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NEW STUDIES OF OPTICAL PUMPING, SPIN RESONANCES, AND SPIN EXCHANGE IN MIXTURES OF INERT GASES AND ALKALI-METAL VAPORS Yuan-Yu Jau A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS JANUARY, 2005 c Copyright by Yuan-Yu Jau, 2005. All rights reserved. ° Abstract In this thesis, we present new studies of alkali-hyperfine resonances, new optical pumping of alkali-metal atoms, and the new measurements of binary spin-exchange cross-section between alkali-metal atoms and xenon atoms. We report a large light narrowing effect of the hyperfine end-resonance signals, which was predicted from our theory and observed in our experiments. By increasing the intensity of the circularly polarized pumping beam, alkali-metal atoms are optically pumped into a state of static polarization, and trapped into the hyperfine end-state. Spin exchange between alkali-metal atoms has minimal effect on the end-resonance of the highly spinpolarized atoms. This new result will possibly benefit the design of atomic clocks and magnetometer. We also studied the pressure dependence of the atomic-clock resonance linewidth and pointed out that the linewidth was overestimated by people in the community of atomic clock. Next, we present a series study of coherent population trapping (CPT), which is a promising technique with the same or better performance compared to the traditional microwave spectroscopy. For miniature atomic clocks, CPT method is thought to be particularly advantages. From our studies, we invented a new optical-pumping method, push-pull optical pumping, which can pump atoms into nearly pure 0-0 superposition state, the superposition state of the two ground-state hyperfine sublevels with azimuthal quantum number m = 0. We believe this new invention will bring a big advantage to CPT frequency standards, the quantum state preparation for cold atoms or hot vapor, etc. We also investigated the pressure dependence of CPT excitation and the line shape of the CPT resonance theoiii retically and experimentally. These two properties are important for CPT applications. A theoretical study of “photon cost” of optical pumping is also presented. Finally, we switch our attention to the problem of spin exchange between alkali-metal atoms and xenon gas. This mechanism is very important to the spin-exchange optical pumping. We report the first measurements of binary spin-exchange rate at high magnetic field (9.4 tesla). We present the first calculation of the magnetic decoupling curves of the binary spin-exchange rates by using two different methods, semi-classical approach (SCA) and distorted-wave Born approximation (DWBA). iv Acknowledgements Life is a road of learning. Sincerely, I would like to thank my advisor, William Happer, a brilliant and respected scientist. With his help, I learned more physics and the attitude of study. Because of his knowledge and intuition, my sight of physics became wider. It is a great privilege to study physics with him. I want to thank Nick Kuzma, our postdoc, for helping me on writing paper, doing experiments, and generously providing his assistance on everything. Some of his interesting stories also delighted me during some dry hours in the office. I am very grateful to Eli Miron, a scientist from Israel. He joined our lab during his sabbatical. His was a very kind, responsible, and work-hard person. Because of him, many experiments can be carried out. He cheered people up and animated everything in the lab. I was very fortunate to work with him. I am grateful to Dan Walter and Warren Griffith for their company of my first two years in Princeton. With their help, I was able to catch up on the lab works and get use to the new environment. It is my fortune to join the atomic physics group with such good people as Mike Romalis, Brain Patton, Amber Post, Andrei Baranga, Tom Kornack, Micah Ledbetter, Igor Savukov, and back to the early time, Ioannis Kominis and Kumar Raman. I learned lots of things from them, and they made lab an interesting place to work. I especially appreciate Mike Romalis, a knowledgeable person, who shared great knowledge and experience with me. The staff of the Physics Department have been unfailing helpful. I would like to thank Laurel Lerner, Mary De Lorenzo, Claude Champagne, John Washington, Kathy Warren, v Ellen Webster, Mike Peloso, Helen Ju, and Angela Qualls. They constantly helped whenever I encountered problems in our department. I am also indebted to Charles Sule of Physics Department and Mike Souza of the Chemistry Department. Without their expertise, many experiments this thesis can not be carried out. It has been very nice to correspond with Jacque Vanier, an expert of atomic clocks. He is a professor in Montreal University in Canada. I learned many new things through the discussions with him. Finally, I want to thank my parents. Because of them, I entered college and then obtained Ph.D. education in Princeton University. vi Contents Abstract iii Acknowledgements v Contents vii List of Figures x List of Tables xx 1 Introduction 1 1.1 Advantage of Alkali-Metal Atoms . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Content Brief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Strong Resonances of Alkali Hyperfine Sublevels from Optical Pumping 2.1 2.2 Density Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9 2.1.1 Evolution in Liouville Space . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Ground-State Relaxation . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Excited-State Relaxation . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.4 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.5 Microwave and RF Fields . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Microwave Spectroscopy and CPT Spectroscopy vii . . . . . . . . . . . . . . . 23 2.3 2.4 2.5 2.2.1 Coherence Induced by Oscillating Fields . . . . . . . . . . . . . . . . 23 2.2.2 Optically Induced Coherence . . . . . . . . . . . . . . . . . . . . . . 24 2.2.3 Inhomogeneous Linewidth Broadening . . . . . . . . . . . . . . . . . 28 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1 Microwave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.2 CPT Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1 Light Narrowing Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.2 Transient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4.3 Pressure Broadening of Microwave Resonance . . . . . . . . . . . . . 47 2.4.4 Push-Pull Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . 51 2.4.5 Pressure Dependent CPT Excitation . . . . . . . . . . . . . . . . . . 61 2.4.6 Line Shape of CPT Resonance . . . . . . . . . . . . . . . . . . . . . 65 2.4.7 Photon Cost of Optical Pumping . . . . . . . . . . . . . . . . . . . . 72 New Simple-Compact Frequency Standard . . . . . . . . . . . . . . . . . . . 83 3 High-Field 3.1 3.2 3.3 3.4 129 Xe 129 Xe-Alkali-Metal Spin Exchange 88 Spin Relaxation Rates Equation . . . . . . . . . . . . . . . . . . . . . 89 3.1.1 Detailed Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.1.2 Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Magnetic Decoupling of 129 Xe-Alkali-metal Spin Exchange . . . . . . . . . . 93 3.2.1 Semi-Classical Approach (SCA) . . . . . . . . . . . . . . . . . . . . . 93 3.2.2 Distorted-Wave Born Approximation (DWBA) . . . . . . . . . . . . 95 Experiments and Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.1 NMR Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3.2 Faraday Rotation Measurements . . . . . . . . . . . . . . . . . . . . 105 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.4.1 Binary Spin-Exchange Rate Coefficients . . . . . . . . . . . . . . . . 111 3.4.2 High-Field Contribution of van der Waals Molecules . . . . . . . . . 116 viii A Evolution of the Density Matrix of Alkali-Metal Atoms 119 A.1 Ground-State Relaxation Due to Weak Collisions . . . . . . . . . . . . . . . 120 A.1.1 S-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A.1.2 Carver Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A.2 Ground-State Relaxation Due to Strong Collisions and Other Mechanism . 123 A.2.1 Spin Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.2.2 Spin Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.3 Excited-State Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.3.1 J-Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.3.2 Spontaneous Decay and Quenching . . . . . . . . . . . . . . . . . . . 126 A.4 Microwave and RF Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.5 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.5.1 D1 Depopulation Pumping . . . . . . . . . . . . . . . . . . . . . . . 131 A.5.2 D1 Repopulation Pumping . . . . . . . . . . . . . . . . . . . . . . . 134 B Circuit Diagram of Signal Divider 137 References 138 ix List of Figures 2.1 In our modelling, we calculated the ensemble behaviors of the alkali-hyperfine sublevels with D1 pumping and relaxation mechanisms, such as S-damping, spin exchange, Carver damping, diffusion, spontaneous decay, quenching, and J-damping. 2.2 12 The population of the sublevel b is pumped out. When a oscillating field turns on at the resonant frequency, the coherence is generated. Some population transfers from the sublevel a to b, and therefore it increases the light absorption or decrease the light transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 23 Two monochromatic fields with frequencies Ω1 and Ω2 couple between two groundstate sublevels and the excited state. Two optical coherences between the ground state and the excited state are generated. When the difference of two optical frequencies is equal to ground-state sublevel splitting, a strong ground-state coherence can be produced. Under the same condition, photons begin scattering between two monochromatic fields, and are less absorbed by atoms. The light absorption is therefore reduced and observed as a transmission peak. When the optical detuning is present, it is equivalent to stimulated Raman scattering. x . . . . . . . . . . . . 25 2.4 We use a hypothetic atom, which has a nuclear spin I = 1/2, to illustrate the CPT coherence in the picture of spin oscillation. Different Λ schemes generate different ground-state coherences. The coherence of 0-0 CPT is equivalent to the electron spin oscillating along the z-direction. The end-state coherence is equal to spin precessing on the x-y plane. The dotted circles represent the dark states for different CPT schemes. The 0-0 CPT excitations from σ+ and σ− pumpings have 180◦ phase difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . . . . . . . . . . . . . . . . . . . . . 30 . . . . . . . . . . . . . . . . . 31 2.5 Lollipop (left) and rectangle (right) cells. 2.6 Experimental setup for microwave spectroscopy. 2.7 The noise spectrum of the transmission light in the experiments. Our homemade divider circuit can suppress the background noise level by 15dB, and can increase the SNR by a factor of 6 ∼ 7. This divider circuit was especially useful when we used the ring laser as the pumping source. Because the laser was conducted by a long fiber (20 m), lots of mechanical vibration noise was converted into the intensity fluctuations of the output light. 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The top panel shows the transient of the transmission light when the microwave field was on and off. The second panel is the zoom-in of the circle on the top panel. The third one is the Fourier’s spectrum of the damping oscillation. The bottom one shows the same resonance measured by scanning the microwave detuning (two sec for each point). Its linewidth is broader due to the microwave power broadening. . 2.9 34 The probing beam for CPT experiments was generated by a Mach-Zenhder Modulator. By modulating the effective light propagation lengths from two crystals, the output beam can be intensity modulated. The optical sidebands of the probing beam can be measured from a Fabry-Perot spectrometer. . . . . . . . . . . . . . xi 35 2.10 Two example CPT signals were found by setting the central frequencies of the modulation to their resonant frequencies. The scanning range was ±50 kHz. The ripple tails were due to the fast scanning, which can be eliminate by slowing down the sweeping speed or shorten the scanning range. The end-resonance signal was obtained by setting the probing beam with 45◦ to the magnetic field. . . . . . . . . 36 2.11 The light narrowing effect is shown by two different end resonances. The linewidth gets narrower by increasing pumping power at beginning, because the optical pumping traps more atoms into the end state. The minimum point is where the spinexchange broadening is comparable to the pumping broadening. The linewidths start increasing after the minimum point, because the optical pumping start dominating the linewidth broadening. However, the circularly polarized pumping can only make the linewidth of 0-0 resonance worse. . . . . . . . . . . . . . . . . . . 2.12 The first light narrowing data of 87 40 Rb hyperfine end-resonance was found from the high pressure lollipop cell. The data agrees with the theory very well. Two insets show the resonance signals from the detuning scan. The scanning step was 400 Hz. The linewidth decreases with increasing pumping power to a minimum value. After this point, the linewidth begins to be dominated by the pumping rate. However, the signal amplitude is always improved by the pumping power. . . . . . . . . . . . . 41 2.13 The top panel shows the calculated transient signal with frequency detuing = 7 kHz, and its multi-exponential background has been subtracted. The middle and bottom panels show the computational results of the complex damping rate at different frequency detunings by solving the evolution equation and using Eq. (2.59). The unit in their vertical axes is 1000/sec. The calculations show the distinction between two different methods is only a few parts per thousand. . . . . . . . . . . . . . . 2.14 An experimental data of the end-transient damping rates from a 87 44 Rb cell with 1 atm N2 buffer gas. The data was fit into Eq. (2.59). The Rabi frequency can be extracted from the fitting of the imaginary decay rates. . . . . . . . . . . . . . . xii 46 2.15 The experimental data of microwave and Zeeman resonance damping rates as the number density of nitrogen. Our experimental data are consistent with D.K. Walter’s measurements, which is plotted as two shadowed areas. The hyperfine resonance data is affected by both S-damping rate and Carver rate. The Zeeman resonance data is only affected by S-damping rate. J. Vanier’s data predicts a much larger values, which is plotted as a dark-gray curves on the top left corner. 2.16 Absolute hyperfine frequency shifts of 85 . . . . . . . 49 Rb induced by the buffer gases at a con- stant pressure PB = 760 torr, inferred from the data of Bean and Lambert [7]. Also shown are the spreads in frequency ∆νp that would be caused by a spread in temperature ∆T = 20 ◦ C centered at 30 ◦ C. The line broadening due to temperature inhomogeneities in Ar is about ten times less than that in He, Ne, or N2 . . . . . . 50 2.17 The oscillation of the electron spin produces time dependent absorption cross sections for different circularly polarized pumping lights. The interlacing σ+ and σ− light pulses repeated every 2π/ω00 can maximize the spin oscillation, and therefore trap the atoms to the 0-0 superposition state. At this point, the transmission light has the maximum transparency. . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.18 Alkali hyperfine sublevels with a hypothetical nuclear spin quantum number I = 1/2. 0-0 state is the only dark state by using a complementary Λ scheme, which has two Λ pumpings with a 180◦ modulation phase difference between each other. Atoms will be eventually trapped into the 0-0 state, the dark state. . . . . . . . . . . . . 53 2.19 The optical sidebands of a pulsed light is like a comb in the spectrum. With high buffer-gas pressure, atoms see all optical sidebands. With low buffer-gas pressure, only two sidebands are seen by atoms. . . . . . . . . . . . . . . . . . . . . . . . 54 2.20 The Michelson type interferometer converts an intensity modulated laser beam to a polarization modulated laser beam. A λ/4 wave plate was needed in one of the arms. The difference of the arms’ length is equal to c/(4ν00 ), where ν00 is the 0-0 resonance frequency. The details of other parts can be found in Fig. 2.6 and Fig. 2.9. 55 xiii 2.21 Comparison of the signal contrasts of conventional CPT and push-pull CPT. The inset shows the modulation spectrum from a Fabry-Perot spectrometer. At the same pumping rate, the CPT signal from push-pull pumping has a better signal contrast, which is about 77 times larger than the conventional CPT signal. The conventional CPT signal on the top panel was taken with 16 times more averages than the pushpull CPT for a better SNR. This experimental data was taken from a spherical cell with diameter = 1.9 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.22 Experimental data and theoretical curves from our density-matrix modelling agree very well to each other. The conventional CPT has an optimized value of pumping rate for maximizing the signal contrast. Higher pumping rate degrades signal of the conventional CPT due to the population trapped to the end state. However, push-pull pumping has no such problem. The right diagram shows a very strong CPT signal by using push-pull pumping of one circled data point on the left diagram. 57 2.23 The relative linewidth contributed from spin-exchange broadening. As the prediction from Eq. (2.49), push-pull optical pumping can reduce the linewidth broadening due the spin-exchange collisions, but the optical pumping used in conventional CPT can not. However, in push-pull pumping, the decrease of the spin-exchange broadening can not compensate the linewidth broadening due to the optical pumping. This tiny light narrowing effect is hardly to be observed. . . . . . . . . . . . . . . . . . . . 58 2.24 The FOMs of the plots in Fig. 2.22. Push-pull CPT has larger FOM than conventional CPT. Here, FOM=contrast(%)/linewidth(kHz)×10. . . . . . . . . . . . . . 60 2.25 By changing the ratios of pumping intensities of σ+, σ−, I1 , and I2 with a selected modulation frequency, we can put lots of population into some hyperfine sublevels. The top-left panel shows 50% on |2, 0i and 50% on |1, 0i states. The top-right panel shows 99% on |2, 0i state. The bottom-left panel shows 95% on |1, 1i state. The bottom-right panel shows 100% on |2, 2i state. . . . . . . . . . . . . . . . . . . . xiv 61 2.26 The amplitude of the optical coherence ρge is like a FM signal passing through a resonator filter g(Ω) = 1 i(Ωge −Ω)+γop , where Ω is the optical angular frequency. Fre- quencies of the signal E(t) higher or lower than the resonance frequency have lower output amplitudes. The lower buffer gas pressure has a narrower filter bandwidth, and can produce a stronger AC component in pumping rate to excite a larger CPT signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.27 The ground-state CPT excitation from the three correlated quantum levels can be approximately analogous to an electronic-mechanical system. The CPT excitation is equivalent to the mechanical oscillation of the two metallic plates on the right diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.28 Calculated excitation efficiency of Cs 0-0 CPT resonance in the nitrogen buffer gas. Here, the modulation index m is defined by E(t) = E0 exp(−iΩ0 t + im sin(ωm t)), where Ω0 is the carrier frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.29 The left and the middle panel show the probing lights along two orthogonal directions, parallel or perpendicular to the spin at zero detuning δω = 0. The signal amplitudes plotted as functions of δω. The in phase signal shows a symmetric curve, and the out phase signal shows a completely asymmetric curve. By using a modulated pumping light, coherence can be excited as a spin along the x-direction by the coherence optical pumping rate RΓ . If there is a light shift (δEv 6= 0), an effective field Rv is generated. This emulated field produces a spin component along the y-direction. Therefore the probing light both probes in phase and out phase signals. The detuning curve becomes asymmetric. . . . . . . . . . . . . . . . . . . . . . 2.30 Numerical calculation of the asymmetry of the resonance of 87 67 Rb at 0.1 atm as a function of Ω−Ω0 . The subplot shows the intensity spectrum of the optical sidebands of the modulated light and the curve of the absorption cross section. The ground state hyperfine splitting is resolved. When the optical detuning is small, only two strong optical sidebands are important. Hence, it can be approximated by two-wave pumping at low optical detuning. . . . . . . . . . . . . . . . . . . . . . . . . . xv 69 2.31 Experimental data shows the line shapes of the end CPT resonances as a function of the laser frequency from a 87 Rb cell with 0.96 atm N2 buffer gas. By using the numerical model, we can extract the parameter of optical pressure broadening and the pressure shift. The two wave model can not fit to data, because in the high pressure cell, atoms can see not only two optical sidebands but also high order sidebands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.32 The calculation result shows a smaller dδA /d∆ around the point of Ω−Ω0 ∼ 0 by using a different pumping spectrum. The inset shows the modulation spectrum, which is generated by a Mach-Zehnder type modulator. The amplitude of the pumping electric field is proportional to sin(φ0 + φ1 sin ωm t). . . . . . . . . . . . . . . . . 71 2.33 The left and middle diagrams show the σ+ pumping with different deexcitation mechanisms. Quenching deexcitation causes higher pumping efficiency than spontaneous decay. The right diagram shows the hyperfine sublevels with an assumption of nuclear spin I = 1/2. The electron spin has chance to be transferred to the nuclear spin through the hyperfine coupling in the excited state for the pumping transitions of region (1), but not of region (2). . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.34 Photon cost for pumping potassium-39 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. . . 76 2.35 Photon cost for pumping rubidium-85 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. . . 77 2.36 Photon cost for pumping rubidium-87 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. . . xvi 78 2.37 Photon cost for pumping cesium-133 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. . . 79 2.38 . Photon costs for two different optical pumpings and nuclear spins with different relaxation mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.39 Two different cavity configurations of hyperfine-modulated lasers for two different types of laser diodes. The regular edge emitting laser diode outputs linearly polarized laser light, and therefore it needs two λ/4 wave plates to produce alternating circular polarization inside the cavity. The VCSEL diode is designed to have almost no preference of the light polarization, so the light polarization can be directly modulated by the vapor cell. The steady lasing point of the two lasers is when the modulation period of the output light is equal to 2π/ω00 . . . . . . . . . . . . . . 84 2.40 An illustration of the spectral responses due different causes. The optical comb in the lasing spectrum is grown to produce the maximum transparency of the vapor cell, and therefore obtain the maximum gain of photons. . . . . . . . . . . . . . . 85 2.41 The normalized amplitudes of the maximum 0-0 coherence with different nuclear spins by using different fast push-pull pumping. D1 push-pull pumping can excited a much stronger CPT coherence than D2, and also make the vapor cell more transparent. 86 3.1 An example of 129 Xe-Rb using SCA and DWBA numerical calculations. The SCA curve behaves like the mean value of the DWBA curve. . . . . . . . . . . . . . . 3.2 A block diagram shows the total setup of the xvii 129 100 Xe spin-exchange rate measurements. 101 3.3 The NMR probe was designed to fit into a 9.4-T Oxford superconducting magnet with bore diameter ∼ 3.5”. The internal space was heated resistively with noninductive wiring. The heat conducting copper tube suppressed the thermal gradient to about 1◦ C per inch. The temperature was stabilized by the feedback of two thermal sensors. The interlayer between inner and outer tubes was stuffed by a porous foam for good heat insulation. The sample cell was held by the NMR coil and hung by two supports. The NMR coil was set to the resonance frequency by adjusting a variable capacitor. The tuning work was done by using a sweep generator (WAVETEK 1062). A probing beam for Faraday rotation measurement passed through the two windows and the center of the cell. The field inhomogeneity inside the 1 inch cell was about 0.3 ppm. 3.4 . . . . . . . . . . . . . . . . . . . . . 102 Arbitrary unit for y-axes. The top panel shows two orthogonal FID signals. The bottom panel is the Fourier’s spectrum of the FID signal. This example signal was the result of eight times average FID from a 4 amagat Xe cell. . . . . . . . . . . . 3.5 The amplitude of the FID signal (filled circles) is proportional to the 129 103 Xe nuclear spin polarization hKz i as it recovers towards its thermal-equilibrium value hKz iT = µK B/2kT . The relaxation time T1 is extracted from the fit of Eq. (3.57), solid curve. Each data point was taken over 16 – 96 averages of the FID signal at the same delay time τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 104 Faraday rotation signal sin2θ (open circles), fit to the calculation of Eqs. (3.63–3.65) (solid line) using the fitting parameters [Cs], θ0 , and ωJ . The optical path length is 2.5 cm and B = 9.4 T. The dashed line shows the mean photon spin s measured from the first harmonic of the photodetector output. The residual non-zero photon spin at high laser frequency (far detuning from the D2 line) is ascribed to the effect of optical guiding mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 108 3.7 Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured Rb vapor number densities [Rb] for the four cells listed in Tab. 3.1. The experimental temperature were between 160 to 200◦ C. The spin-exchange rate coefficients κ are the slopes of the straight-line fits to the data. . . . . . . . . . . . . . . . . . . . 3.8 112 Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured Cs vapor number densities [Cs] for the five cells listed in Tab. 3.2. The experimental temperature were between 110 to 150◦ C. . . . . . . . . . . . . . . . . . . . . . . 3.9 113 Theoretical magnetic decoupling curves used to extrapolate our from B=9.4T to B=0T. Since the experimentally determined potential V had about 5% error, we made 3 to 4% adjustments to the position of the minimum of the interatomic potential V can bring a good agreement between experimental data and calculations. The shape of decoupling curves changes less than 0.1% by applying adjustments on V . 114 B.1 Because of this divider circuit, we were able to obtain the first light narrowing data from the fiber-coupled and noisy Ti-sapphire laser light. . . . . . . . . . . . . . . xix 137 List of Tables 2.1 Experimental and theoretical results of the key parameter of S-damping rate and Carver rate. All numbers listed above have about 10 % errors. For helium gas, the values measured by D.K. Walter were for 3 He. The Carver rates from Vanier’s measurements, which are tagged by “∗”, are much larger than other values in the same block. The numbers from our work are marked by parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 47 Experimentally measured values of κ for the four Rb cells used in our experiment. (Note: 1 amg=2.69 × 1019 cm−3 , is equal to the number density of an ideal gas at 0◦ C and 760 torr) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Experimentally measured values of κ for the five Cs cells used in our experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 111 111 A summary of binary spin-exchange rate coefficients at different magnetic fields obtained by different groups. The zero-field rates quoted for our work are slightly larger than the values measured at B = 9.4 T because of adjustments for the magnetic decoupling discussed in connection with Fig. 3.9. xx . 115 Chapter 1 Introduction In 1950, A. Kastler proposed the idea of using photons to transfer order to atoms by scattering the resonant light [32]. By appropriately choosing the frequency and the polarization of the pumping light, it is possible to generate a desired distribution of populations in atomic energy sublevels of the atoms. In the following decade, this basic idea of optical pumping [22] had been quickly developed with many breakthroughs. The first spin-polarized optical pumping was succeeded by Brossel et al. [14] and Barrat et al. [6] between 1952 to 1954, which were done with sodium atomic beam and sodium vapor cells respectively. The spinexchange optical pumping was first proposed by H.G. Dehmelt in 1957, and was observed by M.A. Bouchiat in 1960 [13]. In 1960, T.H. Maiman used optical pumping to produce the population inversion, and therefore successfully achieved the first laser, ruby laser [37]. In 1961, Bell and Bloom used intensity-modulated light to pump Rb and Cs atoms into a state of continuous spin precession [8]. This was the first demonstration of optically induced Zeeman coherences. In the intervening decades, these important achievements continued to be improved, and have been extended to many scientific studies. Nowadays, to study the atomic energy levels, optical pumping is not just used in microwave or radio frequency (RF) spectroscopy. Optical pumping with multi-coherent lights has become an important technique in the studies of atomic spectroscopies and other atomic physics. CPT (coherent population trapping) spectroscopy [4] is one example. In this thesis, we probe into details 1 2 of hyperfine evolution of alkali-metal atoms through the density-matrix modelling and hyperfine spectroscopy. We investigate CPT phenomena including important details, which were ignored in many previous studies. From the studies, we have developed a new optical pumping technique, which can pump the atoms into nearly pure 0-0 superposition state. Besides, we also study the problem of spin-dependent collisions between alkali-metal and noble-gas atoms. This mechanism plays an important role in spin-exchange optical pumping to produce hyper-polarized noble gases, which are very useful in studies of physics and biomedical applications. 