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arXiv:1809.06789 · Physical Review A

Nonequilibrium time evolution and rephasing in the quantum sine-Gordon model

Dávid X. Horváth, Izabella Lovas, Márton Kormos, G. Takács, Gergely Zaránd

Published Jul 12, 2019• 59 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.100.013613

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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.
We consider two weakly interacting quasi-1D condensates of cold bosonic atoms. It turns out that a time-dependent variation of the tunnel-coupling between those condensates is equivalent to the spatial expansion of a one-dimensional toy-Universe, with regard to the dynamics of the relative phase field. The dynamics of this field is governed by the quantum sine-Gordon equation. Thus, this analogy could be used to 'quantum simulate' the dynamics of a scalar, interacting quantum field on an expanding background. We discuss how to observe the 'freezing' of quantum fluctuations during an accelerating expansion in a possible experiment. We also analyze an experimental protocol to study the formation of sine-Gordon breathers in the relative phase field, seeded by quantum fluctuations.
We study inhomogeneous quantum quenches in the attractive regime of the sine--Gordon model. In our protocol, the system is prepared in an inhomogeneous initial state in finite volume by coupling the topological charge density operator to a Gaussian external field. After switching off the external field, the subsequent time evolution is governed by the homogeneous sine--Gordon Hamiltonian. Varying either the interaction strength of the sine--Gordon model or the amplitude of the external source field, an interesting transition is observed in the expectation value of the soliton density. This affects both the initial profile of the density and its time evolution and can be summarised as a steep transition between behaviours reminiscent of the Klein--Gordon, and the free massive Dirac fermion theory with initial external fields of high enough magnitude. The transition in the initial state is also displayed by the classical sine--Gordon theory and hence can be understood by semi-classical considerations in terms of the presence of small amplitude field configurations and the appearance of soliton excitations, which are naturally associated with bosonic and fermionic excitations on the quantum level, respectively. Features of the quantum dynamics are also consistent with this correspondence and comparing them to the classical evolution of the density profile reveals that quantum effects become markedly pronounced during the time evolution. These results suggest a crossover between the dominance of bosonic and fermionic degrees of freedom whose precise identification in terms of the fundamental particle excitations can be rather non-trivial. Nevertheless, their interplay is expected to influence the sine--Gordon dynamics in arbitrary inhomogeneous settings.
We study inhomogeneous quantum quenches in the attractive regime of the sine-Gordon model. In our protocol, the system is prepared in an inhomogeneous initial state in finite volume by coupling the topological charge density operator to a Gaussian external field. After switching off the external field, the subsequent time evolution is governed by the homogeneous sine-Gordon Hamiltonian. Varying either the interaction strength of the sine-Gordon model or the amplitude of the external source field, an interesting transition is observed in the expectation value of the soliton density. This affects both the initial profile of the density and its time evolution and can be summarised as a steep transition between behaviours reminiscent of the Klein-Gordon, and the free massive Dirac fermion theory with initial external fields of high enough magnitude. The transition in the initial state is also displayed by the classical sine-Gordon theory and hence can be understood by semi-classical considerations in terms of the presence of small amplitude field configurations and the appearance of soliton excitations, which are naturally associated with bosonic and fermionic excitations on the quantum level, respectively. Features of the quantum dynamics are also consistent with this correspondence and comparing them to the classical evolution of the density profile reveals that quantum effects become markedly pronounced during the time evolution. These results suggest a crossover between the dominance of bosonic and fermionic degrees of freedom whose precise identification in terms of the fundamental particle excitations can be rather non-trivial. Nevertheless, their interplay is expected to influence the sine-Gordon dynamics in arbitrary inhomogeneous settings.
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.