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PHYSICAL REVIEW A

VOLUME 56, NUMBER 2

AUGUST 1997

Ground-state Ramsey fringes B. Dubetsky and P. R. Berman Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120 ~Received 27 December 1996; revised manuscript received 2 April 1997! A theory of ground-state Ramsey fringes is developed. The fringes are produced when two pulses, time delayed with respect to one another and each consisting of a pair of traveling waves, interact with an ensemble of atoms. Advantages of this technique for meaurements of inertial effects and atomic recoil are discussed. @S1050-2947~97!50508-7# PACS number~s!: 42.50.Vk, 42.50.Md, 32.70.Jz, 42.62.Eh

I. INTRODUCTION

The method of separated oscillating fields, as introduced by Ramsey @1#, consists of sending an atomic beam through two zones in which atoms interact with rf fields. If the rf fields drive a magnetic dipole transition, the resulting excitation probability varies as cos(fR), where f R 5DT is the Ramsey phase ~difference in phase accumulated by the radiation field and the atomic magnetic dipole in a time T), D is the atom-field detuning, and T is the time it takes for an atom to move between the interaction zones. In an atomic beam, there is a distribution of T, owing to the longitudinal velocity distribution of the atoms. When this distribution is averaged over, one finds that the excitation probability vs D consists of a central fringe centered at D50, having width G R ;1/T 0 , where T 0 is the average transit time between the two interaction zones. Note that the average over T is essential. If all the atoms had the same T, the signal would vary as cosDT and it would be impossible to identify a central fringe. As a consequence, if one wants to apply the Ramsey fringe technique to atoms in a cell by subjecting the atoms to two pulses separated in time by T ~which is the same for all atoms!, it is necessary to repeat the experiment with different T, record the transition probability as a function of T, and average the result over T @ 2 # . In this manner one reproduces the Ramsey fringe pattern. The above considerations are valid only if one can neglect the Doppler phase, f D 5k•vT, where k is the propagation vector of the field and v the atomic velocity. In the rf domain, u f D u !1 and can be dropped. On the other hand, in the optical domain, u f D u @1 is typical and, on averaging over the velocity distribution, the Ramsey fringe signal is destroyed. The signal can be restored by the use of echo techniques involving multiple pulses of standing-wave and/or counterpropagating optical fields ~if one uses copropagating traveling waves, the Ramsey phase and the Doppler phase both go to zero at the time the echo signal is generated!. We propose a generalization of the Ramsey technique @1# in which the radiation fields drive transitions between the same internal ground state of the atom ~for simplicity assumed to be a single, nondegenerate level!, but between different center-of-mass momentum states. Since the spectroscopy is carried out on atoms in their ground states, we refer to this process as the ground-state Ramsey fringe ~GSRF! 1050-2947/97/56~2!/1091~4!/$10.00

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process. We consider atoms interacting with pulses of radiation, although the calculations are equally valid for spatially separated fields ~as in the original Ramsey configuration! or time-separated fields ~as for pulsed radiation incident on atoms in a cell or trap!. In the case of spatially separated fields, the pulses refer to the fields as they appear in the atoms’ rest frame. The detuning D is replaced by the frequency difference d of the two traveling-wave optical fields that comprise each of the pulses. Although this Ramsey fringe technique cannot be used to measure an optical frequency, it can provide unprecedented accuracy for measurements of h/M ~where M is the atomic mass!, gravitational acceleration, and rotation rates. The manner in which this accuracy can be achieved is the subject of this paper. The GSRF technique combines and is complementary to methods developed using cw fields @3# and atom interferometry @4–10#. II. RAMSEY FRINGE FORMATION

We consider a two-level atom interacting with a pulsed radiation field E ~ x,t! 5Re

El ( j, l

j

~ t2T j ! exp~ ikl j •x2iV l t !