1.1 Advantage of Alkali-Metal Atoms Alkali-metal atoms are useful for studying atomic physics, because they are relatively simple, and they are more controllable in experiments. These hydrogen-like atoms have only one valence electron. Their simpler quantum properties provide a good connection between quantum theories and experiments. For experiments, probing sources, such as light sources and microwaves, for alkali-metal atoms are readily available and convenient. Besides, alkalimetals are relatively volatile. Therefore they are easy to be optically probed in the gas phase. Because of the reasons above, different types of experiments with alkali-metal atoms have been constructed for testing of fundamental physics, and many useful inventions are based on alkali-metal atoms. For example, hyperfine and Zeeman spectroscopies of alkalimetal atoms provide ways to measure many fundamental constants; spin-polarized and spin-exchange optical pumping offer a way to study the magnetic properties of atoms. In the past decade, atom cooling and trapping techniques provide a cleaner environment to study many quantum behaviors, such as Bose-Einstein condensation, quantum gas in optical lattice, cavity quantum electrodynamics, quantum computation, and quantum information. In addition, the important inventions related to alkali-metal atoms are atomic clocks, vaporcell magnetometers, noble-gas spin polarizers, etc. 3 1.2 Content Brief In chapter 2, we discuss our detailed studies of the atomic-clock resonances and the optical pumping with multi-coherent light. For deeper investigation, we developed a complete density matrix model to theoretically calculate the relevant phenomena. Our modelling was capable of calculating the full evolution of ground-state and excited-state density matrices. This enabled us to have a completed picture of microwave and CPT spectroscopy. In the past, people have often used three-level or four-level models to describe the CPT phenomena. Such simplified pictures can give a good qualitative explanation for the experimental results, but they are not precise. Moreover, because the several-level modelling is too simplified, it can miss some important phenomena. In order to examine the idea and the theory, we set up the experimental system for microwave spectroscopy and CPT spectroscopy. We used rubidium and cesium vapor cells as the probing targets, and mainly focused on D1 pumping. For atomic-clock resonances, the hyperfine spin relaxations plays a critical part in the quality of the resonance signal. Many of those key relaxation rates [48] have been measured by J. Vanier et al., who did lots of pioneering works related to vapor-cell atomic clocks. However, from our studies, we discovered that there seem to have been systematic errors in some of Vanier’s measurements. We believe those errors were due to temperature gradients in the alkali vapors [29, 38]. For the measurements of the pressure-dependent resonance linewidth, we employed the measurement of a damped transient oscillation of the transmission light, and used thin cells to avoid possible temperature gradient contributions to the measurements. Our results agreed with theoretical calculations and with another experimental measurement done by D.K. Walter [53], who acquired those key parameters by different means. For the optical pumping with modulated light, we present the calculation and experimental data regarding CPT excitation under different buffer gas pressures, CPT line shapes, and the ”photon cost” for optical pumping. The CPT signal is excited by a modulated light beam. The pumped atoms respond differently to the modulated light at different buffer-gas pressure, because of pressure broadening of the optical absorption lines. Therefore the CPT excitation efficiency is not the same for different modulation 4 schemes, such as AM (amplitude modulation) or PM (phase modulation), and for different gas pressures. Besides, the population of ground-state multiplets is also affected by the buffer gas pressure. The common several-level model can not reflect those important details. The CPT line shape is affected by light shifts, which include virtual and real transitions. The virtual shift is controlled by the optical detuning and the pressure broadening. The real shift is affected by the evolution in the excited state. Again, the common simple CPT model can not really calculate those detailed features. In addition, we further discuss some details of the evolution of the excited state, which was mostly ignored in previous CPT studies, through the calculations of ”photon cost”, the mean number of photons per atom that must be absorbed for pumping to a certain quantum state. The photon cost is affected by the excited-state relaxation mechanisms, such as J-damping, quenching, and spontaneous decay. One of the important results from this research is that we confirmed the theoretically predicted “light narrowing effect” of the hyperfine end resonances with our experiments [29]. This effect is due to the conservation of angular momentum. Spin exchange collisions, the dominant relaxation mechanism in dense alkali-metal vapors, conserve angular momentum. Since atoms in the end state have more angular momentum than any potential final state of the collision, spin exchange collisions do not affect end state atoms. To obtain light narrowing, the atoms have to be pumped by D1 circularly polarized light. By increasing the pumping power, we can increase the fraction of atoms in the end state, and therefore suppress the spin-exchange broadening. The light narrowing effect promises to generate a end-resonance signal with a narrower linewidth and stronger amplitude compared to the traditional 0-0 resonance used by most atomic clocks. In general, the performance of an atomic clock is proportional to SNR/∆ν (the signal to noise ratio divided by the linewidth). By using light narrowing effect, we might be able to build a miniature atomic clock or magnetometer with a good performance. The miniature device requires a vapor cell with a small size down to millimeter or sub-millimeter scale. Hence, a high working temperature is necessary to increase the saturated vapor pressure of alkali-metal atoms and to produce 5 sufficient light absorption. However, the spin exchange collision rate between alkali-metal atoms increase at higher atomic densities. The spin-exchange broadening increases the linewidth and decreases the amplitude of the conventional resonance signal, but it has much less influence on the end resonance of a highly spin-polarized atom. The end resonance can be a potential atomic-clock resonance. Besides using the static spin-polarized optical pumping to enhance the signal of the end resonance, we report a new optical pumping method, “push-pull optical pumping”, to greatly improve the 0-0 resonance signal [28]. This is the other significant result from our research. Push-pull optical pumping is a way to optically pump atoms into a pure coherent atomic state. The key requirement is to use D1 pumping light, which alternates between two orthogonal circular polarizations at the Bohr frequency of 0-0 state. This oscillating spin-polarized optical pumping can concentrate the nearly all the atoms into a pure 0-0 superposition state, analogous to the static spin-polarized optical pumping, which concentrates the entire population to the end state with no pressure dependence. The idea of push-pull optical pumping was inspired by Bell-Bloom experiment and the experiments of coherence population trapping. The basic picture of push-pull pumping is to use the alternating photon spin to drive the electron spin, analogous to “push” and “pull”. In the CPT language, push-pull pumping is a special CPT excitation, which has only one dark state (un-pumped state). Therefore, the population will eventually be trapped into the only dark state. In conventional 0-0 CPT, the ground-state 0-0 coherence is excited from a modulated beam with a fixed circular polarization. However, the hyperfine population prefers to be pumped into end state by the circularly polarized light. Therefore its 0-0 resonance signal is weaker. Because of the exceptional capability of push-pull optical pumping, we believe that push-pull optical pumping can enhance the performance of all kinds of CPT frequency standards, and will have further applications of alkali masers, state preparation for studies of cold atoms, quantum computation, and quantum information. In the end of this chapter, we talk about a new possible way to build a compact and simple frequency standard, which is the extension of using the concept of push-pull pumping and combining the technology 6 of semiconductor lasers. In chapter 3, we switch our attention to the spin-exchange collisions between alkalimetal atoms and xenon gas. We designed a high-field experiment to determine the binary spin-exchange rate coefficient of 129 Xe-Rb [25] and 129 Xe-Cs [27], which has been measured by some other groups previously with results that disagreed by large factors due to various systematic errors. The previous measurements were flawed by misestimates of the atomic number density and by misestimates of non-binary relaxation due to van der Waals molecules. People mostly used empirical vapor pressure formulas to calculate the atomic number density. Such estimates can be incorrect for many reasons, such as impurities in the alkali metal. More and more experimental data tell us that we do not understand the spin exchange between xenon and alkali-metal atoms through van der Waals molecules at high pressure (≥ 1 atm) and low field. It is hard to extrapolate the binary spin-exchange rate from the spin exchange in the presence of van der Waals molecules. We solved these two problems by using Faraday rotation measurement to determine the number density of rubidium or cesium experimentally and by using high magnetic fields to eliminate the contribution from van der Waals molecules. Although we artfully avoided the trouble of van der Waals molecules by using high field, the experimental evidence shows that the three-body model is incomplete. Further investigation has to be carried out. A better understanding of the spin exchange will ultimately improve the performance of xenon polarization system. Our measurements were carried out by using NMR techniques to measure thermally polarized 129 Xe nuclear spin. We discuss the rate equations of 129 Xe nuclear spin with detailed balancing in the first section. Since the experiment was done in a very high field, 9.4T, magnetic decoupling of the binary spin-exchange become non-negligible. We compare two different methods, semi-classical approach (SCA) and distorted-wave Born approximation (DWBA), for calculating the magnetic decoupling [26]. These two methods show a good agreement with each other for conditions of experimental interest. The same calculations can be applied to many other spin relaxation processes. In the last section, we summarize our experimental result. We conclude that there was a negligible van der Waals molecules 7 contribution in our measurement. Cesium has twice larger spin-exchange rate coefficient than rubidium, which agrees with our theoretical calculations. Therefore, cesium is a better candidate for xenon polarizer, as long as a high-power Cs pumping laser is available. Chapter 2 Strong Resonances of Alkali Hyperfine Sublevels from Optical Pumping The study of the ground-state resonances of alkali-metal atoms is still an interesting topic today. It directly benefits the design of atomic clocks and magnetometers. These two different devices tell us how precise we can measure “time” and the strength of “spin-coupled fields”. By pushing these two devices to a better precision, we can have a better understanding of the theoretical structure of the fundamental physics. Besides, many advantages can be produced from those two devices to our daily life, such as GPS (global positioning system), high-speed digital systems, biological or medical diagnostics, etc. We have implemented a series of studies of resonances of hyperfine sublevels. We basically focused on the atomic-clock resonances, and several important results were found from our studies. For the atomic-clock resonances, the spin-relaxation mechanisms play a crucial role, and affect the performance of the atomic clocks. From our experiments, one important relaxation mechanism, the Carver rate, which is due to the collisional hyperfine modulation, was found to have been overestimated by J. Vanier, who has made significant contributions to the development of vapor-cell atomic clocks. Our new results of pressure-dependent 8 9 linewidth of the hyperfine resonances might change the idea of the pressure working range of the vapor-cell atomic clocks. To enhance the signal of the atomic-clock resonances, we developed two procedures for different clock resonances. We took the advantage of light narrowing effect to obtain intense end resonances. The light narrowing effect was achieved by using a fixed circularly polarized pumping light. The increase of the atomic polarization suppressed the spin-exchange broadening and concentrated the population to the end-state. The linewidth became narrower, and the signal amplitude became larger. This effect is especially useful at high temperature, where the alkali vapor is dense and the spin exchange dominates the linewidth broadening. For the traditional 0-0 atomic-clock resonance, we contrived a new pumping method, push-pull optical pumping. This new pumping method is capable of concentrating the population into the 0-0 states and producing a tiny suppression of the spin-exchange broadening. Therefore it can definitely improve the performance of conventional atomic clocks, which are based on the 0-0 transition of the alkali-metal atom. Push-pull optical pumping also has other applications, such as for alkali masers and the state preparation for alkali-metal atoms. Its capability of state preparation can be useful for experiments ranging from cold atoms to hot vapor. The detailed studies required the methods of microwave and CPT spectroscopies. In order to understand the relative phenomena, we used density-matrix modelling to calculate the effects of important relaxation mechanisms on the evolution of the ground-state and the excited-state atoms. 2.1 Density Matrix Equations In many experiments of atomic physics, instead of one atom, we look at an ensemble of many atoms. A well-developed method, density matrix, gives an easier and a beautiful way for calculating the ensemble behaviors. For our studies, we took advantage of the density matrix to calculate the relevant phenomena of the experimental observations. The density 10 matrix of an ensemble is given by N 1 X |ψn ihψn |, ρ= N (2.1) n=1 where N is the number of alkali-metal atoms, and |ψn i denotes the quantum state of each atom. Hence, we have Tr(ρ) = 1. The dynamic equation of the density matrix is basically P described by ρ̇ = n [Hn , ρ]/ih̄, where Hn is the total Hamiltonian of the nth atom. We can verify that Tr(ρ̇) = 0, which gives the conservation of the atom number. For an operator P or an observable O, the expectation value hOi = N −1 n hψn |O|ψn i is equal to Tr(Oρ). By calculating the evolution of the density matrix, we can obtain different observables as functions of time, which can be measured with experiments. For the total Hamiltonian, we separate it into three parts, Hn = H H (0) is the time-independent “unperturbed” Hamiltonian, H to pumping lights and oscillating magnetic fields. Here, H (1) (0) (0) +H (1) (2) +Hn , where is the interaction term due and H (1) are identical for all (2) atoms. The interaction of each atom due to random collisions is described by Hn . For (0) the ground state, S-state, the spin-related unperturbed Hamiltonian, Hg , of the valence electron of an alkali-metal atom can be expressed by (0) Hg = AI · S + (µB gS Sz − µI Iz /I)B0 , (2.2) where A is the hyperfine coupling coefficient, I = Ix x+Iy y+Iz z is the nuclear spin operator, S = Sx x+Sy y +Sz z is the electron spin operator, gS is the electron g-factor, µB is the Bohr magneton, µI is the nuclear magneton, I is the nuclear spin quantum number, and B0 is the background DC magnetic field. The ground state has 2[I] = 2×(2I +1) hyperfine sublevels. We usually use the vector |f, mi, which is the eigenstate of the hyperfine interaction, as the bases to compose the density matrix, where f = I ± 2 and m = −f, · · · , f . Those (0) eigenstates remain good for Hg in a weak magnetic field (µB B0 ¿ A). Ground-state relaxation is mainly caused by random interactions between atoms. Con(2) sidering a few important mechanisms, each atom has its own Hamiltonian, Hg , which has a form (2) Hg = γN · S + JS0 · S + δAI · S. (2.3) 11 Here, the collisional Hamiltonian describes S-damping, spin exchange, and Carver damping interactions. For the excited state, we only consider the P1/2 excited state because of our experimental (0) interests. Therefore we find He as (0) He = Aj I · J + (µB gj Jz − µI Iz /I)B0 (2.4) where Aj is the hyperfine coupling coefficient, J is the total electron angular momentum, gj = 2/3 is the Landé g-factor. If we count only spontaneous decay, quenching, and Jdamping, we can write the Hamiltonian for the random interactions as (2) He = γj N · J + H d , (2.5) where γj is the coupling coefficient of the angular momentum interaction. The last term H d appeared in Eq. (2.5) is the Hamiltonian for deexcitation, which includes spontaneous decay and quenching mechanisms. For atoms interacting with lights and oscillating fields, we write the interaction Hamiltonian as H (1) = −D · Eop + (µB gS S + µB gj J − µI I/I) · B. (2.6) Here, D is the electric dipole operator, Eop = Eop (t) is the electric field of the pumping light, and B = B(t) is the oscillating magnetic field. The transition between the S and the P states is only caused by the electric dipole interaction. In our calculations, the evolution of the density matrix is determined by the Hamiltonian described above. Figure 2.1 sketches an overall picture of our density-matrix modelling. In order to reduce the complexity, the numerical calculations were carried out in Liouville space by using MATLAB programming. Our full model is capable of calculating the evolution of the atomic ensemble with D1 optical pumping. With several modifications, it can be extended for the calculation of the density matrix with D2 pumping. The main equations of the density matrix are stated in the following subsections. As for the optical pumping, we mainly discuss the D1 transition, since our experiments were carried out by using D1 12 Excited-state hyperfine sublevels n J-damping P1/2 Quenching + Spontaneous decay Repopulation and D1 pumping a f = a = I+1/2 -a n S1/2 f = b = I-1/2 -b -1 0 -1 0 1 1 b Ground-state hyperfine sublevels S-damping Spin exchange Diffusion Carver rate Optical pumping Repopulation Figure 2.1: In our modelling, we calculated the ensemble behaviors of the alkali-hyperfine sublevels with D1 pumping and relaxation mechanisms, such as S-damping, spin exchange, Carver damping, diffusion, spontaneous decay, quenching, and J-damping. pumping. Detailed derivations of the evolution equations of the density matrix is discussed in appendix A. 2.1.1 Evolution in Liouville Space For calculations of the density matrix in Lioville space, each element of the density matrix becomes an orthogonal vector. Therefore, an N by N density matrix can be represented as a vector with N2 components. To do this, we find ρ −→ |ρ), ρ̇ −→ d |ρ) = −R|ρ), dt (2.7) where |ρ) is the Liouville vector, and R is the relaxation matrix, which has a dimension equal to N2 by N2 . The elements of R can be found by calculating Rij = (i|R|j) = −hα|ρ̇(|µihν|)|βi = Rαβ,µν , (2.8) where |j) = ρµν and |i) = ραβ . In practice, the number of elements of R is much less than N4 , because most off-diagonal elements are zeros. For example, considering a simplest case 13 with only one coherence in the ground state. The size of the ground-state relaxation matrix R is only equal to g + 2 by g + 2, where g = 2[I]. To write the relaxation equation into a more general form, we have d |ρ) + R|ρ) = |Υ), dt (2.9) where |Υ) = |Υ(t)) represents the possible source term. For a steady solution, if |Υ) is independent of time, we find |ρ) = R−1 |Υ). For a dynamic solution, we find XZ t 0 |ρ(t)) = |ρk )e−γk (t−t ) (ρk |Υ(t0 ) + ρ(0)δ(t0 ))dt0 , k (2.10) (2.11) 0 where |ρk ) is the normalized eigenvector of R, and γk is the corresponding eigenvalue of R. When spin-exchange rate is not equal to zero, R is a function of the electronic polarization, and Eq. (2.9) becomes nonlinear. Equation (2.11) is only approximately accurate if spin exchange dominates the relaxation process. However, if |Υ) = 0, we can use an iteration method to find the exact steady solution of Eq. (2.9). In Liouville space, the evolution equation of the density matrix has a simpler equation formalism. Therefore, it is easier to be applied in the numerical calculations. 2.1.2 Ground-State Relaxation S-Damping or Spin Destruction The spin-rotation interaction, γ(r)N · S, causes atoms lose their electronic spin-polarization to the angular momentum of the colliding atom pairs. Here, N is the angular momentum of the colliding pair and S is the electron spin operator. We find the corresponding equation of the density matrix due to the spin-rotation interaction is ρ̇sd = Γsd (ϕ − ρ), (2.12) where ϕ = ρ/4 + S · ρS is the density matrix without electronic polarization. Here, Γsd is called S-damping rate or spin destruction rate, because this interaction destroys the spin 14 polarization; and Γsd is proportional to the number density of buffer gas. By looking at the time derivative of total atomic polarization dhFi/dt, where F = I + S, we find d hFisd = Tr(Fρ̇sd ) = −Γsd hSi. dt (2.13) The result here is a little bit subtle. When a spin-rotation collision occurs, a small amount of the hyperfine coherence is generated. Due to the hyperfine splitting and Zeeman splitting, these zero-frequency coherences quickly decay because of the spin precession in the different phases. Therefore, the off-diagonal elements usually remain zero through the S-damping process. It can be comprehended as that the electron spin transfers its polarization to the nuclear spin by precessing with the nuclear spin through the I · S interaction. The nuclear spin is just like a “spin sink” or “spin reservoir”, which stores the angular momentum acquired from the electron [3]. Spin Exchange The spin-spin coupling interaction, J(r)S0 · S, accounts for the spin exchange between the two colliding atoms of electron spins S0 and S. Its corresponding equation of the density matrix is ρ̇ex = 1 [ δEex , ρ ] + Γex [ϕ(1 + 4hSi · S) − ρ] , ih̄ (2.14) where δEex is the energy shift operator due to the spin-exchange collisions. The spinexchange rate Γex is proportional to the number density of alkali-metal atoms and therefore increases with the environmental temperature. By checking Tr(Fρ̇ex ), we can verify the conservation of the spin polarization from spin exchange. d hFiex = Tr(Fρ̇ex ) = 0. dt (2.15) Carver Rate The coupling coefficient, A, of the hyperfine interaction, AI · S, can be perturbed by the atomic collisions. We separate the perturbation term, δA(r)I · S, from the unperturbed interaction. The corresponding equation of the density matrix due to the random modulation 15 of the hyperfine interaction is ρ̇C = η 2 [I]2 1 [ δEC , ρ(m) ] − I ΓC ρ(m) , ih̄ 8 (2.16) where δEC denotes the pressure shift of the hyperfine splitting frequency, and ρ(m) represents the density matrix with only off-diagonal elements of the coherences between upper f = I + 1/2 = a and lower f = I − 1/2 = b hyperfine multiplets. We use letter “m” to label it, because the hyperfine transitions are mostly in microwave frequencies. The Carver rate ΓC was first introduced by D.K. Walter [53], which is proportional to the number density of the buffer gas. The Carver-rate relaxation can cause the dephasing of hyperfine coherences, but can not affect the hyperfine population and the coherence of Zeeman sublevels for the relatively small magnetic fields. Diffusion In experiments, instead of tracking a cluster of atoms, we probe the atoms from a fixed volume. The atoms can randomly move from one place to others. This spatial diffusion of atoms is equivalent to the diffusion of atomic spins. When the atom diffuses to the wall, both nuclear and electron spin can be completely destroyed. Therefore the population of the hyperfine ground state is relaxed toward equilibrium. The corresponding equation of the density matrix for diffusion mechanism is ρ̇d = Γd (1/g − ρ), (2.17) where 1 is the unit matrix, and the relaxation rate Γd = 3D(π/l)2 if we assume a cubic cell with side length l and the pumping light covers the entire cell. Here D is the diffusion coefficient, which is inversely proportional to the buffer gas pressure. We can check that d hFid = Tr(Fρ̇d ) = −Γd hFi. dt (2.18) 16 2.1.3 Excited-State Relaxation J-Damping The J-damping mechanism occurred in the P1/2 state is very similar to the S-damping in ground state. The relaxation is dominated by the interaction γj N · J, which causes angular momentum transferring between N and J. Here, J = L + S is the operator of the total electron angular momentum. We can find the corresponding equation of the excited-state density matrix ρ(e) is 1 (e) ρ̇jd = Γjd (J · ρ(e) J − {J · J, ρ(e) }), 2 (2.19) where Γjd is the J-damping rate, and { } denotes the anti-commutator. J-damping is an important depolarization mechanism in the excited state. Here, we use the superscript (e) to distinguish from the part of the ground-state density matrix. Spontaneous Decay & Quenching Atoms return to the ground state from the excited state by either spontaneously emitting photons or colliding with quenching atoms. The corresponding relaxation of the density matrix of the excited state is (e) ρ̇(e) − Γq ρ(e) , sq = −Γs ρ (2.20) where Γs is the spontaneous decay rate and Γq is the quenching rate. More detailed discussions about this subject is in appendix A. 2.1.4 Optical Pumping The optical pumping includes depopulation and repopulation mechanisms. The depopulation pumping transfers some population from the ground state to the excited state. The repopulation pumping sends the population back from the excited state to the ground state due to the decay mechanisms (quenching and spontaneous decay). 17 D1 Depopulation Pumping (1) The depopulation pumping is caused by the interaction Hop = −D · Eop . For the buffergas pressure above a few tens of torr, the hyperfine splitting of the excited state becomes unresolved due to the optical pressure broadening. Therefore we can approximate energies of all excited hyperfine sublevels to their center of gravity. For unsaturated optical pumping, we find the effective Hamiltonian of D1 depopulation pumping for the ground state as δHop = (op) Here, the pumping rate, Γ h̄ (op) (1 − 2s · S)Γ . 2i (2.21) (op) = Γαβ δα,β , is a time-dependent complex matrix with only (op) diagonal elements. Taking the matrix elements of Γ , we find Z Z ∞ ∗ (Ω − ω)Ē (Ω) |D|2 Ēop (op) op −iωt Γαα = dω e , dΩ 2 2h̄ i(Ωα − Ω) + γop 0 (2.22) where D is the dipole strength between P1/2 and S1/2 , Ēop (Ω) is the Fourier’s spectrum of the pumping electric field, Ωα is the optical transition frequency from the ground-state sublevel α to the excited state, and γop /π is the optical linewidth. The time-dependent pumping rate can be produced by modulating a CW pumping light, and its dependence of time is determined by the spectrum of the pumping field, optical detuning of the main carrier, optical linewidth, etc. The detailed derivation of the pumping rate can be found in appendix A. The vector s denotes the photon spin of the pumping light, which has values |s| ≤ 1. The effective Hamiltonian δHop is not Hermitian, which can be written as the sum of a Hermitian and an anti-Hermitian operators. Then, δHop = δEv − δEv = 1 2 (δHop † + δHop ) and δΓ = i h̄ (δHop ih̄ 2 δΓ, where † − δHop ). We find the equation of the density matrix for depopulation pumping of ground state as ρ̇dp = 1 1 1 † [δHop ρ − ρδHop ] = [δEv , ρ] − {δΓ, ρ}. ih̄ ih̄ 2 (2.23) Here, we can define δEv as the light shift operator due to the virtual transition, since it has nonzero values only when there is an optical detuning between ground-state and excitedstate sublevels. The operator δΓ is the absorption operator. The averaged light absorption rate of an atom is equal to the total depopulation rate, −Tr(ρ̇dp )=Tr(δΓρ) = hδΓi. 18 For a simpler case, if all hyperfine sublevels see the same pumping rate, the pumping(op) rate matrix Γ h̄ 2 (1 = Γop becomes a pure time-dependent complex number. Therefore, δEv = − 2s · S)Im(Γop ), and δΓ = (1 − 2s · S)Re(Γop ). D1 Repopulation Pumping There is no equation in a simple form for repopulation pumping by using the regular densitymatrix formalism. The simplest expression is in Liouville space as d |ρ)rp = R(rp) |ρ), dt (2.24) where R(rp) is the repopulation matrix in Liouville space, and its detail can be found in appendix A. For a common experimental condition, repopulation pumping is dominated by quenching. Assuming that all ground-state hyperfine sublevels see the same pumping rate, and only if the quenching rate is much larger than the excited-state hyperfine frequency, the change of the ground-state population due to the quenching dominated D1 repopulation pumping is µ ρ̇rp = Re(Γop ) ¶ 1 {1 − 4s · S, ρ} + (S · ρS − iS × ρS) . 8 (2.25) It is understood that Eq. (2.25) is only valid for the calculation in the manner of ρ̇αα = (rp) Rαα,µν ρµν . Full Optical Pumping Equations for Ground-State Population In our experiments, we use nitrogen as the buffer gas, which is very good for quenching. With enough amount of quenching gas, we can assume the decay from the excited state happens much faster than spin relaxation and spin precession in the excited state. Therefore, the nuclear spin is conserved in the excited state. Under this condition, we find the evolution equations of the complete optical pumping (depopulation+repopluation) for the groundstate population, ραα , as the summarization in Eq. (2.26) to Eq. (2.29). Here, we also include (op) D2 = − ih̄ (1 + s · S)Γ two cases of D2 pumping. The effective Hamiltonian for D2 is δHop 2 . 19 Assuming all hyperfine sublevels see the same pumping rate, from Eq. (2.23) and Eq. (2.25), we find D1 pumping + strong quenching: ρ̇op = Re(Γop ) [ϕ(1 + 2s · S) − ρ] . (2.26) D1 pumping + pure spontaneous decay: (assuming the excited hyperfine frequency = 0) µ ρ̇op = Re(Γop ) ¶ 2 1 2 [ϕ(1 + 2s · S) − ρ] + Sz ρSz − ρ . 3 3 6 (2.27) D2 pumping + strong quenching: ρ̇op = Re(Γop ) [ϕ(1 − s · S) − ρ] . (2.28) D2 pumping + pure spontaneous decay: (assuming the excited hyperfine frequency = 0) µ ρ̇op = Re(Γop ) ¶ 1 1 1 [ϕ(1 + 2s · S) − ρ] + Sz ρSz − ρ . 3 3 12 (2.29) We have to notice that Eq. (2.27) and Eq. (2.29) are only qualitatively correct in the real cases, because the excited hyperfine splitting is usually larger than the spontaneous decay rate. Therefore, the electron has enough time to transfer its polarization to the nuclear spin through the precession with the nuclear spin in the excited state before the spontaneous decay. The polarization transfer between electron and nuclear spin cause the optical pumping to polarize atoms more efficiently. Equation (2.26) and (2.28) are good by using the quenching gas such as nitrogen in the pressure near or above an atmosphere, where the quenching rate much exceeds the excited-state hyperfine splitting frequency. Some buffer gases, like helium or argon, are not good for quenching but causing larger J-damping relaxation in the excited state. In this situation, the evolution equation of the density matrix for optical pumping is more complicated and can not be described by the equations listed above. Basically, the population changed by optical pumping are affected by ways of depopulation, repopulation, and the evolution in the excited state. More detail of optical pumping will be discussed in section 2.4.7. 