expected atomic density in the cell is cm x Atomic Density (# / cm 3 ) Cell Temperature (C) Figure 3.6: Predicted potassium atomic density as a function of cell temperature. As stated earlier, the actual atomic density within a Pyrex glass cell is less than the expected density because of surface interactions of the glass with alkali atoms. Two different methods are used to measure the actual atomic density of the K vapor inside the Pyrex cells. 61

74 Method 1 The atomic density can be calculated from the measured magnetometer linewidth in a regime where the transverse relaxation time T 2 is dominated by the K-K spin-exchange rate, R SE. At very low pump power, where the K spin polarization P z 1, an expression for T 2 is given by [14] as 1 1 = R SE T 2 T SE 8, (3.10) where T SE is the transverse relaxation due to K-K spin-exchange [38], and R SE is the K-K spin-exchange rate [51] given by R SE = σ SE n K v K, (3.11) where σ SE = cm 2 is the experimentally determined K-K spin-exchange crosssection [11] and v K = 8 k T/(π M red ) is the relative velocity of the K to K atoms, as a function of temperature T [67]. Here, k is Boltzmann s constant and M red = 39/2 amu (atomic mass unit) is the reduced mass of the two potassium atoms. Using Eqn. 3.11, an expression of the atomic density in this regime can be derived as n K = 8 π Γ z σ SE v K, (3.12) where Γ z = 1 πt 2 is the magnetometer linewidth. Cells 1 and 3 contain different amounts of K metal, where the supply in 1 is less than the amount in 3. The number refers to the order in which the cells were made. A series of magnetometer linewidth measurements are made at different pump beam intensities for both cells, Fig The linewidths are the FWHM widths from the response spectrum of the magnetometer to an applied RF field (while sweeping the RF frequency around the 62

75 Cell #1 Temperature = 181 ± 1 C Cell #3 Temperature = 180 ± 1 C Magnetometer Linewidth (Hz) Total Pump Power at K Cell (mw) Magnetometer Linewidth (Hz) Total Pump Power at K Cell (mw) Figure 3.7: Plot of measured magnetometer linewidth as a function of pump power for K cells 1 and 3, obtained from the magnetometer response spectrum centered around the resonance frequency of 100 khz, with probe power of 1 mw at the cell. The measurements were made at C. The spin-exchange dominated linewidth, T SE, at very low pump powers, reach a maximum of 1303 ± 31 Hz for cell 1 and 2029 ± 43 Hz for 3. magnetometer resonant frequency of 100 khz). The measurements were made at 180±1 C with a probe power of 1 mw at the cell. For very low pump powers, the linewidth of the magnetometer is primarily due to the spin-exchange collisions between K atoms. The measured maximum linewidths of 1303 ± 31 Hz and 2029 ± 43 Hz yield atomic densities of 2.6± /cm 3 and 4.0± /cm 3 for cells 1 and 3, respectively. Compared to the expected densities, calculated using Eqns. 3.7 and 3.9, the relative measured densities of 43% for cell 1 and 66% for cell 3, indicate the dependence of atomic density on the available supply of K metal in the cell. 63

76 Cell #3 T = 180 C 5.5 ln(i 0 /I) (arb. units) ln(measured transmittance) Fit to y = A + Bx A = 2.78 ± 0.01 B = 3.62 ± 0.07 x /cm x x x x x10-13 v x l x (cm3 ) Figure 3.8: Plot of the natural log of the measured transmittance (black dots with error bars) of a linearly-polarized resonant light as a function of wavelength for cell 3, at 180 C. The measured data is fitted to a linear line (red line), where the slope B = 3.62± cm 3 is equivalent to the atomic density. Method 2 Another method, which is less exact but more convenient, for measuring the atomic density utilizes the transmittance of off-resonant light through the atomic vapor. The weakness in this measurement is in its reliance on the Lorentzian line-shape extending far out in the offresonance wings. Use of near resonant light is not possible as the sample is optically thick at normal operating temperatures, see Fig Comparison with method 1 will permit us to gauge the validity of the Lorentzian assumption. Using Eqn. 3.6, atomic density can be calculated by measuring the light transmittance ( I/I 0 ) as a function of wavelength and calculating σ ν. Figure 3.8 shows the results of this measurement for K cell 3 at a temperature of 180 ± 1 C and off-resonant wavelengths below the D 1 line for K. A linear fit (red line) gives 64