~1!

consisting of two pulses ( j51,2) of duration t p , nearly resonant with the transition between ground and excited atomic states, centered at t5T j ~T 1 50, T 2 5T!. Each pulse consists of two traveling-wave modes ( l 51,2) having wave vectors, frequencies, and amplitudes @ V l ,kl j ,E l j (t2T j ) # , where V 1 5V, V 2 5V1 d , k1 1,256k1 , k2 1,256k2 ~see Fig. 1!. When the frequency detuning D5V2 v ~v is the transition frequency between the nondegenerate ground state and the excited state! is larger than the upper state width and the Rabi frequencies of the fields, one can adiabatically eliminate the excited-state atomic wave function to obtain an interaction Hamiltonian V that contains terms proportional to u E 1 j u 2 , u E 2 j u 2 , and u E 1 j E 2 j u . Only the last term is important for the present discussion and leads to an effective interaction Hamiltonian V(x,t)5 ( j51,2V j (t)cos(q j •x1 d t), where V j (t)5 u m 2 E 1 j (t)E 2 j (t) u /2\D, m is a dipole moment matrix element associated with the optical transition, and q1,25 6q, with q5k1 2k2 . In the Raman-Nath approximation ( t p \q 2 /M !1) and with the assumption u d u t p !1, one can R1091

© 1997 The American Physical Society


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B. DUBETSKY AND P. R. BERMAN

56

r m ~ t, d ! 5 ~ 21 ! m21 f m ~ t m ! ˜ F m ~ d ,T ! ,

~3a!

f m ~ t m ! 5J m21 @ 2 u 1 sin~ v q t m !# ^ exp~ 2iq•vt m ! & , ~3b! ˜ F m ~ d ,T ! 5exp~ 2im d T ! J m $ 2 u 2 sin@~ m21 ! v q T # % , ~3c!

FIG. 1. Pulse sequence for creating and observing ground-state Ramsey fringes. A modulated ground-state density ~‘‘atomic grating’’! having period 2 p /q (q5k1 2k2 ) is produced in a gas by a pulse consisting of two modes, $ k1 ,V % and $ k2 ,V1 d % . A second pulse consisting of modes $ 2k1 ,V % and $ 2k2 ,V1 d % interacts with the atoms following a time delay T. Scattering of a readout mode $ k1 ,V % from the grating formed at the echo time t52T leads to emission into the mode $ k2 ,V1 d % , which exhibits Ramsey fringe structure.

relate the wave function c 1 (x) following the jth pulse to the wave function c 2 (x) just before the pulse by c 1 (x) 5 h j (x) c 2 (x), where

h j ~ x ! 5exp@ 2i u j cos~ q j •x1 d T j !# u j 5 * `2` dtV j (t)

~2!

is a transmission amplitude and is a pulse area. It is clear that the atomic density following the first pulse, r 1 (x)5 u h 1 (x) u 2 , equals unity ~the pulse acts as a phase grating!; however, as a result of the interaction each component of the wave function in momentum space acquires sidebands at 6n 1 \q, for integer n 1 . As these components evolve in time after the interaction with the pulse the phase grating is converted into an amplitude grating and the atomic density becomes spatially modulated. Spatial harmonics of the density acquire in the time between pulses Doppler phases, which are integral multiples of q•vT. If quT@1, as is assumed ~where u is the thermal speed!, all spatial modulation of the atomic density is washed out as a result of Doppler dephasing owing to the different Doppler shifts for atoms having different velocities. The second pulse introduces additional sidebands 6n 2 \q on the momentum components and can start a rephasing process. In the vicinities of the echo points t5(m/n)T, where m and n are integers with m.n, a rephasing occurs for the atomic density components, varying as exp(6inq•x) @10#. For n51 ~the case considered below! one can probe the atomic grating having spatial period 2 p /q by scattering a readout pulse (k1 ,V) into field mode (k2 ,V1 d ) @11,5#. If one expands the atomic density as r (t, d ) 5 ( s r ms (t, d )exp(2isq•x2i d t!, it follows that the readout pulse selects that component of the atomic density corresponding to r m (t, d )[ r m1 (t, d ). Using Eq. ~2! to account for each pulse’s action and allowing for the free evolution of the system between the pulses and following pulse 2, one finds

