Measurement of Berry's Phase for Partial Cycles ... - Semantic Scholar
864
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
Measurement of Berry’s Phase for Partial Cycles Using a Time-Domain Atomic Interferometer Makoto Yasuhara, Takatoshi Aoki, Hirotaka Narui, and Atsuo Morinaga
Abstract—The Berry phase of atom for partial cycles of rotating magnetic field in parameter space was successfully measured free from the dynamical phase shift by using a time-domain sodium atom interferometer. The obtained phase shift was in an agreement with the prediction of Berry’s phase for partial cycles. Index Terms—Atom interferomtry, Berry’s phase, rotating magnetic field.
I. INTRODUCTION
I
N 1984, Berry predicted that the wave function of an adiabatically changing quantum-mechanical system acquires the topological (geometrical) phases for cyclic evolution of the Hamiltonian [1] and proposed an experiment for particles with nonzero spin under whole turn of the magnetic field, which subtends a solid angle of . Many experimental demonstrations of Berry’s phase have been carried out using polarized light, neutron interferometers, nuclear magnetic resonance and so on [2]–[5]. In 1992, Miniatura et al. demonstrated the first atomic topological phase by using a longitudinal Stern-Gerlach atomic interferometer [6]. However most of those experiments were done using moving particles, so that the topological phase was obscured with the dynamical phase shift. On the other hand, time-domain atom interferometers with cold atoms were pointed out as being suitable for measuring the topological phase that is free from the dynamical phase, because the cold atoms could be assumed to be at rest during the interaction time [7]. Up to now, one experiment on Berry’s phase using cold atoms was demonstrated in an atom-light system, where the dark state was adiabatically changed via the parameters of the light field [8]. This system had no dynamical phase shift, but an effective magnetic quantum number must be identified. Very recently, the topological phase was imprinted in the atoms of the Bose–Einstein condensate by adiabatically inverting the magnetic bias field and succeeded in generating vortices in a Bose–Einstein condensate [9]. However, there has been no demonstration of Berry’s original proposal for bosonic atoms with a magnetic moment with a directionally rotating magnetic field, except for a conical magnetic field configuration. It is also interesting to investigate the topological effect on atoms for partial cycles, i.e., not for a complete turn of the magnetic field in parameter space, as Weinfurter and Badurek have demonstrated under the interaction of neutrons with a time-dependent magnetic field [10], which was rather complicated to
Manuscript received July 2, 2004; revised October 26, 2004. The authors are with the Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan. Digital Object Identifier 10.1109/TIM.2005.843564
understand. Berry’s phase for partial cycles was discussed theoretically by Samuel et al. [11] and Jordan [12]. It is also important to investigate Berry’s phase because it might be one of the noise sources of future atomic frequency standards. Atom interferometers using the interaction of atoms with light as a beam-splitter, which were realized in 1991, have become a powerful tool to measure quantum phases, because it measures the phase difference between two states with a different quantum number [13], [14]. One of authors (Morinaga) has developed the Mach–Zender type atom interferometer using copropagating laser beams [15]. Previously, the Aharonov-Casher phase was measured using a space-domain atom interferometer [16], [17] and the scalar Aharonov–Bohm effect was demonstrated using the time-domain atom interferometer and has verified that the phase shift was in proportion to the variation of the strength of the resultant magnetic field during two light pulses [18]. In that experiment, the phase change was found to be due only to the variation in magnitude of the resultant magnetic field, not to the rotation of the magnetic field which rotated by a certain angle in one direction and returned to the original position. Therefore, it will be interesting to examine how the phase of the state with magnetic quantum number m changes due to the rotation of the magnetic field with constant amplitude in one direction, which corresponds to Berry’s phase. In this paper, we demonstrated the measurement of Berry’s phase according to Berry’s proposal using a time-domain atom interferometer and measured Berry’s phase for partial cycles.
