ABSTRACT RADIATION TRAPPING IN OPTICAL MOLASSES. By Soo Y. Kim

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1 ABSTRACT RADIATION TRAPPING IN OPTICAL MOLASSES By Soo Y. Kim Using three orthogonal pairs of counterpropagating red-detuned lasers and a magnetic field gradient, we have constructed a trap of 85 Rb atoms at temperatures reduced to the order of 100 microkelvins and densities of 10 9 atoms/cm 3. When the atoms interact with the light, they are absorbing and spontaneously emitting photons. The emitted photons from atoms can be absorbed by neighboring atoms. This phenomenon, called radiation trapping, limits the density and temperature of the trap and also destroys the atomic coherence. We can noninvasively measure the effects of radiation trapping by measuring the intensity correlation function of the light radiated from the trap. We have found that theoretically, the effects of radiation trapping can be detected in dilute atomic samples with an on-resonance optical depth as low as 0.1, more than an order of magnitude less than seen before. Experimental progress is reported.

2 RADIATION TRAPPING IN OPTICAL MOLASSES A Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science Department of Physics by Soo Y. Kim Miami University Oxford, Ohio 003 Advisor Samir Bali Reader Perry Rice Reader S. Douglas Marcum

3 Table of Contents 1. Introduction and Motivation. Laser Cooling and Trapping.1 Laser Cooling. Magnetic Field Gradient Trapping 3. Radiation Trapping 3.1 First-order and Second-order Coherence 3. Optical Bloch Equations with Radiation Trapping for MovingAtoms 3.3 Second-order Coherence for Moving Atoms with Radiation Trapping 4. The Apparatus 4.1 Atomic System 4. Vacuum Chamber 4.3 Frequency Tuned Laser Diode External Cavity Diode Laser 4.3. Saturated Absorption Locking Mechanisms 4.4 The Trapping Setup 4.5 Magnetic Coils Magnetic Field Gradient Coils 4.5. Nulling Coils Switching Circuit 5. Characteristics Measurements 5.1 Number Measurement 5. Loading Curve 5.3 Density Measurements 5.4 Temperature Measurement 6. Intensity Correlations 7. Conclusion and Future Goals References Appendices Appendix A: Parts and Suppliers Appendix B: Circuits *note that references are in brackets [ ], and parts/supplies are superscript. ii

4 Acknowledgements I would like to thank my advisor, Dr. Samir Bali, for taking me into his lab and giving me the opportunity to experience building a full lab from nothing but an empty optics table. He has truly been an integral part of my graduate career at Miami University. I would also like to thank the people I have closely worked with in lab for the past three years: Matthew Beeler, Ronald Stites, Brian Pollock, Charlie LaPlante, Laura Feeney, and Kristy Kallback-Rose. Without them, the lab would not be where it is today. I would also like to thank our collaborating professors, Dr. Perry Rice and Dr. S. Douglas Marcum for all their help and for being the readers for this thesis. Financial support for this project was provided by the College of Arts and Science, the Office for the Advancement of Scholarship and Teaching (OAST), and Miami University. Also, supports from the Cottrell Foundation (Research Corporation), the Petroleum Research Fund of the American Chemical Society are appreciated. A paper entitled Sensitive Detection of Radiation Trapping in Cold-Atom Clouds has been accepted for publication in Physical Review A from work done for this thesis. iii

5 Dedicated to the future Jedis iv

6 1. Introduction and Motivation Cold atoms have been a hot area of research in recent years. It started in 1933 when R. Frisch at the University of Hamburg showed that light can exert a force on atoms [1]. But it took almost 40 years (1970) for A. Ashkin at Bell Laboratories to show trapping by using a focused laser beam, or optical tweezers, to trap small latex spheres immersed in a medium []. He proposed that atoms could be trapped in the same way with resonant light. Five years later, in 1975, the idea of laser cooling was independently presented by two groups: Hänsch and Schawlow at Stanford University [3] and Wineland and Dehmelt at the University of Washington [4]. Both looked at the use of three orthogonal pairs of counterpropagating beams to slow down atoms. The beams are tuned below the resonant frequency of the atoms, so that when the atoms approach the beams, some portion of them are Doppler shifted into resonance and interact with the laser. Any beams that are propagating in the direction of the motion of the atom would be Doppler shifted out of resonance, and not affect the atom. Thus the atoms are slowed down and cooled, given that average speed is related to temperature. Then ten years later, in 1985, S. Chu et al at AT&T Bell Laboratories, created an optical molasses with laser beams [5]. In 1987, again at AT&T Bell Laboratories, Raab et al developed the magneto-optical trap (MOT) using a magnetic field gradient to cool and trap atoms [6]. Finally in 1997, S. Chu, C. Cohen-Tannoudji, and W. Phillips received the Nobel Prize for the atom trap. Also in 001, E. Cornell, W. Ketterle, and W. Phillips were awarded the Nobel Prize for creating Bose-Einstein condensates (BEC). Cold atoms have been used to create atomic clocks [7], further BEC research [8,9,10] and many other applications. We use the MOT to observe the effects of radiation trapping. Atoms trapped in a MOT are continuously interacting with laser light and absorbing and emitting photons. In a dense atom trap, the likelihood that an atom could absorb the emitted photons from another atom increases. The absorption of the photon pushes the two atoms apart, increasing the temperature and decreasing the density of the trap. This also decreases the coherence of the trap due to the random phase and polarization of the spontaneously emitted photon [11,1]. This feature, called radiation trapping, was first seen in cold atom clouds by Walker et al 1

7 [13,14] in They found that for traps of ~10 11 atoms/cm 3 and an optical depth of 3, radiation trapping became evident. In 1996, Bali et al [15] studied the intensity correlations of scattered light from laser-cooled atoms and encountered preliminary effects of radiation trapping, but further study was not done on the effects. In 001, Matsko, Novikova, Scully, and Welch [16] noticed that in their electromagnetically induced transparency (EIT) experiments, radiation trapping effects are present. To explain them, they developed a simple theory for radiation trapping. However, this applies specifically to EIT experiments, which involve three-level atoms, usually hot. They neglect Doppler broadening and direct interactions with the laser. We present a theory using intensity correlations to express the effects of radiation trapping, including Doppler broadening and direct interactions, for an optical molasses. We find that radiation trapping is prevalent in traps with a density of 10 9 and an optical depth of 0.1, which is more than one order of magnitude less than seen before [13,14] and can be measured with a non-invasive technique.

