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arXiv:2005.05155 · Journal of Physics A Mathematical and Theoretical

Trigonometric SU ( N ) Richardson–Gaudin models and dissipative multi-level atomic systems

Sergio Lerma-Hernández, Álvaro Rubio-García, J. Dukelsky

Published Jul 31, 2020• 8 citations
arXivHTMLPDF

Abstract

Abstract We derive the exact solution of a system of N -level atoms in contact with a Markovian reservoir. The resulting Liouvillian expressed in a vectorized basis is mapped to an SU ( N ) trigonometric Richardson–Gaudin model whose exact solution is given by a set of non-linear coupled equations. For N = 2 ( SU (2)) we recover the exact solution of (2019 Phys. Rev. Lett. 122 010401). We then study the SU (3) case for three-level atom systems and discuss the properties of the steady state and dissipative gaps for finite systems as well as for the thermodynamic limit.

DOI
10.1088/1751-8121/abab54

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Abstract We derive the exact solution of a system of N -level atoms in contact with a Markovian reservoir. The resulting Liouvillian expressed in a vectorized basis is mapped to an SU ( N ) trigonometric Richardson–Gaudin model whose exact solution is given by a set of non-linear coupled equations. For N = 2 ( SU (2)) we recover the exact solution of (2019 Phys. Rev. Lett. 122 010401). We then study the SU (3) case for three-level atom systems and discuss the properties of the steady state and dissipative gaps for finite systems as well as for the thermodynamic limit.
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In specific open systems with collective dissipation the Liouvillian can be mapped to a non-Hermitian Hamiltonian. We here consider such a system where the Liouvillian is mapped to an XXZ Richardson-Gaudin integrable model and detail its exact Bethe ansatz solution. While no longer Hermitian, the Hamiltonian is pseudo-Hermitian/PT\text{pseudo-Hermitian}/\mathcal{P}\mathcal{T}pseudo-Hermitian/PT symmetric, and as the strength of the coupling to the environment is increased, the spectrum in a fixed symmetry sector changes from a broken pseudo-Hermitian phase with complex conjugate eigenvalues to a pseudo-Hermitian phase with real eigenvalues, passing through a series of exceptional points and associated dissipative quantum phase transitions. The homogeneous limit supports a nontrivial steady state, and away from this limit this state gives rise to a slow logarithmic growth of the decay rate (spectral gap) with system size. Using the exact solution, it is furthermore shown how at large coupling strengths the ratio of the imaginary to the real part of the eigenvalues becomes approximately quantized in the remaining symmetry sectors.
We determine the exact asymptotic many-body wavefunction of a spin-sss Richardson-Gaudin model with a coupling inversely proportional to time, for time evolution starting from the ground state at t=0+t = 0^+t=0+ and for arbitrary sss. Contrary to common belief, the resulting wavefunction cannot be derived from the spin-1/21/21/2 case by merging spins, but instead requires independent treatment for each spin size. The steady state is non-thermal and, in contrast to the spin-1/21/21/2 case, does not conform to a natural Generalized Gibbs Ensemble. We show that mean-field theory is exact for any product of a finite number of spin operators on different sites. We discuss how these findings can be probed in cavity QED and trapped ion experiments.
An open quantum system with multiple levels coupled to a bosonic environment at zero temperature is investigated systematically using the non-Markovian quantum-state-diffusion (QSD) method [W. T. Strunz, L. Di\'osi, and N. Gisin, Phys. Rev. Lett. 82, 1801 (1999)]. We have established exact time-local QSD equations for a set of interesting multilevel open systems, including high-spin systems, multiple-transition atomic models, and multilevel atomic models driven by time-dependent external fields. These exact QSD equations have paved the way to evaluate the dynamics of open multilevel atomic systems in the general non-Markovian regimes without any approximation.