II. PRINCIPLE The time domain atom interferometer is composed of cold ensembles of a two-level atom whose magnetic quantum numrespectively and has two coherent laser pulses bers are m and with a pulse width of and a frequency , which is near resobetween two states at a nant with the transition frequency , as shown in Fig. 1. The atom is initially magnetic field of in the lower state of m under the quantization magnetic field. After two irradiations of laser pulses which are separated by in time ( should be sufficiently short before the atoms start to fall down due to gravity), the Ramsey resonance occurs in the population of the upper levels as a function of the detuning frequency . If the quantization magnetic field rotates during the of time interval between two pulses, Berry’s phase would be observed as a product of the rotating angle of the field multiplied by the difference of magnetic quantum numbers in the phase of
0018-9456/$20.00 © 2005 IEEE
YASUHARA et al.: MEASUREMENT OF BERRY’S PHASE FOR PARTIAL CYCLES
Fig. 1. Schematic diagram of the measurement of Berry’s phase. A time domain atom interferometer is composed of two laser pulses whose time interval is T . The frequency of the laser with circular polarization is tuned to the transition frequency between two states whose magnetic quantum numbers of m and . During T , the magnetic field rotates by an angle of '.
m
the Ramsey fringes. The interference fringes in the population of the upper level are described as follows:
(1) where the first term denotes the Ramsey fringes, the second term denotes the scalar Aharonov–Bohm phase and the third term is the Berry’s phase. For partial cycles, Berry’s phase corresponds to (2) where denotes the rotation angle of the magnetic field during . With a whole turn of , it produces a phase of [19]. If the amplitude of the magnetic field remained at during the interval , no scalar Aharonov–Bohm effect occurs and we could observe unambiguously the Berry’s phase as the rotation angle varies during . Our strategy to observe Berry’s phase for partial cycles is to use the stimulated Raman transitions between the ground hyperfine states of sodium atom. The Zeeman splitting of the two and are shown in Fig. 2. ground hyperfine states of , For example, we could select only the states of and , by using the two-photon Raman transitions, polarwhich are detuned by 500 MHz in frequency from the to and ized transition of to from the polarized transition of (or and transitions between these via ). The propagation direction of the two-photon Raman beams is orthogonal to the rotation plane of the magnetic field, which also defines the quantization axis. The polarization of the Raman beam is a right-handed circular polarization, hence the electric field can be resolved into two components, one parallel ( polarization) and one orthogonal ( polarization ) to the magnetic field at any time during the rotation. The two optical phases
865
Fig. 2. Energy diagram of the ground hyperfine states of sodium atoms under weak magnetic field and a two-photon stimulated Raman transition.
which the atom acquires from the two Raman beams at absorption and emission simultaneously will be cancelled out. After two Raman pulses and a rotation of magnetic field between them, we can observe the phase shift in the population of the , which is the difference between the Berry state and . phases generated in the states of III. EXPERIMENTAL The experimental setup for the time-domain atom interferometer used was almost the same as that used for the study of the scalar Aharonov–Bohm effect using cold sodium atoms [9]. Here we describe the essential points of the apparatus. For a state were rotation magnetic field sodium atoms in the trapped in a magneto-optical trap at a temperature of 1 mK. At 3 ms after the free expansion of the trapped atoms, sodium atoms state. were initialized by optical pumping to the ground Then the quantization magnetic field was applied to the sodium atoms and the two right-handed circular polarized Raman laser pulses with a pulse width of 20 s were applied to them with a pulse separation of 80 s in order to make the atom interferom100 s. The propagation direction of the Raman eter with laser beams was orthogonal to the direction of the quantization magnetic field and the plane where the magnetic field rotates. state by the polarized The atom was excited to the Raman laser beam and polarized laser beam with frequencies and . One of the two Raman laser beams was generof ated from a frequency stabilized dye laser whose frequency was tuned by a frequency 500 MHz lower than the resonance freand and the frequency of the states of quency of the other Raman beam was shifted from that through a phase-modulated electro-optic modulator which was driven by a frequency synthesizer at around 1.77 GHz. The frequency was swept by a frequency synthesizer in order to generate the Ramsey fringes. The rotating magnetic field was produced by two mutually orthogonal pairs of Helmholtz coils which were driven by alternating currents with a relative phase shift of 90 and with the same field strength. In order to maintain a constant strength of magnetic field during rotation, the amplitude of each alternating
866
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
Fig. 3. Zeeman splittings of stimulated Raman transitions for four directions. (a) x, (b) x, (c) y , and (d) y .
