Temporal Talbot effect in interference of matter waves from arrays of Bose Einstein condensates and transition to Fraunhofer diffraction

Transcription

1 Physics Letters A 34 (004) Teporal Talbot effect in interference of atter waves fro arrays of Bose Einstein condensates and transition to Fraunhofer diffraction A. Gaal a, A.M. Kachatnov b, a Instituto de Física, Universidade de São Paulo, C.P , São Paulo, Brazil b Institute of Spectroscopy, Russian Acadey of Sciences, Troitsk 14190, Moscow Region, Russia Received 18 January 004; accepted 18 February 004 Counicated by V.M. Agranovich Abstract We consider interference patterns produced by coherent arrays of Bose Einstein condensates during their one-diensional expansion. Several characteristic pattern structures are distinguished depending on value of the evolution tie. Transforation of Talbot collapse revival behavior to Fraunhofer interference fringes is studied in detail. 004 Elsevier B.V. All rights reserved. 1. Introduction The interference easureents [1,] on two expanding Bose Einstein condensates (BECs) have created new iportant field of research where the density profile of gas, iaged after releasing fro the trap, provides iportant inforation about the phase of the ground-state wave function. Expansion of coherent arrays of BECs provides new opportunities to test the phase properties of the syste [3 6]. For exaple, Fraunhofer interference patterns observed in [4] deonstrate strong coherence of BECs confined in separate traps, and experient [6] shows that this coherence can be anipulated by eans of collapses * Corresponding author. E-ail address: kach@isan.troitsk.ru (A.M. Kachatnov). and revivals of wave functions due to nonlinear interaction of BECs in tightly confined states fored by three-diensional periodic trapping potential. It is well known (see, e.g., [7]) that entioned above collapse revival behavior of quantu-echanical wave functions [8,9] is a teporal counterpart of optical Talbot effect [10,11] in which interference pattern behind the grating restores at distances ultiple of the so-called Talbot distance d /λ (d is the slit spacing in the grating and λ is the wavelength of light). Siilar Talbot effect was also observed in ato optics [1, 13]. Analogy between spatial Talbot effect and teporal collapse revival behavior of wave functions suggests that such collapses revivals should exist in interference of atter waves eitted fro arrays of BECs provided evolution tie is sall enough, and indeed such effect was observed in [14]. In this connection it is natural to ask how this short-tie Talbot behav /$ see front atter 004 Elsevier B.V. All rights reserved. doi: /j.physleta

2 8 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) 7 34 ior evolves for finite array of condensates to long-tie Fraunhofer behavior observed in [4]. This Letter is devoted to consideration of this proble. In Section we present general forulas for the wave function produced by a linear array of BECs. We confine ourselves with one-diensional theory under supposition that condensate reains confined in radial direction after turning off a periodic optical potential and evolution takes place only along the axial direction of the BECs array. This forulas perit us to distinguish characteristic stages of evolution shorttie Talbot stage with revivals of the wave function in the central part of the array, interediate tie stage when Fraunhofer fringes already fored with Fresnel diffraction pattern inside each of the, and long-tie Fraunhofer stage with standard density distribution along fringes. These stages of evolution of the wave function are studied in detail in Section 3 (Talbot stage) and Section 4 (transition to Fraunhofer stage). The last Section 5 is devoted to conclusions.. General forulas After switching off the periodic optical potential the condensate density decreases and under condition that the initial size σ of each BEC is uch less than the spacing d between sites, the interatoic interaction can be neglected during ost tie of the evolution and, hence, the wave function obeys the linear Schrödinger equation i ψ t = (1) ψ xx. If the initial state is given by ψ(x,0) = ψ 0 (x), then after tie t it evolves into G(x x,t)ψ 0 (x )dx, () where G(x x,t) is well-known Green function of Eq. (1) (see, e.g., [15]) [ i(x x G(x x,t)= (3) πit exp ) ]. t To siplify calculations, we suppose that the initial wave function of BEC in the site of the array with the coordinate kd can be approxiated by a Gaussian function and, hence, we represent the initial state of BEC as 1 ] ψ(x,0) = π 1/4 A k e iφ k (x kd) exp [ σ σ, k (4) where N k = A k is equal to nuber of atos in kth condensate (we suppose that σ d) andφ k is its phase. Then Eq. () yields the solution 1 π 1/4 σ(1 + i t/σ ) [ A k e iφ k (x kd) ] exp σ (1 + i t/σ. ) k (5) This forula should be specified in accordance with the proble under consideration. In the case of large nuber of condensates in the array confined in axial direction by a parabolic potential, the Thoas Feri approxiation can be used for calculation of nuber of atos in kth condensate which gives [4] N k = A k = 15N ) (1 k (6) 16k M km, where ( ω 15 k M = (7) ωx d 8 π N a ) d 1/5, a ho σ N = N k is the total nuber of atos, ω = (ω x ω )1/3 is the geoetric ean of the agnetic trap frequencies, a ho = / ω is the corresponding oscillator length, and a>0isthes-wave scattering length. In the experient [4] there was k M = 10 1, and this large paraeter suggests that there are different stages of evolution. For t d (8) each condensate evolves independently of each other and there is no their interference effects. For t d (9) k d M we have interference between condensates, but in the central part of the array the influence of its finite

