Energy level splitting for weakly interacting bosons in a harmonic trap
arXiv:1903.04974 · doi:10.1103/PhysRevA.100.023605
Abstract
We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap. When the interactions are turned off, the energy levels are equidistant and highly degenerate. At linear order in the coupling parameter, these degenerate levels split, and we study the patterns of this splitting. It turns out that the problem is mathematically identical to diagonalizing the quantum resonant system of the two-dimensional Gross-Pitaevskii equation, whose classical counterpart has been previously studied in the mathematical literature on turbulence. Our purpose is to explore the implications of the symmetries and energy bounds of this resonant system, previously studied for the classical case, for the quantum level splitting. Simplifications in computing the splitting spectrum numerically result from exploiting the symmetries. The highest energy state emanating from each unperturbed level is explicitly described by our analytics. We furthermore discuss the energy level spacing distributions in the spirit of quantum chaos theory. After separating the eigenvalues into blocks with respect to the known conservation laws, we observe the Wigner-Dyson statistics within specific large blocks, which leaves little room for further integrable structures in the problem beyond the symmetries that are already explicitly known.
arXiv commentv2: improved published version
References in corpus (36)
- From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
- Random Matrix Theories in Quantum Physics: Common Concepts
- On weakly turbulent instability of anti-de Sitter space
- Breathing Modes and Hidden Symmetry of Trapped Atoms in 2D
- The unitary gas in an isotropic harmonic trap: symmetry properties and applications
- Holographic Thermalization, stability of AdS, and the Fermi-Pasta-Ulam-Tsingou paradox
- Renormalization group, secular term resummation and AdS (in)stability
- Power law violation of the area law in quantum spin chains
- Renormalization, averaging, conservation laws and AdS (in)stability
- Resonant dynamics and the instability of anti-de Sitter spacetime
- Critical nonlinear Schroedinger equations with and without harmonic potential
- Distinguishability, degeneracy and correlations in three harmonically trapped bosons in one-dimension
- Symmetries of Three Harmonically-Trapped Particles in One Dimension
- Quantum Theory in AdS and its CFT Dual
- A pseudoconformal compactification of the nonlinear Schrödinger equation and applications
- Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates
- On the Cubic Lowest Landau Level Equation
- AdS (in)stability: an analytic approach
- Unfolding of the Spectrum for Chaotic and Mixed Systems
- Conformal flow on and weak field integrability in AdS
- Quantum correlations and degeneracy of identical bosons in a 2D harmonic trap
- The two-body problem of ultra-cold atoms in a harmonic trap
- Exactly solvable model of two trapped quantum particles interacting via finite-range soft-core interactions
- Two infinite families of resonant solutions for the Gross-Pitaevskii equation
- Solvable cubic resonant systems
- Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes
- Quantum Gross-Pitaevskii Equation
- Maximally rotating waves in AdS and on spheres
- A nonrelativistic limit for AdS perturbations
- Quantum resonant systems, integrable and chaotic
- Complex plane representations and stationary states in cubic and quintic resonant systems
- Harmonically Trapped Four-Boson System
- On the continuous resonant equation for NLS: I. Deterministic analysis
- Area Law Violations and Quantum Phase Transitions in Modified Motzkin Walk Spin Chains
- Spectroscopy instead of scattering: particle experimentation in AdS spacetime
- Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping
Cited by in corpus (10)
- Bounds on quantum evolution complexity via lattice cryptography
- Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
- Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
- Time-periodic quantum states of weakly interacting bosons in a harmonic trap
- Complex plane representations and stationary states in cubic and quintic resonant systems
- Time-periodicities in holographic CFTs
- Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
- A protocol to characterize errors in quantum simulation of many-body physics
- Phase-space localization at the lowest Landau level
- A superintegrable quantum field theory