where t m 5t2mT, v q 5\q 2 /2M is a recoil frequency, M is the atom mass, ^ ••• & means averaging over velocities v 5p/M , and J m (x) is a Bessel function. Note that each value of m corresponds to a specific temporal echo plane, t 'mT. By heterodyning the scattered signal with a reference laser, one can determine both the amplitude and phase of ˜ F m ( d ,T). On averaging ˜ F m ( d ,T) over T, one obtains the experimental signal of interest. By varying d , one maps out the GSRF signal. As a function of the pulse separation T the Ramsey fringes ~3c! contain two types of oscillations: an oscillation, having period ;1/u d u , associated with the Ramsey phase d T; and an oscillation, having period ;1/v q , associated with quantum additions to the phase resulting from atomic recoil @12#. If the Bessel function in Eq. ~3c! is expanded and the result is averaged over pulse separation T ~as it must to isolate the central fringe!, one finds F m~ d ! [ ^ ˜ F m ~ d ,T ! & T 5

Ij ( j m

m

~ u 2 ! ^ exp@ 2im ~ d 2 n j m ! T # & T ,

I j m ~ x ! 5J j m 2m/2~ x ! J j m 1m/2~ x ! ,

n j m 5 j m @~ m21 ! /m # v q ,

~4!

where j m is an integer ~half-integer! for even ~odd! m. The average over T can be realized using spatially separated fields @1# for which the time delay T differs for atoms having different velocities or by using time separated fields and varying the pulse separation T @2#. III. DISCUSSION

Equation ~4! is the ground-state Ramsey fringe line shape and the principle result of this paper. Let us discuss some applications. (a) Recoil splitting. Expression ~4! shows that GSRF’s consist of an infinite number of recoil components centered at d 5 n j m ~j m 50,61,62, . . . , or j m 561/2,63/2, . . . !. For example, at the echo point t52T ~m52!, one gets components centered at d 50,6 21 v q ,6 v q , . . . , with weighting functions I j 2 ( u 2 ), plotted in Fig. 2. Measuring the frequency separation between spectral components allows one to determine the recoil frequency from which one can extract a precise value for the ratio \/M @6#. To achieve a high accuracy in this measurement one needs to increase the frequency separation between recoil components, to diminish any dephasing not caused by the recoil effect and to determine d accurately and precisely. To increase the frequency separation between recoil components, instead of using multiple scattering by a set of p pulses @6#, one can use higher-order recoil components.


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GROUND-STATE RAMSEY FRINGES

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FIG. 2. Dependence of the amplitude of the recoil components centered at various d on the pulse area u 2 .

FIG. 3. Line-shape function G 2 ( d ) reflecting the influence of gravitational acceleration on ground-state Ramsey fringes.

From Fig. 2, one sees that the maximum amplitude of the recoil components decreases slowly as a function of component number j m . In Ref. @6# a frequency separation between components of 16v q was achieved. The same separation is obtained in our case for components centered at d 568 v q ( j 2 5616) and the maximum amplitude of these components is only 2.2 times less than the amplitude of the components at d 56 v q . When discussing dephasing caused by effects other than recoil, it is important to note that the initial and final states of the two-quantum transition leading to GSRF’s are identical. As a result, the position of the line center is insensitive to any spatially homogeneous perturbations of the internal wave function, such as those produced by permanent magnetic fields. With regard to controlling the frequency offset d , one can lock this frequency using a highly stabilized quartz oscillator. In this case the accuracy of the detuning d measurement is given by a n , where n and a are the oscillator’s frequency and stability. When the oscillator has the same stability as the Cs frequency standard, the accuracy can be better in our case by the ratio n / n Cs , where n Cs is the transition frequency between hyperfine sublevels in Cs. This factor can be as small as 1022 21023 ~there is a lower bound on n /2p ;10 MHz owing to the fact that this frequency must lie outside the noise bandwidth of the experiment!. (b) Gravitational effects. The gravitational field changes the phase of the echo signals @7,13#. The change has been used to measure the gravitational acceleration g @8,5#. The GSRF technique allows one to carry out this measurement in the frequency domain. When atoms move in a gravitational field, ˜ F m ( d ,T) acquires an additional phase factor exp@if(m) @5#, where f (m) # g g 2 5q•gm(m21)T /2. One sees that gravity and the detuning d affect only the phase, in contrast to the recoil effect,