77 a slope of 3.62 ± cm 3, which from Eqn. 3.6, is equivalent to the atomic density in the cell. This value is comparable with the value obtained in method 1, for the same cell, but indicates that method 2 underestimates the number density by about 10%. The benefit of this method is that it requires a fraction of the time of the previous method and is used on a day-to-day basis while keeping this underestimation in mind. Using the second method we found the number density was stable for cells with significant amount of K on the order of weeks. 3.2 The Oven The oven is an air flow chamber designed to heat the K cell through thermal conduction. It is constructed using G10 fiberglass, high temperature epoxy, thermal insulation, Kapton tape, and thermal Gap Pad (from Bergquist), which are all non-magnetic. All but the last item are thermally insulating; Gap Pad is a thermally conductive soft pad and can be used to make better thermal contact between the K cell and the oven walls. The oven chamber is heated with circulating air that is heated externally by a heating element. As shown in Fig. 3.9, the K cell sitting in the middle of the oven is heated through its top and bottom surfaces, which are in thermal contact with the oven. A 1/2 inch thick layer of a microporous silica insulation (from Armil CFS), wrapped in Kapton tape, covers the entire outside surface of the oven, except on the top where 1/8 inch thick insulation is used to minimize the distance between the cell and the sample holder. Once the cell is sufficiently heated, the vaporized K will start to migrate and accumulate, through a continuous vaporization and condensation process, on the coolest sections of the cell walls forming a K film that blocks the light. Accumulation of this film on the optical surfaces can adversely affect the response and sensitivity of the magnetometer as discussed earlier. Therefore, a major design challenge for the oven is to ensure that the optical surfaces are not only uniformly heated but also kept at a higher temperature relative to the nonoptical section of the cell, particularly the stem. This task is made more difficult by the fact that the optical surfaces are the only parts of the cell that are unable to make direct 65

78 12.5 cm 13.5 cm Insulation Fiberglass and epoxy Oven 11 cm Thermal gap pad Fiberglass divider for controlling air flow Teflon tube for isolating Cell stem Non-magnetic thermocouple Hot air inlet outlet Open to outside atmosphere Figure 3.9: Schematic of the oven showing the flow of hot air through the fiberglass chamber. A section of teflon tubing is used to isolate the cell stem from the circulating hot air and help keep it cooler than the rest of the cell. A barrier is placed between the inlet opening and the outlet to force the incoming air through the oven. The isolation chamber is placed on the cooler side. A type K non-magnetic thermocouple passing through a small opening at the bottom of the isolated chamber measured the temperature of the cell stem. thermal contact with the oven. To prevent, or at least minimize, K film build up on the optical surfaces, the cell stem is isolated inside the oven and somewhat insulated from the circulating hot air by a short section of teflon tubing, see Fig A small pinhole on the bottom of the oven permits cooler air to circulate inside this isolated chamber thereby allowing the stem to remain a few degrees cooler than the rest of the cell. A second small pinhole into the stem chamber provides access for a non-magnetic (type K) thermocouple to monitor the stem temperature. All temperature measurements of the cell will refer to the stem temperature since the atomic vapor density inside the cell is controlled by the temperature of the coolest part of the cell. A PID controller, monitoring and controlling the temperature of the heating element, can 66

