0.7 mw. 1.4 mw 2.2 mw [10 6 ] laser power [mw]

Transcription

1 EUROPHYSICS LETTERS 23 December 1997 Europhys. Lett., (), pp. 1-1 (1997) Inseparable time evolution of anisotropic Bose-Einstein condensates H. Wallis 1 and H. Steck 1;2 1 Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Strae 1, D Garching, Germany 2 Sektion Physik, Ludwig-Maximilians-Universitat Munchen, Theresienstr.37, D Munchen, Germany (received ; accepted ) PACS ?Fi{. PACS ?Jp{. Abstract. { We present the dynamics of expanding anisotropic Bose-Einstein condensates initially in a double-minimum potential well. Analytical solutions for the transverse motion are combined with numerical solutions of the timedependent Gross-Pitaevskii equation (GPE) to calculate curved interference patterns as observed in recent experiments. Such patterns are a signature of the non-separable time evolution of interacting atoms in the mean-eld approximation. Introduction. { Recent experiments with ultracold trapped atoms [1, 2] have revealed the coherence properties of Bose-Einstein condensates. The possibility of interference of independent condensates with denite initial atom numbers has been explained theoretically [3, 4, 5]. The inuence of atom-atom interactions on the interference patterns has been calculated using the Gross-Pitaevskii equation [6, 7, 8]. The results concerning the fringe contrast, spacing and dependence on initial conditions [7, 8] are in quantitative agreement with the experiment [9]. In that experiment two condensates in a double minimum potential well expanded freely after switching o the external potential and formed interference fringes in the overlap region. The subject of the present paper is the curvature of the interference patterns occuring when the initial separation of the condensate is incomplete. In the following it will become clear that the actual curvature is not a diractive eect but originates from the self-interactions of the expanding Bose gas leading to non-separable solutions of the timedependent GPE. To calculate this feature, a semi-analytical method dierent from the one of [4, 7, 8] is developped to obtain the three-dimensional density distribution of a condensate released from a highly anisotropic trap. The approximation used in [4, 7, 8] is based on a product ansatz for the wavefunction and yields accurate results for the distribution along the axes of the coordinate system, but shows inherently parallel interference fringes. By the semi-analytical treatment presented here we do not only speed up the numerical calculation by nearly two orders of magnitude, but also gain insight into the dynamics by reduction to its Typeset using EURO-LATEX

2 2 EUROPHYSICS LETTERS relevant features. The agreement of the calculated distributions with the experimental results is only limited by the experimental uncertainty in the particle number. Expansion of anisotropic condensates with arbitrary longitudinal distribution. { In the conguration under study the atoms were collected in a cigar-shaped trap which was divided into two halfs by a sheet of far-blue detuned, o-resonant laser light [9]. The total potential is sketched in g. 2a) of ref. [7]. The repulsive light shift potential of the Gaussian laser beam is modelled by a potential of height V 0 that is proportional to the laser power, and of width = p 2. The total conning potential reads ( 1 ) V (r; t) = V k (z; t) + V? (; t) V k (z; t) = V? (; t) = h m 2!2 k z2 + V 0 e?z2 = 2i (?t) h m 2!2? 2i (?t); (1) where m denotes the atomic mass. At time t = 0 the magnetic and the optical potentials are suddenly switched o and the condensates are released which is accounted for by the step function (?t). Note that the gravitational potential need not be considered because the initial potential remains harmonic in presence of an additional linear one. After switching o the trap the gravitational acceleration can be fully absorbed into an accelerated reference frame. The interference patterns evolve through atomic motion along the z-axis which is perpendicular to gravity in the experimental setup, and are not aected by it. The self-consistent wave function of the interacting condensate can be calculated as the solution of the time-dependent GPE (r; t) =? 2m + V (r; t) + Uj ~ (r; t)j 2 (r; t) (2) with interaction strength ~ U = 4h 2 an0 =m, N 0 being the condensate particle number, a the s-wave scattering length, and being normalized to 1. For a time-independent potential a stationary solution (r; t) = 0 (r) exp(?it=h) is obtained analytically if the kinetic energy term can be neglected compared to the mean-eld energy (Thomas-Fermi approximation, TFA). The resulting density distribution n TF (r) = (? V (r;?1))= ~ U (3) has the shape of a cut cigar for sucient barrier height V 0, and serves as our initial condition from R t =?1 to t = 0. The value of is determined from the normalization condition drn TF (r) = 1. Explicit calculation of for the above potential requires inversion of an error function which has to be done numerically. A general analytical solution for the dynamics in the time-dependent potential does not exist to the best of our knowledge. The numerical solution on a xed grid in three dimensions involves a high amount of computation time which scales in an infavourable way with the extension of the system. This is due to the fact that an initially prolate distribution expands into an oblate one after the potential is switched o, and due to the reciprocal relation between condensate separation d and fringe-separation x / 1=d. ( 1 ) In the absence of light, the longitudinal and transverse frequencies are! k = 2 19 Hz and!? = 2 250Hz, respectively [9]. Note that in contrast to [4, 7, 8] we denote the \longitudinal axis" as the z-axis, i.e. the axis of rotational symmetry of the initially prolate distribution. The longitudinal axis is perpendicular to gravity acting along the?x direction.