which affects only the amplitude of the atomic matter grating. Using the heterodyne technique, one can measure @5# the grating phase f m 52m d T1 f (m) g , and define the ~complex! line shape as G m ( d )5 ^ exp(ifm)&T . For a typical value, q ;105 cm21 , the gravitational phase becomes on the order of 21/2 unity for a time separation T (m) g 5 @ q•gm(m21)/2# ;100 m s. Averaging the signal in a time interval T P @ T min ,Tmax#, where T min!T(m) g !Tmax , one finds G m ~ d ! 5 Ap i/4~ T ~gm ! /T max! w @ Aim d T ~gm ! # ,

~5!

where w(x) is a plasma dispersion function. The line shape ~5! is plotted in Fig. 3. Fitting experimental records using the line shape ~5!, one measures g. The accuracy of this measurement can be better than that achieved for the echo on a Raman transition @8#, since the influence of the magnetic field considered in @8# plays no role in GSRF’s and the accuracy of the detuning measurement can be higher than the accuracy of the time separation between pulses measurement. (c) Rotational effects. The effects of rotation on echo signals @7# has been observed using optical Ramsey fringes @9# and scattering from microfabricated structures @14#. In a system rotating with frequency Vr , a Coriolis force arises. When the rotation frequency is small, V r T!1, one can re¢ place g in f (m) g by the Coriolis acceleration 2v3Vr to obtain a phase addition due to rotation given by @7#

f r m 5 ~ Vr 3q! •vm ~ m21 ! T 2 .

~6!

The rotational dephasing ~6! and Ramsey dephasing d T have the same dependence on atomic velocity for fields spatially separated by a displacement L @7#. When Vr is perpendicular to the plane (q,L) the sum of these dephasings is equal to ( d 1aqLV r )L/u, where u is the velocity projection


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on L and a is a constant of order unity. This implies that the dephasings can cancel one another, independent of atomic velocity. A shift of the Ramsey fringes at the point of cancellation, d ;qLV r , occurs. This shift can be used to measure Vr and has been observed for optical Ramsey fringes in Ca @9#. The GSRF allows one to increase the accuracy of the measurement. For example, suppose that the stability of the quartz oscillator used to lock the traveling-wave fields is the same as that of the laser used in Ref. @9#. It follows that the detuning d can be measured by a factor n / n Ca better than the detuning D between the optical field frequency and the transition frequency in Ca, n Ca . When one uses fields separated in space by a distance L;1 cm and an atomic velocity spread Du;20 cm/s, the GSRF half-width could be 1.6 Hz @15#. If the relative stability of the line center is 1023 , one can measure a shift d ;1.631023 Hz, which implies that, for the transition in Cs (q51.53105 cm21 ), one can measure a rotation rate with an accuracy of 1028 Hz or 0.014°/h. Since this accuracy is three times better than that achieved using a fiber-optic gyroscope @16#, one can use GSRF’s to create an alternative type of gyroscope. If one uses time-separated rather than spatially separated pulses, a cancellation of the Ramsey and the rotational phase is no longer possible for all velocity subclasses, since the time separation T is the same for all atoms regardless of their velocity ~in contrast to the spatially separated field case for which T5L/u!. As a consequence, the echo signal in Eq. ~6! vanishes on averaging over velocity unless V r &1/quT 2 . Nevertheless, it is possible to show that such small rotation