20 There is an interesting feature of D2 pumping. By looking at Eq. (2.28) and Eq. (2.29), we find that the different repopulation mechanisms have different pumping polarizations by using the same photon spin. We can check Tr(Fρ̇op ) from D2 pumpings. ´ dhFiop Re(Γop ) ³ s = − hSi . dt 3 2 ³ ´ dhFiop s = Re(Γop ) − − hSi . Strong Quenching: dt 4 Spontaneous Decay: (2.30) (2.31) Since the quenching rate is proportional to the pressure of the quenching gas and the spontaneous decay rate is nearly constant, it implies that there is a magic pressure where has no spin-polarized optical pumping. At that pressure, we can not use circularly polarized D2 light to spin-polarize atoms. Although, Eq. (2.30) underestimates the polarization pumping rate for many real cases, but it does not change the direction of the spin polarization. Ground-State Coherence Created by Optical Pumping The ground-state coherence can be generated through ρ̇dp + ρ̇rp . The quasi-steady solution of the ground-state coherence can be obtained by using the secular approximation (rotation wave approximation) and the slowly varying approximation. We find ρeµν = − (1) 1 2 (δΓµν ρνν (1) (1) (1) + ρµµ δΓµν ) + h̄i (δEv,µν ρνν − ρµµ δEv,µν ) (1) + R(rp) µν,αα ραα . i(ωµν − ω) + γµν (2.32) where ωµν and γµν are the Bohr frequency and the relaxation rate of the coherence. Here, δΓ (1) (0) is defined by δΓ = δΓ (1) + (δΓ e−iωt + δΓ (−1) (1) eiωt ) + · · · , and the same for δEv and (1) R(rp) . We define the amplitude of the coherence, ρeµν , by ρµν = ρeµν (t)exp(−iωt), and ω (1) is the frequency near to ωµν . In Eq. (2.32), the light shift term Ev introduces a quadrature complex phase into the coherence. This causes the phenomenon of the asymmetric line (1) shape, which will be discussed in section 2.4.6. The repopulation term R(rp) can also introduce a quadrature complex phase into the coherence from the “real ”-transition light shift. The difference between these two light shifts to the coherence is that the virtual light shift requires a population difference (ρµµ − ρνν 6= 0), but the real light shift does not need the population difference. Optically induced atomic coherence is the central part 21 of coherent population trapping (CPT) and the electromagnetically induced transparency (EIT). More detail will be discussed in later sections. 2.1.5 Microwave and RF Fields Traditionally, we use microwave or RF (radio frequency) fields to excite the coherences of hyperfine sublevels if the frequencies of the oscillating fields match to the hyperfine splitting or Zeeman splitting. Assuming the oscillating field B(t) = Bof (eiωt + e−iωt ), we find the interaction Hamiltonian, (1) Hof = B(t) · (µB gS S − µI I/I). (2.33) For two different hyperfine sublevels |µi and |νi, the Rabi frequency produced by the oscil(1) lating field is equal to ωR = h̄2 hµ|Hof |νi. For a quasi-steady solution, we find the amplitude of the coherence due to the oscillating field is given by ρeµν (t) = iωR /2(ρµµ (t) − ρνν (t)) . i(ωµν − ω) + γµν (2.34) The rate equation of the density matrix due to the oscillating field in the quasi-steady state is described by ½ ρ̇of = ¾ δEof [(ρµσ |µihσ| − ρνσ |νihσ|) − (ρσµ |σihµ| − ρσν |σihν|)] + ih̄ {−Γof [(ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|]} , (2.35) where σ 6= µ and σ 6= ν. The AC Stark shift energy Eof and the effective pumping rate Γof are defined by δEof = 2 /4(ω h̄ωR µν − ω) , 2 2 (ωµν − ω) + γσ(ν,µ) Γof = 2 /2γ ωR µν . 2 (ωµν − ω)2 + γµν (2.36) From Eq. (A.37), one can see that the oscillating field tends to equalized the population of |µi and |νi. 22 2.1.6 Summary A complete evolution equation of the density matrix is summarized by ρ̇ = ρ̇d + ρ̇sd + ρ̇ex + ρ̇C + ρ̇of + ρ̇dp + ρ̇rp = (2.37) 1 1 1 1 1 [ H (0) , ρ ] + [ δEex , ρ ] + [ δEC , ρ ] + [ δEof , ρ ] + [ δEv , ρ ] ih̄ ih̄ ih̄ ih̄ ih̄ η 2 [I]2 (m) +Γd (1/g − ρ) + Γsd (ϕ − ρ) + Γex [ϕ(1 + 4hSi · S) − ρ] − ΓC I ρ 8 n o 1 (rp) + {−Γof [(ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|]} − {δΓ, ρ} + Rαβ,µν ρµν |αihβ| . 2 In the quenching dominated repopulation, we find the longitudinal evolution equation (ρ̇αα = Rαα,µν ρµν ) as ρ̇L = ρ̇d + ρ̇sd + ρ̇ex + ρ̇op + ρ̇of = Γd (1/g − ρ) + Γsd (ϕ − ρ) + Γex [ϕ(1 + 4hSi · S) − ρ] + Re(Γop ) [ϕ(1 + 2s · S) − ρ] + {−Γof [(ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|]} . (2.38) Note: The density matrix evolution due to the oscillating field in Eq. (2.37) and Eq. (2.38) is only correct for the late-time evolution after the field turns on. 23 Excited state Turn on the oscillating field a coherence pumping light b Ground state oscillating fields b Ground state Transmission a 0 Frequency detuning of the oscillating field Figure 2.2: The population of the sublevel b is pumped out. When a oscillating field turns on at the resonant frequency, the coherence is generated. Some population transfers from the sublevel a to b, and therefore it increases the light absorption or decrease the light transmission. 2.2 Microwave Spectroscopy and CPT Spectroscopy Optical hyperfine spectroscopy uses a probing or pumping light to detect the atomic coherence. The on-resonance signal is observed as a transmission dip or a transmission peak of the probing light. 2.2.1 Coherence Induced by Oscillating Fields According to Eq. (2.34), the strongest ground-state coherence can be generated when a oscillating magnetic field couples to the two of hyperfine sublevels, and its oscillating frequency matches to the splitting frequency of the two sublevels. For the steady solution, the oscillating field not only produces the coherence, but also change the relative population of the two sublevels. From the pseudo-spin picture, the oscillating field rotates the pseudo spin from the longitudinal direction to the transverse plane. The coherence is equivalent of the component of the pseudo spin on the transverse plane. More discussions of pseudo-spin picture is in section 2.4.6. To simplify the discussion, we approximate a multi-level problem to a two-level system. Assuming that we label the two selected sublevels as “a” and “b”, 24 from Eq. (2.38), we find ρ̇a = Υa − Γa ρa − Γof (ρa − ρb ) ρ̇b = Υb − Γb ρb − Γof (ρb − ρa ) , (2.39) where Υa and Υb are the equivalent transfer rate from other states to state a and b; Γa and Γb represent the total relaxation rates of state a and b. By using the definition of Γof (the equivalent pumping rate of the oscillating field) in Eq. (A.39), we can find the steady solution by solving ρ̇a = ρ̇b = 0. Therefore 2 −1 b )−(1+Γa /Γb )Υa ](Γa Γb ) ρa = ωR γab [(Υa +Υ + (ω −ω)2 +γ 2 +(Γ−1 +Γ−1 )γ ω 2 ab ρb = ab a b ab R 2 γ [(Υ +Υ )−(1+Γ /Γ )Υ ](Γ Γ )−1 ωR a a a b ab b b b 2 +(Γ−1 +Γ−1 )γ ω 2 (ωab −ω)2 +γab a ab R b + Υa Γa , (2.40) Υb Γb where ωab and γab are the Bohr frequency and decay rate of the coherence. Equation (2.40) can be written into a more general form (ρa or ρb ) = 2A ωR 2 + T /T ω 2 + B, (ωab − ω)2 + γab 1 2 R (2.41) −1 where A and B are some numbers; T1 = Γ−1 a + Γb and T2 = 1/γab . RF or microwave spectroscopy is to selectively probe(or pump out) the sublevel a or b, the light absorption is proportional to the population on the selected sublevel. An illustration is shown in Fig. 2.2. According to Eq. (2.41), the absorption curve is a Lorentzian function. The peak absorption occurs when ω = ωab , and the linewidth is equal to ∆ν = q 2 + T /T ω 2 (FWHM). The second term inside the square root is called the power π −1 γab 1 2 R 2 is proportional to the power of the oscillating field. broadening effect, since ωR Although Eq. (2.41) is only an approximated result, it has a good physics picture. For most of cases, it is a good approximation. For a precise calculation, the absorption is proportional to Tr(δΓρ), where ρ is the steady solution of Eq. (2.38). The calculation can be easily carried out numerically. 2.2.2 Optically Induced Coherence Unlike microwave spectroscopy, CPT spectroscopy uses modulated light to excite the coherence. To simplify the discussion, we consider two monochromatic optical fields, which 25 Figure 2.3: Two monochromatic fields with frequencies Ω1 and Ω2 couple between two groundstate sublevels and the excited state. Two optical coherences between the ground state and the excited state are generated. When the difference of two optical frequencies is equal to groundstate sublevel splitting, a strong ground-state coherence can be produced. Under the same condition, photons begin scattering between two monochromatic fields, and are less absorbed by atoms. The light absorption is therefore reduced and observed as a transmission peak. When the optical detuning is present, it is equivalent to stimulated Raman scattering. couple the ground-state sublevels to the excited-state sublevels, each optical field can induce the coherence between the corresponding ground-state sublevels and the excited state. When the frequency difference of the two monochromatic fields matches to the splitting frequency of the two ground-state sublevels, a ground-state coherence can be produced through a process of two-photon transitions. The CPT phenomena can also be represented by the picture of pseudo spin, which is described in section 2.4.6. Assuming there is no optical detuning and the two sublevels are optically resolved, from Eq. (2.22), we can find the relevant time-dependent pumping rate to be (0) (1) Γop (t) = Γop + Γop (eiωt + e−iωt ), (2.42) (0) (1) where ω = Ω1 − Ω2 , the difference of two optical frequencies, and Γop = Γop = Dop is a real number. We can therefore find the light absorption rate for D1 pumping, which is equal to the total depopulation rate. (0) (0) (1) (1) Tr(δΓρ) = Tr(δΓ ρ ) + Tr(δΓ ρ ) (0) (1) (0) = Dop (1 − 2s · hS i − 2s · hS i) = Dop (1 − s · P (1) − s · P ). (2.43) 26 (0) Here, we use superscript (0) to label the static components, where ρ (0) sity matrix with only diagonal elements, and P represents the den- (0) = 2hS i represents the static spin(1) polarization. We use superscript (1) to label the oscillating components, where ρ (1) sents the density matrix with only off-diagonal elements, and P repre- (1) = 2hS i represents the amplitude of the oscillating spin-polarization. The oscillating spin comes from the atomic coherence. From Eq. (2.32), we find the amplitude of the oscillation spin-polarization to be (1) P = γµν N Dop (|s · Pµν | ŝ), 2 (ωµν − ω)2 + γµν (2.44) where N = ρµµ + ρνν is the total population of the two sublevels, Pµν = 2hµ|S|νi is the equivalent spin-polarization of the coherence, and ŝ is the unit vector along the direction of the photon spin. D1 optical pumping tends to generate the electron spin aligned to the photon spin. The expression (1 − s · P (0) (1) − s · P ) appeared in Eq. (2.43) is usually called the “specific absorption”. Therefore, the intensity of the transmission light affected by the (1) oscillation polarization is proportional to s · P . By checking Eq. (2.44), we find there is a maximum transparency when ω = ωµν . This is the basic principle of CPT spectroscopy. An illustration is shown in Fig. 2.3. People usually like to call this phenomena as coherent population trapping (CPT), because when the probing light reach the peak transmission, lots of atoms tend to be trapped into the superposition state, the dark state or non-coupled state. The two-wave optical pumping is called Λ configuration, because of its look. In Fig. 2.3, the dark superposition state is written by α|1i + β|2i, and we find R1 α + R2 β = 0 (The transition amplitude to the excited state is zero). Here, R1 and R2 are the Rabi frequencies or the coupling strengths of the two optical transitions Ω1 and Ω2 . The socalled bright state or coupled state therefore is β|1i − α|2i. The two-wave pumping light pumping out the atoms in bright state and produce atoms in dark state. Figure 2.4 shows the spin oscillation picture to represent the CPT excitation. The 0-0 CPT coherence is equal to the spin oscillation along the z-direction. The end CPT coherence is equal to the spin precession on the x-y plane. The period of the spin oscillation or precession is equal to the inverse of the splitting frequency of the two sublevels. For the conventional 0-0 CPT, there are two dark states, 0-0 superposition state and the end state. When the ) ( ' & ! " . / 1 0 ) # ( ' ! " & ) , ' & $ +& %!"' ) * 27 Figure 2.4: We use a hypothetic atom, which has a nuclear spin I = 1/2, to illustrate the CPT coherence in the picture of spin oscillation. Different Λ schemes generate different ground-state coherences. The coherence of 0-0 CPT is equivalent to the electron spin oscillating along the z-direction. The end-state coherence is equal to spin precessing on the x-y plane. The dotted circles represent the dark states for different CPT schemes. The 0-0 CPT excitations from σ+ and σ− pumpings have 180◦ phase difference. pumping intensity increases, atoms prefer to be trapped into the end state. This causes 0-0 CPT signal to be weak. For the end-resonance CPT, there is only one dark state, the end state. Therefore, we can obtain a stronger signal from end-resonance CPT. A new pumping method, push-pull optical pumping, can solve the problem of the conventional 0-0 CPT, which will be discussed in section 2.4.4. In the discussion above, we ignore the effect of the optical detuning, optical pressure broadening, multiple optical sidebands from the modulated light, and the repopulation pumping. The optical detuning causes virtual light shift and therefore introduce an extra phase to the coherence. It turns out the line shape of the CPT signal to be asymmetric. The optical pressure broadening and the optical sidebands can affect both the line shape and the strength of CPT excitation. The repopulation pumping has only negligible influence to the ground-state microwave coherence when the buffer gas pressure is lower than 10 atm 28 (depends on the real quenching and J-damping rate). The detailed discussions will be in the later sections. For real cases, a numerical calculation is required to precisely calculate the CPT phenomena. 2.2.3 Inhomogeneous Linewidth Broadening Assuming the line shape of the resonance signal from microwave or CPT spectroscopy is described by a Lorentzian function, and the linewidth broadening is due to the intrinsic decay rate, we find Resonance Signal = (ωµν Aγµν + B, 2 − ω)2 + γµν (2.45) where A and B are two constants. We usually measure the linewidth in FWHM (full width in half maximum). Therefore, the linewidth ∆ν = γµν /π in Hz. From Eq. (2.37), we know that the resonant frequency can be shifted by different influences, such as magnetic field, temperature, pressure, light intensity, etc. Atoms in the different positions of the light path can have different frequency shifts due to the inhomogeneities from different causes. The real resonance signal is a convolution of the Lorentzian function with the distribution of the central frequency. Thereby, the effective linewidth can be much broader. This is called the inhomogeneous broadening. We can consider a 1st order approximation. Assuming the central frequency of the resonance is spread uniformly from −∆/2 to ∆/2, the new line shape becomes · µ ¶ µ ¶¸ (ωµν − ω) + ∆/2 (ωµν − ω) − ∆/2 A arctan − arctan +B. Resonance Signal = γµν ∆ γµν γµν (2.46) We find that the FWHM linewidth ∆ν = (γµν + ∆)/π for this inhomogeneous broadening. When ∆ −→ 0, Eq. (2.46) turns back to Eq. (2.45). Therefore, if there is a magnetic field gradient, a temperature gradient, and a light intensity gradient, we can find that the inhomogeneous broadening caused by these three different sources is given by ∆ν 0 = 1 π µ ¶ ∂ωµν ∂ωµν ∂ωµν ∆B + ∆T + ∆I , ∂B ∂T ∂I (2.47) 29 where ∆B, ∆T , and ∆I represent the total difference of the magnetic field, the temperature, and the light-intensity. The total line-width becomes ∆ν = γµν /π + ∆ν 0 . The temperature gradient can cause a inhomogeneous broadening, because the pressure shift is temperature dependent. The related subject is in section 2.4.3. 30 Figure 2.5: Lollipop (left) and rectangle (right) cells. 2.3 Experiments In order to do the microwave and CPT spectroscopy of the alkali-metal atoms, we prepared cells filled with different alkali-metal and the desired gas mixture. The alkali metal was released by using a gas-oxygen torch to heat alkali-chloride salt, mixed with small chips of calcium metal, in an evacuated glass retort that was attached to a glass tube, which connected with glass cells. A torch was used to distill the free metal into the cells. After the metal had been distilled into the cells the buffer gas, for example, nitrogen and argon, was first mixed and then condensed into the sample cells. Details of cell preparation method is described in Ref.[25]. By using that method, we were able to make cells with pressures from zero to several atmospheres. We used spherical cells and two different flat cells, the “lollipop” type and the rectangle type. Lollipop cells were assembled by two circular Pyrex glass plates in parallel. The separation of two glass plate was about 2 mm. Rectangle cells were made from a piece of rectangular Pyrex tubing, which was 1 cm wide and 1 mm thick, and connected with a ballast compartment. Figure 2.5 shows a picture of the two different M]N^OdicPkh_eQP̀RoljfgnmSTUaVWX>YZ;[R<JI?\K F:@AIG@HL(7)E$89%&'#=AB":!;D*4<+5,Cp-.bq6rs/0123 31 Figure 2.6: Experimental setup for microwave spectroscopy. types of cells. The prepared cells were eventually placed into an oven, which was made by using O-10 plastic material as a frame and 1 cm thick glass plates as window walls. The internal space of the oven was heated by the hot air flow. A feedback loop constructed from a thermal sensor and an OMEGA temperature controller (CN76000) stabilized the oven temperature. We could adjust the oven temperature from room temperature to 160◦ C to obtain different vapor density inside the cell. 2.3.1 Microwave Setup Figure 2.6 shows the apparatus for the experiments with microwave spectroscopy. We used three Helmholtz coils to compensate the external field in x- and y-direction and also produce the Bz field. Each coil was about 70 cm in diameter and had about 200 turns of enamelled wires. The Bz field was adjustable to ±15 G. On the z coil, we added two extra coils, not shown in Fig. 2.6. One is for compensating the field gradient along the z-direction, and the 32 other was to cancel the 60-Hz field. A commercial 60-Hz cancellation system suppressed the low frequency AC field actively by using an inductive coil, which was wrapped on the two sides of the oven, to sense the field strength and feedback to its controller circuit. The controller then generated a compensated field by sending currents to the cancelling coil. In order to reduce the influence of the 60-Hz field, we also shut down the electrical power for nearby room outlets. This minimized the amount of the AC current flowing around the room. After these two steps, we reduced the AC field amplitude to a value less than 30 µG. The microwave field was generated by a microwave horn. Two horns, rated at 6.8 GHz and 9.2 GHz, were used in the experiments. The horn was connected to a frequency synthesizer (HP83752B) with a high frequency coaxial cable. We used two laser systems as the pumping or probing sources, a Ti-sapphire ring laser system (Coherent 899-29) and a diode laser system (Toptica DL100). The ring laser was fiber coupled from the next room. It had higher optical power (up to 1 watt) and wider tuning range (from 785 nm to 910 nm). The diode laser has lower power (up to 50 mW), shorter tuning range (∼ 50 GHz), narrow linewidth, and much less optical noise. We had two diode laser heads for 795 and 894 nm experiments. The precise laser frequency was measured by Burleigh WA-1000 wavemeter. The probing light was linearly polarized first, and we used a quarter-wave plate to switch the light polarization between linear and circular. A small fraction of the light was sent to a background photodetector before the quarter-wave plate. The probe beam passed through the cell and was focused on the transmission photodetector. A divider circuit was made for reducing the background noise level, which is mostly due to the intensity fluctuation of the probing light. The circuit was composed of two commercial ICs, TL082 and AD734. Two photodetectors detected the light intensities before and after propagation through the vapor cell. These two signals were pre-amplified by OP-Amps TL082 and then sent to the analog divider AD734, which generated an output a voltage signal proportional to transmission/background. In this way, the common mode noise of the probing light can be cancelled out to a certain degree, as shown in Fig. 2.7. The output signal from the divider circuit was connected to a voltage amplifier (SRS SR560). Its output was linked 33 10 -3 8 6 Raw transmission signal Divider output 4 Noise level (A.U.) 2 10 -4 8 6 4 2 10 -5 8 6 4 0 100 200 300 Frequency (Hz) 400 500 Figure 2.7: The noise spectrum of the transmission light in the experiments. Our homemade divider circuit can suppress the background noise level by 15dB, and can increase the SNR by a factor of 6 ∼ 7. This divider circuit was especially useful when we used the ring laser as the pumping source. Because the laser was conducted by a long fiber (20 m), lots of mechanical vibration noise was converted into the intensity fluctuations of the output light. to a lock-in amplifier (SRS SR530) and to the computer’s A/D converter (NIDAQ 6052E). The output of the lock-in amplifier then connected to a digital oscilloscope (TDS 3304). All of instruments with available GPIB interfaces were linked together with a PC computer. The job of instruments control, data acquisition, and data processing was done by IGOR program installed in the computer. Signal Measurement of Microwave Spectroscopy We chopped the microwave field at a frequency between 25 to 100 Hz and used the lock-in amplifier to read out the steady intensity of the transmission light at different microwave detuning. Because of the low chopping frequency, we chose stepping the frequency instead of continuously scanning. The computer A/D interface was used for reading the transient signal when the microwave field was switching on and off. We applied a differential input method on the A/D interface to reduce the electrical noise. The transient signal was 34 microwave OFF microwave ON A.U. 10 Damping oscillation 0 -10 Transient trace 0 10 20 30 40 msec Raw transient data Baseline fitting curve After subtraction of the baseline A.U. 20 10 0 512 Average 0 A.U. 3.0 1 2 3 fc = 6.825397600 GHz Offset = 16.7 ± 0.3 kHz Linewidth = 2.2 ± 0.1 kHz 2.0 4 msec Fourier's spectrum of the modified transient trace Lorentzian fitting curve 1.0 0.0 A.U. 8 2.0 1.5 1.0 0.5 0.0 10 12 14 16 18 20 22 24 fc = 6825381.500 kHz Offset = 0.67 kHz Width = 2.96 kHz -10 -5 26 kHz Lock-in signal Lorentzian fit 0 Microwave detuning (kHz) 5 10 Figure 2.8: The top panel shows the transient of the transmission light when the microwave field was on and off. The second panel is the zoom-in of the circle on the top panel. The third one is the Fourier’s spectrum of the damping oscillation. The bottom one shows the same resonance measured by scanning the microwave detuning (two sec for each point). Its linewidth is broader due to the microwave power broadening. triggered in synchronism with the chopping frequency. Therefore, we were able to do the signal averaging to increase the SNR. To measure the equivalent resonance linewidth, we subtracted baseline from the transient data and then Fourier’s transformed it to a spectrum. In Fig. 2.8, the transient method took about 10 second, but the detuning scanning took about 2 minutes. For the linewidth measurements, we prefer to use the transient method. It needs less data acquisition time and had no problem of microwave power broadening. 2.3.2 CPT Setup For CPT spectroscopy, we kept the same experimental setup and added a few parts in addition to those sketched in Fig. 2.6. In order to do the light modulation, we used a 4 ) + , ( * 2 . / 0 1 3 %&' $ !"# 35 Figure 2.9: The probing beam for CPT experiments was generated by a Mach-Zenhder Mod- ulator. By modulating the effective light propagation lengths from two crystals, the output beam can be intensity modulated. The optical sidebands of the probing beam can be measured from a Fabry-Perot spectrometer. Mach-Zehnder type intensity modulator (EOspace, Model AZ 0K1-12-PFU-SFU-800). Its modulation frequency can reach up to 20 GHz. The modulator worked by splitting the input beam and sending each sub-beam through parallel electrooptic crystals (LiNbO3 ). The refraction indices of the crystals were controlled by a DC-biased electric field and the microwave electric fields. The control signal was arranged to generate an opposite optical phase retardation from each crystal. Two sub-beams then recombined to produce the intensity modulation from the interference. The modulation envelope is proportional √ to sin(φ0 + φ1 sin ωm t), where φ0 ∝ V0 (bias voltage), and φ1 ∝ Pm (square root of microwave power). Here, ωm is the modulation frequency. We used an advanced microwave source (Agilent E8257C) for light modulation. We first coupled the laser beam into a singlemode, polarization-maintaining (PM) fiber, which connected to the modulator. The input laser beam was linearly polarized. The orientation of linear polarization should align to the main optical axis of the PM fiber, because the modulation device was only designed 36 60 40 AU 20 0 -20 Ramp signal End resonance CPT signal 0-0 resonance CPT signal -40 -60 0 2 4 6 Scanning time (sec) 8 -3 10x10 Figure 2.10: Two example CPT signals were found by setting the central frequencies of the modulation to their resonant frequencies. The scanning range was ±50 kHz. The ripple tails were due to the fast scanning, which can be eliminate by slowing down the sweeping speed or shorten the scanning range. The end-resonance signal was obtained by setting the probing beam with 45◦ to the magnetic field. for the selected polarization. We used a single-mode fiber to couple the light out from the modulator. A fraction of the output beam was sent to a Fabry-Perot spectrometer (Coherent Model 240), which had 30 GHz free spectral range, for monitoring the modulation pattern. Finally, the modulated beam was guided passing through the sample cell. The detailed schematics is shown in Fig. 2.9. To measure CPT signals, we directly used the nice internal function of frequency scanning of the synthesizer. The CPT resonance signal was observed as a transmission peak when the modulation frequency or its integer multiple swept through the resonant frequency. The example results from a fast ramping rate is shown in Fig. 2.10. To record the signal traces, the data acquisition interface was triggered by the ramping signal. The trace average was required if the SNR is poor. Unlike the microwave spectroscopy, for the end-resonance CPT, a angle between the probing beam and magnetic field was needed to excite the coherence. 37 2.4 Results and Analysis We carried out a series of experiments, including the examination of light narrowing effect, the measurements of the pressure-dependent hyperfine relaxation rates, the idea of push-pull optical pumping, and the line shape of CPT resonances. The theoretical calculations for the evolution of the density matrix were carried out by the MATLAB programming. We compiled a few individual MATLAB codes to numerically compute the results of microwave and CPT spectroscopies, including all experimental observations. Some further studies were also included, such as the transient analysis, CPT excitation with different modulation schemes and at different buffer gas pressure, and the photon costs for optical pumping. 2.4.1 Light Narrowing Effect The light narrowing effect is a narrowing of the magnetic resonance or CPT resonance linewidth caused by increasing the intensity of pumping or probing lights. The basic idea is to use the optical pumping to change the hyperfine population and therefore suppress the spin-exchange broadening. Assuming a magnetic field is present to define the z-direction, we can calculate the total relaxation rate with D1 pumping by using Eq. (2.37). γµν η 2 [I]2 |∆f | + (Γsd + Γex ) = Γd + ΓC I 8 µ ¶ 3 − hµ|Sz |µihν|Sz |νi + Dop − Rop 4 (0) −(sz (Dop − 2Rop ) + hSz iΓex )(hµ|Sz |µi + hν|Sz |νi) − 4Rop hµ|Sz |µihν|Sz |νi ! à X (2.48) hµ|S|αi × hα|S|νiραα , −Γex hµ|S|νi · hµ|S|νi(ρµµ + ρνν ) − 2i α (0) where the effective pumping rate for depopulation and repopulation are Dop = Re(Γop ) and Rop = (0) Γ2q 1 4 ∆ω 2 +Γ2q Re(Γop ), (g) (e) where ∆ω = ωhf − ωhf , and Γq is the quenching rate. Here, (0) hSz i denotes the static part of the longitudinal electronic spin. By definition, the intrinsic linewidth is equal to γµν /π. According to the Eq. (2.48), the relaxation rate due to the spin-exchange mechanism is a function of the population of the hyperfine sublevels. Since the population of hyperfine sublevels is affected by the optical pumping, when the spin exchange dominates the linewidth broadening, the linewidth is no longer a linear function 38 of the optical pumping rate. For further discussions, we focus on three selected resonances: 0-0 resonance, |a, 0i ←→ |b, 0i and |∆f | = 1. Therefore, η 2 [I]2 3 Γ00 = γa0b0 = Γd + ΓC I + (Γsd + Γex ) + Dop − Rop (2.49) 8 4 · ¸ 1 [I] + 2 [I] − 2 −Γex (ρa0a0 + ρb0b0 ) + (ρa1a1 + ρa−1a−1 ) + (ρb1b1 + ρb−1b−1 ) . 4 8[I] 8[I] End resonance, |a, ai ←→ |b, bi and |∆f | = 1. Therefore, µ ¶ η 2 [I]2 3 ([I] − 1)2 − 1 Γend = γaabb = Γd + ΓC I + (Γsd + Γex ) + + Dop − Rop 8 4 4[I]2 P Γex /2 + sz (Dop − 2Rop ) ([I] − 1)2 − 1 − + 4Rop [I] 4[I]2 · ¸ (2[I] − 1)2 − 1) 1 2[I] − 1 −Γex ρ + ρ ) . (2.50) (ρaaaa + 4[I]2 2[I] bbbb 2[I] abab Zeeman end resonance, |a, ai ←→ |a, bi and |∆f | = 0 . Therefore, µ ¶ 3 ([I] − 1)2 − 1 ΓZend = γaaab = Γd + (Γsd + Γex ) − + Dop − Rop 4 4[I]2 (P Γex /2 + sz (Dop − 2Rop ))([I] − 1) ([I] − 1)2 − 1 − 4Rop − [I] 4[I]2 ¸ · 2[I] − 1 1 1 (ρaaaa + ρbbbb + ρ ) . (2.51) −Γex [I] 2[I] 2[I] abab (0) In Eq. (2.50) and Eq. (2.51), P = 2hSz i represents the static electronic polarization. When there is no or only a weak oscillating field, the steady electronic polarization can be calculated by solving Tr(Fρ̇) = 0. Hence, using Eq. (2.38), we got d sz hFz i = −Γd hFz i − Γsd hSz i + Dop ( − hSz i) = 0. dt 2 (2.52) By using the spin-temperature approximation [3], we find hFz i/hSz i = 1 + ²(I, P ), where ²(I, P ) = hIz i/hSz i is a function of the nuclear spin and the electronic polarization. Therefore, P = sz Dop . Dop + Γsd + (1 + ²)Γd (2.53) Equation (2.53) is a very good approximated result for Γd 6= 0. When there is no diffusion relaxation and the oscillating field (Γd = 0 and Γof = 0), the spin-temperature distribution is the exact solution of Eq. (2.38), and therefore Eq. (2.53) is exact. 