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arXiv:2203.01211

Multi-Quantum Dark Solitons in One-Dimensional Bose Gas

Yuki Ishiguro, Jun Sato, Katsuhiro Nishinari

Published Mar 2, 2022• 0 citations
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Abstract

Quantum and classical integrable systems share common mathematical structures, and the phenomena appearing in them are interrelated. Solitons, which universally appear in classical integrable systems, also appear in quantum integrable systems. Here, we consider quantum-classical correspondence in a one-dimensional Bose gas with repulsive delta-function interaction and present quantum states corresponding to multi-dark solitons. Using an exact method, we compute the time evolution of the density profile in the multi-quantum dark soliton states. Localized solitary waves that behave like classical dark solitons are observed in the density profile. We observe collisions of quantum dark solitons and show that they exhibit the properties of classical solitons: stability against scatterings and position shifts due to interactions.

DOI
10.48550/arxiv.2203.01211

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Quantum and classical integrable systems share common mathematical structures, and the phenomena appearing in them are interrelated. Solitons, which universally appear in classical integrable systems, also appear in quantum integrable systems. Here, we consider quantum-classical correspondence in a one-dimensional Bose gas with repulsive delta-function interaction and present quantum states corresponding to multi-dark solitons. Using an exact method, we compute the time evolution of the density profile in the multi-quantum dark soliton states. Localized solitary waves that behave like classical dark solitons are observed in the density profile. We observe collisions of quantum dark solitons and show that they exhibit the properties of classical solitons: stability against scatterings and position shifts due to interactions.
A soliton is a class of solitary-wave solutions having specific mathematical properties, which universally appears in integrable systems.A quantum dark soliton is an emerging concept having theoretical and experimental relevance.Although recent studies have proposed quantum states with density profiles similar to those of classical dark solitons, the concept of quantum dark solitons is not yet fully established in that their collision dynamics have not been examined.Here, we first construct dark soliton states that possess stable scattering properties that are unique to solitons using the Lieb-Liniger model, establishing a direct link between quantum and classical dark soliton theories.
Abstract We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross–Pitaevskii equation) for the one-dimensional Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.
Abstract Dark soliton solutions in the one-dimensional classical nonlinear Schrödinger equation have been considered to be related to the yrast states corresponding to the type-II excitations in the Lieb–Liniger model. However, the relation is nontrivial and remains unclear because a dark soliton localized in space breaks the translation symmetry, while yrast states are translationally invariant. In this work, we construct a symmetry-broken quantum soliton state and investigate the relation to the yrast states. By interpreting a quantum dark soliton as a Bose–Einstein condensation to the wave function of a classical dark soliton, we find that the quantum soliton state has a large weight only on the yrast states, which is analytically proved in the free-boson limit and numerically verified in the weak-coupling regime. By extending these results, we derive a parameter-free expression of a quantum soliton state that is written as a superposition of yrast states with Gaussian weights. The density profile of this quantum soliton state excellently agrees to that of the classical dark soliton. The dynamics of a quantum dark soliton are also studied, and it turns out that the density profile of a dark soliton decays, but the decay time increases as the inverse of the coupling constant in the weak-coupling limit.
Dark and gray soliton-like states are shown to emerge from numerically constructed superpositions of translationally invariant eigenstates of the interacting Bose gas in a toroidal trap. The exact quantum many-body dynamics reveals a density depression with ballistic spreading that is absent in classical solitons. A simple theory based on finite-size bound states of holes with quantum-mechanical center-of-mass motion quantitatively explains the time-evolution and predicts quantum effects that could be observed in ultracold gas experiments. The soliton phase step is found relevant for explaining finite-size effects in numerical simulations. An invariant fundamental soliton width is shown to deviate from the Gross-Pitaevskii predictions in the interacting regime and vanishes in the Tonks-Girardeau limit.
We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.