Many-particle Sudarshan-Lindblad equation: mean-field approximation, nonlinearity and dissipation in a spin system

Abstract

A system of N spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system S interacting with a thermal reservoir R. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of N −1 equations (N > 1). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of S), although the superoperator responsible for the dissipation and decoherence of S remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected. That means that by changing t by −t, reverting the velocities, and replacing the initial conditions in the solution of the equation by the final ones does not lead us to the initial conditions after a time t has elapsed.

Cooperative spontaneous emission from indistinguishable atoms in arbitrary motional quantum states

“…However, as N gets large, this one can be neglected in comparison to ( 51) and ( 52) provided γ = 0 and N − 1 can be replaced by N . Equation (50) can then be related for pure states σ ψ (t) = |ψ(t) ψ(t)| to a non-linear Schrödinger equation for |ψ(t) of the form (in the interaction picture) [54,56,57]…”

Traffic models and traffic-jam transition in quantum (N+1)-level systems

“…Because of that, the master equation approach usually relies on the mean field approximation that reduces the hierarchical set of equations to a closed set of N < D nonlinear equations. For instance, the evolution equations for the expectation values of an operator acting on site j depend only on single site operator expectation values [56], i.e.…”

Lindblad dissipative dynamics in the presence of phase coexistence

“…In this work we shall address right this issue in the prototype quantum first order phase transition displayed by a fully connected quantum Ising model with p-spin exchange, where p > 2. Quantum spin models, besides being paradigmatic systems for studying quantum phase transitions, also constitute a good playground to investigate, both theoretically and experimentally, the driven dissipative dynamics [16][17][18][19][20][21][22][23][24] , including its realization in pertinently designed quantum impurity models realized at junctions of spin chains [25][26][27][28][29] . We shall model the dissipative dynamics of our case study in the framework of Markovian dynamics, through the rather general master equation derived by Lindblad back in 1976 30,31 , but still widely used [32][33][34][35][36][37][38][39] .…”