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arXiv:2105.10515 · Quantum

Quantum-classical correspondence of a system of interacting bosons in a triple-well potential

Erick R. Castro, Jorge Chávez-Carlos, I. Roditi, Léa Ferreira dos Santos, Jorge G. Hirsch

Published Oct 19, 2021• 16 citations
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Abstract

We study the quantum-classical correspondence of an experimentally accessible system of interacting bosons in a tilted triple-well potential. With the semiclassical analysis, we get a better understanding of the different phases of the quantum system and how they could be used for quantum information science. In the integrable limits, our analysis of the stationary points of the semiclassical Hamiltonian reveals critical points associated with second-order quantum phase transitions. In the nonintegrable domain, the system exhibits crossovers. Depending on the parameters and quantities, the quantum-classical correspondence holds for very few bosons. In some parameter regions, the ground state is robust (highly sensitive) to changes in the interaction strength (tilt amplitude), which may be of use for quantum information protocols (quantum sensing).

DOI
10.22331/q-2021-10-19-563

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We study the quantum-classical correspondence of an experimentally accessible system of interacting bosons in a tilted triple-well potential. With the semiclassical analysis, we get a better understanding of the different phases of the quantum system and how they could be used for quantum information science. In the integrable limits, our analysis of the stationary points of the semiclassical Hamiltonian reveals critical points associated with second-order quantum phase transitions. In the nonintegrable domain, the system exhibits crossovers. Depending on the parameters and quantities, the quantum-classical correspondence holds for very few bosons. In some parameter regions, the ground state is robust (highly sensitive) to changes in the interaction strength (tilt amplitude), which may be of use for quantum information protocols (quantum sensing).
Systems of interacting bosons in triple-well potentials are of significant theoretical and experimental interest. They are explored in contexts that range from quantum phase transitions and quantum dynamics to semiclassical analysis. Here, we systematically investigate the onset of quantum chaos in a triple-well model that moves away from integrability as its potential gets tilted. Even in its deepest chaotic regime, the system presents features reminiscent of integrability. Our studies are based on level spacing distribution and spectral form factor, structure of the eigenstates, and diagonal and off-diagonal elements of observables in relationship to the eigenstate thermalization hypothesis. With only three sites, the system's eigenstates are at the brink of becoming fully chaotic, so they do not yet exhibit Gaussian distributions, which resonates with the results for the observables.
In this work, we investigate the semiclassical limit of a simple bosonic quantum many-body system exhibiting both integrable and chaotic behavior. A classical Hamiltonian is derived using coherent states. The transition from regularity to chaos in classical dynamics is visualized through Poincar\'e sections. Classical trajectories in phase space closely resemble the projections of the Husimi functions of eigenstates with similar energy, even in chaotic cases. It is demonstrated that this correlation is more evident when projecting the eigenstates onto the Fock states. The analysis is carried out at a critical energy where the eigenstates are maximally delocalized in the Fock basis. Despite the imperfect delocalization, its influence is present in the classical and quantum properties under investigation. The study systematically establishes quantum-classical correspondence for a bosonic many-body system with more than two wells, even within the chaotic region.
This study explores the semiclassical limit of an integrable-chaotic bosonic many-body quantum system, providing nuanced insights into its behavior. We examine classical-quantum correspondences across different interaction regimes of bosons in a triple-well potential, ranging from the integrable to the self-trapping regime, and including the chaotic one. The close resemblance between the phase-space mean projections of classical trajectories and those of Husimi distributions evokes the principle of uniform semiclassical condensation of Wigner functions of eigenstates. Notably, the resulting figures also exhibit patterns reminiscent of Jason Gallas's "shrimp" shapes.
We present an approach for the analysis of Bose-Einstein condensates in a few mode approximation. This method has already been used to successfully analyze the vibrational modes in various molecular systems and offers a perspective on the dynamics in many particle bosonic systems. We discuss a system consisting of a Bose-Einstein condensate in a triple well potential. Such systems correspond to classical Hamiltonian systems with three degrees of freedom. The semiclassical approach allows a simple visualization of the eigenstates of the quantum system referring to the underlying classical dynamics. From this classification we can read off the dynamical properties of the eigenstates such as particle exchange between the wells and entanglement without further calculations. In addition, this approach offers insights into the validity of the mean-field description of the many particle system by the Gross-Pitaevskii equation, since we make use of exactly this correspondence in our semiclassical analysis. We choose a three mode system in order to visualize it easily and, moreover, to have a sufficiently interesting structure, although the method can also be extended to higher dimensional systems.
We study the ground state properties of bosons in a tilted double-well system. We use fidelity susceptibility to identify the possible ground state transitions under different tilt values. For a very small tilt (for example 10−1010^{-10}10−10), two transitions are found. For a moderate tilt (for example 10−310^{-3}10−3), only one transition is found. For a large tilt (for example 10−110^{-1}10−1), no transition is found. We explain this by analyzing the spectrum of the ground state. The quantum discord and total correlation of the ground state under different tilts are also calculated to indicate those transitions. In the transition region, both quantities have peaks decaying exponentially with particle number NNN. This means for a finite-size system the transition region cannot be explained by the mean-field theory, but in the large-NNN limit it can be.
In this work, we study an effective three-mode Hamiltonian describing interacting bosons. These bosons can be considered as exciton-polaritons in a semiconductor microcavity at the magic angle or cold atoms optically trapped in a triple-well potential with a population imbalance conservation. We observe that the quantum properties of the spectrum characterize a quantum phase transition. We realize a thermodynamic limit of the Hamiltonian and we study the classical properties of the phase space. We observe a close analogy between quantum and classical properties: an inflection point (a level approximation) in spectrum corresponds to a separatrix (a trajectory that separates librations and rotations) in phase space. This analogy allows us to characterize the system phases, the order of the transition, and also to construct a phase diagram.