3 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) size is negligibly sall and local interference pattern can be approxiated by that of an infinite periodic lattice of condensates which leads to teporal Talbot effect. For d k M (10) t d k M the Fraunhofer fringes begin to for. Indeed, their positions are given by (see, e.g., [4]) x n (t) = ±n π t, n= 0, 1,,..., (11) d and if t satisfies the condition (10), then distances between neighboring fringes π t/d are uch greater than the size of each fringe k M d (see below). At the sae tie, the interference pattern inside each fringe is fored by only soe part of the array and hence we get Fresnel diffraction pattern along the fringe. At last, for t k d M (1) we arrive at usual Fraunhofer diffraction when the whole array contributes into interference pattern inside each fringe. To illustrate these stages of evolution of the wave function, we have shown in Fig. 1 the distributions of density ρ = ψ calculated fro forulas (5) (7) with φ k = 0 (coherent condensates). We see that for t k M (d /) the density distribution reproduces periodically in tie with period t r d / (see exact forula (0)), the side fringes begin to for at t k M (d /), andfort k M (d /) there are peaks of density at the coordinates given by (11) and profiles of fringes take Fraunhofer for for t (k M d) /. The solution (5) perits us to investigate these stages of evolution analytically. 3. Talbot revivals of wave function For tie values in the region (9), we can approxiate the array by infinite lattice of equidistant condensates so that for coherent condensates their wave function is given by A π 1/4 σ(1 + i t/σ ) [ (x kd) ] exp (13) σ (1 + i t/σ. ) k= With the use of definition of θ 3 -function (see, e.g., [16]) θ 3 (z, ) = exp [ iπ ( k + zk )] (14) k= the wave function (13) can be presented in the for π 1/4 ) ( A σi exp ( iπx x d d θ 3 d, 1 ), (15) where ( = πiσ (16) d 1 + i ) t σ. By eans of transforation forula (see [16]) ( z θ 3 (17), 1 ) = ( iπz ) i exp θ 3 (z, ) we transfor (15) into ( ) πσa x (18) π 1/4 d θ 3 d,. Then the periodicity property of θ 3 -function, θ 3 (z, ± ) = θ 3 (z, ), leads at once to the periodicity of the wave function, ψ(x,t + t r ) = ψ(x,t), with the period (19) t r = d (0) π. If the array of BECs is realized in optical periodic potential with light wavelength λ, then the spacing between neighboring lattice sites is equal to d = λ/ and the revival tie can be expressed in the for t r = 1 π, 4 E R where E R = q is the recoil energy (q = π/λ). (1) ()

