Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins

Abstract

In this review we demonstrate how the algebraic Bethe ansatz is used for the calculation of the energy spectra and form factors (operator matrix elements in the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As examples we apply the theory to several models of current interest in the study of Bose-Einstein condensates, which have been successfully created using ultracold dilute atomic gases. The first model we introduce describes Josephson tunneling between two coupled Bose-Einstein condensates. It can be used not only for the study of tunneling between condensates of atomic gases, but for solid state Josephson junctions and coupled Cooper pair boxes. The theory is also applicable to models of atomic-molecular Bose-Einstein condensates, with two examples given and analysed. Additionally, these same two models are relevant to studies in quantum optics. Finally, we discuss the model of Bardeen, Cooper and Schrieffer in this framework, which is appropriate for systems of ultracold fermionic atomic gases, as well as being applicable for the description of superconducting correlations in metallic grains with nanoscale dimensions. In applying all of the above models to physical situations, the need for an exact analysis of small scale systems is established due to large quantum fluctuations which render mean-field approaches inaccurate.

49 pages, 1 figure, invited review for J. Phys. A., published version available at http://stacks.iop.org/JPhysA/36/R63

References in corpus (2)

• Entanglement of two-mode Bose-Einstein condensates
• Integrable coupling in a model for Josephson tunneling between non-identical BCS systems

Cited by in corpus (96)