8 . Laser Cooling and Trapping To trap atoms, first they must be slowed down to very low velocities. This can be achieved by having resonant counterpropagating laser beams bombard an atom in three orthogonal directions. Since for a monatomic ideal gas, temperature is related to kinetic energy, 3 1 k B T = mv, (.1) the reduction of velocity leads to a reduction of temperature and therefore slow atoms are said to be cold. The situation of the cooled atoms and six cooling laser beams is called an optical molasses. However, the lasers do not exert a position-dependent force upon the atoms. A magnetic field gradient is necessary to provide the position-dependent force to create an atom trap..1 Laser Cooling At room temperature, gas atoms are moving with average speeds around half a kilometer per second. The kinetic energies of the atoms can be described by a Maxwell-Boltzmann distribution. The atoms in the low velocity tail of the distribution can be slowed down to speeds of centimeters per second using photons. The atom can be imagined as a big bowling ball and the little photons of light as ping pong balls. Conservation of momentum shows that one ping pong ball may be insufficient to significantly affect the bowling ball, but with enough ping pong balls, the bowling ball can be slowed down. Thus, a gas atom can be slowed down to low velocities with a massive quantity of photons, which a laser can supply. For example, a 1mW beam of a 780 nm laser provides photons/sec. Although the bowling ball and ping pong ball model provides a nice picture, it does not accurately depict what actually occurs. When a photon encounters an atom, the energy from the photon excites an electron in a ground state of the atom to an excited state. Therefore, only photons with the correct energy, or matching frequency ω o, will be absorbed by the atom. However, since the atom is moving at a velocity v, the atom perceives the frequency of the photon ω to have been Doppler shifted by k v, where k is the wave propagation vector of the light. Since the momentum 3

9 of a photon is k, this multiplied with the average photon absorption rate of an atom gives the average force [17] on the atom, F cool γ I I o = k, (.1.1) 1 + I I + γ o [ ( δ + k v) ] where γ is the decay rate of the atom, I is the intensity of the laser light, and δ is the laser detuning, δ = ω o ω. We will assume, without any loss in generality, that the laser is reddetuned, i.e. I. The recoil due to emission is not taken into account because the direction of emission from the atom is random and the average of this recoil force would sum to zero. In 1D, the net force exerted by counterpropagating laser beams on an atom moving in the positive direction is kγ I = 1 1 Fcool I o 1 + I I o + γ [ ( δ + kv) γ ] 1 + I I + [ ( δ kv) ] o, (.1.) assuming that the two light fields do not interfere and act on the atom independently. In the approximation that to kv << γ, δ and using the binomial series expansion, the force simplifies F cool ( δ γ ) + ( δ γ ) I = 4k v (.1.3) I o [ 1 + I I ] for three pairs of orthogonal counterpropagating beams. As the equation shows, the force is directed opposite of velocity and therefore clearly is a damping force. If the laser is red-detuned, the atom is Doppler shifted into resonance with light propagating opposite its direction of motion and shifted further out of resonance with light propagating in its direction of motion. Consequently, the atom mostly interacts with the opposing beam and is slowed down. For the case of counterpropagating blue-detuned beams, δ < 0, the atom would be further accelerated. Therefore, to slow an atom in all three spatial dimensions, six counterpropagating red-detuned lasers in three orthogonal directions are used. However, as one can see, the lasers only exert a damping force. The atoms are slowed wherever they are, but there is no preferred spot where the atoms want to gather. Below, we see how a magnetic field gradient is used to provide a position-dependent force. o 4

10 . Magnetic Field Gradient Trapping The electron spin and angular momentum, S and L, and nuclear spin I, produce a net magnetic moment of an atom, characterized by the total angular momentum F, where F = S + L + I. When an external magnetic field is introduced, the energy levels of the atom split, known as the Zeeman effect. The energy shift of a level is dependent on m F, the projection of the total angular momentum of the atom on the magnetic field axis [18], mf eb E =, (..1) m c where e is the electron charge, m e is the electron mass, and B is the external magnetic field, given by µ o NIR B x = 1 e ( R + ( A + x) ) 3 R + ( A x) 1 ( ) 3, (..) where R is the radius of the coil, A is the half the distance between the two coils, NI is the number of ampturns, and x is the distance away from the midpoint between the coils. The resonant frequencies of the levels shift by mf eb υ =. (..3) 4πm c e For simplicity, we will consider an atom with a ground state of F = 0 and an excited state of F = 1. In the presence of a magnetic field, the ground state is unperturbed, but the excited state splits to three levels, m = 1,0, 1. F For a magnetic field gradient, the energy levels split as shown in Fig...1. For a lasercooled atom at a distance z from the center, it experiences a force towards the center of the trap from Eq..1.1 (using v = 0 for simplicity), kγ I 1 1 Ftrap = I o 1 + I I o + [ ( δ υ ) γ ] 1 + I I o + ( δ + υ ) γ Similarly to the cooling force, we use the approximation that [ ]. (..4) v << γ, δ and the binomial expansion to see that the trapping force simplifies to F trap I 4k I o ( δ γ ) + ( δ γ ) = [ 1+ I I ] o v. (..5) 5