39 The spin-temperature distribution describes the population of the hyperfine sublevels by ρ= eβIz eβSz eβFz = . Z ZI ZS (2.54) Here β is the spin temperature parameter, and it can be determined by β = ln( 1+P 1−P ). For P any spin number J, ZJ = Jm=−J eβm . By carefully inspecting Eq. (2.50) and Eq. (2.51), we also find that the repopulation pumping rate Rop actually plays no role if the photon spin sz = 1. In this situation, the state |a, ai is a dark state of D1 pumping, and ground states |a, bi and |b, bi are pumped to |f = a, m = ai of P1/2 excited state. No end-state coherence can be generated in the excited state, and therefore no coherence with the right phase can be repopulated to the ground state. If sz = 1, and using spin-temperature distribution, we can further reduce Eq. (2.50) and Eq. (2.51) into Γend µ ¶ ηI2 [I]2 1 + (Γsd + Γex ) 1 − + Dop = Γd + ΓC 8 2[I] P Γex /2 + Dop (2[I] − 1)2 − 1) − Γex − Qm̄ (P ), [I] 4[I]2 (2.55) and µ ΓZend ¶ 1 1 = Γd + (Γsd + Γex ) + + Dop 2 2[I] (P Γex /2 + Dop )([I] − 1) 1 − − Γex Qm̄ (P ). [I] [I] (2.56) Here Qm̄ = eβ m̄ /ZI , and m̄ = I for the end state. Considering the spin-exchange broadening, we find that when the population is uniform in the ground state, the total relaxation rates contributed by the spin-exchange rate are 58 Γex for 0-0 resonance, (1 − 3 2[I] + 1 )Γex [I]2 for end resonance, and ( 12 + 1 2[I] − 1 )Γex [I]2 for Zeeman end resonance. We also find the minimum values of the spin-exchange rate. They are 12 Γex for 0-0 resonance when all population is in 0-0 states, and no spin-exchange broadening for both end resonance and Zeeman end resonance when all population is in |a, ai state, which is the same as P = 1. By using Eq. (2.53), Eq. (2.49), Eq. (2.55) and Eq. (2.56), we plot the relative linewidths of three different resonances as a function of the relative pumping rate in Fig. 2.11. For comparison, we use D1 pumping with sz = 1 and assume that I = 3/2, 40 Relative linewidth ( Γ / Γsd ) 120 D1 pumping, and sz = 1 Nuclear spin I = 3/2, and Γex/Γsd = 100 100 80 60 0-0 resonance Hyperfine end resonance Zeeman end resonance 40 20 0 0 5 10 15 Relative pumping rate ( Dop / Γsd ) 20 Figure 2.11: The light narrowing effect is shown by two different end resonances. The linewidth gets narrower by increasing pumping power at beginning, because the optical pumping traps more atoms into the end state. The minimum point is where the spin-exchange broadening is comparable to the pumping broadening. The linewidths start increasing after the minimum point, because the optical pumping start dominating the linewidth broadening. However, the circularly polarized pumping can only make the linewidth of 0-0 resonance worse. Γex = 100Γsd , and there are no other relaxation mechanisms. We can see a strong light narrowing effect for two different end resonances. Optical pumping increase the atomic polarization and traps more population into the end state. Therefore, it suppresses the spin-exchange broadening to end resonances. For 0-0 resonance, the spin-polarized optical pumping can only make the linewidth broader, because the optical pumping evacuates the population from 0-0 states and its adjacent states. This disadvantages the conventional 0-0 CPT atomic clocks, which use a fixed circularly polarized laser beam to excited the CPT resonance. Because the 0-0 state population is also evacuated, the signal is much weaker. However, if we use push-pull optical pumping, which will be discussed in the later section, the population can be concentrated into 0-0 state. We can obtain a strong CPT resonance signal, and the maximum suppression of the spin-exchange broadening is from 58 Γex to 12 Γex (20% suppression). 41 87 12 10 Experimental data, -3 8 x10 End resonance linewidth (kHz) o Cell T108 (N2 700 torr + Rb ) at 140 C, thickness 2 mm Laser frequency 377104 GHz, Beam DIA ~ 6mm Microwave fc = 6.8253815 GHz, Rabi freq. ~ 110 Hz Bz ~ 4.6 Gauss 14 6 Fitting curve 10 5 0 -5 -10 60 40 20 0 -10 -5 0 kHz 5 -10 -5 4 0 kHz 5 2 0 1 2 3 4 5 6 7 2 Laser power ( mW/cm ) Figure 2.12: The first light narrowing data of 87 Rb hyperfine end-resonance was found from the high pressure lollipop cell. The data agrees with the theory very well. Two insets show the resonance signals from the detuning scan. The scanning step was 400 Hz. The linewidth decreases with increasing pumping power to a minimum value. After this point, the linewidth begins to be dominated by the pumping rate. However, the signal amplitude is always improved by the pumping power. Figure 2.12 shows an experimental result of light narrowing effect by using microwave spectroscopy. By using Eq. (2.55), we can fit data to the theoretical curve very well. At low laser power, the pumping was too weak to pumping atoms to the end state, the spin-exchange broadening dominates the linewidth. By increasing the pumping power, we observed a strong narrowing effect and also more intense signals. From examination of p Eq. (2.55), one can show that the minimum linewidth is given by ∆ν = c Γex Γ∗sd +constant, where c ≈ 0.4 slightly depends on P and I, and Γ∗sd = Γsd + (1 + ²)Γd . Here, the constant contribution to linewidth broadening are due to other relaxation mechanisms and also the inhomogeneous broadening. Comparing the data point with the narrowest linewidth to the initial value of the linewidth, we obtained about 1/5 factor of the narrowing, and also the 42 SNR was increased by a factor more than 3000. The equivalent FOM (figure of merit) = SNR/∆ν is increased by a factor more than 15000. These enhancing factors are not absolute but depend on the spin-exchange rate. Higher temperature, which has higher spin-exchange rate, has larger enhancing factors. The light narrowing of the end resonances is a result of the conservation of the angular momentum. This special effect will possibly benefit the design of small vapor-cell atomic clocks. For a miniature cell atomic clock, higher working temperature is required to obtain more light absorption. The spin-exchange rate is increased by the temperature. By using spin-polarized optical pumping, the end resonances not only eliminate the spin-exchange broadening, but also have stronger resonance signal, because the population is trapped to the end state. The resonance frequency of the end resonance is more sensitive to the magnetic field than the 0-0 atomic-clock resonance. For constructing an end-resonance atomic clock, an additional field-stabilizing feedback loop is required. We propose a “double lock” method. By using both hyperfine and Zeeman end resonances, we will be able to lock the hyperfine frequency and the magnetic field [29, 56]. Our collaborators in Sarnoff corporation have successfully demonstrated the double-lock technique. 2.4.2 Transient Analysis In microwave spectroscopy, we measure the steady intensity of the transmission light as a function of the frequency detuning of the microwave field. If we suddenly turn on the microwave field and look at the transient of the transmission light, we would expect to see the damping oscillation. The damping rate of the oscillation is determined by the longitudinal and the transverse relaxation rates. Equation (2.38) is a quasi-steady solution for the presence of the oscillating field. We can obtain a correct result of late-time evolution of the density matrix from Eq. (2.38), but not the transient at the beginning. To calculate the transient signal, we can solve the evolution equation in Liouville space. Since there is only one coherence excited by the oscillating field, for the ground-state evolution, we only 43 need to consider a Liouville vector with g + 2 components, where g = 2[I]. Hence, ρ11 .. . |ρ) = ρgg ρ̃µν ρ̃νµ . (2.57) Here, the first g components are the diagonal elements of the ground-state density matrix, and the last two components are the selected off-diagonal elements in the rotation coordinate of frequency ω. We define ρµν = ρ̃µν e−iωt , and we know that ρνµ = ρ∗µν . The order of the vector components is not important. The evolution equation is simply as |ρ̇) = −R|ρ), and we find .. . ··· R= ··· 0 0 .. . .. . 0 Γµµ Γµν −iωR /2 0 iωR /2 Γνµ Γνν iωR /2 −iωR /2 . −iωR /2 iωR /2 γ2 + iδω 0 iωR /2 −iωR /2 0 γ2 − iδω (2.58) Here, Γαβ denotes the transition rate from ραα to ρββ . The transverse decay rate γ2 = γµν , and the frequency detuning δω = (ωµν − ω), where ω is the frequency of the oscillating field. Both Γαβ and γ2 can be obtained from Eq. (2.38) and Eq. (2.37) by setting Γof = 0. The spin exchange is the only nonlinear term in the evolution equation. If the temperature is sufficiently low, the spin exchange has small contribution to the relaxation rate. We can treat it as a linear system. Therefore. the evolution the of the vector |ρ) can be acquired by solving the eigenvalues and eigenvectors of matrix R. The complete solution can be obtained by using Eq. (2.11). The analytical result shows that there are g real eigenvalues and 2 complex eigenvalues. These two complex eigenvalues are complex conjugate and govern the damping oscillation on the transient signal. Further analysis shows that when the Rabi frequency is larger than all the relaxation rates, ωR /Γαβ > 1 and ωR /γ2 > 1, the AU 44 Solution of the equation Solution by using eigenvectors 10 5 0 -5 -10 Re(Γ)/π 1.1 1.0 0.9 0.8 Im(Γ)/2π 0 12 8 4 1 2 3 ms 4 5 Solution of the equation Analytical approximation Solution of the equation Analytical approximation -10 -5 0 Detuing (kHz) 5 10 15 Figure 2.13: The top panel shows the calculated transient signal with frequency detuing = 7 kHz, and its multi-exponential background has been subtracted. The middle and bottom panels show the computational results of the complex damping rate at different frequency detunings by solving the evolution equation and using Eq. (2.59). The unit in their vertical axes is 1000/sec. The calculations show the distinction between two different methods is only a few parts per thousand. solution of the complex eigenvalue can be very good approximated by r 2 γ2 (δω)2 + 21 (γ2 + γ1 )ωR 2 − 1 (γ − γ )2 , Γ= ± i (δω)2 + ωR 2 1 2 2 4 (δω) + ωR (2.59) where γ1 = 12 (Γµµ + Γνν + Γµν + Γνµ ). This approximation has a better precision when the ratio between Rabi frequency and the relaxation rates becomes larger. For the hyperfine end resonance, γ2 is given by Eq. (2.55), and γ1 is given by µ γ1 = (Γex + Γsd + Γop ) 1 − 1 2[I]2 ¶ − P Γex + Γop + Γd . 2[I] (2.60) In order to understand how accurate the Eq. (2.59) is, we consider a real case of 87 Rb. We choose similar ωR that we used in the lab for transient measurements. We try to make γ1 and γ2 close to ωR to examine the limit of Eq. (2.59). Assuming we have a 4 amagat N2 87 Rb cell at 60◦ C. The approximate rates are Γop = 500s−1 , Γsd = 550s−1 , Γex = 300s−1 , 45 ΓC = 1580s−1 , Γd = 31s−1 , and the Rabi frequency ωR = 2π · 2000s−1 . According to the rates above, we find ωR /γ2 ∼ 3.5 and ωR /γ1 ∼ 5.2 for the hyperfine end resonance. For comparison, computational results are shown in Fig. 2.13 obtained by numerically solving the differential equation and by using the analytical approximation in Eq. (2.59). One can see the results from Eq. (2.59), which is shown as the solid curve, go through the circles, which are the numerical solutions of the evolution equation. The relative error between two different methods is only a few parts per thousand. Usually the experimental errors of the measurements of relaxation rates are more than a few percent. Therefore, Eq. (2.59) provides a quick way to measure the transient damping oscillation. Some relaxation key parameters can be measured by using the transient method if we properly change some relative variables to the relaxation rates. Those key parameters play a crucial part of making an atomic clock. Considering buffer-gas pressure-dependent relaxation rates, Γsd , ΓC , and Γd . We can define the key parameters as relaxation rate per unit atomic number density of the buffer gas. Therefore, Γsd = γsd N, ΓC = γC N, and Γd = γd /N, (2.61) where N is the atomic number density of the buffer gas, which is in unit, amagat [1]. By using Eq. (2.55), Eq. (2.60), and Eq. (2.59), we can find the decay rate as a function of the buffer-gas number density at two frequency detunings for the hyperfine end resonance. Γ(δω=0) Γ(δωÀ0) µ ¶ ηI2 [I]2 1 1 = γd /N + γC N + 1 − − γsd N + const. 16 4[I] 4[I]2 µ ¶ ηI2 [I]2 1 = γd /N + γC N + 1 − γsd N + const. 8 2[I] (2.62) Here, we assume that other contributions to the relaxation rates are independent of N . If the diffusion decay parameter γd is known, we can extract γsd and γC from experiments by fitting the data of pressure-dependent damping rates Γ into Eq. (2.62) at two different detuning frequencies . In practice, we prefer using end-resonances for measuring the S-damping rate and Carver rate. Since these two relaxation rates linearly depend on the buffer gas pressure, we can Re(Γ/π) (Hz) 46 600 550 500 450 400 350 Im(Γ/2π) (Hz) 6.825360 6.825362 6000 5000 4000 3000 2000 6.825364 GHz 6.825366 6.825368 6.825370 6.825368 6.825370 fc = 6.825364474 GHz ωR = 1997 Hz 6.825360 6.825362 6.825364 GHz 6.825366 Figure 2.14: An experimental data of the end-transient damping rates from a 87 Rb cell with 1 atm N2 buffer gas. The data was fit into Eq. (2.59). The Rabi frequency can be extracted from the fitting of the imaginary decay rates. reduce the uncertainties of the relaxation measurements by using high pressure cells. The optical pumping efficiency for end resonances is not affected by the buffer gas pressure. For 0-0 resonance, when the buffer gas pressure is high enough that the splitting of the optical absorption line due to the ground-state hyperfine structure is no longer resolved, the optical pumping becomes poor and can not produce a good signal. The transient method with end resonances can provide a good measurement of the relaxation key parameters within the experimental precision. We applied this method to measure S-damping and Carver rate key parameters of Rb and Cs, which will be discussed in the next subsection. Figure 2.8 shows the real signal of the transient measurement. By using the Fourier’s analysis, we can obtain the real and imaginary damping rates from the linewidth and the offset of the resonance peak on the spectrum. Figure 2.14 shows an example result from a 87 Rb cell with about 1 atm N2 buffer gas. We used Eq. (2.62) to approximately retrieve two key parameters first. After that, we assigned previous obtained numbers into the numerical code to have a better data fitting by solving the evolution equation. Usually, this refinement step did not improve the experimental precision, since we know Eq. (2.59) already has a precision better than 1 % for all our experimental conditions. The only requirement of using Eq. (2.59) and 47 Eq. (2.62) is that the Rabi frequency should exceed all relaxation rates. 2.4.3 Pressure Broadening of Microwave Resonance Gas & Worked by He: J. Vanier et al. [48] D.K. Walter et al.[53] N. Beverini et al. [9] T.G. Walker [51] Our work [38] Ne: J. Vanier et al. N. Beverini et al. T.G. Walker Ar: J. Vanier et al. D.K. Walter et al. N. Beverini et al. T.G. Walker Kr: N. Beverini et al. M.A. Bouchiat et al.[11] T.G. Walker N2 : J. Vanier et al. D.K. Walter et al. N. Beverini et al. Our work [29] (s−1 γsd Rb Exp. Thy. Key Parameters (at 27◦ C) amg−1 ) γC (s−1 amg−1 ) Cs Rb Cs Exp. Thy. Exp. Thy. Exp. Thy. 3093∗ 191 ≈0 8.5 96 26 208 (135) (332) 2828∗ ≈0 151 30 32 637 610 14 ≈0 1270 687 964 21244 28108 24985 27382 3062∗ 395 118 115 (122) 884 (820) (430) (1632) Table 2.1: Experimental and theoretical results of the key parameter of S-damping rate and Carver rate. All numbers listed above have about 10 % errors. For helium gas, the values measured by D.K. Walter were for 3 He. The Carver rates from Vanier’s measurements, which are tagged by “∗”, are much larger than other values in the same block. The numbers from our work are marked by parentheses. For atomic-clock resonances, buffer gas causes frequency shift and ground-state spin relaxation. Both results affect the performance of atomic clocks. By carefully selecting the mixture of the buffer-gas, we can optimize the performance of atomic clocks. There are two 48 relaxation mechanisms, S-damping and Carver rate, causing the pressure broadening of the atomic-clock linewidth. Some prior measurements of these two rates were done by different experiments. Most of measured S-damping rates in different buffer gases are consistent. For Carver rates, however, they were seldom measured and had an inconsistent problem. Before our work, the only two experimental data, which give the information of Carver rates of Rb, were done by J. Vanier and D.K. Walter. Two results have a large discrepancy. We believe that Carver rates measured by D.K. Walter are reliable, but Vanier’s results are flawed. In order to have more comparisons, we carried out the theoretical calculations for He buffer gas and also the experimental measurement for nitrogen buffer gas. We obtained consistent results with Walter. Figure 2.15 shows the transient data of hyperfine and Zeeman end resonances of 87 Rb with different N2 pressures. In Table 2.1, we summarize γsd and γC of Rb and Cs in different buffer gases obtained by different groups. We would expect close values in the same blocks on Table 2.1. However, we find a big difference of Rb Carver rates from Vanier’s measurements. From our further study, we believe that unusual large damping rates measured by Vanier were due to the inhomogeneous broadening caused by the temperature gradients. The temperature shift of the hyperfine frequencies are related to the pressure shift. The perturbation to the hyperfine interactions is a function of both temperature and buffer-gas pressure. Bean and Lambert [7] have studied the temperature-dependent pressure shifts of 23 Na, 39 K, and 85 Rb in the buffer gases He, Ne, Ar, and N2 . According to their results, the frequency shift can be expressed by ∆νs = (ν − ν0 ) = β(T )NB , (2.63) where, ν0 is the zero pressure-shift frequency, ν is the shifted frequency, β is a temperaturedependent coefficient, and NB is the number density of the buffer gas. If we assume the buffer-gas pressure is equal inside a cell everywhere, we find the local number density as NB = PB /kT , where PB is the buffer-gas pressure. There the hyperfine frequency shift due 49 Damping rate, Γ/π (1000/sec) 2.0 Hyperfine end resonance data fitted with: ∆ν = (272 ± 16)⋅[Ν2] + (185 ± 40) + 42/[Ν2] Hz Zeeman end resonance data fitted with: ∆ν = (30 ± 2)⋅[Ν2] + (127 ± 5) + 42/[Ν2] Hz 1.5 Hyperfine end resonance (J. Vanier) Hyperfine end resonance (D. K. Walter) 1.0 0.5 0.0 0 1 2 3 Nitrogen buffer-gas number density [N2] (amg) 4 Figure 2.15: The experimental data of microwave and Zeeman resonance damping rates as the number density of nitrogen. Our experimental data are consistent with D.K. Walter’s measurements, which is plotted as two shadowed areas. The hyperfine resonance data is affected by both S-damping rate and Carver rate. The Zeeman resonance data is only affected by Sdamping rate. J. Vanier’s data predicts a much larger values, which is plotted as a dark-gray curves on the top left corner. to the local temperature becomes ∆νs = β(T )PB . kT (2.64) Here, PB acts as a constant, because of the uniform pressure inside the cell. The buffer gas R pressure can be determined from NT = PB (kT (r))−1 dV , where NT is total atom number, T (r) is the local temperature, and dV is the local volume. By using linear approximation, PB = NT k T̄ /V , where T̄ is the average temperature. By using Eq. (2.47), we know the inhomogeneous broadening cause by the temperature gradient is ∂(∆νs )/∂T ·∆T . Figure 2.16 shows the linewidth broadening of 85 Rb in different buffer gases caused by the temperature gradient. 50 Figure 2.16: Absolute hyperfine frequency shifts of 85 Rb induced by the buffer gases at a constant pressure PB = 760 torr, inferred from the data of Bean and Lambert [7]. Also shown are the spreads in frequency ∆νp that would be caused by a spread in temperature ∆T = 20 ◦ C centered at 30 ◦ C. The line broadening due to temperature inhomogeneities in Ar is about ten times less than that in He, Ne, or N2 . For Vanier’s results, we found that the required temperature gradients for producing their linewidth data were only from 0.6 to 0.8◦ C. By carefully inspecting Vanier’s experiment, they used a big cell about 7 cm long. The cell was heated from its bottom. It was not hard to have such a small temperature gradient in their cell. From Fig. 2.16, we know Ar has a much smaller temperature shift. This explains the consistent results of Ar from Vanier’s and Walter’s measurements. There are still many key parameters have not been determined. For designing vapor-cell atomic clocks, it is important to fill out Table 2.1. This will require future experiments or calculations. 51 2.4.4 Push-Pull Optical Pumping For regular spin-polarized optical pumping, we are able to concentrate the ground-state population to the end-states, |a, ai or |a, −ai, by using a pure circularly polarized pumping light. For concentrating the population to other ground-state hyperfine sublevels, such as m = 0 states, some techniques have been developed, but they are complicated. For example, while the hyperfine sublevels are optically unresolved, we can use regular optical pumping combining with the microwave or RF fields to prepare the population to the selected sublevel, but the static magnetic field must be large enough to resolve the Zeeman resonance between different pairs of sublevels. The oscillating field swaps the population between the sublevels. However, this kind of state preparation is laborious especially when we need to move the population from the far end state to the 0-0 state. Without the oscillating magnetic field, we would be able to concentrate the population to the 0 states, but it requires a resolved excited hyperfine splitting of P1/2 . One can use two linearly polarized pumping beams, which are perpendicular to the magnetic field. The pumping lights connect the ground-state hyperfine sublevels to one of the excited hyperfine multiplet. The transitions, |a, 0i(g) ↔ |a, 0i(e) and |b, 0i(g) ↔ |b, 0i(e) , are forbidden. Therefore, |a, 0i(g) or |b, 0i(g) can become a dark state (unpumped state) if the optical transition occurs only between f = a or f = b of the excited state and the ground state. The population will eventually be trapped into one of the 0 states. Obviously, this method can only be applied in the low buffer gas pressure. We report a new method, “push-pull optical pumping”, which can pump atoms into nearly pure 0-0 superposition state in the very low to very high buffer gas pressure. There are only two essential requirements: (i) the time-averaged optical pumping rate must greatly exceed the ground-state relaxation rates and (ii) the use of alternating polarization of D1 pumping light. This new idea can be understood through the following discussion. Let the 0-0 superposition state be described at time t by |ψi = |a, 0ie−iEa0 t/h̄ + |b, 0ie−iEb0 t/h̄ √ . 2 (2.65) 52 Sz 0 Intensity of σ+ Absorption probability of σ−, Intensity of σ− AU Absorption probability of σ+, 0 0 0 1 2 Period (2π/ω00) 3 4 Figure 2.17: The oscillation of the electron spin produces time dependent absorption cross sections for different circularly polarized pumping lights. The interlacing σ+ and σ− light pulses repeated every 2π/ω00 can maximize the spin oscillation, and therefore trap the atoms to the 0-0 superposition state. At this point, the transmission light has the maximum transparency. Here Ea0 and Eb0 are the energies of the basis states |a, 0i and |b, 0i. The phases of the basis states can be chosen such that ha, 0|Sz |b, 0i = 1/2. Then the expectation value of the longitudinal electron spin is hSz i = hψ|Sz |ψi = 1 cos ω00 t. 2 (2.66) The Bohr frequency is ω00 = (Ea0 − Eb0 )/h̄ = 2π/T00 . The 0-0 superposition state is equivalent to the oscillation of the electron spin along the z-direction at Bohr frequency, which has been shown in Fig. 2.4. In the high-pressure limit, the probability for a spinpolarized alkali-metal atom to be excited by a light pulse is proportional to (1 − 2s · hSi), where s is the expectation value of the photon spin [3]. Then the relative probabilities p+ and p− of absorbing RCP(right circular polarization σ+) and LCP(left circular polarization σ−) pulses by the superposition state of Eq. 2.65 are p+ = sin2 ω00 t 2 and p− = cos2 ω00 t . 2 (2.67) As sketched in Fig. 2.17, we assume that RCP pulses hit the atoms at the times t = 0, T00 , " # ! 53 Figure 2.18: Alkali hyperfine sublevels with a hypothetical nuclear spin quantum number I = 1/2. 0-0 state is the only dark state by using a complementary Λ scheme, which has two Λ pumpings with a 180◦ modulation phase difference between each other. Atoms will be eventually trapped into the 0-0 state, the dark state. 2T00 , etc., when p+ = 0. LCP pulses hit the atoms at the times t = 21 T00 , 32 T00 , 52 T00 , etc., when p− = 0. Atoms in the superposition state |ψi of Eq. (2.65) will absorb very little light from pulses of either polarization. Atoms in any other state will be excited by the pulse train. No matter what the initial state of the atoms is, nearly all of them will eventually be pumped into the superposition state of Eq. (2.65), the “dark state” of the system. The pumping light with alternating circular polarization forces electron spin to oscillate at the alternation frequency along the direction of the light. This is an analogous to “push” and “pull” the electron spin. For low buffer-gas pressures, where the hyperfine structure of the optical absorption lines is well resolved, it is sufficient to use a simple electro-optic modulation of linearlypolarized monochromatic laser light to produce two equal-intensity spectral sidebands with a frequency difference of ω00 . One sideband should be approximately resonant for pumping out of the lower hyperfine multiplet and the second should be approximately resonant for pumping out of the upper hyperfine multiplet. If the resulting “two-wave” light is separated into right- and left-circularly polarized sub-beams, and if one sub-beam is delayed by T00 /2 with respect to the other, the combined beam can pump nearly all of the atoms into the 0-0 superposition state |ψi of Eq. (2.65). In the language commonly used in CPT research, 54 Intensity (AU) Light pulses Time Intensity (AU) Spectrum of light pulses Normailized absorption cross section of 87Rb with 5 atm N2 Normalized absorption cross section of 87Rb with 0.05 atm N2 Optical comb Optical frequency Figure 2.19: The optical sidebands of a pulsed light is like a comb in the spectrum. With high buffer-gas pressure, atoms see all optical sidebands. With low buffer-gas pressure, only two sidebands are seen by atoms. this is equivalent to a combination of two Λ-pumping schemes, one with σ + (RCP) and the other with σ − (LCP) light, where a 180◦ modulation phase difference is maintained between the two as shown in Fig. 2.18. By using pulsed light, push-pull optical pumping can work at any gas pressure. In the optical spectrum, the pulsed train produces widely-spread optical sidebands (as the optical comb shown in Fig.2.19), which is separated by the inverse of the pulsing period. When the buffer gas pressure is low, only two optical sidebands can be seen by the atoms. Therefore, two-wave pumping is sufficient for low pressure regime. Whatever the gas pressure is, the alternating circular polarization of the pumping light is needed. In order to generate the alternating circular polarization, we employed a Michelson type interferometer to convert an intensity modulated laser beam to a polarization modulated probing beam. The detail is shown in Fig. 2.20. The intensity modulated laser beam was linearly polarized before entering the Michelson interferometer. A quarter-wave plate was placed in one of the = = 2 3 79 $(!%'0%)#/. : 1"#!,)+*&- 6> 6 6 8 6 7 5 4 ; < 3 ? 