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Open Bose-Hubbard chain: Pseudoclassical approach

A. A. Bychek, P. S. Muraev, D. N. Maksimov, and A. R. Kolovsky
Phys. Rev. E 101, 012208 – Published 13 January 2020
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  • INTRODUCTION
  • PSEUDOCLASSICAL APPROACH
  • CHAIN OF COUPLED OSCILLATORS
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We analyze the stationary current of bosonic carriers in the Bose-Hubbard chain of length L where the first and the last sites of the chain are attached to reservoirs of Bose particles acting as a particle source and sink, respectively. The analysis is carried out by using the pseudoclassical approach which reduces the original quantum problem to the classical problem for L coupled nonlinear oscillators. It is shown that an increase of oscillator nonlinearity (which is determined by the strength of interparticle interactions) results in a transition from the ballistic transport regime, where the stationary current is independent of the chain length, to the diffusive regime, where the current is inversely proportional to L.

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    • Received 17 October 2019

    DOI:https://doi.org/10.1103/PhysRevE.101.012208

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Coupled oscillatorsOpen quantum systemsTransport phenomena
    1. Physical Systems
    Bose-Einstein condensatesStochastic dynamical systems
    1. Techniques
    Semiclassical physics
    Nonlinear DynamicsStatistical Physics & Thermodynamics

    Authors & Affiliations

    A. A. Bychek1, P. S. Muraev2, D. N. Maksimov1,2, and A. R. Kolovsky1,2

    • 1Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia
    • 2School of Engineering Physics and Radio Electronics, Siberian Federal University, 660041 Krasnoyarsk, Russia

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    Issue

    Vol. 101, Iss. 1 — January 2020

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    Images

    • Figure 1
      Figure 1

      Dash-dotted line: Dynamics of I(t)=N(t)/n¯ according to Eq. (4). Parameters are γ=0.5 and n¯=10. The initial condition corresponds to the oscillator ground state, i.e., to the Fock state |n〉 with n=0. Dashed line: Solution of master equation (7) for D=0.5 and γ=0. Solid blue line: Solution of the Langevin equation (20) averaged over 4000 realizations of the random process ξ(t). The inset shows the oscillator spectral density P(ν) for ω=1 and different values of nonlinearity g=0,0.5,1,2, from top to bottom.

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    • Figure 2
      Figure 2

      Comparison between the exact (dash-dotted lines) and pseudoclassical (solid lines) results for L=3, J=1, n¯1=2, n¯L=1, γ=0.5 (D1=γ, DL=γn¯L/n¯1), and U=0.5 (g=1). In the quantum case the Fock basis is truncated to ∑lnl≤20, which gives the total dimension of the Hilbert space, N=1771. In the pseudoclassical case the average is taken over 16 000 realizations of the stochastic processes.

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    • Figure 3
      Figure 3

      Comparison between the cases g=0 and g=2 where the latter case is indicated by dashed lines. Left column, dynamics of the mean actions Il(t); upper right, dynamics of the mean current; and lower right, stationary values of the actions Ĩl along the chain. The other parameters are γ1=γL=0.5, D1=0.5, and DL=0.25. Average over 4000 runs.

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    • Figure 4
      Figure 4

      Oscillator spectral density for g=0 (top) and g=2 (bottom). The other parameters are the same as in Fig. 3.

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    • Figure 5
      Figure 5

      Left: The mean oscillator actions along the chain of length L=40 for DL=D1/2 (open circles) and DL=0 (asterisks). The other parameters are g=2, D1=0.5, and γ1=γL=0.5. Average over 4000 realizations. Right: The stationary current as a function of the inverse chain length for two considered cases DL=0.25 (open circles) and DL=0 (asterisks). The bars show statistical error due to the finite number of realizations.

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