11 The position dependence of the force is not easily seen in Eq...5 because it is hidden in the magnetic field in v. Thus, the polarized laser light and magnetic field gradient provide the cooling and position-dependent forces that result in an atomic magnetooptical trap. e> (F = 1) E m f =1 0 σ + g> (F = 0) x ω B x x -1 σ - x = 0 B = 0 Fig...1. The excited energy level of an atom splits with respect to the B-field. At point x, the laser light ω is closer to resonance with the m 1 level, which is pumped by σ - -polarized light, and as a result, f = pushed toward x = 0. Likewise, at point x, the laser light is closer to resonance with the = + 1 which is pumped by σ + -polarized light, and pushed again towards x = 0. m level, f 6

12 3. Radiation Trapping As an atom interacts with the laser light, it absorbs and emits photons. As will be explained in chapter 6, these photons exit the trap and can be detected for measurements on the trap or can be just lost in space. However, as the density in a trap increases, the photon mean free path decreases and it is more probable that an emitted photon could be absorbed by a neighboring atom instead of just leaving the trap. This phenomenon, called radiation trapping, prevents the trap from getting denser, raises the temperature of the trap, and reduces coherence. As the photon is emitted from the atom, the atom recoils in the opposite direction of the movement of the photon. When the photon is absorbed by another atom, this atom recoils in the direction of the photon movement. Therefore, the two atoms are pushed away from each other, decreasing the density of the trap. The recoil introduces motion to the atoms, increasing temperature. Because the polarization and phase of the emitted photon is random, the coherence of the trap decreases when atoms start absorbing these incoherent photons. By looking at the density and temperature limits and the decrease in coherence, radiation trapping can be detected in the trap. 3.1 First-order and Second-order Coherence The density and temperature can be directly measured, as is shown in Ch. 5, but the decrease in coherence is more difficult to observe. We can look at the first order coherence g (1) (τ) of the trap, which is defined [19] as g (1) ˆ * E ( t) Eˆ( t + τ ) ( τ ) =, (3.1.1) ˆ * E ( t) Eˆ( t) which looks at the electric field Ê at different times. As one can easily see, for τ = 0, ( ) ( 0) 1 g 1 =. According to the Weiner-Khintchine theorem, this is related to the Fourier transform of the normalized frequency spectrum [19], F 1 π ( 1 ( ω) g ) ( τ ) i = e ωτ dτ 7. (3.1.) But rather than measuring the field, we find it simpler to measure the intensity of the radiating photons. The intensity correlation, or second-order coherence g () (t), is defined as

13 g () Iˆ( t) Iˆ( t + τ ) ( τ ) =. (3.1.3) Iˆ( t) where I(t) is the intensity from the radiating atoms at a time t and I(t+τ) is the intensity of the radiating atoms at a time delay τ later. For chaotic light from a large number of spatially coherent sample, we have the relation of the second-order correlation to the first order correlation g (1) (τ), [19] () (1) g ( τ ) = 1+ g ( τ ). (3.1.4) Knowing that ( ) ( 0) 1 g 1 =, we see that ( g ) ( 0) =. This equation does not work for a coherent light source, such as a laser beam, where ( ) ( 0) 1 g = as can be seen from the definition of g () (τ). Since radiating atoms emit their radiate independently of each other, the radiation acts as a chaotic light source. Since g () (τ) only observes photons radiated from the atoms, it is a non-invasive measurement. There is not just one radiating frequency but a range of frequencies because of Dopplerbroadening caused by the range of velocities of the atoms. The general relation is [19] g () D ( τ ) = 1 + exp( τ ), (3.1.5) where D is the Doppler-broadened full width at half-maximum height in the frequency 1 k ln B T D = = ωo. (3.1.6) τ Mc c ω o is the resonant frequency of the atom, M is the mass of the atom, T is the temperature, k B is the Boltzmann constant, and τ c is the coherence time. As one can see in Eq , the temperature of the trap can be found from g () (τ). Bali, et al. found that for clouds under ~10 5 atoms, with densities where radiation trapping should be insignificant, the temperature measurements from the intensity correlations (from Eq ) agreed with temperature measurements obtained by use of the time-of-flight (TOF) method [17]. These two measurements are compared because the TOF is a simple and very direct approach to measuring the temperature of the atoms, while the intensity correlations have many parameters that could affect the inferred temperature value. But, for clouds of more than ~10 5 atoms, it was found that the two temperature measurements no longer agree. For example, for atoms (detuning = 1.85Γ), the correlation method of measuring 8

14 temperature suggested 10 µk but the TOF method measured 50 µk. Bali, et al. speculated that perhaps radiation trapping could have caused this difference due to the increase in spectral width via radiation trapping. However, no attempt was made to include radiation trapping in the theory. 3. Optical Bloch Equations with Radiation Trapping for Moving Atoms We have constructed optical Bloch equations (OBE) for a two-level atom excited by a laser, including spontaneous emission and radiation trapping. First, we see that the number of collisions a photon encounters while moving through the cloud of density n and length l, is nl σ, where σ is the photon cross-section for absorption. This is also known as the optical depth [0]. Since in our case, the number of collisions is <<1, we can treat it as a probability that a photon is absorbed by an atom. If we multiply this by the rate of atoms decaying from the excited state, γρ ee, where γ is the decay rate and ee is the population in the excited state, we obtain the rate of absorption, R, of scattered photons by the atoms. R can also be described by the probability of available radiated photons, as γ nth [16], where we use n th to represent the probability of absorption of a fluorescent photon. equations for R equal, we can obtain the formula for n th The probability of the atom of being in excited state is found by [17] ρ ee n th 1 = 1 + I I ee Thus, by setting the two = nlσρ. (3..1) I / I o + o ( δ / γ ), (3..) where I is the intensity of all six trapping beams, δ is the detuning of the laser and I o is the saturation intensity which is 1.64 mw/cm for 85 Rb. For a trap with a density around atoms/cm 3 I, l 1 mm, =, σ= m, γ = (6.63 ns) 1 [17], and δ is one linewidth, I o we find the optical depth to range from 0.01~0., and n th from about 0.001~0.0. This treatment is only valid for small optical depths and n th <<1. Our starting point is the OBEs for a two-level atom. Including spontaneous emission and radiation trapping, they are 9

