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Physical Review A

Dynamic quantum depletion in phase-imprinted generation of dark solitons

C. K. Law

Published Jul 22, 2003• 30 citations
Publisher

Abstract

We investigate the quantum fluctuations associated with dark-soliton generation in Bose-Einstein condensates. With the phase imprinting procedure treated as a part of the dynamical process, we present numerical solutions of time-dependent quantum depletion. Our results provide evidence of a rapid growth of noncondensate atoms at dark-soliton notches without making assumptions of final quasiparticle states.

DOI
10.1103/physreva.68.015602

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We investigate the quantum fluctuations associated with dark-soliton generation in Bose-Einstein condensates. With the phase imprinting procedure treated as a part of the dynamical process, we present numerical solutions of time-dependent quantum depletion. Our results provide evidence of a rapid growth of noncondensate atoms at dark-soliton notches without making assumptions of final quasiparticle states.
Generation, interaction, and detection of dark solitons in Bose-Einstein condensates are studied. In particular, we focus on the dynamics resulting from phase imprinting and density engineering. We show that solitons slow down significantly when the trap is opened and that soliton phase shifts after binary interactions cannot be observed with present experiments. Finally, motivated by the recent experimental results of Cornish et al. [Phys. Rev Lett. 85, 1795 (2000)], we analyze the stability of dark solitons under changes of the scattering length and thereby demonstrate a new way to detect them. Our theoretical and numerical results compare well with the existing experimental ones and provide guidance for future experiments.
Quantum depletion from an atomic quasi-one-dimensional Bose-Einstein condensate with a dark soliton is studied in a framework of the Bogoliubov theory. Depletion is dominated by an anomalous mode localized in a notch of the condensate wave function. Depletion in an anomalous mode requires a different treatment than depletion without anomalous modes. In particular, quantum depletion in the Bogoliubov vacuum of the anomalous mode is experimentally irrelevant. A dark soliton is initially prepared in a state with minimal depletion, which is not a stationary state of the Bogoliubov theory. The notch fills up with incoherent atoms depleted from the condensate. For realistic parameters the filling time can be as short as 10 ms.
Experimental and numerical studies of the velocity field of dark solitons in Bose–Einstein condensates are presented. The formation process after phase imprinting as well as the propagation of the emerging soliton are investigated using spatially resolved Bragg spectroscopy of soliton states in Bose–Einstein condensates of 87 Rb. A comparison of experimental data to results from numerical simulations of the Gross–Pitaevskii equation clearly identifies the flux underlying a dark soliton propagating in a Bose–Einstein condensate. The results allow further optimization of the phase imprinting method for creating collective excitations of Bose–Einstein condensates.
The dark soliton created in a Bose–Einstein condensate becomes grey in the course of time evolution because its notch fills up with depleted atoms. This is the result of quantum mechanical calculations which describe the output of many experimental repetitions of creation of the stationary soliton, and its time evolution terminated by a destructive density measurement. However, such a description is not suitable to predict the outcome of a single realization of the experiment where two extreme scenarios and many combinations thereof are possible: one will see either (1) a displaced dark soliton without any atoms in the notch, but with a randomly displaced position, or (2) a grey soliton with a fixed position, but a random number of atoms filling its notch. In either case the average over many realizations will reproduce the mentioned quantum mechanical result. In this paper we use N -particle wavefunctions, which follow from the number-conserving Bogoliubov theory, to settle this issue.
We analyze depletion of the dark soliton in a Bose-Einstein condensate in the limit of weak interaction between atoms. The condensate initially prepared in the excited dark soliton state is losing atoms because of spontaneous quantum depletion. These atoms are depleted from the soliton state into single-particle states with nonzero density in the notch of the soliton. As a result the image of the soliton is losing contrast. This quantum depletion mechanism is efficient even at zero temperature when a thermal cloud is absent.
Quantum fluctuations of coupled dark solitons in a trapped BosendashndashndashEinstein condensate