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arXiv:1603.09197 · Physical review. D/Physical review. D.

Quantum sine-Gordon dynamics on analogue curved spacetime in a weakly imperfect scalar Bose gas

Tyler J. Volkoff, Uwe R. Fischer

Published Jul 26, 2016• 4 citations
arXivHTMLPDF

Abstract

Using the coherent state functional integral expression of the partition function, we show that the sine-Gordon model on an analogue curved spacetime arises as the effective quantum field theory for phase fluctuations of a weakly imperfect Bose gas on an incompressible background superfluid flow when these fluctuations are restricted to a subspace of the single-particle Hilbert space. We consider bipartitions of the single-particle Hilbert space relevant to experiments on ultracold bosonic atomic or molecular gases, including, e.g., restriction to high- or low-energy sectors of the dynamics and spatial bipartition corresponding to tunnel-coupled planar Bose gases. By assuming full unitary quantum control in the low-energy subspace of a trapped gas, we show that (1) appropriately tuning the particle number statistics of the lowest-energy mode partially decouples the low- and high-energy sectors, allowing any low-energy single-particle wave function to define a background for sine-Gordon dynamics on curved spacetime and (2) macroscopic occupation of a quantum superposition of two states of the lowest two modes produces an analogue curved spacetime depending on two background flows, with respective weights continuously dependent on the corresponding weights of the superposed quantum states.

DOI
10.1103/physrevd.94.024051

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Using the coherent state functional integral expression of the partition function, we show that the sine-Gordon model on an analogue curved spacetime arises as the effective quantum field theory for phase fluctuations of a weakly imperfect Bose gas on an incompressible background superfluid flow when these fluctuations are restricted to a subspace of the single-particle Hilbert space. We consider bipartitions of the single-particle Hilbert space relevant to experiments on ultracold bosonic atomic or molecular gases, including, e.g., restriction to high- or low-energy sectors of the dynamics and spatial bipartition corresponding to tunnel-coupled planar Bose gases. By assuming full unitary quantum control in the low-energy subspace of a trapped gas, we show that (1) appropriately tuning the particle number statistics of the lowest-energy mode partially decouples the low- and high-energy sectors, allowing any low-energy single-particle wave function to define a background for sine-Gordon dynamics on curved spacetime and (2) macroscopic occupation of a quantum superposition of two states of the lowest two modes produces an analogue curved spacetime depending on two background flows, with respective weights continuously dependent on the corresponding weights of the superposed quantum states.
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.
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.
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.
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.
The sine-Gordon field theory emerges as the low-energy description in a wealth of quantum many-body systems. Recent efforts have been directed towards realizing quantum simulators of the model, by interfering two weakly coupled one-dimensional cold atomic gases. The weak interactions within the atomic clouds provide a sine-Gordon realization in the semiclassical regime. Furthermore, the intricate microscopic dynamics prevents a quantitative understanding of the effective sine-Gordon validity realm. In this work, we focus on a spin-ladder realization and observe the emergent sine-Gordon dynamics deep in the quantum regime. We use matrix-product state techniques to numerically characterize the low-energy sector of the system and compare it with the exact field-theory predictions. From this comparison, we obtain quantitative boundaries for the validity of the sine-Gordon description. We provide encompassing evidence for the emergent field theory by probing its rich spectrum and by observing the signatures of integrable dynamics in scattering events.
Sine-Gordon model coupled with a free scalar field emergent in the low-energy phase dynamics of a mixture of pseudospin-12\frac{1}{2}21​ Bose gases with interspecies spin exchange
Using the approach of low-energy effective field theory, the phase diagram is studied for a mixture of two species of pseudospin- Bose atoms with interspecies spin exchange. There are four mean-field regimes on the parameter plane of g e and g z , where g e is the interspecies spin-exchange interaction strength, while g z is the difference between the interaction strength of interspecies scattering without spin exchange of equal spins and that of unequal spins. Two regimes, with | g z | > | g e |, correspond to ground states with the total spins of the two species parallel or antiparallel along the z direction, and the low-energy excitations are equivalent to those of two-component spinless bosons. The other two regimes, with | g e | > | g z |, correspond to ground states with the total spins of the two species parallel or antiparallel on the x y plane, and the low-energy excitations are described by a sine-Gordon model coupled with a free scalar field, where the effective fields are combinations of the phases of the original four boson fields. In (1 + 1)-dimension, they are described by Kosterlitz–Thouless renormalization group (RG) equations, and there are three sectors in the phase plane of a scaling dimension and a dimensionless parameter proportional to the strength of the cosine interaction, both depending on the densities. The gaps of these elementary excitations are experimental probes of the underlying many-body ground states.