15 * ( Ωρ Ω ρ ) i ρ ee = ρ gg = γ ρ ee + nthγ + ge eg (3..3) γ iω ρ eg = ρ * ge = iδρ eg ( ρ ee ρ gg ). (3..4) th where we define γ ( n + 1) γ and γ is the population decay rate of the excited state 3 eg ω µ γ =. (3..5) 3 3πε c 0 Here, ω eg is the resonant frequency of the atom and µ is the dipole moment [17]. Knowing that ρ ρ = 1, we see that ρ ρ = 0. ρ ee and ρ gg are the atomic populations in the ee + gg ee + gg excited and ground states, respectively, and ρ eg and ρ ge relate to the coherence of the atomic dipole [17]. Thus, ρ eg and ρ ge decrease due to the incoherent process of radiation trapping. ρ ee decreases and ρ gg increases from the spontaneous emission process. Ω is the Rabi frequency, defined as where E o is the amplitude of the electric driving field and Eo Ω = e ( r ˆ eg ε ), (3..6) er eg is the induced dipole moment. Since the atoms are moving, we need to account for the velocity of the atoms into the Rabi frequency. For a moving atom with velocity v, we find that the OBEs become i ik vt * ik vt ( t) = γ ρ + n γ + ( Ωe ρ Ω e ρ ) ρ ee ( t) = ρ gg ee th ge eg (3..7) * γ ρ eg ( t) = ρ ge ( t) = iδρ eg iω e ik vt ( ρ ρ ) ee gg. (3..8) 3.3 Second-order Coherence for Moving Atoms with Radiation Trapping If we treat the field radiated by an atomic dipole as an operator, we have ˆ ( r, t) = K ( r ) ˆ σ ( t), (3.3.1) E + where ˆ σ ( t) are defined as usual as the atomic raising and lowering operators [0,1] and ± 10

16 K( r ) ω d r ( d r ) = 3 4πε oc r r, (3.3.) is the usual spatial dipole pattern at point r radiated by an electric dipole d at the origin, oscillating at ω, The spatial dependence cancels in Eq and we have g ˆ σ ( t) ˆ σ ( t + τ ) (1) + ss ( τ ) =, (3.3.3) ˆ σ ( t) ˆ σ ( t) where the subscript ss denotes the steady-state. ρ eg and ρ ge can be expressed as the expectation values of the raising and lowering operators, + ρ ss σ i t eg ˆ e ω and ρ ge ˆ σ i t e ω + [0]. However, as Eq shows, we need to calculate the two-time correlation function of the operators. By use of the quantum regression theorem, we can use the single-time expectation values to find the two-time correlation function for g () (τ) [1,]. The quantum O t + τ a τ O t, then = regression theorem states that if there is an operatoro, and ( ) ( ) ( ) O i = ( t) O( t + τ ) a ( τ ) O ( t) O ( t) j j i j. Or in words, the fluctuation (two-time correlation) j` j j regresses in the same manner [i.e. using the same factors a j ] as the mean. The quantum regression theorem can be used in certain conditions described in Refs. 1 and. In our σ +τ case, by solving the OBEs in Eqs and 3..8, we can write an equation for ( ) terms of σ ( t) and ( t) +. Then the quantum regression theorem allows us to premultiply the terms of the equation by ( t) σ σ + to obtain t in ˆ σ ( t) ˆ σ ( t + τ ) e + iωt = ρ ( t) e ee ( γ + iδ ) τ + iω ( γ + iδ ) * ρ eg ik v ( t+ τ ) ( t) e ( γ + iδ ) τ ( 1 e ) (3.3.4) where k v δ δ, assuming weak excitation. For the steady state solution, where t, we simply replace ρ ( t) and ρ ( t) by their steady state solutions. ee eg ss ρ and ρ can be found ss ee eg by making Eqs and 3..4 equal to zero. We find that ρ ρ ss eg iω = δ, where ( γ i )( 1 + s) I I s = o 1+ γ ( δ ) ss ee 1 I I = 1+ I I + o o ( δ γ ) and [17]. We find that Eq then becomes 11

