Speedometer based on atomic coherence

2 October 2000

Physics Letters A 275 Ž2000. 20–24 www.elsevier.nlrlocaterpla

Speedometer based on atomic coherence A.B. Matsko 1 Department of Physics, Texas A & M UniÕersity, College Station, TX 77843-4242, USA Received 26 June 2000; accepted 4 August 2000 Communicated by V.M. Agranovich

Abstract A new technique of Quantum Nondemolition Measurement of a projection of canonical momentum of a three level L atom is proposed. The scheme of measurement is based on the detection of a Stokes component of two counterpropagating probe electromagnetic waves, interacting with the atom. The measurement allows to infer the velocity of the atom after the interaction with an accuracy better than the Standard Quantum Limit. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50.L; 06.20.D

Quantum Nondemolition ŽQND. Measurement of the canonical momentum of a freely moving body calls for an interaction independent on space coordinates. The well known physical phenomenon which fulfills this requirement is the interaction of a moving charge with a homogeneous magnetic field w1,2x. Recently it has been shown, however, that the interaction of a two-level atom with running electromagnetic wave also does not depend on the space coordinate in the direction of the wave propagation w3x, which makes such an interaction be suitable for the QND measurement of the projection of the atomic momentum. This measurement is based on the Doppler effect, that gives rise to a velocity dependent AC-Stark shift, which can be registered by detection of the phase shift of the electromagnetic wave.

1

E-mail address: [email protected] ŽA.B. Matsko.. Fax: q1-979-845-2590.

Doppler measurements of the momentum has been first discussed by Von Neumann w4x, who pointed out that the error of momentum measurement using a single photon is determined by the relation mc D P0 f , Ž 1. v 0t where m is the mass of the body, c is the speed of light in vacuum, v 0 is the photon frequency, t is the duration of the one-photon wave packet. However, if the wave-packet frequency modulation resulting from the body acceleration and radiation reaction is taken into account, the maximum measurement sensitivity is determined by Standard Quantum Limit ŽSQL. w5x

D PSQL f

(

"m

t

,

Ž 2.

which exceeds Ž1. for comparably large t . In this Letter a new scheme of Doppler based QND measurement of a projection of the momentum

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 5 5 9 - 4

A.B. Matskor Physics Letters A 275 (2000) 20–24

of a three-level L-type atom is proposed. The information about the momentum can be obtained by detection of the Stokes components of two running electromagnetic waves interacting with the atom. Because the absorption and spontaneous emission can be significantly reduced for an atom in L configuration w6x, the sensitivity of the measurement is no more restricted by the decoherence processes, unlike to the two-level atom w3x. Moreover, due to the resonant nature of the interaction, the coupling in the case of the three-level atom can be much stronger than the coupling in the case of two-level atom. Let us consider a three-level atom placed into field of two counterpropagating plane electromagnetic waves Eq and Ey. Each wave couples one dipole-allowed transition and does not interact with the other, say, Eq is applied to < a: < bq : and Ey to < a: < by : ŽFig. 1.. For example, it can be two fields with opposite circular polarizations and the same frequencies, while ground state levels < bq : and < by : have all equal quantum numbers except magnetic numbers m s "1, and < a: is level with m s 0. For the sake of simplicity, we describe the fields in terms of Rabi frequencies E "s ` " E "r", where E " is the actual field, ` " is the dipole moments of the transitions, which we assume be the same ` "s `. Any atomic movement parallel to the wave propagation axis Z leads to Doppler shift of the frequencies of probe fields in the atomic frame of reference. In turn, in the laboratory frame of reference, the waves see an anti-symmetric shifts of atomic levels, which results in, due to the large linear dispersion of the two-photon resonance, to an opposite change in the index of refraction for Eq and Ey, which can be detected experimentally. One of the main problems of any quantum measurement is connected with decay processes and associated noises. For example, only one act of

l

l

Fig. 1. Idealized three level L scheme

21

spontaneous emission destroys the initial momentum state of the atom in scheme under consideration. To avoid the spontaneous emission the L-type atom should be in the coherent superposition of the ground states which corresponds to the uncoupled Ž‘dark’. state w6x. In approximation of classical electromagnetic fields the ‘dark’ state can be presented as


Eq Ž t . < by : y Ey Ž t . < bq :

(< E

q

Ž t . < 2 q < Ey Ž t . < 2

Ž 3.

