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

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.

Report on scipost_202005_00003v1

“…In addition to this, energy starts to oscillate between the sectors. Depending on the initial density profile imprinted on the gas, Josephson oscillations of density and phase are affected by the presence of the additional term, showing modulations of the amplitude that differ from the ones observed in the SCTDHA treatment of isolated sine-Gordon dynamics [67].…”

“…To date, no satisfying theoretical explanation of this damping is known [66]. The damping seems incompatable with a description in terms of a translationally invariant sine-Gordon model, which fails to provide a mechanism for the observed strong and rapid damping in both a self-consistent harmonic treatment [67] and in a combination of truncated Wigner and truncated conformal space approaches [68].…”

“…This is lower than typical energy scales realized in current experimental set-ups, which operate above 18 nK, but sufficiently close to make this regime interesting in light of possible future experiments. Our strategy is to treat the resulting perturbed sine-Gordon model in the self-consistent timedependent harmonic approximation (SCTDHA) as described in [67]. We consider the dynamics after initializing the system in a state in which the sectors are uncorrelated and observe how the new coupling term causes correlations between the two sectors to develop over time.…”

“…The non-equilibrium dynamics of this model was analyzed for the translationally invariant case in the framework of a SCTDHA in our recent work [67]. The additional σ-term in (11) couples the sine-Gordon model to the Luttinger liquid Hamiltonian H s .…”

Formation probabilities and statistics of observables as defect problems in free fermions and quantum spin chains

“…Being a very natural concept FCS has been studied for a long time in different communities. It has been studied in the context of charge fluctuations 45,46 , Bose gases [47][48][49][50][51] , particle number fluctuations [52][53][54][55][56] , quantum spin chains [57][58][59][60][61][62][63][64] and out of equilibrium quantum systems 51,65,66 .…”