17 γ γ Ω γ τ iω τ γ eg ik vτ iωτ ˆ + ( t) ˆ σ ( t + τ ) = n ss the e + e e. (3.3.5) σ γ 4δ + γ + Ω We also take the average over the Maxwell-Boltzmann distribution of velocities, e ikvτ v ikv m τ ( mv kbt ) k τ kbt m = e e dv = e to get πk B T τ iω egτ ss k τ k B T m iωt [ nthe e ee e e ] γ ρ,0 γ ˆ σ + ( t) ˆ σ ( t + τ ) = + (3.3.6) ss γ, 0 4 ee = Ω δ + γ + Ω. For this average, we replace k v with kv since the where ρ ( ) ss dominant force comes from the laser beams that counter the motion of the atom. Also, we assume that the detuning isδ δ, since kv is small. For molasses at a temperature of 50 K, k is m -1 and v is 0.1 m/s, where as the detuning, at one linewidth is 5.98 MHz. Using the two approximations of detuning and direction in Eq , we can find that g With Eq , we obtain our final form, g () γ τ / iω aτ ss k B Tk τ / m iwτ ( nthe e + ρ ee,0e e ) ( 1) ( τ ) = ss nth + ρ ee,0 ( τ ) = 1+ n 1 + ρ ss. (3.3.7) th ee,0 τ / τ τ τ [ ρ ρ δτ ] γ γ ss k B Tk m ss / k B Tk / m ee,0 e + nth e + nth ee,0e e cos. (3.3.8) Since n th is <<1, the n th term is negligible. We only looked at the atom in one position, but if we average over atomic positions over the whole cloud (assuming the cloud to be much more than one wavelength in size), the cosine term may cancel owing to the random atomic locations [3]. The only term that is left in the equation then, is the dominant ρ term, ss ee, 0 which does have n th in it. We plot Eq for three different values of n th in Fig As can be seen, as n th increases, the coherence decreases. The curve with n th =0.01, optical depth of 0.15, corresponds to a trap with a density of atoms/cm 3 and the curve with n th =0.0, optical depth of 0.3, corresponds to a trap with a density of 10 9 atoms/cm 3. From this, we expect to see effects of radiation trapping at traps with low optical densities, more than an order of magnitude less than in Refs. 13,14. Figures 3.3. and plot the three terms separately. 1

18 Fig Theoretical plot of g () (τ) for three different values of n th. The inset shows the effect of the cosine and n th terms. 13

19 Fig Theoretical plot of ( ) ( τ ) ss g with only the dominant term with ρ for three different values of n th. ee, 0 Fig The cosine and n th terms for a value of n th =0.0. As one can see, these two terms have a much smaller effect than the dominant term shown in Fig

20 4. The Apparatus By using an external cavity, we can tune the frequency of a laser diode to match the resonant frequency of atomic rubidium. The tuned laser light is red-detuned and sent in three orthogonal directions into a rubidium-filled vacuum chamber. With the addition of a magnetic field gradient, we can cool atoms and create a magneto-optical trap (MOT). 4.1 Atomic System The alkali metals are popular candidates for laser cooling and trapping because the excitation frequency of the transition between the ground state and the first excited state is in the visible region [17]. Because it is easy to obtain inexpensive infrared laser diodes, we choose rubidium, whose frequency is in the infrared. There are two natural isotopes, 85 Rb and 87 Rb, but we choose to trap 85 Rb because it is more abundant (7%). In the ground state, the valence electron in atomic 85 Rb lies in the 5s state, where 5 is the principal quantum number which describes the energy level and size, and s describes the orbital angular momentum of the electron L, which in this case, L = 0. When the atom interacts with the light, the electron jumps to the excited state, 5p, where L = 1. The interaction between the spin of the electron S and the angular momentum L causes a splitting, called the fine splitting, of the 5p state, into 5p 1/ and 5p 3/. The half-integer number is the total angular momentum J of the electron, where J = L + S. Because for the 5s state L = 0, the level does not split and is simply 5s 1/. If the interaction of the angular momentum of the electron and the nuclear spin is considered, an even smaller splitting, called the hyperfine splitting, of the energy levels occurs. For 85 Rb, the magnitude of the nuclear spin I, is 5/. Thus, the different levels are described by F = J + I, the total atomic angular momentum, where F ranges from J I to J + I. 15

21 The trapping transition desired is the one from the 5s 1/ level to the 5p 3/ state. When excited, an atom obeys the dipole transition selection rule, where F = 0, ± 1. We choose to keep the trapping laser frequency at the F=3 F =4 transition. However, as can be seen in Fig and by the selection rule, the atom can also be off-resonantly excited to the F = and 3 levels as well. Then when the atom decays from either the F = or 3 state, it may fall back into the F= level, which will not be in resonance with the trapping laser and within a few cycles, each lasting the upper state lifetime of 7 ns, the atom quickly becomes unaffected by the laser. To prevent this, a repumping laser is added, which is tuned to one of the F= F = or 3 transitions. Then when the atom decays again, it can fall into either of the two ground states which are both being excited. The atomic spectrum of the two transitions can be seen in Fig Thus, the atom will continue to stay in resonance with the trapping laser, as well as the repumping laser. Both lasers are also slightly red-detuned to account for the Doppler effect of the moving atoms, as is explained in Ch.. 16

22 Fig The Hyperfine Levels of Rubidum 17

23 F= F F=1 F 85 Rb 87 Rb F=3 F F= F Fig The two transitions for both isotopes of rubidium. 4. Vacuum Chamber The chamber is a steel sphere with 7 pairs of windows protruding from the body. Three are in orthogonal positions, used for the trapping beams to enter and leave the chamber. Three different pump systems are used to bring down and maintain the pressure within the chamber to approximately 10-9 Torr. The first pump is the roughing pump, which mechanically pumps the chamber from atmospheric pressure to the millitorr range. The second is a turbo pump, which mechanically pumps the pressure down to near 10-7 Torr. The final pump is the ion pump, which ionizes the gas and magnetically pulls out ions, allowing us to reach 10-9 Torr. After the chamber is pumped down to low pressures, rubidium is introduced through a leak valve attached to the vacuum chamber. The valve attaches to a short flexible vacuum tube which contains a glass vial of rubidium. After breaking the vial, the tube is heated and the 18

24 valve opened to allow the rubidium to enter the chamber. After many days of filling and heating, the rubidium had saturated the walls of the chamber enough so that we did not have to continuously fill the chamber to get a decent sized trap volume. 4.3 Frequency Tuned Laser Diode External Cavity Diode Laser To use laser light to trap atoms, the atoms must be able to interact with the laser light, which happens when the laser is on resonance with the atom. Because there are no laser diodes made specifically to match the frequency of rubidium, we create an external cavity around a laser diode to tune it to the exact frequency we desire. The external cavity setup is set on an aluminum base plate, Fig , mounted on Sorbothane. Fig Baseplate 19