55 + − + / − /2 Figure 2.20: The Michelson type interferometer converts an intensity modulated laser beam to a polarization modulated laser beam. A λ/4 wave plate was needed in one of the arms. The difference of the arms’ length is equal to c/(4ν00 ), where ν00 is the 0-0 resonance frequency. The details of other parts can be found in Fig. 2.6 and Fig. 2.9. arms, which converted the horizontal linear polarization to vertical linear polarization of the sub-beam. The length difference of the two arms of the interferometer was adjusted to produce a correct retardation between two sub-beams. The shortest time delay from one sub-beam to the other must be equal to the half of the oscillation period of the hyperfine frequency. For 87 Rb, the ground-state hyperfine frequency is about 6.8 GHz, hence, the path difference of two arms is c/(6.8 × 109 )/4 ≈ 1.1 cm, where c is the speed of light in air. Two sub-beams was therefore recombined into one beam with alternation between two linear polarization states on the output of the Michelson interferometer. The intermediate polarized state between the two orthogonal linear polarizations can be circular polarization, linear polarization, or elliptical polarization. The result depends on the difference of the optical phases of two combining sub-beams. The wavelength of the laser beam was about 1 micron. The stability of the optical phase relied on mechanical stability of the Michelson 56 Figure 2.21: Comparison of the signal contrasts of conventional CPT and push-pull CPT. The inset shows the modulation spectrum from a Fabry-Perot spectrometer. At the same pumping rate, the CPT signal from push-pull pumping has a better signal contrast, which is about 77 times larger than the conventional CPT signal. The conventional CPT signal on the top panel was taken with 16 times more averages than the push-pull CPT for a better SNR. This experimental data was taken from a spherical cell with diameter = 1.9 cm. interferometer. In order to guarantee the production of a probing beam with alternating circular polarization, another quarter-wave plate, placed on the output beam, was required. Finally, this probing beam was guided to pass through the sample cell. Figure 2.21 shows the very large signal enhancement of the 0-0 CPT resonance produced by push-pull pumping in comparison to the CPT resonance induced by intensity modulated light of fixed circular polarization. The signals were obtained from a cell containing isotopically-enriched 87 Rb and nitrogen gas at a pressure of 730 torr (defined at room temperature). The very small conventional CPT signal of Fig. 2.21 is mostly due to the optical pumping of atoms into the end state |a, ai, or |a, −ai, depending on the sign of the fixed circular polarization. Push-pull pumping eliminates this cause of signal degradation, because the time averaged photon spin is equal to zero. One can see the baseline of 57 Figure 2.22: Experimental data and theoretical curves from our density-matrix modelling agree very well to each other. The conventional CPT has an optimized value of pumping rate for maximizing the signal contrast. Higher pumping rate degrades signal of the conventional CPT due to the population trapped to the end state. However, push-pull pumping has no such problem. The right diagram shows a very strong CPT signal by using push-pull pumping of one circled data point on the left diagram. the conventional CPT signal is higher than the push-pull CPT in Fig. 2.21, because atoms in conventional CPT were polarized by using fixed circular polarization and became more transparent. More systematic studies of push-pull and conventional CPT resonances are summarized by the experimental points in Fig. 2.22. The measurements were done with Rb cells containing much lower gas pressures than the cell of Fig. 2.21. Modelling calculations of the expected CPT signals, shown by the continuous lines in Fig. 2.22, are in good agreement with the experiments. Typical population distributions from the modelling are shown as insets. Conventional CPT signals first grow with increasing light intensity, as faster pumping overcomes relaxation processes. After reaching a peak, the signals decrease as the light pumps more and more atoms into one of the end states, the dark states for the light of fixed circular polarization [47]. Faster push-pull pumping, however, always increases the 58 contributed from spin exchange Relative linewidth ( Γ (ex) / Γsd ) 80 75 70 D1 pumping, and sz = 1 Nuclear spin I = 3/2, and Γex/Γsd = 100 65 Conventional CPT Push-Pull CPT 60 55 50 0 100 200 300 400 Relative pumping rate ( Dop / Γsd ) 500 Figure 2.23: The relative linewidth contributed from spin-exchange broadening. As the prediction from Eq. (2.49), push-pull optical pumping can reduce the linewidth broadening due the spin-exchange collisions, but the optical pumping used in conventional CPT can not. However, in push-pull pumping, the decrease of the spin-exchange broadening can not compensate the linewidth broadening due to the optical pumping. This tiny light narrowing effect is hardly to be observed. concentration of atoms in the superposition state |ψi of Eq. (2.65), the only dark state for the light of alternating circular polarization. As shown in the left diagram and top panel of Fig. 2.22, a 30% contrast was measured for the push-pull CPT resonance, a substantially larger figure than has ever been previously reported with similar buffer gas pressure, temperature, thickness of the cell, etc. Because of birefringence in the glass windows of our oven and the glass walls of the vapor cell, the circular polarization was degraded and the peak spin of absorbed photons was measured to be |sz | = 0.87. Taking this into account, our modelling calculations predict that 51% of the atoms were concentrated in the superposition state |ψi of Eq. (2.65), for the left diagram, top-right experimental point in Fig. 2.22. The predicted concentration for fast pumping and for peak |sz | = 1 at these conditions is 100%. Analogous to Eq. (2.53), we can find an equation for the on-resonance population concentration, P0−0 , of the two 0 59 states to be (0) P0−0 = Dop + γ10 ¤ £ (0) , (0) (1) (0) ([I] − 1) Dop + γ10 − (|sz |Dop )2 /(Dop + γ20 ) + Dop + γ10 (2.68) where γ10 and γ20 are the longitudinal relaxation rate and the coherence decay rate of the (0) (1) two 0 states without optical pumping. Here, Dop and Dop are DC and AC components of (1) (0) the depopulation pumping rate of push-pull pumping, where |Dop /Dop | ≤ 1. The absolute value of the z-direction photon spin |sz | ≤ 1. One can check Eq. (2.68), without the light (1) modulation, Dop = 0, or without circular polarization of the pumping light |sz | = 0, we find P0−0 = 1/[I]. The maximum value P0−0 = 1, which represents 100% population in the two 0 states. The signal contrast is basically proportional to the strength of the coherence (1) ρ̃00 . From Eq. (2.32), we find that |ρ̃00 | = 1 |sz |Dop 2 D(0) +γ 0 P0−0 . op 2 Push-pull optical pumping is a special CPT excitation, which has only one dark state. Its exceptional ability of trapping atoms into 0-0 state can greatly improve the 0-0 CPT signal, and it can therefore permit improved design for 0-0 CPT atomic clocks. In addition, since the population is trapped to the 0-0 state, the spin-exchange broadening can be further suppressed according to Eq. (2.49). Figure 2.23 shows the relative spin-exchange broadening in conventional CPT and push-pull CPT as function of the relative pumping rate. The plot is based on numerical calculation with the same conditions used in Fig. 2.11. As we can see, the conventional 0-0 CPT gains more spin-exchange broadening by increasing the pumping intensity, because the population is evacuated from the m = 0, ±1 states as the prediction from Eq. (2.49). By increasing the pumping power, push-pull CPT has less linewidth broadening from spin-exchange collisions because of the population concentration to the 0-0 state. However, this light narrowing effect is not strong enough to be noticed, because the decrease of the spin-exchange broadening is not fast enough to catch up the optical pumping broadening. For atomic clocks, instead of regarding the signal contrast of the resonance, we like to optimize the figure of merit (FOM), which is proportional to the signal contrast dividing by the linewidth. Figure 2.24 shows FOMs as functions of the pumping rate from our experimental data, shown in Fig. 2.22. One can see that push-pull CPT is better than 60 140 500 87 300 87 120 200 o Rb with 15 torr N2 at 45 C Push-Pull CPT Conventional CPT 100 FOM FOM 400 o Rb with 90 torr N2 at 60 C Push-Pull CPT Conventional CPT 80 60 40 100 20 0 0 0 5 10 15 -1 Optical pumping rate (sec ) 3 20x10 0 5 10 15 3 20x10 -1 Optical pumping rate (sec ) Figure 2.24: The FOMs of the plots in Fig. 2.22. Push-pull CPT has larger FOM than conventional CPT. Here, FOM=contrast(%)/linewidth(kHz)×10. conventional CPT. In our experimental condition, the photon spin was only about 0.87. For a 100% circularly polarized pumping light, the enhancement of FOM of push-pull CPT will be more pronounced. Another factor, which degrades the performance of push-pull CPT but improves conventional CPT, is the diffusion relaxation. The diffusion relaxation resists push-pull optical pumping to the m = 0 states and also conventional optical pumping to the end state. Therefore, it benefits conventional 0-0 CPT, but disadvantages push-pull CPT. As we can see from Fig. 2.24, low pressure cell has a smaller difference of the maximum FOMs of two different CPT signals, which is due to the larger diffusion at lower pressure. Push-pull optical pumping can improve not only the performance of the CPT atomic clocks, but others like CPT masers, the state preparation for cold alkali-metal atoms or hot vapors. The state preparation of push-pull pumping can be used for atomic-beam atomic clocks, cesium fountain atomic clock, quantum computation, quantum information, etc. Considering that ground-state hyperfine splitting is resolved, we are able to concentrate the population to different hyperfine sublevels by adjusting the pumping intensity for upper and lower hyperfine multiplets and the ratio of the intensity of the RCP(σ+) and LCP(σ−) components. Figure 2.25 shows an example of 87 Rb from theoretical calculation. The end state concentration can be achieved by using conventional optical pumping. For other sublevels, besides the adjustments of the amplitudes of the two related optical sidebands and the σ + /σ − ratio, we have to adjust the modulation frequency to match the resonance frequency of the two related sublevels. For 0 states and end states, we can achieve nearly 61 Figure 2.25: By changing the ratios of pumping intensities of σ+, σ−, I1 , and I2 with a selected modulation frequency, we can put lots of population into some hyperfine sublevels. The top-left panel shows 50% on |2, 0i and 50% on |1, 0i states. The top-right panel shows 99% on |2, 0i state. The bottom-left panel shows 95% on |1, 1i state. The bottom-right panel shows 100% on |2, 2i state. 100% population trapping. For other states, we can still have high degree of population trapping as shown in Fig. 2.25. 2.4.5 Pressure Dependent CPT Excitation To excite CPT resonances, at least two coherent lights need to be used. A simple way to do this is to modulate a CW laser light. Due to the limitation of the experimental devices, it was easier to produce a phase or frequency modulated (PM or FM) light to GHz range in the past, such as using EOM (electro-optic modulator) to modulate the laser light or modulating the supplied current to the laser diode. However, experiments shows that the CPT signals from PM lights vanish when the buffer gas pressure exceeds a few hundred torr [4, 19]. Because more optical sidebands, which are seen by atoms, can interfere with 62 Figure 2.26: The amplitude of the optical coherence ρge is like a FM signal passing through 1 a resonator filter g(Ω) = i(Ωge −Ω)+γ , where Ω is the optical angular frequency. Frequencies op of the signal E(t) higher or lower than the resonance frequency have lower output amplitudes. The lower buffer gas pressure has a narrower filter bandwidth, and can produce a stronger AC component in pumping rate to excite a larger CPT signal. each other, the CPT excitation degrades when the light is phase- or frequency-modulated. For amplitude modulated light, the degradation of CPT excitation at high gas pressure is minimal. Therefore, we chose intensity modulated laser beam to do the CPT spectroscopy. Ideally, pulsed light is the best choice for CPT pumping at high buffer-gas pressure. According to Eq. (2.32), the strength of CPT excitation is proportional to the AC component of the depopulation pumping rate, whose frequency matches to the resonance frequency. Mathematically, the time-dependent pumping rate can be calculated from Eq. (2.22). For a better picture to understand, we compare the pumping rate at two different pressures in time domain as the plots shown in Fig. 2.26. Firstly, the optical coherence ρge between the ground-state sublevel and the excited-state sublevel is excited by the pumping electric field. Atoms are optical resonators. The time dependence of the coherence ρge is proportional to the pumping electric field passing through the atomic-resonator filter, which has the resonance frequency Ωge and the linewidth γop 63 CPT on Three Correlated Levels (ground-state resonance excited by optical fields) Excited state Electronic-Mechanical Analogy of CPT Phenomena (mechanical oscillation excited by electric fields) Electronic resonance frequency and damping : iΩ1 + γop iΩ2+ γop C1 C2 L L Z0(Ω) R iω0 + γ2 Ground state m C +q d k E(t) R m Ω1 ~ (LC1)-1/2 )-1/2 Ω2 ~ (LC2 γop ~ R/L Ω2 - Ω1 = ω0 -q Mechanical oscillation ω0 ~ (2k/m)1/2 γ2 ~ mechanical damping Figure 2.27: The ground-state CPT excitation from the three correlated quantum levels can be approximately analogous to an electronic-mechanical system. The CPT excitation is equivalent to the mechanical oscillation of the two metallic plates on the right diagram. (proportional to the buffer-gas pressure). The interaction between the optical coherence and the pumping field produces the effective ground-state pumping rate. The time-dependent pumping rate Γop (t) is proportional to the multiplication of the pumping field and the optical coherence. Detailed calculations can be found in Appendix A. Figure 2.26 explains why there is poor CPT excitation at high gas pressure. When the buffer-gas pressure is higher, the filter bandwidth is broader. The optical coherence has a waveform pattern closer to the pumping field. Since the pumping field is phase modulated, its amplitude is constant. Therefore, the pumping rate has weaker AC component at high gas pressure. When the gas pressure is lower, it produces more amplitude modulation on ρge , and therefore it generates a higher AC pumping rate. If the gas pressure is further getting lower and the bandwidth of the resonant filter is narrower than the separation of the optical sidebands, ρge becomes constant amplitude with a single oscillation frequency. The interaction between E(t) and ρge can generate the maximum AC pumping rate. An electronic-mechanical analogy of the CPT phenomena is sketched in Fig. 2.27. The modulated optical field to the atoms is equivalent to the modulated electric field E(t) to the system on the right of Fig. 2.27. The two LCR circuits define the peak response frequencies, Ω1 and Ω2 , which mimic the two 64 optical transition frequencies. The field-induced charge current can flow through the LCR circuits. The electric field E(t) induces a charge difference, 2q, between two metallic plates. Assuming the capacitance, C, of the two plates are large enough, so |Ω̄Z0 C| À 1, where Ω̄ = (Ω1 + Ω2 )/2, and Z0 = Z0 (Ω) is the total impedance of the two LCR circuits. The R iΩt dΩ, which interacts with induced charge q on the one of the plates is q(t) = i1Ω̄ Ẽ(Ω)d Z0 (Ω) e the electric field, and then produces force F (t) = q(t)E(t) to excite the mechanical motion. These two plates have mass m respectively, and they are connected by a spring and have p the resonance frequency ω0 = 2k/m = Ω2 − Ω1 . When E(t) is modulated at frequency ω0 /n, where n is an integer, the oscillation of the plates can be excited. One can see that the resistance R, which serves electronic damping, is equivalent to the buffer-gas pressure. If E(t) has only two frequency components Ω1 and Ω2 , a strong mechanical oscillation of the two plates can be produced. If E(t) is frequency-modulated and R/L À ω0 , the mechanical oscillation of the two plates is hardly to be induced. This is equivalent to the pressure-dependent CPT excitation. For amplitude modulated pumping light, the amplitude modulation is always present on the optical pumping rate. Hence, there is no high pressure degradation of CPT signals from AM pumping lights. Figure 2.28 sketches the CPT excitation efficiency by using different pumping schemes at different buffer gas pressures. For simplifying the problem, we consider zero optical-detuned carrier frequency. To excite the CPT resonance, the multiple of the modulation frequency should match the resonance frequency. When the frequency is matched, the time-dependent pumping rate can be therefore described by (0) Re(Γop (t)) = Re(Γop ) + ´ X³ (n) n Re(Γopm )ei m ωµν t + c.c. , (2.69) n where n and m = ωµν /ωm are integers. Here, ωm is the modulation frequency. Below a few atmospheres of the buffer gas pressure, the repopulation pumping has negligible influence to the ground-state coherence. At high very pumping rate, the ground-state decay rate (0) γµν = Re(Γop ). From Eq. (2.32), the CPT coherence is (1) Kµν Re(Γop ) |ρµν | = (ρµµ + ρνν ), 2 Re(Γ(0) op ) (2.70) (1) (0) Excitation efficiency ( Re(Γop )/ Re(Γop ) ) 65 1.2 1.0 0.8 0.6 Ideal δ pulse Two-wave pumping (sinusoidal AM) PM (modulation index m = 2.45) 0.4 0.2 0.0 0 1 2 3 Number density of buffer gas (amagat) 4 5 Figure 2.28: Calculated excitation efficiency of Cs 0-0 CPT resonance in the nitrogen buffer gas. Here, the modulation index m is defined by E(t) = E0 exp(−iΩ0 t + im sin(ωm t)), where Ω0 is the carrier frequency. where Kµν is a positive constant, which only depends on the angle between probing beam and the magnetic field and the absolute value of the photon spin |s|. It can be proved the (1) (0) maximum of Re(Γop )/ Re(Γop ) ratio is equal to one. For 0-0 coherence, if |s| = 1, and the probing beam is along the magnetic field, we find that Kµν = 1. So the maximum |ρµν | (1) (0) is 0.5. We define the excitation amplitude as Re(Γop )/ Re(Γop ). In Fig. 2.28, one can see that the excitation efficiency of PM or FM pumping decays quickly with increase of the buffer gas pressure, but for AM pumping, it only decays initially and then approaches to a constant. If the pumping light is modulated as a δ-pulse train, the excitation efficiency is not affected by the buffer gas pressure. For PM-FM pumping, smaller modulation index leads to a quicker degradation of the excitation efficiency by increasing gas pressure. 2.4.6 Line Shape of CPT Resonance In the CPT spectroscopy, we usually find the resonance signals with asymmetric line shapes, but we seldom see such phenomena from microwave resonance signals. This asymmetric line shape is more pronounced in end-resonance CPT signals when the light shift is present. A 66 simple explanation for this phenomena is that the light shift mimics a magnetic field, which is similar to the microwave field used for microwave spectroscopy, and therefore introduces an extra component of the coherence, which has 90 degree phase difference to the CPT coherence when there is no light shift. So, the total coherence has a phase difference to the oscillating probing light when the light shift is present. The line shape therefore changes from a pure Lorentzian curve to a combination of a Lorentzian and dispersion curve, and becomes asymmetric. For better understanding, we use a spin precession picture to explain the result. For 0-0 CPT, we can use a pseudo-spin system to understand the asymmetric phenomena. For end-resonance CPT, the pseudo-spin picture is very nearly the same as a real spin picture. Considering that there is a coherence generated between state |µi and state |νi due to optical pumping of modulated light, we find the corresponding rate equations regarding these two sublevels to be ρ̇µµ = 1 γ1 (ρνν − ρµµ ) − iRv /2(ρ̃νµ − ρ̃µν ) − RΓ /2(ρ̃νµ + ρ̃µν ) + Υµ 2 1 γ1 (ρµµ − ρνν ) − iRv /2(ρ̃µν − ρ̃νµ ) − RΓ /2(ρ̃µν + ρ̃νµ ) + Υν 2 ρ̇νν = ρ̇µν = (−iωµν − γ2 )ρµν + iRv /2(ρµµ − ρνν )e−iωt − RΓ /2(ρµµ + ρνν )e−iωt ρ̇νµ = (iωµν − γ2 )ρνµ − iRv /2(ρµµ − ρνν )eiωt − RΓ /2(ρµµ + ρνν )eiωt , (2.71) where γ1 is the longitudinal relaxation rate, γ2 is the transverse relaxation rate, and Υσ is the equivalent pumping rate to state σ. Here, we define two numbers: the coherence (1) pumping rate RΓ = hµ|δΓ |νi, and the Rabi frequency of the pseudo-magnetic field Rv = (1) 2hµ|δEv |νi/h̄. In a common two-level system coupled to an effective field, we can use Eq. (2.71) with substitutions of RΓ = 0 and Rv = ωR . For convenience, we can map these two levels into a pseudo-spin representation. Therefore 1 i 1 Ux = (ρ̃µν + ρ̃νµ ), Uy = (ρ̃µν − ρ̃νµ ), Uz = (ρµµ − ρνν ), 2 2 2 (2.72) where Ux , Uy , and Uz are the pseudo-spin expectation values in the rotation frame at frequency ω. The coherence ρµν is defined by ρµν = ρ̃µν exp(−iωt), and we know that "'(%$)&')$%"! ! "$#&%"'()#$% 67 Figure 2.29: The left and the middle panel show the probing lights along two orthogonal directions, parallel or perpendicular to the spin at zero detuning δω = 0. The signal amplitudes plotted as functions of δω. The in phase signal shows a symmetric curve, and the out phase signal shows a completely asymmetric curve. By using a modulated pumping light, coherence can be excited as a spin along the x-direction by the coherence optical pumping rate RΓ . If there is a light shift (δEv 6= 0), an effective field Rv is generated. This emulated field produces a spin component along the y-direction. Therefore the probing light both probes in phase and out phase signals. The detuning curve becomes asymmetric. ρµν = ρ∗νµ . If we only consider the coherence generated through the depopulation pumping, we can find the Bloch equation for this two-level system from Eq. (2.71). U̇x = −δωUy − γ2 Ux − RΓ N /2 U̇y = δωUx − γ2 Uy − Rv Uz U̇z = γ1 (Uz0 − Uz ) + Rv Uy , (2.73) and its vector form U̇ = (−RΓ N /2 − γ2 Ux )x − γ2 Uy y + γ1 (Uz0 − Uz )z + R × U, (2.74) where R = Rv x + δωz is the vectored angular frequency in the rotating coordinate system. Here, Uz0 = (Υµ − Υν )/(2γ1 ) is the equilibrium value of Uz when Rv = 0, N = ρµµ + ρνν is the total population of two states, and δω = (ωµν −ω) is the frequency detuning. We have to 68 notice that the splitting frequency ωµν = ω0 +∆ωv is shifted by the light shift ∆ωv , and both γ1 and γ2 linearly depend on the DC optical pumping rate if we ignore the spin-exchange mechanism. The CPT signal can be thought as an optically probed signal of a pseudo-spin in a rotation frame. Figure 2.29 shows the probing signal as a function of frequency detuning. A symmetric line shape, in phase signal, is found when the probing light is parallel to the spin at zero detuning. A pure dispersive Lorentzian line shape, quadrature signal, is found if the probing light is perpendicular to the spin at zero detuning. A modulated light is equivalent to a fixed-direction pumping or probing beam in the rotation frame. As shown in Fig. 2.29, the pumping beam produces the atomic coherence as a pseudo-spin along the x-direction. When the light shift is present, a mimic field Rv is generated along x-direction and can rotate the spin from z-direction to y-direction, which is the same phenomenon of spin-flip Raman scattering [23, 46]. Therefore, the spin projection on the x-y plane has both x and y components. Since the probing beam is still along the x-direction, both in phase and out phase signals can contribute to the line shape. It turns out that the line shape becomes asymmetric. To generate the spin component along to y-direction from the emulated field Rv requires the non-zero Uz originally. This means a population difference of the two sublevels is needed to produce an asymmetric line shape. This explains why the end-resonance CPT signal is more likely to be asymmetric. The end states have a larger population difference when they are optically pumped. The analytical result can be calculated by solving the steady light-absorption at different detuning from Tr(δΓρ). Therefore, the absorption due to the oscillation part of the pumping (1) light is proportional to Tr(δΓ ρ) = RΓ Ux , where the steady solution Ux = − RΓ N /2(γ2 + Rv2 /γ1 ) − δωRv Uz0 . δω 2 + γ22 + γ2 /γ1 Rv2 (2.75) For an optically thin cell, the transmission signal is proportional to −ARΓ Ux + B, where A and B are two constants. A transmission peak can be observed when the frequency detuning δω ∼ 0. We can see that when Rv 6= 0 and Uz0 6= 0, the line shape becomes asymmetric. For two-wave pumping, Rv /RΓ = −∆/γop , where ∆ = Ω0 − Ω is the optical detuning of the carrier, and γop /π is the optical linewidth. Here, Ω0 is transition frequency from the 69 Pressure=0.1atm, Modulation frequency=3.4GHz, Numerical Two-wave model Asymmetry factor (δA) 3 Excited state 2 1 γop=0.95GHz. ∆ Ω0 0 -1 Ground state -2 -3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Carrier frequency offset to Ω0/2π (GHz) Figure 2.30: Numerical calculation of the asymmetry of the resonance of 87 Rb at 0.1 atm as a function of Ω − Ω0 . The subplot shows the intensity spectrum of the optical sidebands of the modulated light and the curve of the absorption cross section. The ground state hyperfine splitting is resolved. When the optical detuning is small, only two strong optical sidebands are important. Hence, it can be approximated by two-wave pumping at low optical detuning. middle of the two ground-state sublevels to the excited state, and Ω is the optical carrier frequency of the pumping light. When γ2 γ1 > Rv2 , the line shape can be described by γ − δA (ωµν − ω) , (ωµν − ω)2 + γ 2 p 2R U where the asymmetric factor δA = RvΓ Nz0 , and γ = γ22 + γ2 /γ1 Rv2 . CP T resonance signal ∝ (2.76) Since δEv = h̄2 (1−2s·S)Im(Γop ), and δΓ = (1−2s·S)Re(Γop ), we find that the light shift generates as a real magnetic field along the direction of photon spin s, and the depopulation operator δΓ pumps the electron spin along s. For the end-resonance CPT, the coherence is equivalent to the spin precession on the x-y plane in the real space. The pseudo spin can be mapped to the real spin. In the real experiment, a modulated light can produce not only two optical sidebands. Therefore the asymmetric factor δA is not proportional to −∆/γop but is a polynomial function δA = δA (∆/γop ), because the high order optical sidebands can also affect the CPT excitation. Here, δA (∆/γop ) depends on the pattern of the optical sidebands and the 70 Figure 2.31: Experimental data shows the line shapes of the end CPT resonances as a function of the laser frequency from a 87 Rb cell with 0.96 atm N2 buffer gas. By using the numerical model, we can extract the parameter of optical pressure broadening and the pressure shift. The two wave model can not fit to data, because in the high pressure cell, atoms can see not only two optical sidebands but also high order sidebands. buffer gas pressure. Figure 2.30 shows an example of numerical calculation of the CPT line shape as a function of the optical detuning, which assumes the unresolved excited-state and 100 % population difference of the two ground-state sublevels. Here, we choose the same pumping spectrum with two strong peaks as we used for some of our experiments. Because the ground-state hyperfine splitting is resolved, two-wave model agrees with the numerical calculation very well at small optical detuning. However, it starts deviating when ∆ is larger. This is due to the contribution from high order optical sidebands. Figure 2.31 shows the real experimental data from the end-resonance CPT signals. Since the laser beam was arranged as 45◦ to the magnetic field, there is no way to produce a 100% population difference. Our numerical model was able to fit the data from the experiments. Two parameters, optical pressure broadening and the pressure shift, can be extracted from the fitting of data, which has a good agreement with a previous measurement [42] by other 71 Pressure=0.1atm, Modulation frequency=3.4GHz, γop=0.95GHz. Asymmetry factor ( δA ) 1 Mach-Zehnder type AM modulator: phase φ0 = π/2.7 modulation index φ1 = 3 0.5 0 -0.5 -1 -2 -1.5 -1 -0.5 0 0.5 1 Carrier frequency offset to Ω0/2π (GHz) 1.5 2 Figure 2.32: The calculation result shows a smaller dδA /d∆ around the point of Ω − Ω0 ∼ 0 by using a different pumping spectrum. The inset shows the modulation spectrum, which is generated by a Mach-Zehnder type modulator. The amplitude of the pumping electric field is proportional to sin(φ0 + φ1 sin ωm t). means. The two wave model can not be used for high pressure cell, because atoms see more than two optical sidebands at the same time. As the results show, the CPT resonance signals is more sensitive to the light shift at low gas pressure and is less affected by the light shift at higher gas pressure. However, by changing the structure of the optical sidebands, we would be able to reduce the asymmetry factor δA depending on the optical detuning ∆. Figure 2.32 shows a calculation result with a very small dependence on ∆ around the point of Ω−Ω0 ∼ 0 compared to Fig. 2.30. However, such pumping spectrum sacrifices CPT signal contrast, because the strong high order optical sidebands are less absorbed by the vapor. For CPT atomic clocks, the stability of the laser optical frequency is significant. The change of the resonant line shape and the light shift can affect the determination of the resonant frequency. Therefore, more systematic studies of the asymmetries of CPT line shapes are important, especially for the end-resonance CPT clocks. 72 2.4.7 Photon Cost of Optical Pumping In spin-polarized optical pumping, a photon transfers its angular momentum to an atom through the depopulation pumping. The atom loses its spin polarization through the relaxation mechanisms in the excited state, deexcitation (or repopulation) process, and groundstate relaxation. When the optical pumping rate exceeds all the ground state relaxation rates, only excited-state relaxation and deexcitation process are significant. In this subsection, we discuss how different excited-state evolutions and repopulation processes affect the efficiency of optical pumping. We assume that the optical pumping rate is much larger than all ground-state spin-relaxation rates. Therefore, the spin relaxation in the ground state is virtually zero. We use “photon cost” to denote the average number of photons, which need to be absorbed per atom, for completely polarizing an atom or pumping an atom to a pure quantum state. Lower photon cost represents higher pumping efficiency. The qualitative results can be understood from simple physics principles. The detailed results require numerical calculations. Considering D1 pumping with circularly polarized light, if we ignore the hyperfine structure, the ground state and the excited state have only two levels respectively, as shown in Fig. 2.33. Without hyperfine coupling, Fig. 2.33 shows that optical pumping with quenching-dominated deexcitation has a better pumping efficiency than if there is only spontaneous decay, because by absorbing a photon, there is 50% chance for polarizing an atom with pure quenching, but only one-third probability with spontaneous decay. With hyperfine coupling, the electron spin can always transfer its angular momentum to the nuclear spin through the ground-state hyperfine coupling. Because the groundstate relaxation rates are always smaller than the ground state hyperfine frequencies, the electron spin and the nuclear spin have plenty of time to precess about each other. For the excited state, the relaxation rate can either be larger or smaller than the excited hyperfine frequency. The relaxation rates for excited atoms depend on the pressures of buffer gases. Therefore, the efficiency of angular momentum transfer through the excited state depends on the buffer-gas pressure, which can influence the efficiency of optical pumping. $ * ) + &'( & ! "#$% ! " # % ' , , . , / - 73 Figure 2.33: The left and middle diagrams show the σ+ pumping with different deexcitation mechanisms. Quenching deexcitation causes higher pumping efficiency than spontaneous decay. The right diagram shows the hyperfine sublevels with an assumption of nuclear spin I = 1/2. The electron spin has chance to be transferred to the nuclear spin through the hyperfine coupling in the excited state for the pumping transitions of region (1), but not of region (2). . For better understanding, we can consider three special cases to discuss the static spinpolarized optical pumping for unpolarized atoms. (1) Quenching dominates the deexcita(e) tion, and the quenching rate is much larger than excited hyperfine frequency, Γq À ωhf . Hence, there is no way to transfer the angular momentum between the electron and the nuclear spin in the excited state. When an atom absorbs one photon, there is a 50% probability that the photon spin can be transferred to the atom. The spin of a fully polarized atom is I + 1/2. Therefore, on average, (I + 1/2)/0.5 = 2I + 1 = [I] photons are needed to fully polarize an atom. (2) Spontaneous decay dominates the deexcitation, and the spon(e) taneous decay rate is much larger than the excited hyperfine frequency, Γs À ωhf . Hence, by using very similar arguments of previous one, because, in average, only 1/3 photon spin can be transferred to an atom from each absorption, the photon cost is equal to 1.5[I]. However, we have to realize that this result can not happen for real cases, because usually (e) ωhf À Γs for real atoms. (3) J-damping dominates the excited-state spin relaxation, and (e) ωhf À Γjd À (Γq + Γs ). Hence, J-damping can even out the population distribution in the excited state before the atom decays back to the ground state. In this case, there is an equal probability for all ground-state hyperfine sublevels to be repopulated from the excited state. Assuming that we use σ+ pumping light, to fully spin polarize atoms means to pumping 74 the population to the end state |a, ai. There is a total of g = 2[I] sublevels. Therefore, on average, for each photon absorption, 1/g of ground-state population, excluding the state |a, ai, is transferred to the state |a, ai. Initially, there (g − 1)/g population for each atom needing to be pumped to the end state, so the photon cost is (g − 1)/g × (1/g)−1 = 2[I] − 1. For real cases, atoms with pure spontaneous decay have lower photon cost than 1.5[I]. (e) Since ωhf is always larger than Γs , electron spin transfers part of its angular momentum to the nuclear spin before decay back to the ground state. The deexcitation can destroy the electron spin as the way described in Fig. 2.33, but conserves the nuclear spin. Therefore, if there is no J-damping relaxation in the excited state, the spin transfer in the excited state can always increase optical pumping efficiency. From Fig. 2.34 to Fig. 2.37, we show the calculation results of photon costs of 39 K, 85 Rb, 87 Rb, 133 Cs, as a function of quenching and J-damping rates for optical pumping to the end state. Here, we use the relative quenching rate, γq = Γq /Γs , and relative J-damping rate, γjd = Γjd /Γs to label the axes. The results of photon costs are shown as a contour diagram. If we track along two axes, there is a minimum along the γq axis, and there is a maximum along the γjd axis. These two phenomena can be comprehended with following explanations. By increasing γq , more electron spin polarization can be transferred back to the ground state. Therefore, it reduces the photon cost. However, larger quenching rate means less duration in the excited state. Hence, after a certain point, the higher quenching rate can not further help the optical pumping due to the less transfer of angular momentum from electron to nuclear spin in the (e) excited state. Eventually, the photon cost increases to [I] when Γq À ωhf . This is the same result of case (1) described previously. By increasing the γjd , population in the excited state can spread to more sublevels. Therefore, it increases the photon cost. However, larger Jdamping rate means that the spin precession time between each J-damping collision becomes shorter. Hence, less angular momentum is transferred between the electron and the nuclear spin. For J-damping, it means that the population spreads more slowly. Eventually, when (e) Γjd À ωhf , the J-damping evolution becomes independent of nuclear spin. The photon cost become [I], which is the same as extremely high quenching rate. The result of case (3) 75 described previously can not occur in the real cases, because usually the excited hyperfine frequency is not high enough. The closest result of case (3) is where the maximum on the γjd axis. 76 −1 K−39, Γs=3.85e+007 s . Excited hyperfine frequency=57.8 MHz 3 10 2 10 4 3.6 γq 1 3.8 10 3.2 0 10 3.4 4.6 4.2 4.8 4.4 5 −1 10 −1 10 0 10 1 10 γ 2 10 3 10 jd Figure 2.34: Photon cost for pumping potassium-39 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. 