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arXiv:2507.06820 · Physical Review A

Soliton and traveling-wave solutions in coupled one-dimensional condensates

Zeyu Rao, Xiaoshui Lin, Jingsong He, Guang‐Can Guo, Ming Gong

Published Nov 12, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/1jrc-3dy1

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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.
Motivated by the realization of coupled Luttinger liquids in a split condensate for quantum sine-Gordon (SG) model [Haller et al., Nature 466, 597 (2010); Schweiglet et al., Nature 545, 323 (2017); Nat. Phys. 17, 559 (2021)], here we study the phase transition in coupled condensates using the bosonization method. We show that when these condensates have different velocities ui{u}_{i}ui​ and Luttinger parameters Ki{K}_{i}Ki​, the physics is described by a generalized SG model, which is also exactly solvable. We use various approaches, including the Wilson renormalization group, correlation function, variational wave function, and the equivalent Coulomb gas methods, demonstrating the phase transition at 2=1/(4K1)+1/(4K2)2=1/(4{K}_{1})+1/(4{K}_{2})2=1/(4K1​)+1/(4K2​). The physical meaning of this boundary is explained by the density-phase uncertainty principle, showing how the many-body interactions drive the coupled condensates from two gapless phases to one gapless phase. The relative phase between the condensates may fundamentally affect the band structure, yet it will not affect this boundary. Furthermore, we also discuss the relevance of our result in the two-dimensional models for the Berezinskii-Kosterlitz-Thouless transition, from which the physical meaning of this boundary is self-evident. Our results can be used to study the phase transitions in one-dimensional coupled models with Fermions and Bosons, and even the Kane-Fisher model with impurities, and study their phase transitions, non-Gaussian dynamics, and thermalization, which may be of great interest in quantum simulation.
We study an ultracold and dilute superfluid Bose-Fermi mixture confined in a strictly one-dimensional (1D) atomic waveguide by using a set of coupled nonlinear mean-field equations obtained from the Lieb-Liniger energy density for bosons and the Gaudin-Yang energy density for fermions. We consider a finite Bose-Fermi interatomic strength gbf{g}_{bf}gbf​ and both periodic and open boundary conditions. We find that with periodic boundary conditions---i.e., in a quasi-1D ring---a uniform Bose-Fermi mixture is stable only with a large fermionic density. We predict that at small fermionic densities the ground state of the system displays demixing if gbf>0{g}_{bf}>0gbf​>0 and may become a localized Bose-Fermi bright soliton for gbf<0{g}_{bf}<0gbf​<0. Finally, we show, using variational and numerical solutions of the mean-field equations, that with open boundary conditions---i.e., in a quasi-1D cylinder---the Bose-Fermi bright soliton is the unique ground state of the system with a finite number of particles, which could exhibit a partial mixing-demixing transition. In this case the bright solitons are demonstrated to be dynamically stable. The experimental realization of these Bose-Fermi bright solitons seems possible with present setups.
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.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas of quantum mechanics have been realized and proven to be true in experiments, by studying the Bose-Einstein condensates. Amongst them, the observation of macroscopic matter wave interference, the creation of vortices, dark and bright solitons etc, find special mention. The dynamics of this system in presence of an optical lattice reveals a superfluid to Mott insulator quantum phase transition. This has opened a new avenue for research on BEC and its applications. From a theoretical standpoint, the dynamics of BEC can be described by means of an effective mean-field theory. This formalism is much simpler than the many-body Schr¨odinger equation and based on a classical nonlinear evolution equation, known as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc., which are closely related to nonlinear fiber optics. The present thesis deals with nonlinear excitations and different phases of BEC and is organized as follows: after a brief introduction, in chapter two, we start with BEC in a one-dimensional shallow optical lattice, experiencing both cubic and quintic nonlinearity. A superfluid density wave, of twice the periodicity of lattice potential, is identified. This indicates matter redistribution, where atoms from neighbouring sites have been depleted leading to a density wave behaviour. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In presence of pure two-body interaction, the superfluid density shows a periodicity commensurate with the lattice potential. Apart from the superfluid state, a density wave insulating phase is also found to exist, possessing two frequency modulations commensurate with the lattice potential. A dynamical phase transition connects the superfluid phase with an insulating phase, which in turn, is connected with the density wave insulating phase. The solutions are found to be marginally stable, as per the Vakhitov-Kolokolov (VK) criterion. Chapter three describes the dynamics of the atomic-molecular Bose-Einstein condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into account the atom-molecular interaction, as well as atom-molecule inter-conversion. The two-body atom-atom, atom-molecule and molecule-molecule interactions are analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic nonlinear term arising from the inter-conversion between atoms and molecules is identical to the scenario in strongly coupled BEC in one dimension. The presence of both weak and strong interactions adds further richness to this system. We find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed in order to see the effect of the both weak and strong interactions on Lieb mode profile and compare it with the pure atomic condensate profile. Another class of solutions exist in this system, where both atomic and molecular condensates possess bright solitons. In chapter four, we investigate the effect of a harmonic trap on the dynamics of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear excitations of the cigar-shaped Bose-Einstein condensate have strong coupling with the trap at low momenta and hence can be effectively isolated from the Bogoliubov sound modes, which respond weakly to harmonic confinement. This strong coupling with the trap also makes the grey soliton amenable for control and manipulation through trap modulation and temporal variation of the two-body interaction. We have shown that in presence of an expulsive oscillator trap, the grey soliton gets amplified, while having a compression in its width. Exact solutions are also obtained in case of the two component Bose-Einstein condensates (TBECs), where one component is a grey soliton and the other a bright one. The presence of the trap and inter-species interaction significantly affects the grey soliton dynamics. Similar to the single component, in the present case, we have shown that both grey and bright soliton can be compressed simultaneously. A detailed study of the energy and momentum of the soliton is reported, in order to find the effect of the trap on the Lieb mode. In chapter five, we study a BEC-defect complex in a quasi-one dimensional scenario, where the defect atom moves in the condensate with constant velocity. Explicit solutions of the coupled mean-field equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can lower the grey soliton’s maximum velocity, as compared to the defect free case, where the maximum velocity is the sound velocity. We have also carried out the stability analysis of the obtained solutions using renormalized momentum formalism and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable through both the methods. The grey soliton’s energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics. Chapter six describes a parametrically forced Bose-Einstein condensate in the presence of a general moving optical lattice. The interaction between the atoms in the condensate and the time dependent lattice potential leads to a novel propagating superfluid matter wave, which can be controlled through chirp management. This system, when placed in a trap, accelerates and undergoes rapid nonlinear compression, controlled by the chirp. The density achieves its maximum, when the matter wave changes direction. It is also shown that this system exhibits dynamical superfluid insulator transition, where superfluidity breaks down and the condensate transits to an insulating phase. The stability has been investigated using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well as on the nature of the atom-atom interaction. The exact expression for energy is obtained and analyzed in detail to gain physical understanding of the chirp management of the sinusoidal excitations and also the dynamical phase transition.
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.
Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.