79 maintain a cell temperature of 180 C with less than 1 C fluctuation over many hours. The thermocouple, which extends to a few millimeters below the K cell, Fig. 3.9, is grounded to the RF shield. To insure that the thermocouple is not a source of additional noise for the magnetometer, we made noise measurements with the thermocouple completely removed from the system and found no measurable improvements in magnetometer sensitivity. The oven design also incorporates the evacuated windows to assist in heating the optical surfaces directly. Approximately a one inch section of each oven window, which sits flush with the optical surfaces of the K cell, is in good thermal contact, using Gap Pad, with the oven walls. Good thermal contact between the windows and the oven walls is critical as they can heat the optical surfaces when hot, as well as cool them if not sufficiently heated, which can cause severe K film on the optical surfaces. When trying to ensure good thermal contact between the glass surfaces and the oven walls it is very important not to stress the glass walls of either the K cell or the evacuated windows, with excess pressure from the oven. Pressure on the glass walls can cause stress birefringence in an otherwise isotropic material which can induce circular or elliptical polarization of a linearly polarized light. Stress birefringence results when the refractive index of an isotropic material changes along orthogonal optical axes as a result of mechanical pressure or stress on the material. A linearly polarized probe beam passing through stressed glass can become circularly polarized and impart additional light shift noise by optically pumping the K in the probing (ˆx) direction [14]. 3.3 Coil Assembly In an ideal atomic magnetometer, all the K atoms in the cell experience a perfectly homogenous static magnetic field B 0 aligned precisely along ẑ. In practice, environmental fields, such as Earth s field and any other local residual sources, can be compensated for by either actively using field coils or passively by magnetic shielding. For convenience, we employ mu-metal shielding (as described later in this chapter) to minimize background magnetic 67

80 Barker 22.2 cm Maxwell z B0 Saddle X y dbz/dz 20.3 cm x Saddle Y Bx By Golay Y Golay X dbz/dy dbz/dx Figure 3.10: Schematic of the static and gradient coils wrapped around a hollow cylindrical plexiglass structure. fields. Even still, producing an ideal field in reality requires compensation fields; there exists a degree of inhomogeneity either from residual external fields not blocked by the mu-metal shields, local fields from slightly magnetized materials within the mu-metal shields, or the mu-metals shields themselves. Our coil assemble consists of a number of field coils to try and produce a uniform static magnetic field B0 perpendicular to the probe beam, x ˆ, with minimal inhomogeneity across the volume of the K cell. The coil assembly consists of a set of six coil configurations wrapped around a hollow plexiglass cylinder. Three sets of coils produce static magnetic fields along x ˆ, yˆ, and zˆ, while the other three produce fields in zˆ with linear gradient in the x, y, and z directions. To accommodate these coils, which require exact geometry, radial and axial groves were machined on the outside surface of the cylinder to hold the 68

81 Table 3.2: Magnetic field strength of the field coils. Coil Type Predicted Measured Barker (ẑ) 0.82 G/A ± G/A Saddle X (ˆx) 0.15 G/A ± G/A Saddle Y (ŷ) 0.15 G/A ± G/A Maxwell (db z /dz) G/A/cm ± G/A/cm Golay X (db z /dx) G/A/cm ± G/A/cm Golay Y (db z /dy) G/A/cm ± G/A/cm loops precisely in place. With the coils wrapped within these channels, two layers of high temperature Kapton tape is used to cover the entire outside surface of the cylinder and seal the coils in place. This helps to prevent the coils from coming loose and lose their geometry when repeated heating and cooling of the wires causes them to expand and contract. The main static field B 0 along ẑ, produced by the Barker coil configuration [68], produces a fairly uniform field across the potassium cell. Two transverse coils, called Saddle coils [69], wrapped orthogonally with respect to each other, generate a uniform magnetic field in ˆx and ŷ. They are used to help align B 0 perpendicular to the probe beam and parallel to the pump beam if the need arises. In addition to correcting the alignment of B 0, the saddle coils are also used to generate reference RF fields for the purpose of magnetometer tuning and testing. The gradient fields are used to compensate for any linear magnetic field inhomogeneity in B 0. These field gradients may be caused by residual fields inside the shielding. Fields in ẑ with linear gradients in ˆx and ŷ are produced by Golay coils [70] and in the ẑ by a Maxwell coil configuration [71]. Table 3.2 shows the theoretical and measured magnetic field strengths of these coils as a function of current I. The measured inhomogeneity of these coils across the volume of the cell is sufficiently small. A better indicator of inhomogeneity in B 0 is the broadened K linewidth. Under optimal conditions, we have measured a signal linewidth of 315 ± 4 Hz for resonant frequency of 423 khz, Fig Assuming the linewidth broadening to be caused entirely by field inhomogeneity, the measured linewidth corresponds to a maximum field inhomogeneity of 0.5 mg across the entire K cell. 69