3 H. WALLIS AND H. STECK: INSEPARABLE TIME EVOLUTION OF ANISOTROPIC CONDENSATES 3 In the case of time-dependent harmonic potentials, an analytical extension of the timeindependent TFA is applicable[10, 11]. However, the initial condition according to the potential of eq. (1) strongly diers from a harmonic trap. The basic idea of the present scheme is therefore to solve explicitly for the transverse dynamics of the released condensates by a fast uniform expansion similar to the scaling solution[10, 11]. The remaining evolution along the longitudinal axis of the initial cloud is then calculated numerically for a number of transverse positions. It is assumed that the particle number is large enough such that the Thomas-Fermi approximation for the transverse solution stays applicable within the observation time. A transverse cut through the cloud retains its parabola-like density distribution for all later times. We achieve this using the ansatz im exp 2h (; z; t) = 2 2 _ ; z; t (4) by scaling the solution of the initial transverse potential at t 0. After insertion of Eq.(4) into Eq.(2), several derivatives cancel and one is left with the exact equation (r; t) =? h2 2m? 2 + m2 2 # U +V (r; t) + ~ 2 j(r; t)j2 (r; t) (5) where? denotes the transverse part of the Laplacian. operating on the argument 0 = = of the function. We replace the terms proportional to by choosing as solutions of the dierential equation, m V?(; t) = V?(;?1) 4 : (6) which turns into =! 2? =3 for t > 0. With the initial conditions (0) = 1 and (0) _ = 0 the transverse scaling factor evolves according to q = 1 +! 2? t2 (7) The above scaling is exact for a two-dimensional gas in a harmonic potential [10] and is correct to rst order in! k =!? 1 for the three-dimensional prolate anisotropic case. By assuming two-dimensional scaling the transverse repulsion of the cloud is slightly overestimated since the total density decreases additionally owing to the longitudinal expansion. However, within the interaction time of interest here (! k t 5), the longitudinal size of the cloud has increased only very slightly (see also g. 3 of ref. [7]). We now turn to the approximate equation of motion for ( 0 ; z; t) containing the kinetic energy along the z-axis exactly. As the dominant eect of the expansion is allowed for by the scaling we disregard the remainder of the transverse Laplacian,? = 2, thus admitting an error proportional to the ratio of the residual transverse kinetic energy over the longitudinal one. Note that within this approximation the transverse expansion velocity does not depend on the position along the longitudinal axis. The nal equation for the longitudinal motion ~(0 ; z; t) =? V U k(z; t) + ~ 2 j~(0 ; z; t)j 2 ~( 0 ; z; t); (8)

4 4 EUROPHYSICS LETTERS where we set ( 0 ; z; t) = exp(?i( 0 ; t))~( 0 ; z; t) and solves = V?( 0 ;?1) h 2 (9) for t > 0. Note that the eective non-linearity ~ U= 2 in eq. 8 decreases rapidly during the transverse expansion. The initial condition is a solution of the stationary GPE (? V? ( 0 ;?1))( 0 ; z;?1) =? V k(z;?1) + U ~ j( 0 ; z;?1)j 2 ( 0 ; z;?1); (10) embodying the parabola-like variation of the transverse density at any z. On a given cylinder of expanding radius = 0 the wave function stays normalized to Z Z dz j( 0 ; z; t)j 2 = dz j( 0 ; z;?1)j 2 n( 0 ;?1) (11) at any time. The numerical solution of eq. (8) is performed separately for xed 0 = const. For the integration the well-known split-operator scheme is used for a small number of onedimensional cuts at various initial distances 0. The number of cuts is smaller by one to two orders of magnitude than the number of grid points used for the transverse expansion on a two-dimensional grid. With this method we obtain the density at any point and time. Curved interference patterns. { We now investigate the interference patterns of atoms released q from the two minima of the potential eq. (1). The minima are separated by a distance d = ln[m! 2 k 2 =(4V 0 )] from the plane z = 0 between the condensates. For non-interacting atoms with a Gaussian initial distribution centered around the points (d; 0; 0) with transverse width?0 and longitudinal width k0 the density distribution at time t would read n NI (x; y; z; t) = G? (x; t)g? (y; t) 2(1 + exp(?d 2 = 2 k0 )] Gk (z? d; t) + G k (z + d; t)+ q + G k (z? d; t)g k (z + d; t))2 cos 2mdz ht + m 2 2 k0 =(h2 t 2 )!# : (12) Here we dened the Gaussians G (; t) = ( p?1 ) exp(? 2 =( 2 )), and 2 = h 2 t 2 =(m 2 0). 2 Note that the density n NI is always separable in cartesian coordinates and results in parallel interferences fringes. For interacting atoms we obtain non-separable distributions instead. The example we display in g. 1 is a two-dimensional cut through the condensate as measured by the technique of [9], clearly showing the curvature of the interferences fringes. All interference patterns of incompletely separated condensates show a maximum right in the center, a signature of the phase-locking [8]. The interference patterns depend crucially on the dividing potential barrier. For a smaller barrier the central peak is wider and higher whereas almost no secondary interference fringes are visible. The atoms originate from \slits" separated by a small distance, leading to a wide interference period. With increasing separation of the initial \slits", more and more interference fringes appear around the central one which gets narrower. The initial density distribution of the \cut cigar" now includes atoms with an incomplete separation on the longitudinal axis and a wider separation o-axis. This can be easily visualized since the sum of magnetic and optical potential forms a saddle surface having a maximum along the longitudinal axis but a minimum along the transverse axis. Therefore with increasing distance from the longitudinal axis the

