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arXiv:cond-mat/0701421 · Physical Review B

Linear response theory for a pair of coupled one-dimensional condensates of interacting atoms

Vladimir Gritsev, Anatoli Polkovnikov, Eugene Demler

Published May 15, 2007• 145 citations
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Abstract

We use the quantum sine-Gordon model to describe the low-energy dynamics of a pair of coupled one-dimensional condensates of interacting atoms. We show that the nontrivial excitation spectrum of the quantum sine-Gordon model, which includes soliton and breather excitations, can be observed in experiments with time-dependent modulation of the tunneling amplitude, potential difference between condensates, or phase of tunneling amplitude. We use the form-factor approach to compute structure factors corresponding to all three types of perturbations.

DOI
10.1103/physrevb.75.174511

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We use the quantum sine-Gordon model to describe the low-energy dynamics of a pair of coupled one-dimensional condensates of interacting atoms. We show that the nontrivial excitation spectrum of the quantum sine-Gordon model, which includes soliton and breather excitations, can be observed in experiments with time-dependent modulation of the tunneling amplitude, potential difference between condensates, or phase of tunneling amplitude. We use the form-factor approach to compute structure factors corresponding to all three types of perturbations.
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.
Ultracold condensates provide a unique platform for exploring soliton physics. Motivated by the recent experiments realizing the sine-Gordon model in a split one-dimensional (1D) BEC, we demonstrate that this system naturally supports various density and phase solitons. Many theories of the 1D Bose gas rely on mean-field or quasiparticle frameworks, which may suffer from infrared singularities, often causing the small-fluctuation expansion to lose validity in long-wavelength regimes. Thus, we explore the physics using the bosonization technique as a complement in the long-wavelength regime, in which the phase and density are conjugate pairs, and determine its effective Language equation and the associated equation of motion. We show that in the presence of asymmetry between the two condensates, alternative solutions beyond those in the sine-Gordon model emerge. We calculate the traveling-wave solutions and soliton solutions in this model and determine their corresponding energy densities analytically. Finally, we discuss the relevance of these solutions to the experiments and discuss their observations. Since the physical models have already been realized in experiments, this work opens alternative possibilities for the realization of various soliton and periodic solutions using two coupled condensates.
We investigate the relative phase between two weakly interacting 1D condensates of bosonic atoms after suddenly switching on the tunnel coupling. The following phase dynamics is governed by the quantum sine-Gordon equation. In the semiclassical limit of weak interactions, we observe the parametric amplification of quantum fluctuations leading to the formation of breathers with a finite lifetime. The typical lifetime and density of these "quasibreathers" are derived employing exact solutions of the classical sine-Gordon equation. Both depend on the initial relative phase between the condensates, which is considered as a tunable parameter.
The sine-Gordon model emerges as a low-energy theory in a plethora of quantum many-body systems.Here, we theoretically investigate tunnel-coupled Bose-Hubbard chains with strong repulsive interactions as a realization of the sine-Gordon model deep in the quantum regime.We propose protocols for quantum gas microscopes of ultracold atoms to prepare and analyze solitons, which are the fundamental topological excitations of the emergent sine-Gordon theory.With numerical simulations based on matrix product states, we characterize the preparation and detection protocols and discuss the experimental requirements.
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.
Quantum mechanical many-body systems are inherently hard to describe microscopically due to the exponential scaling of their state space with the number of particles. Progress in many-body physics is therefore often facilitated by identifying the relevant degrees of freedom of a system and constructing an effective field theory containing the dominant physical processes for the phenomena one would like to understand. Universal dynamics far from equilibrium, which has first been experimentally observed in a spinor Bose-Einstein condensate, has emerged as an im- portant concept which so far has eluded a comprehensive theoretical explanation. In this thesis, we experimentally test the validity of the sine-Gordon model, a paradigmatic integrable model in non-linear physics that describes the low energy behavior of a multitude of physical systems, as the effective field theory for our experimental platform, which is a spinor Bose-Einstein conden- sate. We find qualitative agreement in a first set of quench experiments between the late-time dynamics of the condensate and the effective theory for a wide range of parameters. The presence of topological excitations is conjectured to have an influence on the universal dy- namics exhibited by many-body systems. We develop a novel phase imprinting scheme in order to locally manipulate the degree of freedom of the effective sine-Gordon theory. We use the imprinting scheme to deterministically generate sine-Gordon-type solitons in the spinor con- densate, and characterize their behavior. The ability to control the velocity of the solitons with an experimental parameter opens up the study of collisions of the sine-Gordon-type solitons. We characterize the interaction of the solitons during collision, finding quantitative agreement with numerical simulations. We furthermore find that the solitons display even richer behavior than in the integrable sine-Gordon model, opening up the study of non-integrable effects such as energy transfer between internal and external modes. Observing the full spin-1 nature of the solitons during collisions additionally reveals that the solitons are a special case of a larger family of solitonic solutions. These results lay the foundation for the experimental exploration of uni- versal dynamics and its physical mechanisms through an effective field theory. The investigation of the sine-Gordon-type solitons furthermore constitutes the first experimental study of solitons in multicomponent Bose-Einstein condensates that go beyond the so-called Manakov regime.