+
0
+
0
Fig. 5. Ramsey fringes under (a) a constant magnetic field and (b) a rotating magnetic field. The rotating angle is radians. The phase difference is observed at the center frequency of the spectra.
IV. RESULTS AND DISCUSSION
Fig. 4. Timing diagram of measurement of Berry’s phase.
current was adjusted so that the Zeeman frequency shifts of the resonance for both axes of the Helmholtz coil agreed with each other within 1 kHz. The typical Zeeman spectra are shown in Fig. 3 for two orthogonal axis and two counter directions. The constant magnetic field is typically 0.02 mT, which corresponds 140 kHz. Therefore, the rotato a Larmor frequency of tion frequency of the magnetic field was adjusted to be less than 10 kHz, to fulfill the adiabatic condition. During the interval between the two Raman pulses, the resultant magnetic field was rotated with a almost constant frequency of (DC 10 kHz), . then the magnetic field has rotated by an angle of after The transition probability of atoms in the state of two Raman pulses was deduced from the absorption coefficients in the of the laser beams which are resonant from the state to the in the state. The time sequence diagram after magneto-optical trap (MOT) is summarized in Fig. 4. The time sequence was repeated every 10 ms and the transition probabilities for each run were accumulated and averaged in a computer. The population probability of the excited state was monitored as a function of detuning frequency.
With a pulse separation of 100 s, the visibility of the Ramsey and was refringes for the states of duced to only 0.16, because they were sensitive to the residual time-dependent magnetic field. Therefore, in the present experand were selected iment, the states of instead. Fig. 5(a) shows a typical Ramsey fringe under a DC magnetic field, which was obtained with a pulse width of 20 s and a pulse separation of 80 s. Within the spectrum width of 40 kHz, Ramsey fringes with a cycle of 10 kHz can be clearly seen with a visibility of 0.42. Fig. 5(b) shows the Ramsey resonance under a rotating magnetic field with a frequency of 5 kHz under the same pulse trains. The visibilities are almost the same as Fig. 5(a) with the same fringe cycles. But we can see the shift of the center frequency of the envelope and the shift between both phases at the center frequency. This frequency shift was within 10 kHz for each measurement and it may occur due to the fluctuation of the magnetic field at the irradiation times of the two pulses, although we adjusted them in advance. The phase at the center frequency of the envelope was obtained for several rotation frequencies up to 10 kHz for clockwise rotation (CW) and counterclockwise (CCW) rotation of magnetic field. The frequency of CCW was described with a negative signature for convenience. At 10 kHz, the magnetic 100 s. Fig. 6 shows the phase field turns full circle for shift as a function of rotation angle for the resonance between , , and , . The phase the states of changed along a straight line from a positive phase to a negative phase as the frequency changed from a negative to a positive value. It should be noted that the phase changed with a rotation angle for partial cycle according to (2). The slope of the phase shift is reduced to be 0.97 0.13, which is in agreement with the prediction of Berry’s phase for the phase difference between and . Furthermore, the fact that the the states of
YASUHARA et al.: MEASUREMENT OF BERRY’S PHASE FOR PARTIAL CYCLES
867
discussions on Berry’s phase over 10 years. The authors also thank P. Toschek for discussions on atom interferometry.
REFERENCES
Fig. 6. Phase shifts between the wavepackets of the F = 2, m = 1 state and the F = 1, m = 0 state as a function of rotation angle of the magnetic field.