82 1.0 Normalized Signal Real Amplitude Linewidth = 315 ± 4 Hz v - v 0 (Hz) Figure 3.11: Minimum measured magnetometer linewidth corresponds to a B 0 inhomogeneity of 0.5 mg across the cell. These field coils, particularly the transverse saddle coils, could be a source of noise for the magnetometer considering they are located inside the RF shields. Any noise in the current sources can translate into noise for the magnetometer and can reduce the sensitivity. The current sources for the static and gradient coils are laser diode controllers from Thorlabs (models LDC 340s and LDC 201c), capable of provide a stable current with very low noise. A set of model LDC 340 provide up to 4 A of current for B 0 and the gradient coils, with current noise of < 30 µa rms between 10 Hz and 10 MHz, while two model LDC 201c, with current noise of < 0.2 µa rms in the same frequency range, provide up to 100 ma for the saddle coils. To prevent ground loops, we have isolated the AC ground of all current source from the power outlets and grounded each controller to the common ground of the magnetometer. Also, in an effort to minimize all possible current noise from the coils, we have placed custom lowpass filters on all current sources. Experimental tests have shown 70

83 that having a common ground can stabilize our results, but the addition of strong filters did not significantly change the measured noise, hence we do not suspect that the current sources are a significant source of noise. Detuning Coils In addition to the coil assembly, there is a set of coils inside the oven surrounding the K cell called the offset coils. It consists of two sets of Helmholtz coils, as shown in Fig. 3.12, with the inner coils generating a strong field in one direction while the outer coils generate a weaker field in the opposite direction inside the cell. This configuration is designed to minimize the net field outside of the oven and close to the RF shield since the rapid turning on and off of the offset field, as necessary in resonance pulse detection, can cause eddy currents in the nearby RF shields resulting in noise. This combination also produces a highly inhomogeneous field across the K cell, which is used for detuning during resonance detection, by producing an approximately 2 G inhomogeneous static field B offset ẑ to augment B 0. The net field will shift the Larmor frequency ω shift = γ K (B 0 + B offset ) far away from the tuned resonant RF frequency ω L = γ K B 0 during the excitation of the sample by a strong RF pulse. Without the offset field present, the magnitude of the excitation pulse ( 1 mt), which is typically 12 orders of magnitude larger than a typical NQR signal ( 1 ft), will destroy the K polarization. The recovery time of the magnetometer, based on optical depth and the pumping rate [17], is on the order of ms, which will render the magnetometer unable to detect the quickly decaying resonant signals (typically less than a few milliseconds). The highly inhomogeneous field produced by the offset coil serves to increase the K Zeeman linewidth during excitation so that any induced transverse polarization of the K by the RF pulse is quickly dephased [17]. The proximity to the K cell means Johnson noise current from the 22 gauge copper wires of the offset coils may be a considerable factor. Using the development in [64], the set of inner loops, which are approximately 2.5 cm from the center of the cell, can generate magnetic noise on the order of 0.1 ft/ Hz at 180 C. This noise, on the same order of the 71