4 30 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) 7 34 Fig. 1. Evolution of the density profile of array of condensates with tie calculated according to Eqs. (5) (7) with d = 8, σ = 1 (in diensionless units) and k M = 10. At t = 0.3t r,wheret r is given by Eq. (0), we see coplex interference pattern ( collapse of wave function); at t = 0.5t r the central part coincides with that for t = 0 but shifted to a half-period d/; at t = t r the initial distribution is alost copletely restored; at t = 5t r the side fringes start to for, and, finally, at t = 30t r we see Fraunhofer diffraction of atter waves fro finite grating. The evolution tie t r / corresponds to the transforation of θ 3 -function θ 3 (z, + 1) = θ 4 (z, ) = θ 3 (z + 1/,), that is we obtain the wave function shifted to the distance d/ with respect to its initial for: ψ(x,t + t r /) = ψ(x + d/,t). (3) Above calculation explains periodic restoration of initial wave function by eans of transforation properties of θ-functions. To relate this approach with standard one (see, e.g., [9]), let us consider the proble fro a different point of view. The linear Schrödinger equation (1) with periodic initial condition ψ(x,0) = A π 1/4 σ k= ] (x kd) exp [ σ can be solved by the Fourier ethod which gives (4)

5 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) ψ(x,t) π 1/4 A = { d 1 + exp [ π σ ( d 1 + i ] t )k σ k=1 ( ) } πx cos (5) d k. (The relationships between presentations (13) and (5) ofthesaewavefunctionψ(x,t) is expressed by the known identity for these two series; see, e.g., [17].) We see that at t equal to ultiple of Talbot tie t r, t = nt r, all phase factors in (5) becoe equal to unity and (5) reduces to the Fourier series for the initial periodic wave function (4). This ethod of derivation of tie-periodicity of the wave function shows that periodic restoration of the initial state is not a specific feature of the initial state (4) built of Gaussian functions. Indeed, any periodic initial function can be expanded into Fourier series and haronics cos(πxk/d), k = 1,,..., evolve with tie according to factors exp( i π t k ) which d becoe equal to unity at t = nt r. Thus, any periodic initial wave function copletely restores periodically its for. The described above picture of periodic in tie changes of the interference pattern is shown in Fig. where even for relatively sall nuber of condensates first several revivals are clearly seen. Fig.. Evolution of density profiles for BEC arrays with zero relative phase. Two first revivals at t = nt r, n = 1,, are clearly seen as well as fractional revivals at interediate oents t r /8, t r /4, 3t r /8, t r /, etc. The above theory can be generalized on non-zero phases in the initial state and hence in the solution (). For exaple, in the case of alternating phases of condensates, e iφ k = ( 1) k, (6) the wave function can be expressed in ters of θ 4 - function [16], π 1/4 A σi d ) ( exp ( iπx x (7) d θ 4 d, 1 ), or, with the use of the transforation forula [16], ( z θ 4 (8), 1 ) = ( iπz ) i exp θ (z, ), in the for ( ) πσa x (9) π 1/4 d θ d,. Then the property θ (z, + 1) = exp(πi/4)θ (z, ) leads to restoration of the initial state (up to inessential constant phase factor) after revival tie t r = d (30) π = 1 π. 8 E R Let us estiate an order of agnitude of the revival tie for arrays of BECs. In the case [6] of 87 Rb BECs array loaded into optical potential with light wavelength λ = 838 n forula (1) gives t r 75 µs. This is about one order of agnitude less than the revival tie, caused by nonlinear interaction, of single condensate in the experient [6]. In this experient absorption iages were taken after a tie-of-flight period of 16 s which is uch greater (with factor 00) than our estiate of t r. For nuber of sites in 3D lattice 10 3 we have k M 10 and, hence, the observed interference patterns correspond to the Fraunhofer liit (1). In this case the difference of interference patterns was caused by difference in initial states of condensates at different hold ties of evolution of each condensate in strongly confined states fored by 3D periodic trapping potential. In the experient [4] the revival tie is t r 69 µs and a typical iage was taken at t = 9 s, that is for k M 100 again in the Fraunhofer liit (in accordance with the theory developed in this Letter).