slopes for both directions of rotation are the same will mean that the dynamical phase shift is not included in it. In order to test the dependence of the magnetic quantum between the states of number, the phase shift for , and , was also examined. As expected according to theory, the inclination of the phase and the slope is 1.2 shift was reversed to that of 0.2. This result clearly proves that the measured phase is the Berry phase. Furthermore, the present results show that Berry’s phase depends on the sign of the g-factor of the state which we discussed in another paper [20]. V. SUMMARY In conclusion, we demonstrated Berry’s phase for magnetic field rotation using a time-domain atom interferometer free from the dynamical phase shift. The results show that the phase is given by a magnetic quantum number multiplied by the rotation angle for partial cycles, and that the phase shift depends on the direction of rotation and the dependence of the magnetic quantum number. ACKNOWLEDGMENT A. Morinaga would like to thank M. Kitano of Kyoto University and J. Helmcke and F. Riehle of PTB, Germany, for many
[1] M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” in Proc. R. Soc. Lond., vol. A392, 1984, pp. 45–57. [2] A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett., vol. 57, pp. 937–949, 1986. [3] T. Bitter and D. Dubbers, “Manifestation of Berry’s phase in neutron spin rotation,” Phys. Rev. Lett., vol. 59, pp. 251–254, 1987. [4] D. Suter, G. C. Chingas, R. A. Harris, and A. Pines, “Berry’s phase in magnetic resonance,” Mol. Phys., vol. 61, pp. 1327–1340, 1987. [5] A. Shapere and F. Wilczek, Eds., Geometric Phases in Physics, Singapore: World Scientific, 1989. [6] C. Miniatura, J. Robert, O. Gorceix, V. Lorent, S. Le Boiteux, J. Reinhardt, and J. Baudon, “Atomic interferences and the topological phase,” Phys. Rev. Lett., vol. 69, pp. 261–264, 1992. [7] K. Sengstock, U. Sterr, G. Hennig, D. Bettermann, J. H. Muller, and W. Ertmer, “Optical Ramsey interferences on laser cooled and trapped atoms detected by electron shelving,” Opt. Commun., vol. 103, pp. 73–78, 1993. [8] C. L. Webb, R. M. Godun, G. S. Summy, M. K. Oberthaler, P. D. Featonby, C. J. Foot, and K. Burnett, “Measurement of Berry’s phase using an atom interferometer,” Phys. Rev. A, vol. 60, pp. R1783–R1786, 1999. [9] A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard, and W. Ketterle, “Imprinting vortices in a Bose–Einstein condensate using topological phases,” Phys. Rev. Lett., vol. 89, pp. 190 403-1–190 403-4, 2002. [10] H. Weinfurter and G. Badurek, “Measurement of Berry’s phase for noncyclic evolution,” Phys. Rev. Lett., vol. 64, pp. 1318–1321, 1990. [11] J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett., vol. 60, pp. 2339–2342, 1988. [12] T. F. Jordan, “Berry’s phases for partial cycles,” Phys. Rev. A, vol. 38, pp. 1590–1592, 1988. [13] F. Riehle, T. Kisters, A. Witte, J. Helmcke, and C. J. Bordé, “Optical Ramsey spectroscopy in a rotating frame; Sagnac effect in a matter-wave interferometer,” Phys. Rev. Lett., vol. 67, pp. 177–180, 1991. [14] M. Kasevich and S. Chu, “Atomic interferometry using stimulated Raman transitions,” Phys. Rev. Lett., vol. 67, pp. 181–184, 1991. [15] A. Morinaga, M. Nakamura, T. Kurosu, and N. Ito, “Phase shift induced from the dc Stark effect in an atom interferometer comprised of four copropagating laser beams,” Phys. Rev. A, vol. 54, pp. R21–R24, 1996. [16] S. Yanagimachi, M. Kajiro, M. Machiya, and A. Morinaga, “Direct measurement of the Aharonov-Casher phase and tensor Stark polarizability using a calcium atomic polarization interferometer,” Phys. Rev. A, vol. 65, pp. 042 104-1–042 104-7, 2002. [17] K. Honda, S. Yanagimachi, and A. Morinaga, “Three-level atom interferometer with bichromatic laser fields,” Phys. Rev. A, vol. 68, pp. 043 621-1–043 621-11, 2003. [18] K. Shinohara, T. Aoki, and A. Morinaga, “Scalar Aharonov–Bohm effect for ultracold atoms,” Phys. Rev. A, vol. 66, pp. 042 106-1–042 106-4, 2002. [19] M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Optics, vol. 34, pp. 1401–1407, 1987. [20] Dependence of Berry’s phase for atom on a sign of the g-factor in the rotating magnetic field , submitted for publication.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
Measurement of Berry’s Phase for Partial Cycles Using a Time-Domain Atomic Interferometer Makoto Yasuhara, Takatoshi Aoki, Hirotaka Narui, and Atsuo Morinaga
Abstract—The Berry phase of atom for partial cycles of rotating magnetic field in parameter space was successfully measured free from the dynamical phase shift by using a time-domain sodium atom interferometer. The obtained phase shift was in an agreement with the prediction of Berry’s phase for partial cycles. Index Terms—Atom interferomtry, Berry’s phase, rotating magnetic field.