• Quantum spin squeezing
• Fermi gases in one dimension: From Bethe Ansatz to experiments
• Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Finite-Size Scaling Exponents of the Lipkin-Meshkov-Glick Model
• Continuous unitary transformations and finite-size scaling exponents in the Lipkin-Meshkov-Glick model
• 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
• Entanglement dynamics in the Lipkin-Meshkov-Glick model
• Large N Scaling Behavior of the Lipkin-Meshkov-Glick Model
• Quantum quenches from integrability: the fermionic pairing model
• Adiabatic quantum dynamics of the Lipkin-Meshkov-Glick model
• Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence
• Quantum criticality of the Lipkin-Meshkov-Glick Model in terms of fidelity susceptibility
• Few-body Bose gases in low dimensions -- a laboratory for quantum dynamics
• Yang-Baxter integrable models in experiments: from condensed matter to ultracold atoms
• The Heine-Stieltjes correspondence and the polynomial approach to the standard pairing problem
• Integrability vs exact solvability in the quantum Rabi and Dicke models
• Ground-state properties of the attractive one-dimensional Bose-Hubbard model
• An eigenvalue-based method and determinant representations for general integrable XXZ Richardson-Gaudin models
• Bethe Ansatz approach to quench dynamics in the Richardson model
• Exactly-solvable models for atom-molecule hamiltonians
• The 1D interacting Bose gas in a hard wall box
• Exact mesoscopic correlation functions of the pairing model
• Classical and quantum dynamics of a model for atomic-molecular Bose--Einstein condensates
• Quantum phase transitions in fully connected spin models: an entanglement perspective
• Bethe Ansatz for 1D interacting anyons
• Noisy Spins and the Richardson-Gaudin Model
• Exact solution of the p+ip pairing Hamiltonian and a hierarchy of integrable models
• On the structure of typical states of a disordered Richardson model and many-body localization
• Spin polarization through Floquet resonances in a driven central spin model
• Richardson-Gaudin integrability in the contraction limit of the quasispin
• Continuous unitary transformations in two-level boson systems
• Ground state of one-dimensional bosons with delta interaction: link to the BCS model
• Quantifying ground state entanglement of the BCS model
• Solving the Richardson equations close to the critical points
• Multiparticle Entanglement in the Lipkin-Meshkov-Glick Model
• Magnetic ordering and quantum statistical effects in strongly repulsive Fermi-Fermi and Bose-Fermi mixtures
• On the determinant representations of Gaudin models' scalar products and form factors
• The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1) Richardson-Gaudin integrable models
• Exact form factors in integrable quantum field theories: the scaling Z(N)-Ising model
• The extended Heine-Stieltjes polynomials associated with a special LMG model
• Integrability of the russian doll BCS model
• Quantum phase transitions in an interacting atom-molecule boson model
• Inner products in integrable Richardson-Gaudin models
• Non-equilibrum dynamics in the strongly excited inhomogeneous Dicke model
• Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode
• Polynomial algebras and exact solutions of general quantum non-linear optical models I: Two-mode boson systems
• Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation
• Dynamics of entanglement in a dissipative Bose-Hubbard dimer
• Behaviour of the energy gap in a model of Josephson coupled Bose-Einstein condensates
• Engineering First-Order Quantum Phase Transitions for Weak Signal Detection
• Quantum phase transitions in Bose-Einstein condensates from a Bethe ansatz perspective
• The nested SU(N) off-shell Bethe ansatz and exact form factors
• Dynamical correlation functions of the mesoscopic pairing model
• Emergent quantum phases in a heteronuclear molecular Bose--Einstein condensate model
• Avoided crossings and dynamical tunneling close to excited-state quantum phase transitions
• Exactly solvable models in atomic and molecular physics
• Algebraic Bethe Ansätze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models
• The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing
• Thermodynamics, spin-charge separation and correlation functions of spin-1/2 fermions with repulsive interaction
• Quantum integrable multi-well tunneling models
• Many-body reduced fidelity susceptibility in Lipkin-Meshkov-Glick model
• Competing interactions in semiconductor quantum dots
• Quantum Inverse Scattering Method with anyonic grading
• On the Thermodynamic Limit of the Lipkin Model
• Quantum Phase Transition in Finite-Size Lipkin-Meshkov-Glick Model
• Nature of the many-body excitations in a quantum wire: theory and experiment
• Berry phase and fidelity susceptibility of the three-qubit Lipkin-Meshkov-Glick ground state
• Liouvillian skin effects and fragmented condensates in an integrable dissipative Bose-Hubbard model
• Polynomial algebras and exact solutions of general quantum non-linear optical models II: Multi-mode boson systems
• Phase-space mixing in dynamically unstable, integrable few-mode quantum systems
• Integrability of an extended d+id-wave pairing Hamiltonian
• Integrability and quench dynamics in the spin-1 central spin XX model
• BEC-BCS crossover in a p+ip-wave pairing Hamiltonian coupled to bosonic molecular pairs
• A variational method for integrability-breaking Richardson-Gaudin models
• Integrable Models From Twisted Half Loop Algebras
• SU(3) Richardson-Gaudin models: three level systems
• A family of many-body models which are exactly solvable analytically
• Quantum phase transition in a three-level atom-molecule system
• Classical and quantum analysis of a heterotriatomic molecular Bose-Einstein-condensate model
• A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons
• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• Classical-quantum correspondence in bosonic two-mode conversion systems: polynomial algebras and Kummer shapes
• Inelastic collisions in an exactly solvable two-mode Bose-Einstein Condensate
• The Elliptic Gaudin Model: a Numerical Study
• Bethe states for the two-site Bose-Hubbard model: a binomial approach
• Exactly solvable models for multiatomic molecular Bose-Einstein condensates
• Steady-states of out-of-equlibrium inhomogeneous Richardson-Gaudin quantum integrable models in quantum optics
• Quasi-Exactly Solvable Models Derived from the Quasi-Gaudin Algebra
• Energy level statistics for models of coupled single-mode Bose--Einstein condensates
• Some spectral equivalences between Schrodinger operators
• Relative Phase and Josephson Dynamics between Weakly Coupled Richardson Models
• A bosonic multi-state two-well model
• Hiking a generalized Dyck path: A tractable way of calculating multimode boson evolution operators
• Exactly solvable models for triatomic-molecular Bose-Einstein Condensates
• "Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models