6 3 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) Transition to Fraunhofer interference Now we shall turn to the regions (10) and (1). Effects of Fresnel diffraction can be noticed in Fig. 1 for t = 10t r. However, they are not expressed clearly enough because of sooth distribution (6) of density in the array used in our calculations. Therefore it is ore instructive to consider finite array with equal aplitudes A k = 1 of wave functions in each condensate and take φ k = φ k, that is with equal differences φ of phases between neighboring condensates. Then Eq. (5) with t σ / reduces to ψ(x,t) = 1 σ π 1/4 i t e ix t e σ x t k M [ ( dx exp i φ )k + id k ], t t k= k M (31) where we have taken into account only leading real and iaginary contributions in the series expansion of the exponential in powers of σ /t. Thesuhere has axial aplitude when all ters are in phase in linear in k approxiation. This condition defines coordinates x n of the centers of fringes, x n = π ( n + φ ) t, n = 0, ±1, ±,... (3) d π To consider profiles of fringes, we introduce the coordinate δ which is reckoned fro the center of the fringe: x = x n + δ, (33) so that dependence on δ is deterined ainly by the factor Φ(δ,t) = k M k= k M exp ( idδ t k + id k ). t (34) If t satisfies the condition (10), then both ters in the exponential have the sae order of agnitude and, on one hand, the fringe width is of order of agnitude of the array length, δ k M d, and, on the other hand, it is uch less than the distance between fringes. Therefore the coordinate x in the factor exp( σ x / t ) can be replaced by x n. Thus, the wave function in Fig. 3. The central fringe profile for several values of the nuber of sites in the array. Tie t corresponds to the region (10). The plots are calculated for d = 8, σ = 1and(a)k M = 0 at t = 40t r ; (b) k M = 40 at t = 80t r ;(c)k M = 80 at t = 160t r. Foration of the Fresnel pattern is clearly seen. vicinity of the nth fringe is given by ψ n (x, t) = 1 σ π 1/4 i t e ix t exp [ π σ ( d n + φ π ) ] Φ(δ,t), (35) where δ = x x n.now,fork M 1 the su in (34) can be approxiated by integrals which are easily

7 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) Fig. 4. Evolution of density profile of central fringe on tie. Values of the paraeters are equal to d = 8, σ = 1, k M = 80 and (a) t = 500t r ; (b) t = 1000t r ;(c)t = 000t r ;(d)t = 16000t r. Transforation of Fresnel profile shown in Fig. 3(c) to standard Fraunhofer profile is clearly seen. {[ ( ) expressed in ters of Fresnel functions [18]: C t (k Md + δ) ( )] π t Φ(δ,t) = d e iδ + C t t (k Md δ) [ ( ) [ ( ) C t (k + S Md + δ) t (k Md + δ) ( )] ( ) } + C t (k + S (37) Md δ) t (k Md δ). ( ( ) The exponential factor deterines the nuber of + i S t (k Md + δ) atos in the nth fringe: ( ))] + S (36) t (k Md δ). N n = const exp [ 4π σ ( (38) d n + φ ) ]. π This forula reduces to Eq. (6) of Ref. [4] for φ = 0. Thus, distribution of density in the nth fringe is given Dependence on δ deterines fine interference pattern by inside fringes. It is expressed by the factor in curly brackets and deonstrates typical Fresnel for (see, πσ ψ n = d exp [ 4π σ ( d n + φ ) ] e.g., [15, Section 3.3], or [19, Section 8.7]) of diffraction fro a slit with width k M d equal to the whole π array length. Accuracy of this analytical description