I. INTRODUCTION
I
N 1984, Berry predicted that the wave function of an adiabatically changing quantum-mechanical system acquires the topological (geometrical) phases for cyclic evolution of the Hamiltonian [1] and proposed an experiment for particles with nonzero spin under whole turn of the magnetic field, which subtends a solid angle of . Many experimental demonstrations of Berry’s phase have been carried out using polarized light, neutron interferometers, nuclear magnetic resonance and so on [2]–[5]. In 1992, Miniatura et al. demonstrated the first atomic topological phase by using a longitudinal Stern-Gerlach atomic interferometer [6]. However most of those experiments were done using moving particles, so that the topological phase was obscured with the dynamical phase shift. On the other hand, time-domain atom interferometers with cold atoms were pointed out as being suitable for measuring the topological phase that is free from the dynamical phase, because the cold atoms could be assumed to be at rest during the interaction time [7]. Up to now, one experiment on Berry’s phase using cold atoms was demonstrated in an atom-light system, where the dark state was adiabatically changed via the parameters of the light field [8]. This system had no dynamical phase shift, but an effective magnetic quantum number must be identified. Very recently, the topological phase was imprinted in the atoms of the Bose–Einstein condensate by adiabatically inverting the magnetic bias field and succeeded in generating vortices in a Bose–Einstein condensate [9]. However, there has been no demonstration of Berry’s original proposal for bosonic atoms with a magnetic moment with a directionally rotating magnetic field, except for a conical magnetic field configuration. It is also interesting to investigate the topological effect on atoms for partial cycles, i.e., not for a complete turn of the magnetic field in parameter space, as Weinfurter and Badurek have demonstrated under the interaction of neutrons with a time-dependent magnetic field [10], which was rather complicated to
Manuscript received July 2, 2004; revised October 26, 2004. The authors are with the Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan. Digital Object Identifier 10.1109/TIM.2005.843564
understand. Berry’s phase for partial cycles was discussed theoretically by Samuel et al. [11] and Jordan [12]. It is also important to investigate Berry’s phase because it might be one of the noise sources of future atomic frequency standards. Atom interferometers using the interaction of atoms with light as a beam-splitter, which were realized in 1991, have become a powerful tool to measure quantum phases, because it measures the phase difference between two states with a different quantum number [13], [14]. One of authors (Morinaga) has developed the Mach–Zender type atom interferometer using copropagating laser beams [15]. Previously, the Aharonov-Casher phase was measured using a space-domain atom interferometer [16], [17] and the scalar Aharonov–Bohm effect was demonstrated using the time-domain atom interferometer and has verified that the phase shift was in proportion to the variation of the strength of the resultant magnetic field during two light pulses [18]. In that experiment, the phase change was found to be due only to the variation in magnitude of the resultant magnetic field, not to the rotation of the magnetic field which rotated by a certain angle in one direction and returned to the original position. Therefore, it will be interesting to examine how the phase of the state with magnetic quantum number m changes due to the rotation of the magnetic field with constant amplitude in one direction, which corresponds to Berry’s phase. In this paper, we demonstrated the measurement of Berry’s phase according to Berry’s proposal using a time-domain atom interferometer and measured Berry’s phase for partial cycles.