A comparably slow atomic motion does not destroy the ‘dark’ state and, therefore, does not lead to spontaneous emission. However, the spontaneous emission can appear during transient process, when the atom enters the interaction region, if initial state of the atom is different from the uncoupled state. The ‘dark’ state depends on the mutual phase of the applied electromagnetic fields and, therefore, on the atomic position in space. In our scheme phase difference between the fields experiences 2p shift on the distance D z s lr2, where l is the wavelength of the fields. Hence, because of atomic position uncertainty, it is impossible to say which phases the fields have on the point of entrance of an atom for any reasonable quantum state of the atomic beam. To avoid this complication the adiabatic population transfer technique can be used w7–11x. An atom in the ‘dark’ state can adiabatically follow a changing light field, transferring population from one ground state to another, without significant population of intermediate excited states. The adiabatic population transfer is essentially coherent stimulated process. Adiabatic population transfer has been used for realization of atomic mirrors and beamsplitters w10x, which are based on the similar configuration of the field-atom interaction compared with discussed above scheme. The main idea is that the ‘dark’ state is velocity selective in traveling counterpropagating light waves. The incident atoms are deflected from the light waves by absorption and subsequent stimulated reemission of laser photons. Our goal here is, in some sense, the opposite to the idea of manipulation of atoms by light fields. To realize QND measurement of the momentum we

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A.B. Matskor Physics Letters A 275 (2000) 20–24

need to achieve an efficient interaction of the atom with the light fields with subsequent restoration of the initial atomic momentum state, not the atomic deflection as it is in w10x. We propose to use space configuration of fields shown in Fig. 2. Atom, initially prepared in state < bq : enters the interaction region. The amplitude of Eq field is much smaller than that of Ey on the entrance, so the state of the atom corresponds to the ‘dark’ state Ž3.. With atomic propagation along the X direction, the difference between the field amplitudes disappears, and the populations of the atom adiabatically follow the field. In the center of the interaction region the ground states of the atom are equally populated. On the exit from the interaction region the population returns to the initial state. This adiabatic population transfer is accompanied by change of atomic momentum in the direction of propagation of the Ey field. However, because the atomic dipole momenta return to the initial state after the interaction by the coherent way, the mechanical momentum of the atom also becomes the same as it was before the interaction. The change of atomic momentum during the interaction leads to additional uncertainty of atomic position, which, in principle, is necessary condition for momentum measurements according to the Heisenberg uncertainty relations. Let us assume that the atomic motion is slow enough. Then, in the first approximation, we solve the problem of interaction of the atom with electromagnetic fields and find atomic matrix elements as a function of the field amplitudes independently on the atomic motion. After that we eliminate internal atomic variables and find the actual term which describes coupling between the atomic mechanical degree of freedom and electromagnetic waves. Analytic expressions for lowering operators of the atom s b " a can be obtained from the stationary

solution of the Bloch equations for the atomic populations

s˙ by bys gr sa a y i Ž Ey sa byy c.c. . , s˙ bq bqs gr sa a y i Ž Eq sa bqy c.c. . , and polarizations ) s˙a b "s yGa b " sa b "y iE " Ž s b " b "y sa a . ) y iE. sb . b ", ) s˙ by bqs yG bybq s bybqy iEy sa bqq iEq s bya ,

Ž 5. where

Ga b "' gr q i Ž D " d . ,

G bybq' g 0 q 2 i d .

Ž 6.