8 34 A. Gaal, A.M. Kachatnov / Physics Letters A 34 (004) 7 34 depends on the nuber of sites in the array and increases with growth of k M. In Fig. 3 it is shown how exact profile of density along the fringe changes with increase of k M. Its transforation into Fresnel diffraction pattern is clearly seen. Sall ripples are obviously caused by the discrete structure of the array. For larger values of tie (1) Fresnel structure evolves into usual for of density distribution in Fraunhofer diffraction fro finite slit with width k M d. In this liit of very large t the quadratic in k tern in exponentials in Eq. (34) is uch less than unity and can be oitted. Then siple integration gives Φ(δ,t) = ndδ t sin( k M dδ ) and hence distribution t of density inside fringes is proportional to Φ(δ,t) = 4 sin (k M dδ/t) (dδ/t) (39) which is standard Fraunhofer diffraction distribution fro finite slit (see, e.g., [19, Section 8.5]). The described here evolution of profile is illustrated in Fig. 4. The total intensity of nth fringe is still deterined, of course, by Eq. (38). 5. Conclusion We have presented in this Letter analysis of interference of atter waves during one-diensional expansion of finite arrays of condensates. It shows that the interference pattern exhibits quite coplicated evolution with tie fro Talbot collapses and revivals of wave function through interediate region of Fraunhofer fringes with Fresnel patterns inside the, and, eventually, to standard Fraunhofer diffraction fro finite grating. One ay suppose that technique of density iaging will perit one to study experientally all these stages. Acknowledgeents This work was supported by FAPESP (Brazil) and CNPq (Brazil). A.M.K. thanks also RFBR for partial support. References [1] M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee, D.M. Kurn, W. Ketterle, Science 75 (1997) 637. [] Y. Shin, M. Saba, T.A. Paquini, W. Ketterle, D.E. Pritchard, A.E. Leanhardt, cond-at/ [3] C. Orzel, A.K. Tuchan, M.L. Fenselau, M. Yasuda, M.A. Kasevich, Science 91 (001) 386. [4] P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F.S. Cataliotti, P. Maddaloni, F. Minardi, M. Inguscio, Phys. Rev. Lett. 87 (001) [5] M. Greiner, I. Bloch, O. Mandel, Th.W. Hänsch, T. Esslinger, Phys. Rev. Lett. 87 (001) [6] M. Greiner, O. Mandel, Th.W. Hänsch, I. Bloch, Nature 419 (00) 51. [7] M. Berry, I. Marzoli, W. Schleich, Phys. World 14 (6) (001) 39. [8] J.H. Eberly, N.B. Narozhny, J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44 (1980) 133. [9] I.Sh. Averbukh, N.F. Perelan, Usp. Fiz. Nauk 161 (1991) 41, Sov. Phys. Usp. 34 (1991) 57. [10] H.F. Talbot, Philos. Mag. 9 (1836) 401. [11] K. Patorski, Prog. Opt. 8 (1989) 1. [1] J.F. Clauser, S. Li, Phys. Rev. A 49 (1994) R13. [13] M.S. Chapan, C.R. Ekstro, T.D. Haond, J. Schiedayer, B.E. Tannian, S. Wehinger, D.E. Pritchard, Phys. Rev. A 51 (1995) R14. [14] L. Deng, E.W. Hagley, J. Denschlag, J.E. Sisarian, M. Edwards, C.W. Clark, K. Helerson, S.L. Rolston, W.D. Phillips, Phys. Rev. Lett. 83 (1999) [15] R.P. Feynan, A.R. Hibbs, Quantu Mechanics and Path Integrals, McGraw Hill, New York, [16] H. Batean, A. Erdélyi, Higher Transcendental Functions, vol., McGraw Hill, New York, [17] D. Muford, Tata Lectures on Theta I, Birkhäuser, Boston, [18] I.S. Gradshteyn, I.P. Ryzhik, Tables of Integrals, Series, and Products, Acadeic Press, New York, [19] M. Born, E. Wolf, Principles of Optics, Pergaon, Oxford, 1968.