II. PRINCIPLE The time domain atom interferometer is composed of cold ensembles of a two-level atom whose magnetic quantum numrespectively and has two coherent laser pulses bers are m and with a pulse width of and a frequency , which is near resobetween two states at a nant with the transition frequency , as shown in Fig. 1. The atom is initially magnetic field of in the lower state of m under the quantization magnetic field. After two irradiations of laser pulses which are separated by in time ( should be sufficiently short before the atoms start to fall down due to gravity), the Ramsey resonance occurs in the population of the upper levels as a function of the detuning frequency . If the quantization magnetic field rotates during the of time interval between two pulses, Berry’s phase would be observed as a product of the rotating angle of the field multiplied by the difference of magnetic quantum numbers in the phase of
0018-9456/$20.00 © 2005 IEEE
YASUHARA et al.: MEASUREMENT OF BERRY’S PHASE FOR PARTIAL CYCLES
Fig. 1. Schematic diagram of the measurement of Berry’s phase. A time domain atom interferometer is composed of two laser pulses whose time interval is T . The frequency of the laser with circular polarization is tuned to the transition frequency between two states whose magnetic quantum numbers of m and . During T , the magnetic field rotates by an angle of '.
m
the Ramsey fringes. The interference fringes in the population of the upper level are described as follows:
(1) where the first term denotes the Ramsey fringes, the second term denotes the scalar Aharonov–Bohm phase and the third term is the Berry’s phase. For partial cycles, Berry’s phase corresponds to (2) where denotes the rotation angle of the magnetic field during . With a whole turn of , it produces a phase of [19]. If the amplitude of the magnetic field remained at during the interval , no scalar Aharonov–Bohm effect occurs and we could observe unambiguously the Berry’s phase as the rotation angle varies during . Our strategy to observe Berry’s phase for partial cycles is to use the stimulated Raman transitions between the ground hyperfine states of sodium atom. The Zeeman splitting of the two and are shown in Fig. 2. ground hyperfine states of , For example, we could select only the states of and , by using the two-photon Raman transitions, polarwhich are detuned by 500 MHz in frequency from the to and ized transition of to from the polarized transition of (or and transitions between these via ). The propagation direction of the two-photon Raman beams is orthogonal to the rotation plane of the magnetic field, which also defines the quantization axis. The polarization of the Raman beam is a right-handed circular polarization, hence the electric field can be resolved into two components, one parallel ( polarization) and one orthogonal ( polarization ) to the magnetic field at any time during the rotation. The two optical phases
865
Fig. 2. Energy diagram of the ground hyperfine states of sodium atoms under weak magnetic field and a two-photon stimulated Raman transition.
which the atom acquires from the two Raman beams at absorption and emission simultaneously will be cancelled out. After two Raman pulses and a rotation of magnetic field between them, we can observe the phase shift in the population of the , which is the difference between the Berry state and . phases generated in the states of III. EXPERIMENTAL The experimental setup for the time-domain atom interferometer used was almost the same as that used for the study of the scalar Aharonov–Bohm effect using cold sodium atoms [9]. Here we describe the essential points of the apparatus. For a state were rotation magnetic field sodium atoms in the trapped in a magneto-optical trap at a temperature of 1 mK. At 3 ms after the free expansion of the trapped atoms, sodium atoms state. were initialized by optical pumping to the ground Then the quantization magnetic field was applied to the sodium atoms and the two right-handed circular polarized Raman laser pulses with a pulse width of 20 s were applied to them with a pulse separation of 80 s in order to make the atom interferom100 s. The propagation direction of the Raman eter with laser beams was orthogonal to the direction of the quantization magnetic field and the plane where the magnetic field rotates. state by the polarized The atom was excited to the Raman laser beam and polarized laser beam with frequencies and . One of the two Raman laser beams was generof ated from a frequency stabilized dye laser whose frequency was tuned by a frequency 500 MHz lower than the resonance freand and the frequency of the states of quency of the other Raman beam was shifted from that through a phase-modulated electro-optic modulator which was driven by a frequency synthesizer at around 1.77 GHz. The frequency was swept by a frequency synthesizer in order to generate the Ramsey fringes. The rotating magnetic field was produced by two mutually orthogonal pairs of Helmholtz coils which were driven by alternating currents with a relative phase shift of 90 and with the same field strength. In order to maintain a constant strength of magnetic field during rotation, the amplitude of each alternating
866
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 54, NO. 2, APRIL 2005
Fig. 3. Zeeman splittings of stimulated Raman transitions for four directions. (a) x, (b) x, (c) y , and (d) y .
+
0
+
0
Fig. 5. Ramsey fringes under (a) a constant magnetic field and (b) a rotating magnetic field. The rotating angle is radians. The phase difference is observed at the center frequency of the spectra.