77 −1 Rb−85, Γs=3.57e+007 s . Excited hyperfine frequency=363 MHz 6 3 10 6 2 4.6 γq 4.4 4.2 1 5 5.8 10 6.2 4.8 10 7 7.4 6.8 7.2 6.4 8.2 8.8 7.8 8.6 8 9 8.4 9.2 6.2 6.6 6.4 0 10 6.2 6.8 6.4 7 6.6 7.2 7.4 7.6 6.86.6 −1 10 9.4 7 −1 10 0 10 1 10 γjd 2 10 3 10 Figure 2.35: Photon cost for pumping rubidium-85 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. 78 Rb−87, Γ =3.57e+007 s−1. Excited hyperfine frequency=818 MHz s 4 4 3 10 4 8 3. 3.6 3.4 2 10 4 3 3.2 4.2 4.2 4.4 4.6 4.4 5.4 5.2 5 4.84.6 5.8 5.6 6 5.45.25 4.8 6.2 5 5.6 5.2 5.8 5.4 6 4.8 1 γq 4.2 4.4 4.6 10 5 0 10 5.6 6.6 6.2 6.4 5.8 −1 10 6 −1 10 0 10 1 10 γ 2 10 3 10 jd Figure 2.36: Photon cost for pumping rubidium-87 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. 79 Cs−133, Γs=2.94e+007 s−1. Excited hyperfine frequency=1120 MHz 8 3 7. 10 8 8 7. 6 7. 2 8 6. 86 5.4 6. 4 5.8 5.6 7 7.4 8 1 γq 8 .6 10 6.2 6 2 10 9 0 10 8.2 8.2 8.2 8.4 8.6 8.4 8.8 8.4 9 8.6 9.29.6 9.4 9.28.8 8.6 9 8.8 9.8 10 9.4 10.2 9.6 10.4 9.8 9.2 9 10.6 9.4 10.8 10 9.6 11 10.2 9.8 10.4 11.2 10.6 10 10.2 11.4 10.810.4 11.811.6 11 10.6 11.210.8 12 12.2 11.411 12.4 11.6 11.8 11.2 12 7.8 11.4 12.6 12.2 11.6 11.8 12 12.4 −1 10 12.8 −1 10 0 10 1 10 γjd 2 10 3 10 Figure 2.37: Photon cost for pumping cesium-133 to the end state, which is plotted as a function of relative quenching rate and J-damping by using a contour diagram. The difference of values between each contour is 0.2. Darker color represents lower value, and lighter color means higher value. The value of the contour is equal to the photon cost. 80 Figure 2.4.7 summarizes the photon costs for optical pumping to the end state and push-pull optical pumping to 0-0 state. We calculated three different cases, spontaneous (e) (e) (e) decay (ωhf À Γs ), quenching (Γq À ωhf ), and J-damping (ωhf À Γjd À (Γq + Γs )), for different nuclear spin numbers. Photon cost of optical pumping to the end state 20 Number of photons Excited-state relaxation dominated by: 15 Quenching Spontaneous decay J-damping 10 5 0 0 1 2 Nuclear spin ( I ) 3 4 Photon cost of push-pull optical pumping to the 0-0 state 40 Number of photons Excited-state relaxation dominated by: 30 Quenching Spontaneous decay J-damping 20 10 0 0 1 2 Nuclear spin ( I ) 3 4 Figure 2.38: . Photon costs for two different optical pumpings and nuclear spins with different relaxation mechanisms. 81 The numerical results are listed as following with the sequence of the nuclear spin numbers I = 12 , 23 , 25 , and 72 : • Photon cost for pumping to the end state: – Spontaneous decay: 2.25 (exact), 3.52, 4.88, 6.26 – Quenching: 2I + 1 −→ 2, 4, 6, 8 – J-damping: 4I + 1 −→ 3, 7, 11, 15 • Photon cost for pumping to the 0-0 state: – Spontaneous decay: 3.5 (exact), 7.85, 13.3, 19.7 – Quenching: 3 (exact), 9.67, 19.4, 32.1 – J-damping: 4I + 1 −→ 3, 7, 11, 15 We can find that excited-state relaxation dominated by J-damping produces maximum photon costs for optical pumping to the end state, but minimum photon costs for the push-pull optical pumping to the 0-0 state. However, we have to understand that the (e) condition, ωhf À Γjd À (Γq + Γs ), used for calculations is not completely correct. As our previous discussions, the closest results are on the maximum of γjd axes of Fig. 2.34 to 2.37. Therefore, for real cases with relaxation dominated by J-damping, the photon costs are less than the result shown in Fig. 2.38 for end-state pumping, and are more than the result shown in Fig. 2.38 for 0-0 state pumping. Another interesting result is that if we compare the photon costs of the spontaneous decay with the quenching, we find that the spontaneous decay has higher photon cost than quenching when I = 1/2. We can understand this result through the right sub-diagram in Fig. 2.33. The optical pumping in the region (1) has higher efficiency, because of the transfer of angular momentum between the electron and the nuclear spin in the excited state. But there is no such advantage in the region (2), which means the spontaneous decay has poor pumping efficiency than the quenching in the region (2). When I > 1/2, region (1) has more population, the total photon cost is dominated by region (1). Therefore, there is less photon cost for spontaneous decay. When I = 1/2, 82 the region (2) has more population then region (1), the higher pumping efficiency in the region (1) can not compensate the lower pumping efficiency in the region (2). Therefore, the spontaneous decay has higher photon cost than the pure quenching. Different buffer gases can affect optical pumping differently, such as N2 , which is a good quenching gas; He and Ar, which are very bad for quenching but can introduce large Jdamping rate. According to the study of N. D. Bhaskar [10] in 1979, optical pumping can generate a spin-polarized wave front inside the vapor cell. The propagation speed of this wave front is inversely proportional to the photon cost. Therefore, it is possible to measure the quenching and J-damping rate of different buffer gases for P1/2 state by measuring the speed of wave front from a series of vapor cells with different buffer gas mixtures. We can use those experimental data to fit with contour calculations shown in Fig. 2.34– 2.37 to extract quenching and J-damping rates. 83 2.5 New Simple-Compact Frequency Standard Besides the applications of push-pull optical pumping mentioned in section 2.4.4, one of the most promising uses of push-pull optical pumping is to build a new small-size frequency reference, which can have a precision comparable to the conventional atomic clocks but without their complexity. Comparing different types of frequency references, quartz-crystal oscillators are easily miniaturized, but they have poor long-term (days to years) stability; atomic clocks have excellent long-term stability, but they are hard to miniaturize. Today, the best quartz oscillator demonstrates a short-term (seconds to hours) frequency uncertainty as low as 10−14 , but the long-term uncertainty is only ∼ 10−9 ; the very best atomic clocks can provide both short-term and long-term frequency uncertainty down to 10−17 . The fundamental frequency of a quartz oscillator is determined by the mechanical property of the quartz crystal. The “aging” problem of the crystal hinders its frequency stability from extending to a longer period. The fundamental frequencies of atomic clocks are determined by the fundamental atomic constants and by small perturbations of environmental parameters. Therefore, the atomic clock is much more stable. Because of the fast development of digital technology, such as high-speed digital communications, GPS system, etc., a precise, long-term, smallsize, time base becomes more and more important for every digital system. For example, the basic requirement of the time base for a GPS system is to have a daily uncertainty as low as 10−11 . The crystal oscillators are not good enough for those applications. Hence, miniature atomic clocks become the only choice, and several high technology companies develop such clocks. So far, all types of atomic clocks output the time-base signals by passively locking to the hyperfine frequencies of the alkali-metal atoms (usually 87 Rb and 133 Cs). They all require frequency-tunable local oscillators with frequency-locking loops. The microwave circuits increase the complexity of the atomic clocks and consume a large fraction of the operating power. To overcome those difficulties, we propose a new idea to build a simple-compact frequency standard, which is expected to greatly reduce the system complexity and the 84 (1) Edge emitting laser diode (2) Figure 2.39: Two different cavity configurations of hyperfine-modulated lasers for two different types of laser diodes. The regular edge emitting laser diode outputs linearly polarized laser light, and therefore it needs two λ/4 wave plates to produce alternating circular polarization inside the cavity. The VCSEL diode is designed to have almost no preference of the light polarization, so the light polarization can be directly modulated by the vapor cell. The steady lasing point of the two lasers is when the modulation period of the output light is equal to 2π/ω00 . power consumption. Our scheme combines the idea of push-pull optical pumping and the technology of semiconductor lasers, which is shown in Fig.2.39. A laser is a positive feedback amplifier of photons. An alkali-vapor cell inside the laser cavity plays like a photonic filter and converter to generate a special lasing mode, which produces the light modulation. Generally, a laser tends to lase in an optical mode, which has the maximum gain or the minimum loss of photons from their round-trip inside the cavity. Without the vapor cell, the lasing spectrum is determined by the characteristics of the laser cavity and the gain profile. With a vapor cell inside the cavity, a steady lasing point is met while the lasing spectrum produces the maximum efficiency of push-pull optical pumping, which makes the vapor cell become the most transparent. At this point, the output laser light is modulated 85 Spectral response (not to scale) External cavity Gain medium in laser diode Bragg mirror Growth of optical comb Photons scatter between each peak Non-polarized alkali vapor ν00 Optical frequency Figure 2.40: An illustration of the spectral responses due different causes. The optical comb in the lasing spectrum is grown to produce the maximum transparency of the vapor cell, and therefore obtain the maximum gain of photons. at 0-0 hyperfine frequency ω00 , and its optical spectrum is like an optical comb as shown in Fig. 2.40. We have to notice that the vapor cell inside the cavity acts not only as a filter of the photon spin and the frequency of photons, but also the frequency converter of photons. When coherent population trapping is present, a scattered photon from an atom can change its frequency from original ν to ν ± ν00 . Eventually, an optical comb (many peaks separated by ν00 ) is established in the lasing spectrum. Conventional mode-locked lasers can also generate comb spectrum. However, the spacing of the comb peaks is usually determined by the free spectral range of the cavity. In Fig. 2.39, there are two different types of cavity configuration. For edge emitting diode, the output light is linearly polarized. In order to generate the condition of pushpull pumping, two quarter-wave plates placed on the two sides of the vapor cell inside the cavity is needed. The round-trip time of the photon inside the cavity should be about the multiple of 2π/ω00 . The vapor cell should be placed around the middle inside the cavity. It is easy to see that the configuration shown in (1) of Fig. 2.39 is the same idea of producing alternating circular polarization by using Michelson type interferometer as shown 86 1.0 D1 push-pull pumping CPT D2 push-pull pumping CPT 0-0 CPT efficiency 0.8 0.6 0.4 0.2 0.0 0 1 2 3 4 Nuclear spin (I) Figure 2.41: The normalized amplitudes of the maximum 0-0 coherence with different nuclear spins by using different fast push-pull pumping. D1 push-pull pumping can excited a much stronger CPT coherence than D2, and also make the vapor cell more transparent. in Fig. 2.20. The only difference is that we employ the counter propagation of photons to obtain the alternating polarization inside the cell. Because we do not want the cavity mode to dominate the lasing spectrum, the Q value of the cavity should not be too high. In this case, the length of the cavity and the position of the cell do not need to be exact, and can have more tolerance. A simpler version of the hyperfine modulated laser can be found in (2) of Fig. 2.39. The VCSEL diode can be designed to be no preference of output light-polarization. The polarization of photons inside the cavity is controlled by the vapor cell. A previous study of polarization modulation of VCSEL can be found in Ref.[40, 35]. In practice, the size of the cavity can be scaled down by using some transparent material with large dielectric constant. In order to prevent the diode from lasing at the wavelengths outside the absorption line of the vapor, a bragg mirror is placed into the cavity to guarantee the correct lasing wavelength and narrow the lasing linewidth. The vapor cell should have a high buffer-gas pressure to ensure a broad enough optical absorption line. One can see that the push-pull 87 optical pumping is necessary for cavity modulation with a vapor cell. Conventional CPT uses fixed circular polarization, which will bleach out the vapor inside cell and produce a steady circular polarization without any modulation. In addition, it is better to use a laser diode with an output wavelength equal to that of the alkali D1-line. As shown in Fig. 2.41, D1 push-pull CPT has much better efficiency than D2 push-pull CPT. This new idea has not been tested experimentally. However, many studies about the external cavity modulation of the laser diode [57, 40, 35] and the demonstration of an optoelectronic vapor cell oscillator [43] strongly support our idea. We believe this new type of frequency standard can bring lots benefit to the industry of electronics and optics. Chapter 3 High-Field 129Xe-Alkali-Metal Spin Exchange Spin-exchange optical pumping has been widely used in producing highly polarized 3 He and 129 Xe. The basic idea is to use alkali-metal vapor as an intermediary for transferring the angular momentum of photon to the nuclei of noble gases. In practice, the alkali-metal atoms are first polarized by scattering circularly polarized light. Through spin-exchange collisions between noble-gas atoms and alkali-metal atoms, the nuclear spins of the noble-gas atoms are eventually polarized to their maximum angular momentums. Comparing to 3 He, 129 Xe has some advantages for scientific applications, such as much larger abundance on Earth, stronger interaction with chemical and biological compounds, larger chemical shifts of the nuclear resonances, and a much higher liquefaction temperature. Those features make hyperpolarized 129 Xe a powerful probe in many experiments, and it has played an important role in many areas of scientific research [41, 34] as well as in biomedical applications [2, 44]. A better understanding of spin exchange between and alkali-metal atoms can help us produce the hyperpolarized 129 Xe 129 Xe more efficiently and successfully use it in more scientific studies. For 129 Xe and alkali-metal atoms, there are two dominant mechanisms of spin exchange. One is binary collisions and the other is van der Waals molecules. For binary collisions, the 88 89 spin exchange occurs when two atoms collide with each other; the interaction time is about a few picoseconds. There is a weakly attractive van der Waals force between alkali-metal atoms and 129 Xe. The binding energy is around 10 meV. At room temperature, the van der Waals molecules can be generated through the process of three-body collisions. The third particle, which can be another nearby atom or molecule, can carry away the extra energy of the two colliding atoms by bumping with the colliding 129 Xe and alkali-metal atom. Two atoms are therefore trapped into their shallow potential wall. The duration of the spin interaction in van der Waals molecules can be up to a few nanoseconds. This short-lived molecule breaks into two atoms by colliding with another atom or molecule. Since the formation rate and the spin interaction time of van der Waals molecules are determined by the third particle collisions, the spin exchange contributed from the van der Waals molecules has a complicated dependence on the xenon or buffer gas pressures. People usually used to measure the pressure-dependent spin-exchange rate to extrapolate the binary spin-exchange rate. But recent evidences show that the behavior of the spin exchange from van der Waals molecules at high gas pressure is still poorly understood. We believe previous results of binary spin-exchange are incorrect. Because of the longer spin interaction time of van der Waals molecules, the spin exchange of van der Waals molecules can be easily suppressed to negligible values at high magnetic field if the inverse of the electron Larmor frequency is much shorter than the interaction duration. Therefore, the high-field spin-exchange measurements can give spin-exchange rates that are dominated by binary collisions with negligible contribution from van der Waals molecules. 3.1 3.1.1 129 Xe Spin Relaxation Rates Equation Detailed Balancing For binary collisions, spin relaxation is caused by random spin-flipping collisions. When a magnetic field is applied to a spin system, a spin-flipping collision can cause a change of the internal atomic energy and therefore cause a difference between incoming and outgoing 90 kinetic energies. According to the principles of quantum scattering, the collision cross section σ can be described by: σ21 (v1 ) = v1 |f (v1 , v2 )|2 v2 |f (v1 , v2 )|2 and σ12 (v2 ) = . v1 v2 (3.1) Here σ21 (v1 ) is the total cross section for scattering from an initial spin state 1 to a final spin state 2 for which the relative speed of the atoms before the collision is v1 and the relative speed after the collision is v2 . The total scattering amplitude is f (v1 , v2 ) = f (v2 , v1 ). Hence we conclude the condition of its detailed balance: σ21 (v1 )v12 = σ12 (v2 )v22 . (3.2) The spin-exchange rate is proportional to hσvi, the ensemble average of σv. Assuming that the spin flipping energy in the magnetic field is equal to ∆E, due to the conservation of the total energy, we know that ∆E is equal to the difference of kinetic energies. That is 1 1 ∆E = ∆ES + ∆EK = gS µB B + 2µK B = M v22 − M v12 , 2 2 (3.3) where ∆ES is electron spin-flipping energy of the alkali-metal atom, ∆EK is nuclear spinflipping energy of 129 Xe, gS is the g-factor of electron, µK is the nuclear magnetic moment, B is the strength of magnetic field, and M is the reduced mass of a colliding pair. Assuming the system is in a thermal equilibrium at temperature T , We define the spin-exchange rate coefficients κ21 and κ12 by Z κ21 = hσ21 v1 i = dv1 P (v1 )v1 σ21 (v1 ) µ ¶ µ ¶ Z Z M v12 M v22 ∆E 3 = C dv1 exp − v1 σ21 (v1 ) = C dv2 exp − + v23 σ12 (v2 ) 2kT 2kT kT Z = e∆E/kT dv2 P (v2 )v2 σ12 (v2 ) = hσ12 v2 ie∆E/kT = κ12 , (3.4) where µ ¶ M v2 P (v) = Cv exp − , 2kT 2 r µ ¶3/2 2 M C= . π kT (3.5) Here k is the Boltzmann constant, and P (v) is the velocity probability function. Equation (3.4) shows the result of detailed balance in thermal equilibrium, and there is a factor 91 e∆E/kT between hσ21 v1 i and hσ12 v2 i. Here we define the geometric mean of the spinexchange rate coefficient √ κ21 κ12 ≡ hσvi, (3.6) κ21 = κe∆E/2kT and κ12 = κe−∆E/2kT . (3.7) κ= where When ∆E > 0, we have “exothermic” rate coefficient κ21 and “endothermic” rate coefficient κ12 based on the result of Eq. (3.4). Similarly, we can find the spin-relaxation rates for 129 Xe and alkali-metal atom as following: α21 = αe∆EK /2kT , α12 = αe−∆EK /2kT , γ21 = γe∆ES /2kT , γ12 = γe−∆ES /2kT , where α and γ denote the spin-relaxation rates of 129 Xe (3.8) and alkali-metal atom respectively due to other causes. 3.1.2 Rate Equations To obtain the spin-relaxation rate equations, we set the number density of alkali-metal atoms [A] = [A+ ] + [A− ] and [A± ] = [A](1/2 ± hSz i). Similarly, for the number density of (3.9) 129 Xe, [Xe± ] = [129 Xe](1/2 ± hKz i). (3.10) Here we use “+” and “−” to label the spin up and spin down states. The expectation values of electron and nuclear spins are hSz i and hKz i respectively. Thus, d 1 1 [A+ ] = −κ21 [Xe− ][A+ ] + κ12 [Xe+ ][A− ] − γ21 [A+ ] + γ12 [A− ], dt 2 2 d 1 1 [Xe+ ] = κ21 [Xe− ][A+ ] − κ12 [Xe+ ][A− ] + α21 [Xe− ] − α12 [Xe+ ]. dt 2 2 (3.11) 92 Substituting Eq. (3.9) and (3.10) into Eq. (3.11), we find d [129 Xe] d hSz i = γ cosh(∆ES /2kT )(hSz iT − hSz i) − hKz i, dt [A] dt (3.12) and also d hKz i = α cosh(∆EK /2kT )(hKz iT − hKz i) + dt [A]{κ21 (1/2 − hKz i)(1/2 + hSz i) − κ12 (1/2 + hKz i)(1/2 − hSz i)}, (3.13) where the thermal equilibrium values hKz iT = 1 1 tanh(∆EK /2kT ) and hSz iT = tanh(∆ES /2kT ). 2 2 (3.14) Equation (3.12) and (3.13) are nonlinear and their steady solutions are hKz i = hKz iT and hSz i = hSz iT . At a representive experimental temperature T = 400 K and a magnetic field B = 9.4 T, we find the Xe nuclear thermal polarization |2hKz iT | ∼ 6.6 ppm and for alkalimetal electron, |2hSz iT | ∼ 1.6%. Typically, γ/[Xe] À κ, which implies that hSz i ∼ = hSz iT is a very good approximation. Therefore we can simplify Eq.(3.13) into 1 d hKz i = − (hKz i − hKz iT ) , dt T1 (3.15) where the longitudinal relaxation rate cosh(∆EK /2kT ) 1 = [A]κ + α cosh(∆EK /2kT ) ∼ = [A]κ + Γ0 . T1 cosh(∆ES /2kT ) (3.16) The ratio of the hyperbolic cosine functions in Eq. (3.16) gives a factor of 0.9995 in typical conditions of our experiments. An interesting result is that if we artificially make hSz i = 0 and assume α ∼ 0, we can find the steady solution of Eq.(3.13) is hKz i = 1 κ21 − κ12 ' hSz iT . 2 κ21 + κ12 (3.17) Here, the polarization is transferred from the electron spin to the xenon nuclear spin. 93 3.2 Magnetic Decoupling of 129 Xe-Alkali-metal Spin Exchange The typical duration of binary collision is less than 10−11 seconds, so it is difficult to be decoupled by magnetic field. However, when the field strength is larger than a few tesla, the electron Larmor frequency becomes comparable to the inverse of the binary interaction time. The binary spin-exchange can be suppressed. We calculate the spin-exchange rate coefficient as a function of B field by using two different methods, the semi-classical approach (SCA) and the distorted-wave Born approximation (DWBA) [25, 27]. 3.2.1 Semi-Classical Approach (SCA) The Hamiltonian, H, for the spin energy of a colliding pair can be written in the central mass frame as H = H (0) + H (1) , (3.18) where H (0) = gS µB Sz B − 2µK Kz B, and H (1) = α(r)K · S. (3.19) Here gS is the electron g-factor, µB is the Bohr magneton, µK is the nuclear magnetic moment, and B is magnetic field along the z direction. α(r) denotes the coupling coefficient for nuclear spin of the spins for 129 Xe 129 Xe and electron spin of the alkali-metal atom, K and S represent and the alkali-metal atom respectively. The H (1) (r) is time dependent because the internuclear separation r of the two colliding atoms depends on the time t. For the spin state, |qi = |mK , mS i, there are four totally orthogonal combinations, |1i = |mK = 1/2, mS = 1/2i, |2i = |mK = −1/2, mS = 1/2i, |3i = |mK = 1/2, mS = −1/2i, |4i = |mK = −1/2, mS = −1/2i. (3.20) At magnetic field B and without the spin interaction term, H (1) , there are energies [²1 ²2 ²3 ²4 ] for four spin states, where ²q = gS µB mS B − 2µK mK B. The spin states |1i and |4i are the 94 eigenstates of H (0) , and H (1) cause spin flipping between state |2i and |3i, since 1 K · S = Kz Sz + (K+ S− + K− S+ ). 2 (3.21) In SCA, we use the laws of classical mechanics to calculate the trajectory of the two colliding atoms. We ignore the small spin energy, and we express the relative velocity as a function of the separation between two atoms. s µ ¶ 2 E0 b2 vr = E0 − V (r) − 2 , M r (3.22) where vr = |dr/dt| is the relative radial speed, M is the reduced mass, E0 is the initial kinetic energy, V (r) is the interaction potential, and b is impact parameter. Equation (3.22) follows from the conservation of energy and angular momentum. From first-order time-dependent perturbation theory, we find the transition probability between state |2i and |3i is ¯ Z ∞ ¯2 ¯1 ¯ 0 (1) i∆ωt0 ¯ ¯ W = W32 = W23 = ¯ dt h3|H |2ie ¯ , ih̄ −∞ (3.23) where ∆ω = ±(²2 −²3 )/h̄ = ±(gS µB +2µK )B/h̄ = ±∆E/h̄ is the energy difference between two states. The Fermi contact Hamiltonian, H (1) , is a function of r, and r is a function of t. We can calculate the time is a function of the atomic separation, r, by using Eq. (3.22). Z r dr0 , (3.24) t(r) = ± 0 r0 vr (r ) where r0 is the shortest distance between a colliding pair. Therefore the probability W becomes a function of initial conditions of the two colliding atoms and magnetic field. To do the ensemble average of σv, we have to integrate all possible initial velocities and impact parameters. Therefore the binary spin-exchange rate coefficient, κ, which represents the spin-exchange rate for unit number density of the alkali-metal atoms is Z Z κ = 2πbdbdv vP (v)W (b, v) ¯ Z ∞ ¯ Z ∞Z ∞ ¯1 dr ¯¯2 ¯ = dbdv 2πbvP (v) ¯ α · cos(∆ωt) ¯ , ih̄ r0 vr 0 0 (3.25) where α, t, and vr are functions of r, and P (v) is the Maxwell-Boltzmann distribution p appeared in Eq. (3.5). The integral variable v is equal to 2E0 /M . Since ∆ω is propor- 95 tional to the magnetic field, a stronger field makes the oscillation more rapid inside the integral (3.25). Therefore, it decreases the value of κ and causes the magnetic decoupling. The calculation in integral (3.25) ignores the small difference of incoming and outgoing scattering velocities, which is due to the spin flipping at magnetic field. Consider the detailed balancing, we can write the endothermic and exothermic rate coefficients as κ32 = κ e∆E/2kT , κ23 = κ e−∆E/2kT . 3.2.2 (3.26) Distorted-Wave Born Approximation (DWBA) In DWBA method, we treat a colliding event completely quantum mechanically, so the unperturbed Hamiltonian of Eq. (3.19) becomes H (0) = − h̄2 2 ∇ + V (r) + gS µB Sz B − 2µK Kz B, 2M (3.27) where M is the reduced mass, V (r) denotes the atomic interaction potential between two colliding atoms. Starting from the Eq. (3.18), we treat the Fermi contact term as a perturbation term. Hence we can find a Schrödinger equation (H (0) − E)|Ψi = −λH (1) |Ψi, (3.28) where E is the total energy of the system and λ is the perturbation parameter. To solve this equation, we can expand the wave function into a power series in λ. |Ψi = 4 X ³ ´ |qi ψq(0) + λψq(1) + λ2 ψq(2) + · · · , (3.29) q=1 where |qi represents four spin states. Therefore we can rewrite Eq. (3.28) as µ ¶ 4 2M 2M X 2 2 (n) ∇ − 2 V (r) + kq ψq = 2 hq|H (1) |piψp(n−1) , h̄ h̄ p=1 (3.30) where kq2 = 2M (E − ²q )/h̄2 . When n = 0, the righthand side of Eq. (3.30) is zero. For the scattering process, we find the asymptotic form of the wave function, |Ψi ∼ |pieikp ·r + 4 eikq r X |qifqp (r̂), r q=1 (3.31) 96 where p and q denote the incoming and outgoing spin states, and fqp (r̂) is the scattering amplitude, which is a function of the direction r̂ of the scattered wave. We can also expand fqp into the power series in λ, (0) (1) (2) fqp = fqp + λfqp + λ2 fqp + ··· . (0) (3.32) (1) Here, fqp is the zeroth-order scattering amplitude without spin flipping, and fqp gives the first-order scattering amplitude with spin-exchange. By comparing the Eq. (3.29), (3.31), and (3.32), we find asymptotic forms as eikq r (1) eikq r (0) fqq δqp , ψq(1) ∼ f . r r qp ψq(0) ∼ eikq ·r + (0) To get the analytical solution of ψq (3.33) (1) and ψq , we firstly rewrite the lefthand side of Eq. (3.30) into the radial form. Then there are two equations should be solved. 1 D̂lq ulq = 0, D̂lq βlq = δ(r − r0 ), r (3.34) where D̂lq = d2 l(l + 1) 2M − − 2 V (r) + kq2 , 2 dr r2 h̄ (3.35) and where l is the quantum number of angular momentum. There are two linearly independent solutions for ulq in Eqs. (3.34). They are glq and hlq . When r = 0, glq ∝ rl+1 and hlq ∝ r−l . While r is outside the potential wall of V (r), they can be expressed into asymptotic forms, glq (r) ∼ sin(kq r − πl/2 + δlq ), and hlq (r) ∼ ei(kq r−πl/2+δlq ) . (3.36) (0) The wave function ψq (r) can be represented by the products of glq (r)/r and angular wave function Ylm (r̂). ψq(0) (r) = X lm alm glq (r) Ylm (r̂). r (3.37) When r is large, Eq. (3.37) should reduce to the asymptotic form shown in Eqs. (3.33). By comparing two equations, we can determine the coefficient alm , and therefore ψq(0) = 4π X i(πl/2+δlq ) e glq (r)Yl (r̂) · Yl (k̂q ), kq r l (3.38) 97 where Yl (r̂) · Yl (k̂q ) = P m Ylm (r̂) (1) ∗ (k̂ ). For ψ , the solution can be found by using a · Ylm q q method of Green’s function. The Green’s function is defined as µ ¶ 2M ∇2 − 2 V (r) + kq2 Gq (r, r0 ) = δ (3) (r − r0 ). h̄ (3.39) Therefore, from Eq. (3.30), we find Z ψq(1) (r) = d3 r0 Gq (r, r0 ) 4 2M X hq|H (1) |piψp(0) (r0 ). h̄2 p=1 (3.40) The solution of Green’s function is Gq (r, r0 ) = X1 l r βlq (r, r0 )Yl (r̂) · Yl (r̂0 ), (3.41) where βlq (r, r0 ) is one of the solution in Eqs. (3.34), which can be constructed by glq and hlq . In order to make Gq have asymptotic form as outgoing spherical wave, which is ∝ eikq r /r, and also have a finite value when r = 0, we require A(r0 )glq (r), r < r0 0 βlq (r, r ) = B(r0 )hlq (r), r ≥ r0 . (3.42) At the joint point, r = r0 , we ask βlq to be continuous but discontinuous of its 1st order derivative. Hence 0 1/r0 = −glq (r0 ) 0 (r 0 ) −glq hlq (r0 ) h0lq (r0 ) A(r0 ) B(r0 ) . (3.43) The solution of Eq. (3.43) is A(r0 ) B(r0 ) = hlq (r0 ) 1 , r0 W 0 glq (r ) (3.44) 0 (r 0 ). From Eqs. (3.34), we can verify dW/dr 0 = 0, hence where W = glq (r0 )h0lq (r0 ) − hlq (r0 )glq W is a constant. So we can use the asymptotic forms of glq and hlq to calculate W . Hence W = ikq sin θeiθ − kq cos θeiθ = −kq , (3.45) 98 where θ = kq r − πl/2 + δlq . The solution of Green’s function is Gq (r, r0 ) = − X l 1 hlq (r> )glq (r< )Yl (r̂) · Yl (r̂0 ), kq rr0 (3.46) where r> is the larger of r or r0 , and r< is the smaller. The asymptotic form for r À r0 is Gq ∼ − ekq r X i(−πl/2+δlq ) 1 glq (r0 )Yl (r̂) · Yl (r̂0 ). e r kq r 0 (3.47) l By substituting the Eq. (3.47) and Eq. (3.38) into Eq. (3.40) and comparing the asymptotic form in Eq. (3.33), we obtain first-order scattering amplitude of spin flip for initial state |pi to final state |qi, (1) fqp (r̂) = X ll0 − 2M h̄2 Z d3 r0 {Yl0 (r̂) · Yl0 (r̂0 )eiδl0 q 1 4π gl0 q (r0 )hq|H 0 |pieiδlp glp (r0 )Yl (r̂0 ) · Yl (k̂0p )}. 