@1# N. F. Ramsey, Phys. Rev. 76, 996 ~1949!. @2# L. S. Vasilenko et al., Opt. Commun. 53, 371 ~1985!. @3# See, for example, P. R. Berman et al., Phys. Rev. A 38, 252 ~1988!; N. Lu and P. R. Berman, ibid. 36, 3845 ~1987!; J. Guo et al., ibid. 46, 1426 ~1992!; J. Y. Courtois et al., Phys. Rev. Lett. 72, 3017 ~1994!; D. R. Meacher et al., Phys. Rev. A 50, R1992 ~1994!; A. Hemmerich et al., Europhys. Lett. 22, 89 ~1993!. @4# B. Dubetsky et al., Pis’ma Zh. Eksp. Teor. Fiz. 39, 531 ~1984! @JETP Lett. 39, 649 ~1985!#. @5# S. B. Cahn et al., Phys. Rev. Lett. ~to be published!. @6# D. S. Weiss et al., Appl. Phys. B 59, 217 ~1994!. @7# Ch. J. Borde, Phys. Lett. A 140, 10 ~1989!. @8# M. Kasevich and S. Chu, Appl. Phys. B 54, 321 ~1992!. @9# F. Riehle et al., Phys. Rev. Lett. 67, 177 ~1991!.

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rates give rise to measurable changes in the recoil frequency. The atomic velocity changes by d v;\q/M , when an atom scatters from the two fields making up the optical pulse ~Kapitza-Dirac effect @17#!. Replacing v→v1 d v in Eq. ~6! one sees that, to first order in Vr , effects related to d v vanish. Calculations of higher-order corrections will be published elsewhere. They show that the joint action of the rotation and quantum scattering results in a change of the recoil frequency given by

v q → v qr m ' v q @ 11 ~ m/2!~ V r T ! 2 # .

~7!

The requirement that V r &1/quT 2 places an upper bound on the correction to the recoil frequency in Eq. ~7!. For T ;1/v q ~a necessary condition to see atom interference!, one estimates that the relative weight of the correction is ( v q /qu) 2 . For example, in a recent experiment @5# this parameter was ;431024 . Owing to the high accuracy of recoil frequency measurements ~in @5# the recoil frequency was determined to one part in 105 ! one can observe this correction. ACKNOWLEDGMENTS

We are pleased to acknowledge helpful discussions with J. L. Cohen, V. G. Goldort, N. N. Rubtsova, T. Sleator, and L. S. Vasilenko. This research is supported by the U.S. Army Research office under Grant No. DAAH04-93-G-0503 and by the National Science Foundation under Grant No. PHY9414020.

@10# B. Dubetsky and P. R. Berman, in Atom Interferometry, edited by P. R. Berman ~Academic Press, Cambridge, MA, 1997!, pp. 407–468. @11# T. Mossberg et al., Phys. Rev. Lett. 43, 851 ~1979!. @12# B. Dubetsky and V. M. Semibalamut, in Sixth International Conference on Atomic Physics. Abstracts ~Riga, 1978!, edited by E. Anderson et al. ~unpublished!, p. 21. @13# B. Dubetsky et al., Phys. Rev. A 46, R2213 ~1992!. @14# M. K. Oberthaler et al., Phys. Rev. A 54, 3165 ~1996!. @15# B. Dubetsky, Kvant. Elektron. 3, 1258 ~1976! @Sov. J. Quantum Electron. 6, 682 ~1976!#. @16# C.-X. Shi and H. Lizuka, Microw. Opt. Technol. Lett. 9, 233 ~1995!. @17# A. P. Kazantsev et al., Pis’ma Zh. Eksp. Teor. Fiz. 31, 542 ~1980! @JETP Lett. 31, 509 ~1980!#.