IV. RESULTS AND DISCUSSION
Fig. 4. Timing diagram of measurement of Berry’s phase.
current was adjusted so that the Zeeman frequency shifts of the resonance for both axes of the Helmholtz coil agreed with each other within 1 kHz. The typical Zeeman spectra are shown in Fig. 3 for two orthogonal axis and two counter directions. The constant magnetic field is typically 0.02 mT, which corresponds 140 kHz. Therefore, the rotato a Larmor frequency of tion frequency of the magnetic field was adjusted to be less than 10 kHz, to fulfill the adiabatic condition. During the interval between the two Raman pulses, the resultant magnetic field was rotated with a almost constant frequency of (DC 10 kHz), . then the magnetic field has rotated by an angle of after The transition probability of atoms in the state of two Raman pulses was deduced from the absorption coefficients in the of the laser beams which are resonant from the state to the in the state. The time sequence diagram after magneto-optical trap (MOT) is summarized in Fig. 4. The time sequence was repeated every 10 ms and the transition probabilities for each run were accumulated and averaged in a computer. The population probability of the excited state was monitored as a function of detuning frequency.
With a pulse separation of 100 s, the visibility of the Ramsey and was refringes for the states of duced to only 0.16, because they were sensitive to the residual time-dependent magnetic field. Therefore, in the present experand were selected iment, the states of instead. Fig. 5(a) shows a typical Ramsey fringe under a DC magnetic field, which was obtained with a pulse width of 20 s and a pulse separation of 80 s. Within the spectrum width of 40 kHz, Ramsey fringes with a cycle of 10 kHz can be clearly seen with a visibility of 0.42. Fig. 5(b) shows the Ramsey resonance under a rotating magnetic field with a frequency of 5 kHz under the same pulse trains. The visibilities are almost the same as Fig. 5(a) with the same fringe cycles. But we can see the shift of the center frequency of the envelope and the shift between both phases at the center frequency. This frequency shift was within 10 kHz for each measurement and it may occur due to the fluctuation of the magnetic field at the irradiation times of the two pulses, although we adjusted them in advance. The phase at the center frequency of the envelope was obtained for several rotation frequencies up to 10 kHz for clockwise rotation (CW) and counterclockwise (CCW) rotation of magnetic field. The frequency of CCW was described with a negative signature for convenience. At 10 kHz, the magnetic 100 s. Fig. 6 shows the phase field turns full circle for shift as a function of rotation angle for the resonance between , , and , . The phase the states of changed along a straight line from a positive phase to a negative phase as the frequency changed from a negative to a positive value. It should be noted that the phase changed with a rotation angle for partial cycle according to (2). The slope of the phase shift is reduced to be 0.97 0.13, which is in agreement with the prediction of Berry’s phase for the phase difference between and . Furthermore, the fact that the the states of
YASUHARA et al.: MEASUREMENT OF BERRY’S PHASE FOR PARTIAL CYCLES
867
discussions on Berry’s phase over 10 years. The authors also thank P. Toschek for discussions on atom interferometry.
REFERENCES
Fig. 6. Phase shifts between the wavepackets of the F = 2, m = 1 state and the F = 1, m = 0 state as a function of rotation angle of the magnetic field.
slopes for both directions of rotation are the same will mean that the dynamical phase shift is not included in it. In order to test the dependence of the magnetic quantum between the states of number, the phase shift for , and , was also examined. As expected according to theory, the inclination of the phase and the slope is 1.2 shift was reversed to that of 0.2. This result clearly proves that the measured phase is the Berry phase. Furthermore, the present results show that Berry’s phase depends on the sign of the g-factor of the state which we discussed in another paper [20]. V. SUMMARY In conclusion, we demonstrated Berry’s phase for magnetic field rotation using a time-domain atom interferometer free from the dynamical phase shift. The results show that the phase is given by a magnetic quantum number multiplied by the rotation angle for partial cycles, and that the phase shift depends on the direction of rotation and the dependence of the magnetic quantum number. ACKNOWLEDGMENT A. Morinaga would like to thank M. Kitano of Kyoto University and J. Helmcke and F. Riehle of PTB, Germany, for many
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