0 kq r kp r 0 (3.48) For spin exchange, [qp] can only be [32] or [23]. Hence the total cross section Z (1) dΩ|f32 (r̂)|2 σ̄ = (3.49) ¯ Z ∞ ¯2 X π ¯1 ¯ 0 0 0 ¯ 0 2M ¯ = (2l + 1) ¯ α(r )gl3 (r )gl2 (r )¯ , dr 2 h̄ 0 k̄ h̄k̄ l where k̄ = √ √ (1) (1) k3 k2 and f32 = f23 . We define σ̄ = σ32 σ23 , where σ32 = k3 k2 σ̄ and σ23 = σ̄, k2 k3 (3.50) which satisfy the detailed balancing, σ32 k22 = σ23 k32 , appeared in Eq (3.2). Finally, we obtain the binary spin-exchange rate coefficient Z κ= dv vP (v)σ̄(v). (3.51) The value κ32 and κ23 can be evaluated by replacing σ̄ by σ32 and σ23 into Eq. (3.51) correspondingly. 99 Connection Between SCA and DWBA To connect the SCA and the DWBA, we first look at the angular momentum, L ⇒ M vb ⇔ p h̄ l(l + 1), of the two colliding atoms. We can find the first connection as Z Z db 2πb = dl X π πh̄2 (2l + 1) ←→ (2l + 1). p2 k̄ 2 (3.52) l Secondly, by using Wentznel-Kramers-Brillouin (WKB) approximation, we can approximate glq (r) as s glq (r) ≈ q 2M (E − ²q )/h̄2 , θlq (r) = where kq = kq 0 θlq (r) Rr rlq sin (θlq (r)) , (3.53) 0 (r 0 ) + π/4 (note: the phase π/4 can be dr0 θlq obtained by using Airy function when r → rlq ), and s µ ¶ h̄2 l(l + 1) 2M 0 θlq (r) = E − ²q − V (r) − . 2M r2 h̄2 It is easy to find vlq (r) = h̄ 0 M θlq (r) (3.54) from classical analogy. Hence, we find the second connection as Z dr where v̄r = 0 2M h̄k̄ Z 0 0 0 α(r )gl3 (r )gl2 (r ) ←→ dr0 α(r0 ) [cos(θl3 − θl2 ) − cos(θl3 + θl2 )], v̄(r0 ) (3.55) √ v3 v2 . When E À ∆²q , we find the third connection, which is θl3 − θl2 (²2 − ²3 ) d θlq (r) = ≈ (²3 − ²2 ) d²q h̄ Z r r0 dr0 M 0 (r 0 ) = ∆ω · t. h̄θlq (3.56) By using three connections, (3.52), (3.55), and (3.56), and discarding the rapidly oscillating part, cos(θl2 +θl3 ) in Eq. (3.55), we can show the SCA and DWBA are equivalent if the total energy E is much larger than the spin-flipping energy, and the total integration distance is much larger than de Broglie wavelength. Figure 3.1 shows a typical result of SCA and DWBA. 100 v=300 m/s, l=0, B=30T 0.04 1 0.02 0 0 -1 glq(r) (SCA) -5 V(r) (eV), −α (r) ( 10 eV) 0.06 (DWBA) -0.02 0.5 1 Radial distance (nm) 1.5 Figure 3.1: An example of 129 Xe-Rb using SCA and DWBA numerical calculations. The SCA curve behaves like the mean value of the DWBA curve. 3.3 Experiments and Calculations At high temperature, where alkali vapor density is high enough to make spin-exchange dominate the Xe relaxation, the spin-exchange rate coefficient κ of 129 Xe is equal to the slope of longitudinal spin-relaxation rate 1/T1 as a function of the alkali-metal number density [A], namely κ = d(1/T1 )/d[A], (Eq. (3.16)). By independently measuring 1/T1 and [A] under the same conditions, we can retrieve κ. 1/T1 was measured by using NMR method, and the technique of Faraday rotation measurement was used to determine the value of [A]. The detailed information about the experimental apparatus is summarized in Fig. 3.2. A number of spherical sample cells, made of aluminosilicate glass, were prepared for the experiments[25]. Each cell was about 1 inch diameter, and contained Rb or Cs metal Z \+e)]^_`abPQR!&"#'(IJ$%K [Tcd UL),WMNLN* T Y, WV.PXQRfTUWNLI.S I V + . O * + BF5G C8?0DHE:721A6@ C9/;0:1<2=3>4?56789@ :A0481 101 Figure 3.2: A block diagram shows the total setup of the ments. with different mixture of enriched 129 Xe 129 Xe spin-exchange rate measure- (71%) and pure nitrogen gas. The sample cell was set into a probe, shown in Fig. 3.3. The experiments were carrying out in 9.4-T field, which was produced by an Oxford superconducting magnet. For comparisons of the spin-exchange rates from experiments to theories, C and MAT- LAB programs were used for the theoretical rate calculations. Since MATLAB is much more convenient for solving the differential equation, we used C to do the SCA calculations and MATLAB to do the DWBA calculations. With the same accuracy, the SCA calculation performed in C code was about 100 times faster than DWBA in MATLAB code. 3.3.1 NMR Measurements A homemade NMR spectrometer was built for measuring the longitudinal decay time T1 by detecting the nuclear spin signal of 129 Xe. IGOR software installed in a PC computer 102 Laser beam UP B (Scaled) Oxford 9.4T Superconducting Magnet Figure 3.3: The NMR probe was designed to fit into a 9.4-T Oxford superconducting magnet with bore diameter ∼ 3.5”. The internal space was heated resistively with non-inductive wiring. The heat conducting copper tube suppressed the thermal gradient to about 1◦ C per inch. The temperature was stabilized by the feedback of two thermal sensors. The interlayer between inner and outer tubes was stuffed by a porous foam for good heat insulation. The sample cell was held by the NMR coil and hung by two supports. The NMR coil was set to the resonance frequency by adjusting a variable capacitor. The tuning work was done by using a sweep generator (WAVETEK 1062). A probing beam for Faraday rotation measurement passed through the two windows and the center of the cell. The field inhomogeneity inside the 1 inch cell was about 0.3 ppm. with a Windows 2000 operation system took care of device communications, experimental control, and the data analysis. A complete NMR measurement cycle was accomplished by several stages: 1. The probing pulse sequence was originally arranged by the IGOR program and then sent to a pulse generator board (Pulse Blaster). 2. The output of the pulse generator board passed to an analog switch (Mini-Circuit ZASWA-2-50DR), which acted as a shutter to turn on or off the RF field. We used a frequency synthesizer (PTS310) as the RF source. It had a frequency range from 0 to 310MHz with an uncertainty less than 10 ppb. It was set to near 110.48MHz, which was about a few kHz offset from the precession frequency of 129 Xe nuclear spin at 9.4T. 3. After the analog switch, the RF pulse sequence was then amplified by 30–70dB (AMT 3304C with attenuators) and finally sent 103 Figure 3.4: Arbitrary unit for y-axes. The top panel shows two orthogonal FID signals. The bottom panel is the Fourier’s spectrum of the FID signal. This example signal was the result of eight times average FID from a 4 amagat Xe cell. to the NMR resonant coil inside the probe, Fig. 3.3. The probe was temperature stabilized by two OEMGA CNi-3252-C24 controllers. 4. In the part of signal detection, a π/2 pulse was sent to flip the nuclear spin to the transverse plane to excite a free-induction decay (FID) signal. A pre-amp (MITEQ 1467) with 60dB gain magnified the FID signal. 5. The magnified signal was then split into two channels and mixed with two orthogonal sine waves from the RF source. This down converted hundred megahertz FID signal to the kilohertz range. 6. Two channels connected with a low-pass filter (KROHN-HITE 3940) to get rid of the high frequency noise that would be down converted to the low frequency noise by the sampling process. 7. Digitized FID signals were finally acquired by a digital oscilloscope (YOKOGAWA DL708E) then transmitted back to a computer to do the Fourier’s spectrum analysis by IGOR. The example FID signal is shown in Fig. 3.4. We used saturation-recovery NMR technique to measure T1 . Initially, a saturating pulse train was sent to destroy the thermal equilibrium nuclear polarization hKz iT = µK B/2kT 104 Figure 3.5: The amplitude of the FID signal (filled circles) is proportional to the 129 Xe nuclear spin polarization hKz i as it recovers towards its thermal-equilibrium value hKz iT = µK B/2kT . The relaxation time T1 is extracted from the fit of Eq. (3.57), solid curve. Each data point was taken over 16 – 96 averages of the FID signal at the same delay time τ . (here µK is the nuclear magnetic moment of 129 Xe, k is the Boltzmann constant, and T is the absolute temperature). After a delay time τ , a single 90◦ tipping pulse was used to excite an FID signal from the recovering nuclear spin polarization hKz i, described by hKz i = hKz iT (1 − e−τ /T1 ) . (3.57) To evaluate the amplitude of the FID signal, we Fourier transformed the signal traces from time domain to frequency domain. The FID signal appeared as a resonant peak on the spectrum. This spectrum provided the information of the amplitude, the linewidth, and the frequency offset. Using the signal spectrum, we can verify the strength of the RF field, the homogeneity of the magnetic field, and the buffer gas pressure, which caused a chemical shift. By recording the amplitude of the FID signal at different delays τ , the relaxation rate 1/T1 can be retrieved as a parameter of the exponential fit, Eq. (3.57). A typical T1 measurement is shown in Fig. 3.5. 105 3.3.2 Faraday Rotation Measurements To avoid the uncertainty in alkali-metal vapor density, [A], associated with using the empirical vapor-pressure formulas [45, 33], we employed a Faraday rotation technique [25] to determine [A] at different temperatures and buffer-gas pressures. A Ti-sapphire ring laser (model “Coherent 899-29”), pumped by a Nd-Vanadate “Verdi” laser (Coherent), was used to generate the probing beam with a capability of wavelength scanning. Two optics sets were used for the ring laser. The short-wave optics were for scanning from 700 nm to 830 nm, which covers D1 and D2 of rubidium. The mid-wave optics were for scanning from 785nm to 910nm, which covers D1 and D2 of cesium. The probe beam was guided by two mirrors to redirect the beam vertically from horizontal setup. Detail of the measurement In our experiment, we can describe a light beam in a pure polarization state with two parameters: (i) the orientation angle θ of the semimajor axis of the elliptically polarized light relative to the x axis in a reference plane normal to the beam direction, and (ii) the mean photon spin s, a real number ranging from −1 to +1, which describes the relative circular polarization of the light. The ratio a of the minor to the major axis of the ellipse √ is related to the mean spin s by a = (1 − 1 − s2 )/s. In our experiment, the semimajor axis of the elliptically polarized light entering the cell was oriented at an angle θ0 , which we could vary for experimental convenience. The mean spin s0 of the light entering the cell was close to zero. By using a linear polarizer (LP) shown in Fig. 3.2, the probe light had nearly perfect linear polarization. Stray birefringence in lenses and windows and reflections from two turning mirrors, not shown in Fig. 3.2, could change both θ and s, but we verified experimentally that these were negligible effects. The orientation angle of the light emerging from the cell is θ = θ0 + ∆θ, where the Faraday rotation angle ∆θ can be several radians because of large [A] and B. In principle, the circular dichroism of the vapor can induce a corresponding change, ∆s, in the mean photon spin, but unless the laser wavelength is very close to the center of the absorption lines, ∆s 106 will be negligible and the light emerging from the vapor will remain nearly linearly polarized with s ≈ 0. To measure the Faraday rotation angle ∆θ, we set the stress axis of a photo-elastic modulator (PEM) (HINDS PEM-80) along the x axis, at 45◦ to the axes of a linearly-polarizing beam-splitter cube (BS-LP) [20]. The angle ψ of the relative phase retardation between the fast and slow axes of the PEM was varied sinusoidally with time t at the frequency ωp /2π = 83.4 kHz. That is, ψ = ψp sin ωp t, where the peak phase shift ψp is proportional to the driving voltage of the PEM and inversely proportional to the laser wavelength. After passing through the BS-LP, the probe beam was detected with a photodiode, which produced the output voltage, VPD = gIB · (1 + p 1 − s2 sin 2θ cos ψ + s sin ψ), (3.58) where g is the effective gain of the optical detection system. The laser power IB = IB (t) was mechanically chopped at the frequency Ω/2π = 153 Hz. The laser wavelength λ was scanned with time, and this caused large variations in the angle θ, and also some variation of IB because of the wavelength dependence of the laser output power. There were also noise fluctuations of IB . We can decompose the signal (3.58) into different frequency harmonics in the familiar way for the frequency modulation, VPD = V0 + Vωp sin ωp t + V2ωp cos 2ωp t + · · · , where amplitudes p 1 − s2 J0 (ψp ) sin 2θ) , V0 = gIB · (1 + Vωp = 2gIB · sJ1 (ψp ) , p = 2gIB · 1 − s2 J2 (ψp ) sin 2θ V2ωp (3.59) (3.60) (3.61) are chopped at the relatively low frequency Ω. Here Jn are Bessel functions of the first kind. We empirically adjusted the PEM driving voltage to find the smallest amplitude ψp = ψ0 (that is, ψ0 = 2.405) for which J0 (ψp ) = 0. Then V0 = gIB became independent of the linear polarization angle θ, and could be used to normalize the signals Vωp and V2ωp . A 107 small correction was made to the data to account for the slight variation of ψp ∝ 1/λ over the 1% range of the wavelengths used in this work. As sketched in Fig. 3.2, the signal V2ωp was detected with two lock-in amplifiers (LIA). The voltage VPD was applied to the first LIA, which was referenced to the PEM drive frequency 2ωp with a 1 ms integration time. The output of the first LIA was applied to a second LIA, referenced to the probe laser chopping frequency Ω with a 0.1 s integration time. For further analysis, the output voltage of the second LIA is processed with one channel of a digitizing interface (not shown in Fig. 3.2) to obtain a digitized signal S2ωp ∝ V2ωp . This was stored, along with the laser wavelength, in a computer (PC). The signal V0 was detected by applying VPD to a third LIA, referenced to the chopper frequency Ω and with a 0.1 s integration time, and acquired as described above to obtain the digitized signal S0 ∝ V0 . A digitized record of the probe laser wavelength λ was stored along with each pair of values, S0 and S2ωp . For further analysis of the digitized data, we computed the ratio R = R(λ) = S2ωp /S0 . According to (3.59) and (3.61) the ratio should be p R = S2ωp /S0 = G 1 − s2 sin 2(θ0 + ∆θ), (3.62) where the constant G accounts for the differences in LIA gains in the system. In order to acquire the Faraday rotation data sin2θ, we first measured the mean photon spin s by using a lock-in amplifier locked to the signal s sin ψ, contributed by the last term in Eq. (3.58), at the first harmonic of ωp . A typical measured value of s as a function of laser detuning is shown by a dashed line in Fig. 3.6. By locking to the second harmonic of ωp , we measured √ 1 − s2 sin 2θ that appears in the second term of Eq. (3.58). We used the measured mean photon spin s to extract sin2θ in Eq. (3.62) from the second-harmonic signal. 108 Figure 3.6: Faraday rotation signal sin2θ (open circles), fit to the calculation of Eqs. (3.63– 3.65) (solid line) using the fitting parameters [Cs], θ0 , and ωJ . The optical path length is 2.5 cm and B = 9.4 T. The dashed line shows the mean photon spin s measured from the first harmonic of the photodetector output. The residual non-zero photon spin at high laser frequency (far detuning from the D2 line) is ascribed to the effect of optical guiding mirrors. High-field Faraday rotation From the well-known theory of Faraday rotation (e. g. see [55]), the rotation angle for the light of wavelength λ = 2πc/ωλ close to the strong D1 and D2 resonance lines is ∆θ = l [A] X Jmσ AJmσ (ωλ − ωJmσ ) , [(ωλ − ωJmσ )2 + γ 2 /4] (3.63) where l is the optical path length of the laser beam inside the cell and the quantum numbers Jmσ are discussed below. The coefficients AJmσ are h i2 J, m+σ 3π ωλ re c fJ σ C1/2, ρm m; 1, σ AJmσ = , (2J + 1) ωJ (3.64) where re = e2 /me c2 = 2.82 × 10−13 cm is the classical electron radius, e and me are the electron charge and mass in cgs units, c is the speed of light, fJ are the oscillator strengths of absorption lines to the 2 PJ excited states. To a good approximation f1/2 = 1/3 and 109 f3/2 = 2/3. The resonant frequencies of the two D lines are ωJ = 2πc/λJ . These are pressure shifted slightly from the free-atom values of λ1/2 = 7947 Å and λ3/2 = 7800 Å for Rb and λ1/2 = 8943 Å and λ3/2 = 8521 Å for Cs. We assume pressure-broadened Lorentzian lines with the same width at half maximum of γ, as indicated in Eq. (3.63). The quantity in square brackets in Eq. (A.43) is a Clebsch-Gordan coefficient [49]. The index σ = ±1 determines whether the absorbed light is left- or right-circularly polarized. The index m = ±1/2 is the azimuthal electron spin quantum number of the ground-state Rb atom, and ρm = Z −1 exp{−2mµB B/kT } is the probability of a given atom being in a state with the quantum number m. Here Z = 2 cosh µB B/kT ; the cell temperature is T (in kelvin), and µB is the Bohr magneton. The resonant frequencies ωJmσ are ωJmσ = ωJ + µB B [gJ (m + σ) − 2m], h̄ (3.65) where the Zeeman splitting of the excited-state azimuthal levels is determined by the gvalues, g1/2 = 2/3 and g3/2 = 4/3. Of the eight possible combinations for the summation indices in Equation (3.63), Jmσ = (1 ± 1/2, ± 1/2, ±1), the triples (1/2, 1/2, 1) and (1/2, −1/2, −1) are excluded since they refer to non-existent excited-state sublevels with J = 1/2 and mJ = ±3/2. So there are only 6 non-zero terms in the sum (3.63). From inspection of Eq. (3.62), we see that the experimentally-determined ratio R on the left can be made equal to the theoretical expression on the right by setting the gain parameter G to the maximum value of R, and then fitting the unknown number density [A] along with the approximately known initial-orientation angle θ0 and the resonance-line wavelengths λJ . The errors of such [A] measurements are about 3%, mostly due to small uncertainties in l, γ, and ρm (T ). With the relatively large values of magnetic field (B = 9.4 T) and atomic number densities ([A] ∼ 1014 cm−3 ) used in our experiments, we could observe a few oscillations of the signal V2ωp , with negligible attenuation, as the laser was tuned toward the resonance line. The resulting highly-structured signal could be fit very precisely (Fig. 3.6). To obtain the maximum data with this limitation, the initial orientation angle θ0 was chosen to make 110 the signal from the second LIA slightly positive in the limit of large positive detunings, ωλ − ω3/2 À γ. 111 3.4 3.4.1 Results and Analysis Binary Spin-Exchange Rate Coefficients We prepared four Rb cells and five Cs cells for the measurements of spin-exchange rate coefficients. By changing the temperature, we obtained different number density of Rb or Cs. So we were able to measure the nuclear spin relaxation rates depending on the Rb or Cs number density. The measured values of κ are listed in Table 3.1 and 3.2. Figure 3.7 and 3.8 summarize the experimental results of all data points. We obtained our experimentally determined values of κ are κRbXe (180◦ C, 9.4T) = (1.75±0.12)×10−16 cm3 /s and κCsXe (130◦ C, 9.4T) = (2.81 ± 0.20) × 10−16 cm3 /s. Cell No. T094 T095 T096 T097 Xe density (amg) 0.79 0.79 0.79 4.0 N2 density (amg) 3.0 1.0 0 0 κ (10−16 cm3 /s) 1.84 ± 0.17 1.60 ± 0.17 1.76 ± 0.16 1.78 ± 0.10 Table 3.1: Experimentally measured values of κ for the four Rb cells used in our experiment. (Note: 1 amg=2.69 × 1019 cm−3 , is equal to the number density of an ideal gas at 0◦ C and 760 torr) Cell No. T114 T115 T116 T117 T118 Xe density (amg) 1.0 4.0 1.0 1.0 1.0 N2 density (amg) 3.5 0 0 1.0 2.0 κ (10−16 cm3 /s) 2.98 ± 0.14 3.05 ± 0.15 2.72 ± 0.20 2.72 ± 0.25 2.57 ± 0.13 Table 3.2: Experimentally measured values of κ for the five Cs cells used in our experiment. Several previous measurements of spin-exchange rates were done at very low magnetic field. In order to compare our results with the others, we used theoretical decoupling curve to extrapolate the κ values from 9.4T to zero field. The decoupling curves were numerically calculated by using SCA and DWBA methods. For the van der Waals potential V (r), we chose the semiempirical potentials of Buck and Pauli[15], which are based on the results of 112 3 Spin-exchange rate coefficients (cm /s) -1 Measured relexation rate 1/T1 (s ) 0.12 T094 ( 1.84 ± 0.17)×10 0.10 T095 ( 1.60 ± 0.17)×10 T096 ( 1.76 ± 0.16)×10 0.08 T097 ( 1.78 ± 0.10)×10 -16 -16 -16 -16 0.06 0.04 0.02 0.00 0 100 200 300 400 500 600 12 -3 Measured Rb number density [Rb] (10 cm ) Figure 3.7: Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured Rb vapor number densities [Rb] for the four cells listed in Tab. 3.1. The experimental temperature were between 160 to 200◦ C. The spin-exchange rate coefficients κ are the slopes of the straight-line fits to the data. scattering experiments. They are parametrized as follows: h¡ ¢ ¡ ¢ i 4² rm 11 − 11 rm 4 , r ≤ rm 7 r 4 r V (r) = h¡ ¢ ¡ ¢ i 3² rm 14 − 7 rm 6 , r > rm 4 r 3 r , (3.66) where ² is the depth of the minimum potential energy and rm is the distance where the potential energy is equal to ². For Xe-Rb, ² = 13.68 meV and rm = 0.536 nm. For Xe-Cs, ² = 13.62 meV and rm = 0.547 nm. The spin coupling coefficient α for a 129 Xe atom, displaced a distance r from an alkali- metal atom, is α(r) = 8πgS µB µK |ψ(r)|2 , 3K (3.67) where the nuclear spin K = 1/2 and µK = −0.7725 × 5.05 × 10−27 J/G. We use the orthogonalized wave (OW) approximation[54] to write ψ(r), the amplitude of the valence 113 3 Spin-exchange rate coefficients (cm /s) -1 Measured relexation rate 1/T1 (s ) 0.05 T114 ( 2.98 ± 0.14)×10 0.04 T115 ( 3.05 ± 0.15)×10 T116 ( 2.72 ± 0.20)×10 T117 ( 2.72 ± 0.25)×10 0.03 T118 ( 2.57 ± 0.13)×10 -16 -16 -16 -16 -16 0.02 0.01 0.00 0 20 40 60 80 100 120 140 12 -3 Measured Cs number density [Cs] (10 cm ) Figure 3.8: Measured longitudinal nuclear spin-relaxation rates 1/T1 plotted versus measured Cs vapor number densities [Cs] for the five cells listed in Tab. 3.2. The experimental temperature were between 110 to 150◦ C. electron wave function of the alkal-metal atom at the nucleus of the Xe atom as ψ(r) = φ(r) − 5 X Z χn (0) d3 s χ∗n (s)φ(r + s). (3.68) n=1 Here φ(r) is the amplitude of the unperturbed valence electron orbital of an alkali-metal atom (5s for Rb and 6s for Cs). The occupied s-wave orbitals of the Xe core are denoted by χn (s) with principal quantum numbers n. We denote the displacement of an electron from the Xe nucleus by s. For numerical evaluation of (3.68) we used the tabulated orbitals of Clementi and Roetti [17]. The calculated V (r) and α(r) of 129 Xe-Rb are shown in Fig. 3.1. The numerical computation showed that there is less than 2% difference of the results between SCA and DWBA [26]. Figure 3.9 shows the calculated decoupling curves by using SCA method. We obtained the κ values at zero field to be κRbXe (180◦ C, 0T) = (2.2±0.15)× 10−16 cm3 /s and κCsXe (130◦ C, 0T) = (4.1 ± 0.30) × 10−16 cm3 /s from the extrapolations. From the calculations, there was a slight temperature dependence of κ, which was about κ−1 dκ/dT = 240ppm/◦ C around T = 100◦ C and B = 0 tesla. The magnetic decoupling in 3 Spin-exchange rate coefficient κ (10-16 cm /s) 114 5 CsXe theory RbXe theory 4 Experiment at 9.4 T: 3 CsXe: -16 3 (2.81 ± 0.20)×10 cm /s 2 RbXe: -16 3 (1.75 ± 0.12)×10 cm /s 1 0 0 10 20 30 40 Magnetic field B (T) 50 60 Figure 3.9: Theoretical magnetic decoupling curves used to extrapolate our from B=9.4T to B=0T. Since the experimentally determined potential V had about 5% error, we made 3 to 4% adjustments to the position of the minimum of the interatomic potential V can bring a good agreement between experimental data and calculations. The shape of decoupling curves changes less than 0.1% by applying adjustments on V . Xe-Cs is more pronounced than in Xe-Rb, because of the slower relative velocity between xenon and cesium. In Table 3.3, we list κ-values of 129 Xe-Rb and 129 Xe-Cs obtained by different groups. In out experiments, we eliminated two potentially serious sources of systematic error: (1) the number density of alkali-metal atoms, and (2) the poorly-understood contribution of van der Waals molecules to the spin exchange. One or both of these sources of error were present in all of the earlier measurements listed in Table 3.3. We used Faraday rotation measurements to determine the number density of alkali-metal atoms. Under different conditions, we found that the real number density of alkali-metal atoms might be larger or smaller by a factor of two than the prediction from the empirical vapor pressure formula. We chose a high magnetic field to eliminate the contribution of van der Waals molecules to spin exchange. The other groups either used saturated vapor pressure formula to determine the atomic 115 Name Rb experiments: Grover[21] Volk[50] Zeng[58] Cates[16] Augustine[5] Rice[39] Our work[25] Rb calculations: Walker[51] Walter[54] Our work[26, 27] Cs experiments: Zeng[59] Liu[36] Ishikawa[24] Our work[27] Cs calculations: Walker[51] Walter[54] Our work[26, 27] Temperature (◦ C) B-field (Tesla) κ (10−16 cm3 /s) Year ∼ 70 ∼ 65 ∼ 90 ∼ 100 ∼ 85 ∼ 110 ∼ 180 2 × 10−7 2 × 10−7 5 × 10−4 1.1 × 10−5 2.35 4.7 9.4 < 87 < 41 4.1 3.7 ± 0.7 < 2.8 0.6 ± 0.1 1.75 ± 0.12 (1978) (1980) (1985) (1992) (1997) (2002) (2002) 100 100 100 0 0 0 6.3 1.2 2.1 ± 0.15 (1989) (1998) (2004) ∼ 50 ∼ 60 ∼ 25 ∼ 130 1.88 4.7 2.2 × 10−4 9.4 < 5.1 < 6.3 < 10.2 2.81 ± 0.2 (1991) (1992) (2000) (2003) 100 100 100 0 0 0 9.2 2.7 4.0 ± 0.3 (1989) (1998) (2004) Table 3.3: A summary of binary spin-exchange rate coefficients at different magnetic fields obtained by different groups. The zero-field rates quoted for our work are slightly larger than the values measured at B = 9.4 T because of adjustments for the magnetic decoupling discussed in connection with Fig. 3.9. density from temperature [21, 50, 58, 16, 5, 39, 59, 36] or used extrapolation to eliminate the contribution of van der Waals molecules [58, 16, 39]. The recent work of Kadlecek [30] shows that we don’t really understand the spin-exchange mechanism from van der Waals molecules when the pressure is larger than one atmosphere. Their result showed that at high pressure, van der Waals molecules can contribute a substantially larger spin-relaxation rates than we expected. Thus the extrapolation method used by authors, such as Cates [16] and the others, might not be appropriate. From our experimental data, we did not see any noticeable pressure dependence of the κ values. The pressure independence is to be expected if the rates are due to binary collisions, but not if the rates include a substantial contribution from the formation and breakup of 116 van der Waals molecules. We concluded there were no important contributions from van der Waals molecules in our measurements. The results also show that Cs has a larger spinexchange rate coefficient, which agrees with some previous measurements. Therefore, once the high power pumping source is ready for cesium, Cs can be a potential candidate in spin-exchange optical pumping of xenon, not only because of its larger κ, but also its higher vapor pressure. 