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

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 for the complete set of eigenmodes is given by a set of non-linear coupled equations. For N = 2 (SU (2)) we recover the exact solution of Phys. Rev. Lett. 122, 010401 (2019). 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.

Dynamics of fluctuations in quantum simple exclusion processes

“…In this sector and for two replicas, the on-site representation is the [6] representation of sl( 4) or so (6), that of rank 4 anti-symmetric tensors of sl( 4) or that of 6-dimensional vectors of so (6). We have the decomposition rule, [20] as sl( 4) or so( 6) modules, with [15] the rank two antisymmetric so (6) tensors and [20] the rank two traceless symmetric so( 6) tensors. The projectors on these representations are (with d = 6)…”

“…The simplest such class of Lindblad equations can be mapped to imaginary-time Schrödinger equations with non-Hermitian "Hamiltonians" that are quadratic in creation/annihilation operators [8][9][10][11]. More recently it has been shown that there exist Lindblad equations whose evolution operators are related to interacting integrable quantum spin chains [12][13][14][15][16][17][18][19][20][21][22][23][24]. This opens the door to obtaining exact results on the average dissipative dynamics of many-particle Lindblad equations by employing methods from quantum integrability.…”

Exact solution of a quantum asymmetric exclusion process with particle creation and annihilation

“…Another class of solvable Lindblad equations are "triangular" models which add particle loss and dephasing terms to otherwise number conserving integrable models [31,32,33]. Recently a new direction for constructing solvable many particle Lindblad equations was identified through the discovery of Lindblad equations that can be related to interacting Yang-Baxter integrable models [34,35,36,37,38,32,33,39,40,41,42]. The approach of Refs [34,38] is based on a superoperator representation of the Lindblad equations, which gives rise to solvable "two-leg ladder" quantum spin chain models.…”

Exceptional spectral phase in a dissipative collective spin model

“…This system is a particular limit of the Richardson-Gaudin model and has been solved exactly in Refs. [25,26]. It is relevant for the study of quantum cavity electrodynamics [27,28], magnetic grains on a metallic surface [24], or superconducting quantum circuits [29].…”