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arXiv:1812.06690 · Journal of Statistical Mechanics Theory and Experiment

Self-consistent time-dependent harmonic approximation for the sine-Gordon model out of equilibrium

Yuri D van Nieuwkerk, Fabian H L Essler

Published Aug 1, 2019• 26 citations
arXivHTMLPDF

Abstract

Abstract We derive a self-consistent time-dependent harmonic approximation for the quantum sine-Gordon model out of equilibrium and apply the method to the dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time evolution of experimentally relevant observables and in particular derive results for the probability distribution of subsystem phase fluctuations. We investigate the regime of validity of the approximation by applying it to the simpler case of a nonlinear harmonic oscillator, for which numerically exact results are available. We complement our self-consistent harmonic approximation by exact results at the free fermion point of the sine-Gordon model.

DOI
10.1088/1742-5468/ab3579

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Abstract We derive a self-consistent time-dependent harmonic approximation for the quantum sine-Gordon model out of equilibrium and apply the method to the dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time evolution of experimentally relevant observables and in particular derive results for the probability distribution of subsystem phase fluctuations. We investigate the regime of validity of the approximation by applying it to the simpler case of a nonlinear harmonic oscillator, for which numerically exact results are available. We complement our self-consistent harmonic approximation by exact results at the free fermion point of the sine-Gordon model.
We discuss the nonequilibrium time evolution of the phase field in the sine-Gordon model using two very different approaches: the truncated Wigner approximation and the truncated conformal space approach. We demonstrate that the two approaches agree for a period covering the first few oscillations, thereby giving a solid theoretical prediction in the framework of the sine-Gordon model, which is thought to describe the dynamics of two bosonic condensates in quasi-one-dimensional traps coupled via a Josephson tunneling term. We conclude, however, that the recently observed phase-locking behavior cannot be explained in terms of homogeneous sine-Gordon dynamics, which hints at the role of other degrees of freedom or inhomogeneity in the experimental system.
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.
We consider a prime example of simulating interacting relativistic QFT with cold atoms: the realisation of the sine-Gordon model by tunnel-coupled quasi-1D Bose gases. While experiments have shown that it can realise the sine-Gordon model in equilibrium, studies of non-equilibrium dynamics have revealed phase-locking behaviour that contrasts with predictions from sine-Gordon field theory. Here, we examine a one-dimensional field-theoretic model of the system and find that the phase-locking behaviour can be understood in terms of the longitudinal harmonic trap, and that the additional degrees of freedom observed in the experiment do not appear to play a significant role. Therefore, the experimental setup provides a good simulator of the sine-Gordon quantum field theory, even out of equilibrium, if the inhomogeneous background induced by the trap is taken into account. Furthermore, our results support the idea that modifying the longitudinal trap to a box shape should result in agreement with standard sine-Gordon dynamics. The main remaining open issues are accounting for 3D corrections and modelling the effect of the boundaries.
This thesis presents four theoretical studies on pairs of tunnel-coupled one-dimensional Bose gases. It first recapitulates how at low energies, the gases’ symmetric and antisymmetric combinations of density and phase are governed by two Tomonaga-Luttinger Liquids (TLL). Tunneling between the gases is believed to yield a quantum sine-Gordon model for the antisymmetric sector. The theoretical background and various applications of this model are summarized, before presenting the first new theoretical results. These show in detail how matter-wave interference measurements can access eigenvalues of the relative phase operator from the TLL, allowing to study the operator’s distribution functions and multi-point correlators. The derivation clarifies why this construction is limited to short expansion times and weak interactions and what modifications occur away from this limit. This leads to a new formula predicting longitudinal “density ripples”. The second half of the thesis was stimulated by recent and unexplained experimental results in the tunnel-coupled case, where density-phase oscillations were seen to rapidly damp out. The work first studies whether this can be explained within a translationally invariant sine-Gordon model. Treating this model in a self-consistent harmonic approximation leads to a negative conclusion. Second, the next leading perturbation due to the tunnel-coupling is investigated in a box geometry. Although this yields a non-negligible coupling between the (anti)symmetric sectors, the effects are not strong enough to explain the damping. Finally, a new low-energy theory is developed, which does not rely on the TLL and which allows to study the roles of both higher excited levels of the transverse potential and of a realistic longitudinal potential. Strong damping is observed as a result of the longitudinal potential, with a dependence on the particle number that is compatible with experimental results. This indicates that performing the experiments in a hard-wall box potential might eliminate the damping effects.
The two-dimensional classical sine-Gordon model is investigated by using a two-parameter self-consistent renormalization method up to second order of renormalized vertices, where coupled self-consistent equations for renormalized mass m and renormalized temperature x are obtained. In high temperature region (x > 2), the self-consistent equations have a massless solution which stands for the existence of the Kosterlitz-Thouless phase, and lead to an equation for x which is the same as that given by José et al. and is consistent with the result of Wilson's renormalization group method. In low temperature region (x < 2), numerical calculation is carried out, and it is shown that the system is well described by the self-consistent harmonic approximation, which is a one-parameter (m) renormalization treatment, except near x = 2. Near x ≲ 2 our self-consistent equations have no solution, through this can be partly remedied by counting chain diagrams.