3.4.2 High-Field Contribution of van der Waals Molecules At high field, we estimated the contribution of van der Waals molecules to spin-exchange rates. We use Xe-Rb as an example. Similarly arguments hold for Xe-Cs. For simplicity of notation we assume 100% isotopic abundance of 129 Xe. From the similar discussion in section 3.2, we can write the spin Hamiltonian of a Xe-Rb molecule as H = h̄ωSz + α S · K = H (0) + H (1) . (3.69) Here, S is the spin operator of the Rb valence electron, K is the spin operator of the 129 Xe nucleus, a representative value of the hyperfine coupling coefficient averaged over vibrational-rotational states of the molecule is ᾱ/h̄ ≈ 2.0 × 108 s−1 , and ω = 2µB B/h̄ = 1.65 × 1012 s−1 is the Larmor frequency of the electron in a magnetic field of 9.4 T. At this large field, we can ignore the nuclear spin I of the Rb atom in calculating its spin exchange rates with 129 Xe. The zeroth-order part of the Hamiltonian (3.69) is H (0) = h̄ωSz + ᾱSz Kz and the spin-exchange perturbation is H (1) = ᾱ (S+ K− + S− K+ )/2. The 129 Xe-Rb molecule will be formed in one of the four spin states |mS mK i, where mS = ±1/2 and mK = ±1/2 are the azimuthal quantum numbers of the Rb electron spin and the Xe nuclear spin, correspondingly. These are eigenstates of the zeroth-order Hamiltonian, H (0) |mS mK i = (h̄ωmS + ᾱmS mK )|mS mK i ≈ h̄ωmS |mS mK i. We denote with |1i for mS = 1/2, mK = −1/2 and |2i for mS = −1/2, mK = 1/2. One can (1) readily verify that H12 = h1|H (1) |2i = ᾱ/2. From first-order perturbation theory, the probability that a molecule, formed in the state |1i at time t = 0, will have made a transition to the state |2i at time t > 0 is W = (ᾱ/h̄ω)2 sin2 (ωt/2). Previous experimental work 117 [58, 12] has shown that Xe-Rb molecules are broken up by collisions with other xenon atoms or nitrogen molecules. The mean lifetime τ was found to be very nearly 1/τ = ([N2 ] + 3.6 [Xe]) × 5.5 × 109 s−1 amg−1 . A Xe atom is approximately 3.6 times as likely to break up a Xe-Rb molecule as an N2 molecule. Considering a high pressure cell, T094, we find a shorter duration τ = 31 ps. Therefore τ ω = 51. The magnetic field is large enough that the probability of a spin flip from state |1i to |2i is independent of the molecular lifetime τ and is hW i = (ᾱ/h̄ω)2 /2 = 7.3 × 10−9 . The factor of 1/2 is from the mean value of sin2 (ωt/2). A Xe-Rb molecule can form in simultaneous collisions of a Xe atom, a Rb atom and a third atom or molecule. The third body is another Xe atom or a nitrogen molecule N2 in our cells. The number density of molecules will therefore grow at the rate d [XeRb] = ζ [Xe][Rb]. dt (3.70) where ζ = Zr ( [N2 ] + 3.6 [Xe] ). Three-body rate coefficients Zr have been determined from previous experimental work. For nitrogen [58], Zr = 3.9 × 10−32 cm6 /s (after rescaling the original data by the ratio of temperatures 349 K/453 K). For cell T094, this implies that ζ = 6.2×10−12 cm3 /s. Denoting the number densities of spin-up and spin-down atoms with the subscripts + and − respectively, we see the formation of Xe-Rb molecules will cause [Xe+ ] to change at the rate d [Xe+ ] = ζ ([Xe− ][Rb+ ] − [Xe+ ][Rb− ])hW i. dt (3.71) We have neglected the slight difference in the exothermic and endothermic collisions. In keeping with this approximation we have neglected the thermal polarization of the Rb atoms. From Eq (3.9) and Eq (3.10), We find d hKz i = −κ0 [Rb]hKz i, dt (3.72) where the three-body rate coefficient for cell T094 is κ0 = ζ hW i = 4.5 × 10−20 cm3 /s. (3.73) 118 This is 3900 times smaller than the experimental result, κ = 1.75 × 10−16 cm3 /s. Similarly small estimates are found for the other cells. Although the spin exchange from van der Waals molecules is still not understood above an atmosphere pressure. We believe the estimates shown above is valid based on following arguments. The molecule spin-exchange rate is proportional to the molecule formation rate and the transition probability. From the discussion above, we know that in a magnetic field the transition probability can not be larger than (ᾱ/h̄ω)2 . From the calculations, it is smaller than the transition probability of a binary collision in our experiments. Besides, the molecule formation needs some mechanisms involved with other particles rather than just binary collisions. In our experiments, the formation rate should be less than the binary collision rate. Otherwise, we might be able to see a systemic pressure dependence on our experimental data. We conclude that Xe-Rb and Xe-Cs molecules cannot contribute more than 1% to the rates measured in these experiments. The uncertainties in the estimates of the molecular formation and breakup rates are not large enough to change this conclusion. Appendix A Evolution of the Density Matrix of Alkali-Metal Atoms To derive the evolution equation of the density matrix, we can start from the Schröedinger equation for the ith atom of an ensemble, ih̄ d |ψi i = Hi |ψi i, dt (A.1) where |ψi i = |ψi (t)i represents the atomic quantum state of an alkali-metal atom, and Hi = H (0) +H (1) (2) + Hi is the total Hamiltonian for each atom. Here H independent “unperturbed” Hamiltonian, and H (1) =H (1) (0) is a time- (t) is a time-dependent purtur- bation representing the interaction of the atom with the optical pumping, microwave, and RF fields. Both H (0) and H (1) are identical for all atoms. The collisional interaction of the (2) ith atom is represented by the time-dependent Hamiltonian Hi (2) = Hi (t). The density matrix of an ensemble of N identical atoms can be described by ρ= 1 X |ψi ihψi |. N (A.2) i In the interaction representation, we can define the quantum state, |ψei i, in the interaction frame by |ψi i = e−iH (0) 119 t/h̄ |ψei i. (A.3) 120 Then, ih̄ d e e (1) + H e i(2) )|ψei i, |ψi i = (H dt (A.4) where e (n) = eiH H (0) t/h̄ H (n) e−iH (0) t/h̄ . (A.5) The density matrix in the interaction frame, ρ̃ is defined by 1 ρ = e ih̄ H A.1 (0) t 1 ρ̃ e− ih̄ H (0) t . (A.6) Ground-State Relaxation Due to Weak Collisions (2) A weak collision is one for which the peak value of hHi /h̄i is much smaller than the inverse of the collisional duration δt. S-damping and Carver damping come from weak collisions. But, spin exchange from a collision of the two alkali-metal atoms is a strong collision. For each collision event, we can find the evolution for the quantum state |ψei i from Eq. (A.4). ei |ψei (t)i, |ψei (t + δt)i = U (A.7) where the evolution operator is µ ei = exp U 1 ih̄ Z t t+δt ¶ (2) 0 0 e Hi (t )dt . (A.8) In Schröedinger picture, the evolution operator for |ψi i becomes Ui = e−iH (0) (t+δt)/h̄ e iH Ui e (0) t/h̄ . (A.9) Ui is unitary so U U † = 1. If we denote the density matrix of an atom just before the ith collision by ρ̃ and just ei ρ̃U e † . We obtain that the time derivative of after the collision by ρ̃ + δ ρ̃, we find ρ̃ + δ ρ̃ = U i ρ due to H (2) is equal to the product of the collision frequency and the average of the small change, δ ρ̃, from each collision. Therefore · dρ̃ dt ¸(2) D E ei ρ̃ U e † − ρ̃ , =N U i (A.10) 121 where N is the number of collisions occurred per unit time in an ensemble system, and h i denotes the ensemble average over all possible types of collisions, for example, over all ei into impact parameters and orbital directions. We can expand U Z ∞ e i(2) (t0 ) dt0 + 1 H 2 −∞ 1 2 = 1 − iΘi − Θi + · · · . 2 ei = 1 + 1 U ih̄ µ 1 ih̄ Z ∞ −∞ e (2) (t0 )i dt0 H ¶2 + ··· (A.11) Here, Θi is a Hermitian operator. For weak collisions, hΘi i ¿ 1. So we can approximate e , hence Eq. (A.10) to the second order of U · dρ̃ dt ¸(2) · ¸ 1 = N −i[ hΘi i, ρ̃ ] + hΘi ρ̃Θi i − { hΘi Θi i, ρ̃ } . 2 (A.12) Considering the different initial state |νi and the final state |µi, we find 1 hµ|Θi |νi = h̄ Z ∞ −∞ (2) (2) hµ|Hi (t)|νi eiωµν t dt = hµ|Hi (−ωµν )|νi/h̄, (2) (A.13) (2) where Hi (ω) is the Fourier’s spectrum of Hi (t). The result of Eq. (A.13) shows that hµ|Θi |νi depends on the Bohr frequency ωµν . If we can change δtωµν substantially by changing the external field, we can change the nature of the collisional relaxation. For most cases, the effective duration of a collision, δt, is much shorter than 2π/ωµν , so Θi ∼ = (2) Hi (0)/h̄. For our experimental interest, the collisional duration, δt, is so short compared to the ei . Combining H (0) , H (1) , and H (2) , we find the total dynamic inverse of ωµν . Hence, Ui ≈ U equation of the density matrix to be · ¸ dρ 1 1 1 (0) (1) = [ H , ρ ] + [ H , ρ ] + N −i[ hΘi, ρ ] + hΘρΘi − { hΘΘi, ρ } . dt ih̄ ih̄ 2 (A.14) For convenience, we use hΘi to denote hΘi i, and apply the same rule for all other ensemble averages. The evolution equation of the density matrix should conserve the total number of atoms. We can check that Tr(dρ/dt) = 0 from Eq. (A.14), because Tr(Oρ)=Tr(ρO), where O is any operator. By using Eq. (A.14), we are able to calculate the evolution of all observables analytically or numerically. 122 A.1.1 S-Damping The S-damping or spin destruction rate is caused by the transferring of the angular momentum between the electron spin S and the angular momentum N of the two colliding atoms (2) through spin-rotation interaction, Hsd = γ(r)N · S. Here the γ is the coupling coefficient, which is a function of the atomic separation. Assuming that the duration of the collision is much shorter than all 2π/ωµν , we find Θsd Z 1 = h̄ ∞ γN · Sdt = φ · S, (A.15) −∞ where φ = (φx , φy , φz ). The statistical property of φ is that hφi = 0 and hφi φj i = δij hφ2i i. The integration performed in Eq. (A.15) can be carried out by using the trajectory calculation. In this semi-classical approximation, the atomic separation r(t) is a function of time. By using Eq. (A.12), we find the relaxation due to the S-damping as ρ̇sd ¶ µ ¶ µ 3 1 = Γsd S · ρS − {S · S, ρ} = Γsd S · ρS − ρ , 2 4 (A.16) where the S-damping rate Γsd = Nsd hφ 2 i/3. We like to express the S-damping relaxation as ρ̇sd = Γsd (ϕ − ρ), (A.17) where ϕ = ρ/4 + S · ρS is the density matrix without the electronic polarization. We can verify that Tr(ϕ) = 1 and Tr(Sϕ) = 0. A.1.2 Carver Rate The atomic hyperfine structure is ascribed to electron-nuclear spin coupling, AI · S. The coupling strength A is perturbed by the collisions between atoms. We describe the per(2) turbation Hamiltonian of the hyperfine interaction as HC = δAI · S. The perturbation δA = δA(r) is due to the collisionally-induced change in the value of the valence-electron wave function at the nuclei of the alkali-metal atom. Therefore, we can find 1 ΘC = h̄ Z ∞ −∞ δAI · Sdt = φC I · S. (A.18) 123 Similarly, we obtain the evolution of the density matrix due to the random hyperfine modulation by applying Eq. (A.12). ρ̇C µ ¶ hφ2C i hφ2C i = NC −i[ hφC iI · S, ρ ] + 2 I · SρI · S − {I · SI · S, ρ} . h̄ 2h̄2 (A.19) In the weak magnetic field, we can still use |f, mi to represent the eigenstate. We use f = I +1/2 = a to label the upper hyperfine sublevels and f = I −1/2 = b to label the lower hyperfine sublevels. So we have I · S|a, mi = 12 I|a, mi and I · S|b, mi = − 12 (I + 1)|b, mi. We P define the density matrix of hyperfine transition ρ(m) = |fi , ma ihmb , fj |, where fi 6= fj . We can rewrite Eq. (A.19) into ρ̇C = η 2 [I]2 1 [ δEC , ρ(m) ] − I ΓC ρ(m) , ih̄ 8 (A.20) where [I] = 2I + 1. Here, the collision shift or pressure shift δEC = h̄NC hφC iI · S. The ”Carver rate” ΓC = NC hφ2C i/ηI2 , which was first introduced by D.K. Walter [53]. The isotope coefficient is ηI = µI /(2IµN ), where µI and µN are the nuclear magnetic moment and nuclear magneton. The random modulation of hyperfine splitting can only cause the dephasing of the coherence between upper and lower hyperfine sublevels when the magnetic field is weak. It does not transfer population from one sublevel to another. A.2 Ground-State Relaxation Due to Strong Collisions and Other Mechanism A.2.1 Spin Exchange Spin exchange occurs when two alkali-metal atoms collide with each other. Spin-exchange collisions are the extreme opposite of weak collisions. Through the spin-spin coupling, (2) Hex = J(r)S0 · S, electrons can exchange their spins from each other. Usually the strength of the spin-spin coupling between two alkali-metal atoms is comparable to the interatomic potential V0 (r). Hence, we can not use Eq. (A.12). More detailed discussions can be found 124 in Ref [22, 52, 31], and we find the relaxation due to spin exchange as ρ̇ex = = = = µ ¶ 0 ® 3 00 0 0 0 0 [A] −i[ 2hvσ S i · S, ρ ] + hvσ i(4 S · SρS · S − ρ + {hS · Si, ρ}) 4 µ ¶ 1 3 [ δEex , ρ ] + Γex S · ρS + 2ihS0 i · iS × ρS − ρ + {hS0 i · S, ρ} ih̄ 4 ¡ ¢ 1 [ δEex , ρ ] + Γex ϕ − ρ − 2hS0 i · S × ρS + {hS0 i · S, ρ} ih̄ ¡ ¢ 1 [ δEex , ρ ] + Γex ϕ − ρ + 4ϕhS0 i · S , (A.21) ih̄ where [A] is the number density of the alkali-metal atoms, v is the relative velocity of the two colliding atoms, and the spin-exchange cross section σ = σ 0 + iσ 00 is defined by [31] σ0 = σ 00 = π X (2l + 1) sin2 δl k2 l π X (2l + 1) sin2 2δl . 2k 2 (A.22) l Here, δl = 3 δl −1 δl is the difference between phase shifts 3 δl and 1 δl for scattering of the lth partial wave on the triplet and singlet potentials of the alkali-metal dimers. The spatial frequency k, is proportion to the relative velocity v. Some discussions about the partial wave calculations can be found in section 3.2.2. If the kinetic energy in the central mass frame of the two colliding atoms is larger than the interaction strength hJi, we find that δl ∝ hJi. The spin-exchange shift δEex = 2h̄[A]hvσ 00 S0 i · S, which produces an effective magnetic field from the spin polarization, and the spin-exchange rate Γex = [A]hvσ 0 i. Since both S0 and S belong to the same ensemble, we can express the spin-exchange relaxation as ρ̇ex = 1 [ δEex , ρ ] + Γex [ϕ(1 + 4hSi · S) − ρ] . ih̄ (A.23) We can verify that Tr(Sρ̇ex ) = 0. The spin-exchange mechanism conserves the total electron spin. A.2.2 Spin Diffusion In the alkali-metal vapor, atoms keep colliding and changing their position. This causes atoms to diffuse spatially. The relaxation of the density matrix in a certain coordinate 125 caused by diffusion process can be described by ∂ρ(r) = D∇2 ρ(r), ∂t (A.24) where D is the diffusion coefficient. Since each atom carries their own atomic spin, the diffusion of atoms leads to the spin diffusion. When the spin flow hits a boundary, such as the cell wall, a strong depolarization occurs and forces atoms to even out the population of the spin states. (Note: it is very possible that the cell wall traps the incoming atoms and releases new unpolarized atoms) We can write down the evolution of the density matrix due to the spin diffusion as ρ̇d = Γd (1/g − ρ), (A.25) where 1 is the unit matrix, and g = 2[I] is the total number of the ground-state sublevels. We choose Γd to represent the relaxation rate due to the spin diffusion. Considering a simplest case with a cubic cell, the length of each side is l. Hence we can find that ρ(x, y, z) ∝ X Aa sin( a,b,c aπ bπ cπ x)Bb sin( y)Cc sin( z), l l l (A.26) where a, b, c are integer numbers. If we only consider the lowest mode, which is a = b = c = 1, the diffusion relaxation rate Γd = 3D(π/l)2 . A.3 Excited-State Relaxation For our experimental interest, we only discuss the P1/2 excited state of alkali-metal atoms. A.3.1 J-Damping The J-damping rate is very similar to the S-damping. Instead of depolarizing the electron spin in S1/2 ground state, the expectation value of electron angular momentum in P1/2 is (2) diminished due to the interaction Hjd = γj (r)N · J. Here the γj is the coupling coefficient. Similarly, the relaxation due to the J-damping in P1/2 state is ¶ µ 1 (e) (e) (e) ρ̇jd = Γjd J · ρ J − {J · J, ρ } , 2 where Γjd is the J-damping rate. (A.27) 126 A.3.2 Spontaneous Decay and Quenching There are two important deexcitation mechanisms causing atoms to decay from the excited state to the ground state. Spontaneous decay is due to the electromagnetic perturbation from the ground state of the quantized EM field to the atoms. Atoms return to the S1/2 states from the P1/2 states by randomly emitting photons. Quenching decay is caused by atoms colliding with quenching gases. Atoms decay to ground states by transferring the excited-state energy to the internal degrees of freedom of the quenching gases. Usually, the decay process is dominated by the strong coupling with electron. It is fairly good to ignore the tiny contribution from the nuclear spin. Therefore, spontaneous decay or quenching partially or completely destroys the electronic polarization, but they conserved the nuclear polarization in the decay process. The relaxation equation due to spontaneous decay and quenching is (e) (e) ρ̇sq = −Γs ρ (e) − Γq ρ , (A.28) where Γs is the spontaneous decay rate and Γq is the quenching rate. With optical pumping, some population was transferred from the ground state to the excited state. Assuming that Υ is the population injection rate from the ground state by optical pumping, and Γjd = 0, we find (e) ρ̇ = 1 (0) (e) (e) [ H , ρ ] − (Γs + Γq )ρ + Υ. ih̄ e (A.29) (0) Here, He is the unperturbed spin-dependent Hamiltonian of the P1/2 state. For each P (en) (e) (e) component of the density matrix, ρkl , we can expand to ρkl = n ρkl e−inωt . We can find (en) the steady solution of the amplitude ρkl for each frequency component nω by solving d (en) (en) ρkl = −(i(ωkl − nω) + Γs + Γq )ρkl + Υkl = 0. dt (A.30) Hence, (n) (en) ρkl = (n) (0) Υkl , i(ωkl − nω) + Γs + Γq (A.31) (1) where Υkl is defined by Υkl = Υkl + Υkl e−iωt + · · · . The result of Eq. (A.31) will be useful in the discussion of the repopulation pumping with non-zero frequency component of the 127 density matrix. The repopulation pumping due to spontaneous decay and quenching will be discussed in the section of optical pumping. A.4 Microwave and RF Fields Considering the ground-state hyperfine sublevels, the oscillating field interacts with alkalimetal atoms through (1) Hof = B(t) · (µB gS S − µI I/I), (A.32) where B(t) = Bof (eiωt + e−iωt ) is the oscillating magnetic field. Usually, we ignore the interaction to the nuclear spin I, because it is much smaller than the interaction to the electron spin S. Therefore, we find the time evolution equation due to the oscillating field to be ρ̇of = 1 (1) [ Hof , ρ ]. ih̄ (A.33) Considering two hyperfine sublevels |µi and |νi coupled by the oscillating field, we find that (1) the Rabi frequency produced by the oscillating field is equal to ωR = h̄2 hµ|Hof |νi. We can also find the rate equation related to these two sublevels due to the oscillating field as ρ̇µµ = −iωR /2(ρ̃νµ − ρ̃µν ) ρ̇νν = −iωR /2(ρ̃µν − ρ̃νµ ) ρ̃˙ µν = (−i(ωµν − ω) − γµν )ρ̃µν + iωR /2(ρµµ − ρνν ) ρ̃˙ νµ = (i(ωµν − ω) − γµν )ρ̃νµ − iωR /2(ρµµ − ρνν ), (A.34) where ρ̃µν is define by ρµν = ρ̃µν e−iωt . Here, we drop the rapid oscillation term with the frequency component ωµν +ω (secular approximation or rotation wave approximation). The solution of ρ̃µν is iωR ρ̃µν (t) = 2 Z t t0 0 e−[i(ωµν −ω)+γµν ](t−t ) (ρµµ (t0 ) − ρνν (t0 ))dt0 . (A.35) Here, the integral shows that the oscillating field is turned on at t0 , and t ≥ t0 . For late-time evolution, ρµµ (t) and ρνν (t) become slowly varying. We find the quasi-steady solution of 128 ρ̃µν to be ρ̃µν (t) = iωR /2(ρµµ (t) − ρνν (t)) , i(ωµν − ω) + γµν (A.36) which is the exact solution if the ensemble reaches equilibrium. By substituting Eq. (A.36) into Eq. (A.34), we find the evolution equation of the density matrix due to the oscillating field in the quasi-steady state to be ρ̇of = −Γof ((ρµµ − ρνν )|µihµ| + (ρνν − ρµµ )|νihν|) , (A.37) and similarly, we find the equivalent light shift operation δEof 1 [δEof , ρof ] = ((ρµσ |µihσ| − ρνσ |νihσ|) − (ρσµ |σihµ| − ρσν |σihν|)) , ih̄ ih̄ (A.38) where σ 6= µ and σ 6= ν. The AC Stark shift energy Eof and the effective pumping rate Γof are defined by δEof = 2 /4(ω h̄ωR µν − ω) , 2 2 (ωµν − ω) + γσ(ν,µ) Γof = 2 /2γ ωR µν . 2 (ωµν − ω)2 + γµν (A.39) The damping rate γσ(ν,µ) shown in Eq. (A.39) has a swapped subscript, which means γσν for ρσµ and ρµσ ; also γσµ for ρσν and ρνσ . We have to notice that the results of Eq. (A.37) −1 . and Eq. (A.38) are correct only when t À γµν A.5 Optical Pumping For D1 pumping of the alkali-metal atoms, the S1/2 ground state and the P1/2 excited state are coupled to the electromagnetic field of the pumping light. The transition between the S and the P states is caused by the electric dipole interaction with the oscillating electric field. Therefore, the interaction Hamiltonian of optical pumping is (1) Hop = −D · Eop = −D · (E + E∗ ), (A.40) where D = −er is the dipole operator, and Eop = Eop (t), a real-value function, is the electric field of the pumping light. Here, E and E∗ are complex conjugate. We use E to 129 represent the negative-frequency field, and use E∗ to represent the positive-frequency field. The complex field E = E(t) can be expressed by E(t) = 1 2π Z ∞ 0 dΩĒop (Ω)e−iΩt , (A.41) where Ēop (Ω) = Ē∗op (−Ω) is the spectrum of the pumping electric field, and Ω is the optical frequency. For transitions between S1/2 and P1/2 , we use |S, mS i and |J, mJ i to represent the angular-momentum quantum states of the valence electron of the ground state and the excited state. We find that the dipole operator can be expressed by [22, 3] D = D(ge) + D(eg) = DA + D∗ A† , (A.42) where transition operators A† and A can be decomposed by A†+ = |J, 21 ihS, − 12 |, A†− = |J, − 12 ihS, 12 |, A†z = 12 (|J, 12 ihS, 21 | − |J, − 12 ihS, − 12 |) . and A+ = |S, 21 ihJ, − 12 |, A− = |S, − 12 ihJ, 12 |, Az = 12 (|S, 12 ihJ, 21 | − |S, − 12 ihJ, − 12 |) (A.43) In Cartesian coordinate, we find A† = A†x x + A†y y + A†z z = A†+ +A†− x 2 + A†+ −A†− y 2i + A†z z . and A = Ax x + Ay y + Az z = A+ +A− x 2 + A+ −A− y 2i (A.44) + Az z For D1 transition, S = 1/2 and J = 1/2. Therefore we know both A† and A have the same matrix representation as the spin- 12 operator S. In Eq. (A.42), the dipole amplitude D is defined by [18] |D|2 = 2h̄re c2 feg , ω0 (A.45) where re = 2.82 × 10−12 cm is the classical electron radius, c is the speed of light, feg ≈ 1/3 is the oscillation strength of D1 resonance, and ω0 = ωeg is the mean Bohr frequency of D1 transition. Equation (A.40) now can be written as (1) Hop = −DA · E∗ − D∗ A† · E. (A.46) 130 Here, we only need to consider the processes of the excitation from the ground state to the excited state by annihilating photons and the deexcitation from the excited state to the ground state by creating photons. In Eq. (A.46), we throw out the terms of −DA · E and −D∗ A† · E∗ , which make the excitation with photon creation and deexcitation with photon destruction. Although, these virtual transitions can really happen, their effects are extremely small compared to our experimental concerns. The complex electric field E(t) = E(t)ε̂, where E(t) is the complex scalar field, and ε̂ is a complex vector with unitary length to describe the polarization. The photon spin s can be expressed by [3] s= iε̂ × ε̂∗ = iε̂ × ε̂∗ . ε̂ · ε̂∗ (A.47) Optical Coherence The optical coherence is induced by the pumping field. In the interaction picture, we find d 1 e (1) ρ̃ = [ H , ρ̃ ]. dt ih̄ op (A.48) (1) e op Here, H is defined by Eq. (A.5), and ρ̃ is defined by Eq. (A.6). For distinguishing the ground state and the excited state, we use Greek letters to label the ground state, and English letters to label the excited state. From Eq. (A.48), we find d 1 e (1) e (1) ρ̃kν = [ H kµ ρ̃µν − ρ̃kl Hlν ]. dt ih̄ (A.49) (1) e (1) = hk|H e op Here, H |µi. In Eq. (A.49), the two consecutive subindices represent the action kµ of summation. For calculation, we have to sum over the subindices µ and l in Eq. (A.49). The same rule is applied for all following calculations. Typically, the pumping rate, Γop is up to 104 s−1 , and the deexcitation rate, Γsq = Γs + Γq ≥ 108 s−1 . Under the condition of unsaturated optical pumping (Γop ¿ Γsq ), the population in the excited state is negligible. Therefore, we can confidently neglect the part of ρ̃kl in Eq. (A.49). With the introduction of the optical coherence damping rate γop , we find the differential equations for optical 131 coherences ρ̃kµ as d dt ρ̃kµ ∗ = − Dih̄ E · A†kν eiωkν t ρ̃νµ − γop ρ̃kµ , . and d dt ρ̃µk (A.50) D = − ih̄ ρ̃µν E∗ · Aνk eiωνk t − γop ρ̃µk The solution of Eqs. (A.50) can be found in the same manner of Eq. (A.35). Therefore ρ̃kµ D∗ =− ih̄ Z t t0 0 0 e−γop (t−t ) eiωkν t E(t0 ) · A†kν ρ̃νµ dt0 . (A.51) Generally, the maximum changing rate of ρ̃νµ is a few kHz. Therefore, by substituting Eq. (A.41) into Eq. (A.51), we can carry out the integration by using the slow varying approximation, and find ρ̃kµ = − D∗ Ê(t, ωkν ) · A†kν ρ̃νµ eiωkν t , ih̄ where 1 Ê(t, ωkν ) = 2π Z ∞ dΩ 0 Ēop (Ω)e−iΩt . i(ωkν − Ω) + γop (A.52) (A.53) One can see that Ê(t, ωkν ) is equivalent to that the pumping field Eop (t) is filtered by a resonator filter at frequency ωkν with the bandwidth 2γop . Similarly, we can integrate Eq. (A.51) to find ρ̃µk = A.5.1 D ρ̃µν Aνk · Ê∗ (t, ωkν )eiωνk t = ρ̃∗kµ . ih̄ (A.54) D1 Depopulation Pumping To calculate the population, which is pumping out from the ground state to the excited state, we take the matrix elements of Eq. (A.48) to find d ρ̃µν dt 1 e (1) e (1) ] [ Hµk ρ̃kν − ρ̃µk H (A.55) kν ih̄ i |D|2 h = − 2 E∗ · Aµk A†kσ · Ê(ωkσ )ρ̃σν eiωσν t + ρ̃µσ Ê∗ (ωσk ) · Akµ A†kν · Eeiωµσ t eiωµν t . h̄ = In the Schröedinger picture, Eq. (A.55) can be rewritten as d (g) 1 1 (0) † ρ = [ H , ρ(g) ] + [ δHop ρ(g) − ρ(g) δHop ]. dt ih̄ ih̄ (A.56) 132 Here, ρ(g) denotes the density matrix of the ground-state, and the matrix elements of the effective Hamiltonian δHop are hµ|δHop |νi = δHµν = |D|2 ∗ 2h̄ ∗ (op) † E · Aµk A†kν · Ê(ωkν ) = ε̂ · Aµk Akν · ε̂ Γkν (t), ih̄ i (A.57) where the time-dependent optical pumping rate between the ground-state sublevel ν and the excited-state sublevel k is Z (op) Γkν (t) = Z dω e −iωt ∞ dΩ 0 ∗ (Ω − ω)Ē (Ω) |D|2 Ēop op . 2 i(Ω 2h̄ kν − Ω) + γop (op) (A.58) (op) From the calculation of Eq. (A.51), we find that Γνk = Γkν ∗ . The effective Hamiltonian can be written as the sum of a Hermitian and an anti-Hermitian operators, δHop = δEop − ih̄ δΓ. 2 (A.59) The Hermitian light-shift operator is 1 † δEv = (δHop + δHop ), 2 (A.60) and the Hermitian light-absorption operator is δΓ = i † (δHop − δHop ). h̄ (A.61) Here, δEv is called “virtual transition” light-shift operator. It has nonzero value only when the optical detuning of the pumping light is performed. The equation for depopulation pumping is written as ρ̇dp = 1 1 1 † [δHop ρ − ρδHop ] = [δEv , ρ] − {δΓ, ρ}. ih̄ ih̄ 2 (A.62) In our experimental studies, the Zeeman splittings are much smaller than the hyperfine (e) splitting, and the excited hyperfine splitting is not optically resolved (γop > ωhf ). Under these experimental conditions, the optical pumping rate is only determined by the upper or lower ground-state hyperfine multiplets. Therefore, from Eq. (A.57), we find 1 † ε̂∗ · Aµk Akν · ε̂ −→ ε̂∗ · AA† · ε̂ = ε̂∗i (AA† )ij ε̂j = (1 − 2s · S). 4 (A.63) 133 Here, we use one identity of D1 transition, which says (AA† )ij = (SS)ij = 41 (δij − iSk ²ijk ). Hence, the Eq. (A.57) can be further simplified into h̄ (op) (1 − 2s · S)Γ , 2i δHop = (A.64) where the optical pumping rate is a diagonal matrix, and its diagonal matrix elements are Z (op) Γµµ = Z dω e ∞ −iωt 0 ∗ (Ω − ω)Ē (Ω) |D|2 Ēop op . dΩ 2 i(Ω − Ω) + γop 2h̄ µ (A.65) Here, Ωµ is the transition frequency from the selected ground-state sublevel µ to the central gravity of the target of the excited hyperfine sublevels. For the CPT experiments, the probing light is usually modulated at the half frequency of the hyperfine resonance. In this way, the symmetric optical sidebands to the carrier frequency is produced. By calculating Eq. (A.65), the complex pumping rate of the frequency component, which is equal to the microwave resonance frequency, has the same value for both upper and lower hyperfine multiplets. In our experiments, we intensity-modulated the laser to generate the equal intensity two first-order sidebands and suppressed the carrier and high-order sidebands. In this case, when the optical detuning is small, it is a very good approximation to write the scalar pumping rate as (0) (1) (−1) Γop = Γop + Γop e−i2ωm t + Γop ei2ωm t , (−1) where Γop (A.66) (1) = Γop ∗ , and ωm is the modulation frequency, which is equal or slightly de- tuned to the hyperfine resonance frequency. In this two-wave approximation, we find (0) (0) (1) (1) that Im(Γop )/Re(Γop ) = Im(Γop )/Re(Γop ) = −∆/γop , where the optical detuning ∆ = (Ω0 − Ωc ). Here, Ω0 is the transition frequency from the center of the two resonant hyperfine sublevels to the excited state, and Ωc is the carrier frequency of the probing beam. The corresponding light-shift and light-absorption operators are δEv = h̄2 (1 − 2s · S)Im(Γop ) and δΓ = (1 − 2s · S)Re(Γop ). We find the expectation value of the DC photon absorption rate to be (0) (0) Tr(δΓ ρ) = Re(Γop )(1 − 2s · hSi). (A.67) 134 A.5.2 D1 Repopulation Pumping Repopulation is a regeneration of ground-state atoms due to spontaneous decay or quenching decay of excited atoms. The evolution equation of the density matrix of repopulation pumping cause by spontaneous decay is [22] 4 4ω03 d (g) (e) (e) DA · ρ D∗ A† = Γs A · ρ A† . ρ = dt eh̄c3 3 (A.68) The spontaneous decay rate Γs = 2ω02 re feg /c. For quenching decay, we assume that the electron polarization is completely destroyed. Comparing to ϕ, which is the ground-state density matrix without the electronic polarization, we find the evolution equation for quenching process is given by d (g) ρ = Γq dt µ ¶ 1 (e) † (e) † Uρ U + A · ρ A . 4 (A.69) Here, U = |S, 12 ihJ, 12 | + |S, − 12 ihJ, − 12 | is the mapping operator. To evaluate Eq. (A.68) and Eq. (A.69), we need to find the steady solution of the density matrix of the excited state. The injection population rate Υkl appeared in Eq. (A.30) can be calculated by applying Eq. (A.48). Therefore, we find 1 e (1) e (1) ] = d ρ̃kl [ Hlν ρ̃νl − ρ̃kν H (A.70) νl ih̄ dt i |D|2 h † † ∗ ∗ = Ê(ω ) · A ρ̃ A · E + E · A ρ̃ A · Ê (ω ) ei(ωkl −ωνµ )t νµ νµ kν µl µl lµ kν kν h̄2 (op) (op) = 2(Γkν + Γµl )(ε̂ · A†kν ρ̃νµ Aµl · ε̂∗ )ei(ωkl −ωνµ )t . e kl = Υ e kl is defined by Υkl = Υ e kl e−iωkl t . For the modulated light, we only care about the Here, Υ pumping rate with frequency components of ω = 0 and ω = 2ωm ∼ ω0 , where ω0 is the (op) (0) (1) (−1) selected ground-state resonance frequency. Hence, Γkν = Γkν +Γkν e−i2ωm t +Γkν ei2ωm t . In our experiments, we used nitrogen as the buffer gas, which produced minimal J-damping relaxation in the excited state. From Eq. (A.31), we find the steady value of ρkl in Schröedinger frame to be ρkl = X µ,ν;n=(0,±1) = X µ,ν;n=(0,±1) (n) i(ωkl − ωµν Υkl − 2nωm ) + Γs + Γq 2(Γkµ + Γνl )(ε̂ · A†kµ ρµν Aνl · ε̂∗ )e−i2nωm t (n) (n) i(ωkl − ωµν − 2nωm ) + Γs + Γq . (A.71) 135 For our experimental condition, the buffer gas leads to a quenching-dominant deexcitation, which is Γq À Γs . Therefore, by using Eq. (A.69), we find the regeneration of the groundstate population due to the repopulation pumping as d ραβ = dt X µ,ν,n,k,l µ Γq (n) (n) 2(Γkµ + Γνl )e−i2nωm t i(ωkl − ωµν − 2nωm ) + Γq × ¶ 1 † † ∗ † ∗ † Uαk ε̂ · Akµ ρµν Aνl · ε̂ Ulβ + Aαk · (ε̂ · Akµ ρµν Aνl · ε̂ Alβ ) .(A.72) 4 It is easier to write the repopulation equation in Liouville space, (rp) ρ̇αβ = Rαβ,µν ρµν , (A.73) where the matrix elements of the repopulation operator R(rp) is (rp) Rij = (rp) Rαβ,µν = X (n) (n) 2(Γkµ + Γνl )e−i2nωm t × i(ωkl − ωµν − 2nωm ) + Γq n,k,l µ ¶ 1 † † ∗ † ∗ † Γq Uαk ε̂ · Akµ Aνl · ε̂ Ulβ + Aαk · (ε̂ · Akµ Aνl · ε̂ Alβ ) .(A.74) 4 (rp) Here, we find the “real transition” light shift operator ih̄Im(Rµν,µν ) for the density-matrix element ρµν . The light shift here is due to the real transition. The ground-state coherence is optically pumped to the excited state. The difference of the coherence frequencies between the ground-state and the excited state causes a difference of time-phase between the repopulated coherence and the ground-state coherence. Hence, the effective coherence oscillation is dragged or pushed by the repopulation pumping, which leads to a frequency shift. Typically, when the buffer-gas pressure is above hundred torr, the quenching rate is (e) larger than the excited hyperfine splitting (Γq > ωhf ). If all ground-state hyperfine sublevels see the same pumping rate, we can simplified Eq. (A.72) for the diagonal elements of the density matrix into d (g) ρ dt ³ ´ (g) (g) = 4Re(Γop ) U ε̂ · A† ρ A · ε̂∗ U † + A · (ε̂ · A† ρ A · ε̂∗ A† ) µ ¶ 1 = Re(Γop ) {1 − 4s · S, ρ} + (S · ρS − is · (S × ρS)) . 8 (A.75) 136 (rp) It is understood that Eq. (A.75) is only valid for ρ̇αα = Rαα,µν ρµν . Combining depopulation pumping, we find the longitudinal evolution equation of the density matrix to be d (g) ρ dt µ ¶ 1 1 = − {δΓ, ρ} + Re(Γop ) {1 − 4s · S, ρ} + (S · ρS − is · (S × ρS)) 2 8 · ¸ 1 1 = Re(Γop ) − {1 − 2s · S, ρ} + {1 − 4s · S, ρ} + (S · ρS − is · (S × ρS)) 2 8 = Re(Γop ) [ϕ(1 + 2s · S) − ρ] . (e) (A.76) (g) When ωhf < Γq < ωhf , the result of Eq. (A.76) is good for the calculation of ρ̇αα = (rp) (g) Rαα,µν ρµν . When Γq > ωhf , the result of Eq. (A.76) is valid for the calculation of all (rp) density-matrix elements, which is ρ̇αβ = Rαβ,µν ρµν . In this subsection of repopulation pumping, we assume that the buffer gas, such as nitrogen, has strong quenching ability but small J-damping in the excited state. However, many buffer gases, such as helium and argon, can produce large J-damping rates in the excited state. The equations shown above are on longer valid. For the additional J-damping mechanism in the excited state, it is easier to numerically carry out the calculation in Liouville space. There is no simple form that can be found by using the regular formalism of the density matrix. Appendix B Circuit Diagram of Signal Divider C3 1uF R3 200 To B 4 +15V 2 U2A TL082 U1 AD734 1 3 1 2 + BNC PD1 in BNC PD1 out +VCC DD 14 13 C1 1uF 8 To C 8 To A 3 BNC PD2 out 2 W Z1 Z2 U4A TL082 + 1 - 3 4 5 R1 2M 10nF 6 7 4 C4 X1 X2 R4 20k ERREF U0 U1 U2 Y1 Y 2 -VCC 12 11 10 BNC (PD1/PD2) x Gain out 9 C2 1uF 8 To D U3B TL082 BNC Gain out -15V 7 8 4 5 8 - + To C To D D1 20V 6 5 + BNC PD2 in U5B TL082 - 6 7 4 To B R2 200k To A 0.1uF C9 A B C6 1uF C C5 1nF +15V -15V D C7 1uF C8 1nF +15V -15V R5 5k +15V Figure B.1: Because of this divider circuit, we were able to obtain the first light narrowing data from the fiber-coupled and noisy Ti-sapphire laser light. 137 References [1] 1 amagat (amg) = 2.687 × 1019 cm−3 . [2] M. S. Albert, G. D. Cates, B. Driehuys, W. 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