Entanglement pre-thermalization in a one-dimensional Bose gas

Journal name:
Nature Physics
Volume:
11,
Pages:
1050–1056
Year published:
DOI:
doi:10.1038/nphys3478
Received
Accepted
Published online

Abstract

An isolated quantum system often shows relaxation to a quasi-stationary state before reaching thermal equilibrium. Such a pre-thermalized state was observed in recent experiments in a one-dimensional Bose gas after it had been coherently split into two. Although the existence of local conserved quantities is usually considered to be the key ingredient of pre-thermalization, the question of whether non-local correlations between the subsystems can influence pre-thermalization of the entire system has remained unanswered. Here we study the dynamics of coherently split one-dimensional Bose gases and find that the initial entanglement combined with energy degeneracy due to parity and translation invariance strongly affects the long-term behaviour of the system. The mechanism of this entanglement pre-thermalization is quite general and not restricted to one-dimensional Bose gases. In view of recent experiments with a small and well-defined number of ultracold atoms, our predictions based on exact few-body calculations could be tested in experiments.

At a glance

Figures

  1. Schematic illustration of our set-up and time-averaged auto- and cross-correlation functions in the course of pre-thermalization.
    Figure 1: Schematic illustration of our set-up and time-averaged auto- and cross-correlation functions in the course of pre-thermalization.

    a, The field operator in the Lieb–Liniger Hamiltonian before the split consists of the left (L) and right (R) components. Initially, the system is tightly confined by a harmonic trap in the y- and z-directions, so that the system is 1D. Then, the harmonic trap is deformed into a deep double-well potential along the y-direction, where the two potential minima create two 1D systems, which we call the left and the right. After this process, each particle at position x in the original 1D system becomes a superposition of the left and right states. b, The time averages up to time t of two-point correlation functions for the number of particles N = 3. The red curves show the auto (cross)-correlation functions and the blue curves show the corresponding thermal averages at an effective temperature (see text). The cross-correlation exhibits entanglement pre-thermalization, whereas the auto-correlation shows pre-thermalization at an effective temperature that turns out to be lower than the ‘equilibrium’ temperature determined from the energy of the system on the basis of the canonical distribution.

  2. Comparison of the time-averaged auto-correlation functions.
    Figure 2: Comparison of the time-averaged auto-correlation functions.

    The red curve shows the long-term average of the auto-correlation function, which agrees well with the equilibrium auto-correlation function at the inverse effective temperature βeff = 0.151 (blue curve). The dashed curve shows the auto-correlation function at the inverse equilibrium temperature βeq = 0.137, which deviates significantly from the red one at long distance but fits excellently at short distance. The inset shows the enlarged view at short distance.

  3. Pre-thermalization and entanglement pre-thermalization exhibited by two-point correlation functions.
    Figure 3: Pre-thermalization and entanglement pre-thermalization exhibited by two-point correlation functions.

    a, The auto-correlation function shows pre-thermalization at an effective temperature βeff−1 over the entire region. The value of βeff is determined from the χ2 fit of to CLeq(x). b, The cross-correlation function cannot be fitted based on a Gibbs state at any temperature owing to entanglement pre-thermalization. The red curves show the infinite-time averages of the cross-correlation function calculated by the Bethe ansatz method. The blue curves show the thermal equilibrium averages of the cross-correlation function CLReq(x) at the temperature βeff−1. Entanglement pre-thermalization thus manifests itself in the cross-correlation function. The error bars both in the auto-correlations and the cross-correlations are smaller than the width of the lines (see Supplementary Methods for the details).

  4. The system-size dependence of the deviation of the auto-correlation (red) and cross-correlation (blue) functions from the equilibrium ones.
    Figure 4: The system-size dependence of the deviation of the auto-correlation (red) and cross-correlation (blue) functions from the equilibrium ones.

    The dashed lines are the least-square fittings of the data. Entanglement pre-thermalization survives in the thermodynamic limit. Furthermore, the non-vanishing behaviour of the auto-correlation implies that the Gibbs state at the effective temperature does not give the complete description of a single subsystem.

  5. Diagonal (purple dashed) and off-diagonal (green dashed) contributions in the infinite-time average of the cross-correlation function for N = 3.
    Figure 5: Diagonal (purple dashed) and off-diagonal (green dashed) contributions in the infinite-time average of the cross-correlation function for N = 3.

    The calculated cross-correlation function (red) and the thermally averaged cross-correlation function (blue) are also shown for comparison. The lengths of the two orange arrows are the same, indicating that the deviation of the calculated cross-correlation from the thermally averaged one is due to the off-diagonal component.

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Affiliations

  1. Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

    • Eriko Kaminishi,
    • Takashi Mori,
    • Tatsuhiko N. Ikeda &
    • Masahito Ueda
  2. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • Tatsuhiko N. Ikeda
  3. RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

    • Masahito Ueda

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All authors contributed extensively to the work presented in the paper and the writing of the manuscript.

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The authors declare no competing financial interests.

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