25 The Sorbothane absorbs and reduces any vibrations that could destabilize the external cavity system. This is all placed within an aluminum box to reduce any sudden thermal fluctuations and air currents from the outside environment since we also temperature tune the laser to assist in frequency tuning. The walls are held together with screws and the bottom of the box protrudes on two sides so that we can secure the box to the table with clamps. However, the laser light must exit the box so holes are made in the box. Also, we need to be able to adjust mounts in the cavity system, so holes are made in the walls of the box that allow in hex-key screwdrivers. To minimize any external influences, tape is placed over the adjustment holes and opened only when needed. The laser light exit holes are left open to prevent any reduction in power. Inside the box, we have the external cavity laser system. A laser diode is placed in aluminum housing, Fig , so that the longer axis of the elliptical beam is parallel to the table Drill and Tap Ø Ø Ø Drill and Tap Fig Laser mount and Teflon spacer A driver 1 provides current to the laser diode, which has a safety feature that allows us to set a current limit so that we do not destroy the diode. A Teflon spacer, Figure , is set between the housing and the aluminum base plate to thermally isolate the laser diode mount from the rest of the setup. 0

26 Ridges are made on the bottom of the Teflon spacer and diode mount so that there is a better grip onto the surface below. Nylon screws secure the diode mount to the base plate to prevent any heat conduction. A thermoelectric heater/cooler is secured to the housing with adhesive 3 to thermally stabilize the laser diode. By applying a current across the leads, we can see that the side with the leads is the active side. A heat sink is attached to the inactive side of the heater/cooler to rapidly dissipate the heat. A thermistor 4 is semi-secured with heat sink paste in a hole drilled as close as possible to the laser diode to monitor the temperature. The heater/cooler and thermistor allow us to set and control the temperature of the laser diode with a controller 5. The controller has a safety aspect that sets limits on the current so that we do not destroy the heater/cooler by trying to overwork it. After the laser diode, a collimating lens 6 is critically placed to collimate the diverging beam, Fig Fig Collimating lens mount The lens is achromatic, with a wavelength range of nm, where our wavelength of 780 nm falls clearly within the range. The center of the lens should be at the same height as the center of the laser diode to ensure the beam is at constant height. A micrometer screw finely adjusts the spacing between the lens and laser diode. The lens should be placed so that the laser beam has a constant size and shape over a distance of a few meters. Next, a beamsplitter 7 is glued onto a 1 cm 3 aluminum cube and secured onto the baseplate by a screw. It is used to transmit half of the collimated light and reflect the other half, which will later be used as the main trapping beam. 1

27 The last element in the external cavity of the laser diode is a diffraction grating 8, glued onto the side of a mount 9. Placing a small amount on the corners of the grating should secure it in place (but still allow us the possibility to remove it with some ease if we need to). The transmitted beam from the beamsplitter cube above hits the diffraction grating, which disperses different wavelengths into different angles, except for the zero order. We can adjust the diffraction grating to a certain angle by adjusting the grating mount and have the desired wavelength go back through the beamsplitter into the laser diode. This allows us to selectively increase the power of the desired wavelength in our laser diode so that that wavelength will predominantly constitute the trapping beam. Two piezo-electric disks 10 soldered back to back are placed behind the diffraction grating to allow us to even more finely adjust the angle of the grating. These piezos are ones normally found in buzzers. Plastic spacers cut from transparency sheets are placed between the piezo and the mount so that there is no electrical conduction between the two. Grease is applied on the plastic that is in contact with the ball of the micrometer screw so that the screw turns smoothly. By applying a current from a function generator, we can cause the piezo to flex and thus move the diffraction grating by very small amounts. Because the disks protrude below the mount, a deep groove has to be made in the baseplate so that the diffraction grating mount can move. The final setup is shown in Fig In order for us to have frequency tuning, we have to ensure that the light from the diffraction grating must be aligned so that it goes back into the laser diode. When we look at the light reflected out of the beamsplitter, we see two spots. One spot is the beam reflected from the diffraction grating. By adjusting the angle of the diffraction grating mount, we can move one of the spots closer to the other which means the light is returning directly back into the laser diode. When near each other, the moving spot jumps into the stationary one and remains there even while the angle of the mount is still being adjusted. This situation exists only for a small range of angle adjustments, after which the two spots separate again. This short angular range represents the external cavity adjustments when we are tuning the frequency of the laser diode. For the trapping laser, we use a Sanyo laser diode 11. This diode is on resonance when set at C with a current of A, which is the typical operating current. The repumping laser is a Sharp laser diode 1. This diode is on resonance at 6.1 C and has a

28 typical operating current of A. Because everything drifts from day to day, we need to adjust the mounts by a method called threshold tuning so that we ensure maximum feedback. First, the current is brought down to the threshold current of the laser, or when the laser no longer lases. When the diffraction grating is horizontally and vertically aligned and the collimation optimized, the laser will begin to lase at currents lower than the threshold current. By bringing down the current in small increments, we can adjust the mount until the laser is barely lasing. This is the new threshold and the laser is optimized for wavelength selection. For example, the Sanyo laser threshold current is typically 30 ma, but with feedback, the new threshold current is 6 ma. For the Sharp, the typical threshold current is ma, and the new threshold current is 40 ma. We have currently found a new high power, cheap Sharp laser diode 13 that may be used in future applications. 3

29 Fig External Cavity Diode Laser 4

30 4.3. Saturated Absorption To confirm that the laser diode is frequency tunable about the resonant frequency of rubidium, it is placed in a saturated absorption setup, Fig Fig Saturated absorption setup. 5