84 12.5 cm 6.5 cm Inner offset coils 10 loops each Outer offset coils 4 loops each 7.0 cm 4.5 cm B offset 4.5 cm 12.5 cm K cell z 22 gauge copper wire y x Figure 3.12: Schematic of the offset coils surrounding the K cell. These coils produce a inhomogeneous static field B offset in ẑ during resonance detection to offset B 0 and shield the K spins from the excitation pulse by shifting the tuning frequency of the magnetometer away from the frequency of the pulse. photon shot noise in our experiments, can be a considerable source of environmental noise. 3.4 Sample Holder and the Excitation Coils An ultra-sensitive RF atomic magnetometer is well suited for the detection of weak NQR signals. For this purpose, we have designed and built a sample holder to contain and excite the NQR material. The source of our NQR signal is 70 grams of ammonium nitrate in powdered crystalline form. It is placed in a ceramic container that places the bottom of the sample material approximately 2.8 ± 0.2 cm above the top surface of the K cell, Fig The proximity of the sample to the cell is very important for calculating the expected NQR signal amplitude as it is inversely proportional to the separation distance to the third power (1/r 3 ). The sample holder consists of a thin walled non-magnetic G10 fiberglass cylinder, with a 72

85 Boron nitride container Excitation coils Cooling tubes Shielding coils Distance from bottom of sample to top of K cell ~ 2.8 cm Figure 3.13: Schematic of the sample holder and excitation coils. solenoidal RF excitation coil wrapped on the outside surface, and a fiberglass bottom plate that holds the shielding planar coils. The cylinder, approximately 6.5 cm in diameter and 3.5 cm tall, houses a boron nitride sample container. Boron nitride has a very high thermal conductivity and is meant to minimize any thermal gradients across the sample; thermal gradients will cause broadening of the NQR frequency and decrease the signal amplitude [17]. With the oven directly below the holder, circulating cold water through a few loops of tubing wrapped around the outside and top of the sample holder will help to keep the sample at room temperature. Grooves on the outside surface of the cylinder hold in place the loops of the solenoid which generate the excitation pulse for the sample. Using a tuning circuit, Fig. 3.14, the excitation coil, with resistance R, is impedance matched to 50 Ω at the resonance condition with the relation ( 50 Ω = R 1 + C ) 2 2, (3.13) C 1 73

86 Excitation coil L R C 2 C 1 RF input Figure 3.14: Diagram of the tuning circuit for the excitation coil. The coil is tuned to 423 khz and impedance matched to 50 Ω. For R = 7.4 Ω, C 1 = 7.2 nf, and C 2 = 12.7 nf, the Q of the excitation coil is 1.7, with a ring-down time of < 2 µs. and tuned to 423 khz, a characteristic NQR frequency of ammonium nitrate, by ν = 1 1 2π L(C1 + C 2 ), (3.14) where the coil inductance is L = 6.9 µh. For R = 7.4 Ω, C 1 = 7.2 nf, and C 2 = 12.7 nf, the excitation coil has a measured Q of 1.7, where Q ν ν with a broad excitation frequency bandwidth ν = 250 khz. With the signal detection decoupled from the excitation circuit, a low Q with ring-down times 1 π ν of < 2 µs does not limit the start of signal acquisition. The shielding coils, held in place on the bottom surface of the plate, Fig. 3.13, is designed to produce a field that attempts to nullify the field from the excitation coil in the region of the K cell, in effect shielding the K vapor from the excitation pulse. Otherwise, the excitation pulse, which as stated earlier is 12 orders of magnitude greater in amplitude than the NQR signal, can destroy the K spin polarization. Figure 3.15 shows field cancelation of up to 2 orders of magnitude at the center of the K cell. 3.5 The Shielding The K cell is shielded from both static magnetic fields and external RF electromagnetic interference. The static shields consist of a set of three mu-metal enclosures, one inside the other, that attenuate external static fields by shunting the magnetic field around the shielded space. The use of the mu-metal magnetic shielding is largely for convenience, in 74

