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 consider time-of-flight measurements in split one-dimensional Bose gases. It is well known that the low-energy sector of such systems can be described in terms of two compact phase fieldsφ a,s (x). Building on existing results in the literature we discuss how a single projective measurement of the particle density after trap release is in a certain limit related to the eigenvalues of the vertex operator e iφa(x) . We emphasize the theoretical assumptions underlying the analysis of "single-shot" interference patterns and show that such measurements give direct access to multi-point correlation functions of e iφa(x) in a substantial parameter regime. For experimentally relevant situations, we derive an expression for the measured particle density after trap release in terms of convolutions of the eigenvalues of vertex operators involving both sectors of the twocomponent Luttinger liquid that describes the low-energy regime of the split condensate. This opens the door to accessing properties of the symmetric sector via an appropriate analysis of existing experimental data.

We consider a model of two tunnel-coupled one-dimensional Bose gases with hard-wall boundary conditions. Bosonizing the model and retaining only the most relevant interactions leads to a decoupled theory consisting of a quantum sine-Gordon model and a free boson, describing respectively the antisymmetric and symmetric combinations of the phase fields. We go beyond this description by retaining the perturbation with the next smallest scaling dimension. This perturbation carries conformal spin and couples the two sectors. We carry out a detailed investigation of the effects of this coupling on the non-equilibrium dynamics of the model. We focus in particular on the role played by spatial inhomogeneities in the initial state in a quantum quench setup.