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Physical Review Letters

Simulation of a Single Collision of Two Bose-Einstein Condensates

Jan Chwedeńczuk, Paweł Zin, Kazimierz Rza̧żewski, Marek Trippenbach

Published Oct 25, 2006• 24 citations
Publisher

Abstract

We propose a method for simulating a single realization of a collision of two Bose-Einstein condensates. Recently [Phys. Rev. Lett. 94, 200401 (2005)], we introduced a quantum model of incoherent elastic scattering in a collision of two counterpropagating atomic Gaussian wave packets. Here we show that this model is capable of generating data that can be interpreted as results of a single collisional event. We find a range of parameters, including relative velocity, population, and the size of colliding condensates, where the structure of the halo of scattered atoms in a single realization strongly differs from that averaged over many realizations.

DOI
10.1103/physrevlett.97.170404

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We propose a method for simulating a single realization of a collision of two Bose--Einstein condensates. Recently in Ziń {\it et al.} (Phys. Rev. Lett. {\bf 94}, 200401 (2005)) we introduced a quantum model of an incoherent elastic scattering in a collision of two counter-propagating atomic Gaussian wavepackets. Here show that this model is capable of generating the data that can be interpreted as results of a single collision event. We find a range of parameters, including relative velocity, population and the size of colliding condensates, where the structure of hallo of scattered atoms in a single realization strongly differs from that averaged over many realizations.
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We investigate numerically simulated collisions between experimentally realistic Bose-Einstein condensate wave packets, within a regime where highly populated scattering haloes are formed. The theoretical basis for this work is the truncated Wigner method, for which we present a detailed derivation, paying particular attention to its validity regime for colliding condensates. This paper is an extension of our previous Letter [A. A. Norrie, R. J. Ballagh, and C. W. Gardiner, Phys. Rev. Lett. 94, 040401 (2005)], and we investigate both single-trajectory solutions, which reveal the presence of quantum turbulence in the scattering halo, and ensembles of trajectories, which we use to calculate quantum-mechanical correlation functions of the field.
We analyze atom-atom correlations in the sss-wave scattering halo of two colliding condensates. By developing a simple perturbative approach, we obtain explicit analytic results for the collinear (CL) and back-to-back (BB) correlations corresponding to realistic density profiles of the colliding condensates with interactions. The results in the short-time limit are in agreement with the first-principles simulations using the positive-PPP representation and provide analytic insights into the experimental observations of Perrin et al. [Phys. Rev. Lett. 99, 150405 (2007)]. For long collision durations, we predict that the BB correlation becomes broader than the CL correlation.
Bragg diffraction divides a Bose-Einstein condensate into two overlapping components, moving with respect to each other with high momentum. Elastic collisions between atoms from distinct wave packets can significantly deplete the condensate. Recently Ziń {\it et al.} (Phys. Rev. Lett. {\bf 94}, 200401 (2005)) have introduced a model of two counter-propagating atomic Gaussian wavepackets incorporating dynamics of the incoherent scattering processes. Here we study the properties of this model in detail, including the nature of the transition from spontaneous to stimulated scattering. Within the first order approximation we derive analytical expressions for the density matrix and anomalous density which provides excellent insight into correlation properties of scattered atoms.
Superfluid suppression of the elastic scattering loss in atomic Bose condensed systems can be significant when the ratio of the relative velocity of the wave packets to the critical velocity is $\ensuremath{\sim}1.$ We show how to incorporate the effect of such losses and their suppression into the time-dependent dynamics of two colliding condensate wave packets. We illustrate the magnitude of the effect through three-dimensional simulations of the collision dynamics of an ∣F=2,MF=2〉|F=2,{M}_{F}=2〉∣F=2,MF​=2〉 with an ∣F=2,MF=1〉87Rb|F=2,{M}_{F}=1〉{}^{87}\mathrm{Rb}∣F=2,MF​=1〉87Rb condensate in a harmonic trap, as studied in a recent experiment [Maddaloni et al., Phys. Rev. Lett. 85, 2413 (2000)]. These calculations show that the effect of elastic scattering and its suppression should be seen in the oscillatory center-of-mass amplitude of the MF=1{M}_{F}=1MF​=1 wave packet for suitable experimental conditions.