87 1.0 Measured field Predicted field Normalized Excitation field Center of K cell Top of K cell Bottom of sample Top of sample Distance form center of K cell (cm) Figure 3.15: Measured and predicted field from the excitation coils. The area between the blue lines represent the volume of the sample and the region before the orange line represents the K vapor. order to reach a homogeneous field across the cell which is constant over the experiments. Inside the mu-metal shields is a Faraday cage that shields the K spins from external RF noise. Without the AC magnetic shielding the noise floor of the magnetometer will be dominated by laboratory environmental noise. While the static and RF magnetic shielding allow for high sensitivity to be easily achieved, it is also important to ensure that thermal magnetic noise from these shields do not contribute significantly to the noise floor of the magnetometer. RF Shielding The RF shield is a rectangular aluminum box with inside dimensions of 30.5 cm ˆx 25.4 cm ẑ 22.9 cm ŷ and wall thickness of 1.9 mm. Four rectangular opening on the walls of the RF shield accommodate the pump and probe beams and the evacuated windows. The shielding 75

88 around these openings were later extended using sections of aluminum tubes, which act as a waveguide with cutoff frequency given by ν cutoff = c(0.293) r and a decay constant of λc 2π, where c is the speed of light, λ c is the cutoff wavelength, and r is the radius of the waveguide [72]. Extensions with inside radius of r = 4.4 cm and length of 10 cm on the pump side and r = 3.8 cm and 8 cm long on the probe side provide a cutoff frequency which is well above 423 khz. Noise measurements, with and without the extensions present on the RF shield, did not show any measurable difference in sensitivity, which indicates that environmental RF magnetic noise is adequately shielded by the Faraday cage. Pass-through BNC connectors on top surface of the RF shield provide connection points for the field coils. Two small ( 1.5 cm) diameter holes, along with a pinhole, at the bottom provide access for the hot air inlet and outlet tubes and the oven thermocouple. Two small holes on top allow access for the cooling tubes of the sample holder. Thermal motion of electric charge inside the RF shield produces eddy currents resulting in magnetic noise. The magnetic noise from the surfaces of the RF shield can be calculated using the phenomenological model by Varpula and Poutanen [73]. For frequencies which are sufficiently away from the critical frequency ν c, the magnetic noise can be approximated by B ν νc = µ 2 σkt t 2π 1 z (ν/ π 2 ν c), (3.15) 2 where µ is the magnetic permeability, σ is conductivity, k is Boltzmann s constant, T is the temperature of the conductor, t is the thickness of the conductor, and z is the distance from the conductor. The variable ν c = 1 4µσzt by 3 db from the zero-frequency noise given by is the frequency at which the noise is reduced B ν=0 = µ 2 σkt t 2π z(z t). (3.16) 76

89 Using Eqn the noise contribution from the RF shield is approximately 0.01 ft/ Hz at 423 khz [74]. Magnetic Shielding The static shielding consists of a set of three individual cylindrical mu-metal cages, with one inside the other, enclosing the RF shield. The inner shield is 48 cm in diameter and 56 cm long while the outer shield has a diameter of 74 cm with a length of 82 cm. Mu-metal has a very high magnetic permeability and is very effective at attenuating static magnetic fields. It is also more effective in tandem which is the reason for the triple layers. Four round openings, between 6 8 cm in diameter, provide a path for the pump and probe beams. A number of smaller openings provide access for heating and cooling tubing and cables for the various coils. The mu-metal shields are tested using a large set of Helmholtz coils, placed around the outside of the shields. The field produced by the Helmholtz coils is attenuated by a factor of 1000 in the vicinity of the K cell. Considering the local ambient field is less than 1 G, this shielding factor is more than adequate for creating an environment with inhomogeneity which is less than the gradient produced by the field coils. Using the framework provide by Lee and Romalis [64] to calculate magnetic noise from high-permeability magnetic shields, the noise from the mu-metal, as seen by the K, is approximately 0.15 ft/ Hz at 423 khz. This could be a considerable source of external noise for the magnetometer. However, due to the RF shielding, the noise from the mu-metal inside the RF shield is found to be on the order of 10 6 ft/ Hz at 423 khz [74]. Therefore, the total noise contribution from all the shielding is on the order of 0.01 ft/ Hz, which is an order of magnitude smaller than the fundamental photon shot noise. 77