™

gr is the radiative linewidth of the transitions < a: < b " :, d is the two-photon detuning, D is one-photon detuning. We evaluate the stationary solutions of the Bloch-equations by considering D s 0 and taking into account only the lowest order in g 0 and d . In this limit the solution is sa b "s E "

ž

ig 0 < E<2

.

2 d < E. < 2 < E<4

/

,

Ž 7.

where < E < 2 s < Eq < 2 q < Ey < 2 4 g 0 gr . To find coupling between the atomic momentum and electromagnetic fields we present last ones in form E "s aˆ "

(

2p ` 2v 0 V "V

,

Ž 8.

where V is the quantization volume, aˆ " and aˆ†" are annihilation and creation operators, that obey commutation relations aˆ " ,aˆ†" s 1. We assume, that the fields are in the coherent state and present the operators as a sum of large mean and small fluctuation parts aˆ "s '² n: q d aˆ ", '² n: 4 d aˆ ". It is instructive to be mentioned here, that the condition < E < 2 4 g 0 gr can be rewritten now as ² n: 4

Fig. 2. The scheme of the interaction region

Ž 4.

8p 2 g 0 V 3

v 0 l3

,

Ž 9.

A.B. Matskor Physics Letters A 275 (2000) 20–24

where 2pl s v 0rc. For a sufficiently small quantization volume this condition is fulfilled even for a couple of quanta ² n: f 1. Therefore, we neglect by the losses in the following. The interaction energy between the atom and fields is w9x

23

tion of the corresponding Hamiltonian equations in form i Ž d aˆ "y d aˆ†" . , i Ž d aˆ " Ž 0 . y d aˆ†" Ž 0 . . " =

Vˆ s " d

aˆ†y aˆyy aˆ†q aˆq aˆ†y aˆyq aˆ†q aˆq

mc ,

Vz c

Ž 11 .

where Vz is Z component of the velocity of the atom. Using Ž10. and Ž11. we can write the the effective Hamiltonian, which properly describes interaction of the fields and the atom Hˆint s

2m

ž

Pˆz q

" v 0 aˆ†y aˆyy aˆ†q aˆq c

aˆ†y aˆyq aˆ†q aˆq

2

/

,

" v 0 aˆ†y aˆyy aˆ†q aˆq c

2 mc 2'² n:

Ž d aˆy Ž 0.

aˆ†y aˆyq aˆ†q aˆq

Ž 14 . d aˆ "q d aˆ†" s d aˆ " Ž 0 . q d aˆ†" Ž 0 . ,

S s yi Ž d aˆqy d aˆ†q y d aˆyq d aˆ†y . ,

is the canonical momentum of the atom. It is easy to see from Ž12. that P˙ˆz s 0, which means that the momentum does not change during the interaction. The actual velocity of the atom, in turn, changes, due to two-photon recoil effect, as it was mentioned above. However, this change can be totally compensated by the effect of adiabatic passage. Taking in mind, that the interaction Ž12. keeps photon number of each mode unchanged when the atom has entered to the central zone of the interaction region Žsee Fig. 2., we present linearized solu-

Ž 16 .

which in linear approximation corresponds to the detection of the Stokes component iŽ aˆ†q aˆyy aˆ†q aˆy ., allows to measure the momentum with accuracy

(

2 "m

s '2 D PSQL , Ž 17 . t which can be reached for the optimal photon number " v 0 v 0t ² n: 1 s . Ž 18 . mc 2 2 This restriction appears because the signal is masked by two different kinds of fluctuations. These are vacuum fluctuations and so called back action noise. The back action noise results from the electromagnetic wave interaction via atomic nonlinearity. The vacuum fluctuations depend on ² n: by different way compared with the back action noise, which leads to the sensitivity restriction. However, following to the logic of the strategy of the variational measurements w12,13x, we see, that the sensitivity can be much better if one detects not the phase, but optimally chosen linear combination of quadrature amplitudes of the fields

D P1 f

Ž 13 .

