Nonequilibrium time evolution in the sine-Gordon model

We study the nonequilibrium dynamics of the quantum sine-Gordon model describing a pair of Josephson-coupled one-dimensional bosonic quasicondensates. Motivated by experimentally accessible quench procedures where the zero mode of the quasicondensates is weakly coupled to finite momentum modes, we develop a Hamiltonian truncation scheme relying on a mini-superspace treatment of the zero mode (MSTHA). We apply this method to simulate the time evolution after both weak and strong quantum quenches, injecting a low or high energy density into the system, and we demonstrate that MSTHA accurately captures the dynamics from the hard-core boson limit to the experimentally relevant weakly interacting regime for sufficiently mild quenches. In the case of high energy densities, MSTHA breaks down for weak interaction but still extends the range of validity of previous Hamiltonian truncation schemes. We also compare these results to the semiclassical truncated Wigner approximation, and we establish that the dynamics can be well approximated by the semiclassical description in the weakly interacting regime realized in the experiments. In addition, we clarify the importance of the phononic modes depending on the sine-Gordon interaction strength.

Figure 1

The time-dependent expectation value of $:cos\beta \widehat{\phi }:$ (top row) and the phase-phase correlator $\langle {\widehat{\phi }}_k{\widehat{\phi }}_{-k}\rangle$ (bottom row) for various values of $K$ for dimensionless volume $l=10$, starting from the initial state (39). Joined markers correspond to TWA, while solid lines show the MSTHA results. For the two larger values of $K$, the difference between the two approximations is entirely invisible. The dashed red line corresponds to the (numerically) exact solution of the zero-mode quantum pendulum dynamics.

Figure 2

The time-dependent expectation value of $:cos\beta \widehat{\phi }:$ (top row) and the phase-phase correlator $\langle {\widehat{\phi }}_k{\widehat{\phi }}_{-k}\rangle$ (bottom row) for various values of $K$ for dimensionless volume $l=10$, starting from the state (42). Joined markers correspond to TWA, while solid green lines show the MSTHA results. The dashed red lines correspond to the (numerically) exact solution of the zero-mode quantum pendulum dynamics.

Figure 3

Examples on the convergence of the MSTHA for $l=10$ and two different values of $K$. The left panel shows the $\langle :cos\beta \widehat{\phi }:\rangle$ for $K=1$ and different level-cutoffs ${\ell }_{\text{max}}$, where the MSTHA simulations converged. The right panel displays simulations corresponding to $K=9.$ and various ${\ell }_{\text{max}}$ are shown where the MSTHA failed to converge, resulting in remaining truncation errors in the results.