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arXiv:0906.4949 · Physical Review A

Quantum many-body dynamics of dark solitons in optical lattices

Ryan V. Mishmash, Ippei Danshita, Charles W. Clark, Lincoln D. Carr

Published Nov 13, 2009• 54 citations
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Abstract

We present a fully quantum many-body treatment of dark solitons formed by ultracold bosonic atoms in one-dimensional optical lattices. Using time-evolving block decimation to simulate the single-band Bose-Hubbard Hamiltonian, we consider the quantum dynamics of density and phase engineered dark solitons as well as the quantum evolution of mean-field dark solitons injected into the quantum model. The former approach directly models how one may create quantum entangled dark solitons in experiment. While we have already presented results regarding the latter approach elsewhere [R. V. Mishmash and L. D. Carr, Phys. Rev. Lett. 103, 140403 (2009)], we expand upon those results in this work. In both cases, quantum fluctuations cause the dark soliton to fill in and may induce an inelasticity in soliton-soliton collisions. Comparisons are made to the Bogoliubov theory which predicts depletion into an anomalous mode that fills in the soliton. Our many-body treatment allows us to go beyond the Bogoliubov approximation and calculate explicitly the dynamics of the system's natural orbitals.

DOI
10.1103/physreva.80.053612

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We present a fully quantum many-body treatment of dark solitons formed by ultracold bosonic atoms in one-dimensional optical lattices. Using time-evolving block decimation to simulate the single-band Bose-Hubbard Hamiltonian, we consider the quantum dynamics of density and phase engineered dark solitons as well as the quantum evolution of mean-field dark solitons injected into the quantum model. The former approach directly models how one may create quantum entangled dark solitons in experiment. While we have already presented results regarding the latter approach elsewhere [R. V. Mishmash and L. D. Carr, Phys. Rev. Lett. 103, 140403 (2009)], we expand upon those results in this work. In both cases, quantum fluctuations cause the dark soliton to fill in and may induce an inelasticity in soliton-soliton collisions. Comparisons are made to the Bogoliubov theory which predicts depletion into an anomalous mode that fills in the soliton. Our many-body treatment allows us to go beyond the Bogoliubov approximation and calculate explicitly the dynamics of the system's natural orbitals.
Inspired by experiments on Bose-Einstein condensates in optical lattices, we study the quantum evolution of dark soliton initial conditions in the context of the Bose-Hubbard Hamiltonian. An extensive set of quantum measures is utilized in our analysis, including von Neumann and generalized quantum entropies, quantum depletion, and the pair correlation function. We find that quantum effects cause the soliton to fill in. Moreover, soliton-soliton collisions become inelastic, in strong contrast to the predictions of mean-field theory. These features show that the lifetime and collision properties of dark solitons in optical lattices provide clear signals of quantum effects.
We study quantum dynamics of bosonic atoms that are excited to form a phase kink, or several kinks, by an imprinting potential in a one-dimensional trap. We calculate dissipation due to quantum and thermal fluctuations in soliton trajectories, collisions and the core structure. Single-shot runs show weak filling of a soliton core, typically deeper solitons in the case of stronger fluctuations and spreading/disappearing solitons due to collisions. We also analyze a soliton system in an optical lattice that shows especially strong fluctuation-induced phenomena.
Consider the tunneling of a many-body wavefunction through a potential barrier. Specifically we examine quantum many-body tunneling of ultracold bosons in one dimensional optical lattice potentials. Such systems are described by the Bose-Hubbard model, and serve as an ideal testbed for the observation of quantum phenomena on macroscopic length scales. Bright solitons, self-sustaining nonlinear waves, are confined by a thin potential barrier. By decreasing the size of the barrier or by tuning the nonlinearity, we find that the bound states can be transformed into quasi-bound ones as the soliton tunnels collectively out of confinement. The lowly entangled 1D many-body problem can be numerically solved via time-evolving block decimation: a time-adaptive density matrix renormalization group routine. As a sanity check the results can be compared to mean-field theory predictions obtained from numerical analysis of the Nonlinear Schrodinger equation. The goal behind this effort is to explore the previously intractable physics in regimes where the mean-field theory fails, and to provide a stringent test on the validity of quantum mechanics for complex entangled systems.
We study the quantum tunneling dynamics of many-body entangled solitons composed of ultracold bosonic gases in 1D optical lattices. A bright soliton, confined by a potential barrier, is allowed to tunnel out of confinement by reducing the barrier width and for varying strengths of attractive interactions. Simulation of the Bose Hubbard Hamiltonian is performed with time-evolving block decimation. We find the characteristic 1/e1/e1/e time for the escape of the soliton, substantially different from the mean field prediction, and address how many-body effects like quantum fluctuations, entanglement, and nonlocal correlations affect macroscopic quantum tunneling; number fluctuations and second order correlations are suggested as experimental signatures. We find that while the escape time scales exponentially in the interactions, the time at which both the von Neumann entanglement entropy and the slope of number fluctuations is maximized scale only linearly.
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.
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.