31 A mirror is placed after the beamsplitter to steer the reflected beam out of the box. A linear polarizer is used to control the intensity of the beam to prevent power broadening, so we can get more distinct peaks. Figure a. Power broadened peaks. b. non-power broadened peaks. Then the beam encounters a thick glass plate 14. About 4% of the light is reflected on each surface, giving us two weak reflected beams (separated by about cm) and one strong transmitted beam. The two weak beams are reflected into a rubidium vapor cell 15 and then reflected off a glass microscope slide into two photodiodes 16. As the diffraction grating is ramped by the piezo, the laser is ramped through a small range of frequencies. When the beams are on resonance, they are absorbed by the rubidium atoms. Since the atoms are moving inside the cell, the different velocities Doppler shift them into and out of resonance with the beams. Thus, the signal into the photodiodes is a Doppler broadened absorption of the weak beams, as shown in Fig

32 Fig Doppler broadened signal. The strong transmitted beam is reflected off two mirrors 17 and set to coincide with one of the two weak beams, making sure that it is not reflected into the photodiodes. The strong and the weak beam counter-propagate, so that atoms Doppler shifted into resonance with the weak beam are further Doppler shifted out of resonance with the strong beam. But for atoms moving with a velocity perpendicular to beam propagation, both beams are on resonance with the atom. When on resonance, the strong beam causes the atoms to be mostly in the upper state due to saturation. The weak beam is too weak to cause saturation and thus a Doppler-free saturation dip, also known as a Bennet hole, is burned into its absorption profile [4]. By subtracting the other weak beam s Doppler broadened profile, a burned spectrum with no Doppler effects is obtained, as is demonstrated in Fig

33 Fig Removing Doppler effect in signal via saturated absorption. In reality, there are six Bennet holes, but for simplicity, only one is shown. As the trapping laser is ramped, it is on resonance with the transitions from the F=3 F =, 3, 4 states. Hence we expect to see three peaks, not six, corresponding to these transitions. The extra three peaks are called cross-over peaks. They appear because it is possible for an atom to be Doppler-shifted into resonance to a certain transition 1 (say F=3 F =3), such that ω = ω 1 δ, with the weak beam and simultaneously be Doppler-shifted into resonance to a different transition (say F=3 F =), such that ω = ω + δ, with the counterpropagating strong beam, Fig Fig The atom is blue Doppler shifted into resonance with the weak beam, = 1 -, and red Doppler shifted into resonance with the strong beam, = +, where the detuning k v δ =. Again, the strong beam saturates, and a hole is burned in the Doppler spectrum at a frequency, ( ) ω which can easily be seen to be at the arithmetic mean of the two transitions, 1 + ω ω =. 8

34 For the three transitions, there are three corresponding crossover peaks at the arithmetic means, thus resulting in six peaks in the spectrum, as is clearly seen in Fig F =3~4 F =~4 F =4 F =3 F =~3 F = Figure Trapping saturated absorption spectrum. The peaks with two numbers equal to F are the crossover peaks, whilst the atomic transitions have only one. The stable Doppler-free saturated absorption spectrum confirms that our laser can be tuned to the trapping frequency of rubidium. By reducing the ramp on the piezo, we can selectively choose a frequency that we want to use. Knowing that we want to trap on a frequency a few linewidths red-detuned from the F=3 F =4 transition, we can lock onto that specific frequency. A locking circuit and an AOM are used to lock onto the desired trapping frequency Locking Mechanisms Using a locking circuit [5], we can lock onto the side of any desired peak of the spectrum, but not at an arbitrary position in the hyperfine spectrum. This is a problem because to trap moving atoms, the trapping laser must be red-detuned, typically between 1 4 linewidths. Thus, an Acousto-Optical Modulator 19 (AOM), consisting of an RF wave propagating through a crystal, is used. Applying an RF voltage on the piezoelectric transducer inside the AOM, an acoustic wave is sent into the medium (PbMoO 4 ). The acoustic wave perturbs the refractive index of the medium, which results in an optical diffraction grating. When the beam enters the wave front at the Bragg angle, the intensity of the first order is maximized. 9

35 The first order frequency is also shifted because of the sound wave. It can be thought of as a Doppler shift as the wave encounters a surface moving at the velocity of the sound wave [6]. Our AOM can shift the frequency by MHz. The crossover peak, F=3 F =3~4, next to the trapping transition, F=3 F =4, is 60.3 MHz away, and the next crossover peak, F=3 F =~4, is 9 MHz away, Fig Fig The six peaks are schematically drawn separately for clarity with their corresponding detunings, with respect to the trapping peak. Because we want to red-detune the laser and the first crossover peak is at the minimum modulation of the AOM, we find it convenient to lock onto the F=3 F =~4 transition. Then referring to the AOM data sheet, we can adjust the voltage so that we are the detuned from resonance by a desired amount. 30

36 4.4 The Trapping Setup The trapping laser light must undergo a few changes before it is ready to be used to trap atoms. There are four major elements in the trapping beam shaping setup: the anamorphic prism pair 19, the optical isolator 0, the AOM, and the telescope. As mentioned earlier, the light from the laser diode is elliptical. Thus, an anamorphic prism pair is used to circularize the beam by reducing one side of the beam while unchanging the other. Figure shows how the prism pairs are aligned. For different ellipticities, the angles of the prism pairs differ as the figure shows. Fig The alignment of the anamorphic prism pair. The elliptical beam enters in the right and leaves circular from the left. For a :1 ratio, the values for A, B, C, D are 1., 6.0, 5.1mm, and 5.3 mm, respectively. For a 3:1 ratio, the values are 30.4, 0.1, 6.4 mm, and 6.4 mm, respectively. The power of the beam, measured at an aperture set after the anamorphic prism pair, is found to decrease by 10%. Next, the circular beam is sent into an optical isolator. The optical isolator is used to prevent backreflections from feeding back into the laser system and destabilizing the laser. The isolator contains a Faraday rotator, which consists of a magneto-optically active cylindrical crystal mounted inside a concentric permanent ring magnet, and two polarizers on the two ends. The plane of polarization of the input polarizer is set parallel to the linear polarization of the laser beam. Next, the Faraday isolator rotates the polarization of the light by 45 in a direction determined by only the direction of the co-axial magnetic field, regardless of the propagation direction of the beam [6]. The plane of polarization of the 31