5 H. WALLIS AND H. STECK: INSEPARABLE TIME EVOLUTION OF ANISOTROPIC CONDENSATES 5 Fig. 1. { Interference pattern in a horizontal plane for two interfering Bose-Einstein condensates expanding during a time t = 40ms, for a power of 1.4mW. The theoretical plot was calculated for N0 = condensate atoms, the experimental value was N0 = atoms. Left: experimental data courtesy of H. J. Miesner,W. Ketterle and collaborators, right: results from numerical integration of the scaled GPE. Laser power: 1.4mW. a) 60 b) 50 d 0 [µm] laser power [mw] d 0 [µm] 0.7 mw mw 2.2 mw N 0 [10 6 ] Fig. 2. { Distance from the central density maximum to the rst minimum, d0, (on axis); a) as a function of the power of the dividing laser beam; b) as a function of particle number. The dierent curves correspond to N0 = 2; 5; 7; atoms (a); respectively to dierent laser powers (b). interference pattern along the longitudinal axis changes as described above. Consequently, the nal three-dimensional density distribution of the central maximum has the shape of a convex lens (see g. 1). The above eect becomes most visible for large atom numbers when the chemical potential is higher than the saddle point but still not as high as the barrier o-axis. In g. 2a) the

6 6 EUROPHYSICS LETTERS fringe width of the central maximum is plotted as a function of laser power. The fringe width increases sharply once the laser power is smaller than the critical value that is necessary for a complete separation. The dierent curves in the left plot correspond to dierent atom numbers. The increase of the fringe width with the atom number is plotted in g. 2b). In the regime of incomplete separation, the fringe width and the longitudinal extension of the cloud are in principle sucient information to determine both atom number and laser power a posteriori. The curvature of the central density maximum also depends on the laser beam waist which was kept xed here. Generally, a narrower beam waist leads to stronger anharmonic deformation of the potential along the longitudinal axis, and a corresponding initial density distribution. The curved interference fringes are a manifestation of the fact that the dynamics of the condensate wavefunctions cannot be separated into products of wavefunctions in each direction. Conclusion. { Using an ecient combination of scaling solutions and numerical techniques we quantitatively explained unexpected features of recent interference experiments. The nonlinearity of the mean-eld theory leads to non-separable features of the interferences which vanish only in the limits of large initial distance between the condensates or small atom number. In that limit previously employed approximations [4, 7, 8] become exact. Since the width of the interference fringes depends so critically on the amount of incomplete separation a reliable prediction of a particular experimental result requires a precise knowledge of the condensate particle number. *** We thank H. J. Miesner and W. Ketterle for the permission to publish their experimental data in g.1. The contributions of A. Rohrl and M. Naraschewski to the split-operator code are acknowledged. H. W. appreciates a stimulating discussion on the scaling transform with Y. Castin. The paper has been supported by the Deutsche Forschungsgemeinschaft (grant Wa 727/6-1). REFERENCES [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science, 269 (1995) 198. [2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett., 75 (1995) [3] J. Javanainen and S. M. Yoo, Phys. Rev. Lett., 76 (1996) 161. [4] M. Naraschewski, H. Wallis, A. Schenzle, J. I. Cirac, and P. Zoller, Phys. Rev. A, 54 (1996) [5] J. I. Cirac, C. W. Gardiner, M. Naraschewski, and P. Zoller, Phys. Rev. A, 54 (1996) R3741. [6] W. Hoston and L. You, Phys. Rev. A, 53 (1996) [7] H. Wallis, A. Rohrl, M. Naraschewski, and A. Schenzle, Phys. Rev. A, 55 (1997) [8] A. Rohrl, M. Naraschewski, A. Schenzle, and H. Wallis, Phys. Rev. Lett., 78 (1997) [9] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn and W. Ketterle, Science, 275 (1997) 637. [10] Y. Kagan, E. L. Surkov, and G. Shlyapnikov, Phys. Rev. A, 54 (1996) 1753, Y. Castin and R. Dum, Phys. Rev. Lett., 77 (1996) [11] Scaling of Gaussian testfunctions was studied by V. M. Perez-Garca, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. Lett., 77 (1996) 5320.