90 3.6 Pump and Probe Lasers The light sources for the pump and probe beams are from two LION External Cavity continuous-wave tunable diode Lasers in Littman/Metcalf configuration, from [75], producing a linearly polarized Gaussian beam (TEM 00 mode) of up to 100 mw output power and a wavelength tuning range of 757 nm 782 nm. These single mode lasers have a narrow linewidth (< 300 khz) with sidemode suppression of 48 db at 770 nm. For more power, both pump and probe lasers are supplemented by tapered amplifiers [75], in a master-slave configuration as shown in Fig The available total power is up to 1000 mw. The output beam from the tapered amplifiers is 2 mm in diameter with a Gaussian intensity profile. For optimal magnetometer performance and to ensure uniform K polarization across the cell, it is necessary to expand the beams and change the power profile of both the probe and pump beams from a small Gaussian profile to a large collimated and uniformly distribution top-hat profile before the K cell. This requires the expansion of the original beams by 2 orders of magnitude to uniformly illuminate the cm cell. While much work has been done to especially design and manufacture optical elements to shape the beam [76 79], we were able to find off-the-shelf components using the optical design program Zemax [80]. The expansion and the redistribution can be achieved with a Keplerian beam expander, Fig. 3.16, which consists of two positive lenses; a smaller aspheric lens (OptoSigma ) and a larger spherical lens (Thorlabs LA1238 plano-convex lens for the probe and OptoSigma plano-convex lens for the pump), with focal points f 1 = 10.5 mm and f 2 = 100 mm for the probe and f 1 = 10.5 mm and f 2 = 150 mm for the pump. When the focal points of these two lenses are coincident on the same point, as shown in Fig. 3.16, the magnification of the incident beam is f 2 f1 [81]. The limited availability in the size and shape of aspheric lenses limits the magnification by a factor of 10 while also maintaining a collimated beam with a top-hat profile. To achieve the necessary expansion, the pump and probe beams are initially expanded 78

91 Pump Beam Evacuated Windows a K cell y Keplerian beam expansion Uniform beam profile x z Gaussian beam profile f 1 f 2 Small aspheric lens b Large spherical lens Relative power distribution at center of the K cell c Figure 3.16: A Zemax simulation of the pump beam expansion (a) using a Keplerian beam expander (b) resulting in a uniform power distribution across the center of the K cell (c). (1st expansion) with commercially available beam expanders. For the pump beam, a variable beam expander from Special Optics (model X-770) is used to expand the beam by a factor of 5.5 before the second Keplerian expansion (2nd expansion). The probe beam requires a combination of two beam expanders, a fixed 7 expander (model X-770) followed by a variable beam expander (model X-770) set to 2 magnification, before the 2nd Keplerian expansion. These beam expanders produce an expanded collimated beam without changing the beam power profile. Figure 3.17 shows the measured Gaussian power distribution of both the probe and pump beams after the first beam expansion. The measurements were made with a very small ( 0.5 mm) aperture in front of a power meter affixed to a two-axis translation stage. Figure 3.18 shows the fully expanded flat-top power distribution of the pump beam measured at the location of the K cell. Both lasers output linearly polarized light. The polarization of the probe laser is found to be 98% linear. A Glan-Laser Calcite linear polarizer, placed before the aspheric lens, is 79