37 output polarizer is then set parallel to the rotated laser beam. The power of the beam, measured at an aperture set after the isolator, is found to decrease by 15%. If light reflects back into the optical isolator, the polarization of the beam encounters an additional rotation of 45 in the same direction as before and will be perpendicular to the input polarizer. Consequently, no light continues back into the laser system. Next, the light is focused into the AOM, described earlier, so that we can modulate the frequency and precisely detune the light by 1-4 linewidths. A lens is used to focus the light into the diffraction grating in the crystal formed by the RF-generated acoustic wave. The beam size needs to stay constant through the length of the crystal so that it encounters the same number of slits of the diffraction grating evenly. This can be done by making the confocal parameter, or the axial distance for which the beam radius lies within a factor of its minimum, to be much greater than the length of the crystal inside the AOM. The confocal parameter z o, can be found by combining the following equations and Wo λ =, πθ o W o zo =, 4.4. θ o where W o is the beam waist radius, λ is the wavelength of the beam, and θ o is the angle of the light [4]. θ o is found by 1 W tan where W is the initial beam size radius and f is the focal f length of the lens. For a 3mm diameter size beam and a lens with a focal length of 0 cm, the confocal parameter is found to be only 0.88 cm, whereas a lens with a focal length of 30 cm has a confocal parameter of cm. Since the crystal in our AOM is 0.8cm along the beam path, a lens with at least a focal length of 30 cm must be used. To conserve tabletop space, a 30 cm lens is used instead of a lens with a greater focal length. Another 30 cm lens is set focal length away after the AOM to bring the beam back to its original size. Also, a λ/ 1 plate is placed before the AOM to ensure that the polarization of the beam is horizontal, which is preferred by the AOM, i.e. diffraction efficiency is maximized. An aperture is used to select and transmit only the first order beam for the trap. After optimization of power, the beam is reduced by 75% through the AOM. Because the beam is so small in diameter, a 3

38 telescope is used to magnify the beam. A 0cm lens is placed focal length after a microscope objective of 10X magnification. Using an aperture, we cut the beam to a diameter of 15mm. Finally, the total power of the beam has been reduced to 5% of the total initial power out of the laser. In our case, we get a trapping beam of 7-8 mw from an initial laser beam of 30 mw. A λ/ plate is placed before a polarizing beamsplitter so that we can adjust the beam that 0% of the light goes into the x-direction and 80% of the light goes into the y and z- directions. A 50/50 beamsplitter 3 splits the stronger beam equally between the y and z- directions. Finally, before the three orthogonal beams enter the vacuum chamber, they are circularly polarized by λ/4 plates 4, shown in Fig σ - σ - σ + σ - x σ + σ + z y Fig The polarization of the light is determined by the gradient caused by the magnetic coils. As can been seen in the figure, the light in the y and z-directions are polarized opposite of the light in the x-direction. This is because the gradient in the y and z-directions are in the opposite direction of the x-direction. Maxwell s equations give us B = 0 and that shows B B B us that = +, which explains the flip in sign of the gradient in the y and z- x y z directions. 33

39 σ + B σ - x σ - B σ + y, z Fig The gradient is opposite in the x-direction as the y and z-directions. Thus, the polarizations of the incoming laser beams must also be different according to the field. By aligning the beams to a mark on a plastic disk, we center each beam so that the trap will form in the center of the chamber. On the other side of the chamber, another λ/4 plate, in an arbitrary orientation, and a mirror are placed to reflect the beams straight back into the chamber with opposite polarizations. The retro-reflection alignment is done by aligning the reflection back onto the mirrors that were before the chamber. Thus, the trap has six orthogonal circularly polarized beams coinciding at the center of the chamber. The repumping laser setup is not as complex as the trapping laser. An anamorphic prism pair circularizes the beam like the trapping beam. We do not have an optical isolator, but since there are not many components in the setup, each lens and mirror are adjusted so that the backreflections are misaligned as to not enter the laser system. Two lenses are used to increase the bean size, modulated by an aperture. The beam is sent into the same polarizing beamsplitter as the trapping beam to ensure that they coincide. The proportional intensity is not crucial for the repumping as with the trapping since it only needs to be in the chamber, so polarizations and powers are not adjusted. 34

40 4.5 Magnetic Coils Magnetic Field Gradient Coils The lasers act as velocity-damping forces in all six directions and cool the atoms. However, since they do not exert position-dependent forces, they are not enough to create an atom trap. We rely on a magnetic gradient to gather the atoms in one central spot where the magnetic field is zero. By putting two coils in an anti-helmholtz configuration as in Fig , we can create the gradient needed. The coils are parallel to each other, but have currents running in opposite directions. Although the Helmholtz configuration has the distance between the coils the same as the radius, due to spatial restrictions, the radius is smaller. This will give us a nonlinear gradient, but for a centimeter area in the center of the chamber where the trap resides, the gradient is close to being linear. The gradient perpendicular to the axis of the coils is half the magnitude of the gradient along the axis. R x A Fig The magnetic gradient coils are set up in an anti-helmholtz configuration. We found the magnetic field for a point x in Eq.... To find the gradient, the derivative of the field is taken and we obtain Eq B 3µ 0NIR A + x A x = + x 4 ( R + ( A + x) ) 5 R + ( A x) ( ) 5 ( ) 35

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