Exact solution of the two-axis countertwisting Hamiltonian

Abstract

It is shown that the two-axis countertwisting Hamiltonian is exactly solvable when the quantum number of the total angular momentum of the system is an integer after the Jordan-Schwinger (differential) boson realization of the SU(2) algebra. Algebraic Bethe ansatz is used to get the exact solution with the help of the SU(1,1) algebraic structure, from which a set of Bethe ansatz equations of the problem is derived. It is shown that solutions of the Bethe ansatz equations can be obtained as zeros of the Heine-Stieltjes polynomials. The total number of the four sets of the zeros equals exactly to $2J+1$ for a given integer angular momentum quantum number $J$, which proves the completeness of the solutions. It is also shown that double degeneracy in level energies may also occur in the $J\to \infty$ limit for integer $J$ case except a unique non-degenerate level with zero excitation energy.

arXiv commentLaTex 10 pages. Version to appear in Annals of Physics

References in corpus (20)

• Quantum information with continuous variables
• Quantum spin squeezing
• Squeezing and entanglement in a Bose-Einstein condensate
• Implementation of Cavity Squeezing of a Collective Atomic Spin
• Entanglement and Extreme Spin Squeezing
• Fisher Information and entanglement of non-Gaussian spin states
• Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit
• States of an Ensemble of Two-Level Atoms with Reduced Quantum Uncertainty
• Spin-Nematic Squeezed Vacuum in a Quantum Gas
• Exact spectrum of the Lipkin-Meshkov-Glick model in the thermodynamic limit and finite-size corrections
• The Thermodynamical Limit of the Lipkin-Meshkov-Glick Model
• Exactly-Solvable Models Derived from a Generalized Gaudin Algebra
• Spin squeezing of a cold atomic ensemble with the nuclear spin of one-half
• The Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem
• The Heine-Stieltjes correspondence and a new angular momentum projection for many-particle systems
• The extended Heine-Stieltjes polynomials associated with a special LMG model
• Exact solutions for a family of spin-boson systems
• Spin squeezing and entanglement via finite-dimensional discrete phase-space description
• The Quadratic Spin Squeezing Operators
• Analytical solvability of the two-axis countertwisting spin squeezing Hamiltonian

Cited by in corpus (5)

• Inner products in integrable Richardson-Gaudin models
• Exact solution of the two-axis countertwisting Hamiltonian for the half-integer $J$ case
• Generating accessible entanglement in bosons via pair-correlated tunneling
• Exact solution of the two-axis two-spin Hamiltonian
• Polynomial deformations of $sl(2)$ and unified algebraic framework for solutions of a class of spin models