Ž 15 .

where t is the interaction time, d aˆ " Ž0. describes the initial state of the field. Expressions Ž14., Ž15. allows to find the maximal possible sensitivity of the measurement of the atomic momentum. An usual measurement of combination of phase quadrature amplitudes

Ž 12 .

where Pˆz s mVz y

" v0

qd aˆ†y Ž 0 . y d aˆq Ž 0 . y d aˆ†q Ž 0 . . ,

,

1

q

2'² n:

Ž 10 .

where operators aˆ " describe fields in the middle of the interaction region. There is no interaction in the system if the two-photon detuning is absent. However, if the atom is moving in Z direction, d is nonzero because of Doppler effect:

d,v0

Pˆz

v 0t

Sopt s yi Ž d aˆqy d aˆ†q y d aˆyq d aˆ†y . sin f q Ž d aˆyq d aˆ†y y d aˆqy d aˆ†q . cos f .

Ž 19 .

A.B. Matskor Physics Letters A 275 (2000) 20–24

24

For the optimal phase f such as ctg f s

" v 0 v 0t mc 2 2² n:

,

Acknowledgements

Ž 20 .

the term, corresponding to the excess backaction noise, cancels out, and the sensitivity is determined by the vacuum fluctuation of the fields only mc '2² n: s D P0'2² n: . D Popt f Ž 21 . v 0t Therefore, the proposed technique allows to approach the maximum sensitivity of measurement Ž1. if photon number ² n: is small enough. It is worth to be mentioned here, that for reasonable values of the parameters inequality Popt - PSQL may be valid. The possibility of reducing the measurement error below SQL means that the energy of the free particle can be measured with accuracy better than "rt w5,14x. In conclusion, we have proposed a method of QND measurement of a projection of the mechanical momentum of a three-level L-type atom. The method is based on adiabatic passage technique, which allows to preserve the value of the momentum in the case of resonant tunings of the probe fields. The accuracy of the measurement can be better than SQL.

The author gratefully acknowledges the support from the Office of Naval Research and the National Science Foundation. References w1x L.D. Landau, E.M. Lifshitz, Quantum mechanics: non-relativistic theory, Pergamon Press, NY, 1991. w2x Yu.I. Vorontsov, Theory and Methods of Macroscopic Measurements, Nauka, Moscow, 1989 Žin russian.. w3x T. Sleator, M. Wilkens, Phys. Rev. A 48 Ž1993. 3286. w4x J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955. w5x Yu.I. Vorontsov, T.P. Bocharova, Zh. Eksp. Teor. Fiz. 84 Ž1983. 1601 ŽSov. Phys. JETP 57 Ž1983. 933.. w6x M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, UK, 1996. w7x F.T. Hioe, Phys. Lett. A 99 Ž1983. 150. w8x J. Oreg, F.T. Hioe, J.H. Eberly, Phys. Rev. A 29 Ž1984. 690. w9x J.R. Kuklinski, U. Gaubatz, F.T. Hioe, K. Bergmann, Phys. Rev. A 40 Ž1989. 6741. w10x P. Marte, P. Zoller, J.L. Hall, Phys. Rev. A 44 Ž1991. 4118. w11x E. Korsunsky, Phys. Rev. A 54 Ž1996. 1773. w12x S.P. Vyatchanin, A.B. Matsko, Zh. Eksp. Teor. Fiz. 104 Ž1993. 2668 ŽSov. Shys. JETP 77 Ž1993. 218.. w13x S.P. Vyatchanin, A.B. Matsko, Zh. Eksp. Teor. Fiz. 109 Ž1996. 1873 ŽSov. Shys. JETP 83 Ž1996. 690.. w14x Yu.I. Vorontsov, Usp. Fiz. Nauk 133 Ž1981. 351 ŽSov. Phys. Usp. 24 Ž1981. 150..