Abstract We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ‘state-dependent’ rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-) local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.
Abstract We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ‘state-dependent’ rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-) local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.
We consider a class of open quantum many-body Lindblad dynamics characterized by an all-to-all coupling Hamiltonian and by dissipation featuring collective ``state-dependent" rates. The latter encodes local incoherent transitions that depend on average properties of the system. This type of open quantum dynamics can be seen as a generalization of classical (mean-field) stochastic Markov dynamics, in which transitions depend on the instantaneous configuration of the system, to the quantum domain. We study the time evolution in the limit of infinitely large systems, and we demonstrate the exactness of the mean-field equations for the dynamics of average operators. We further derive the effective dynamical generator governing the time evolution of (quasi-)local operators. Our results allow for a rigorous and systematic investigation of the impact of quantum effects on paradigmatic classical models, such as quantum generalized Hopfield associative memories or (mean-field) kinetically-constrained models.
Abstract We consider a class of open quantum many-body systems that evolves in a Markovian fashion, the dynamical generator being in GKS-Lindblad form. Here, the Hamiltonian contribution is characterized by an all-to-all coupling, and the dissipation features local transitions that depend on collective, operator-valued rates, encoding average properties of the system. These types of generators can be formally obtained by generalizing, to the quantum realm, classical (mean-field) stochastic Markov dynamics, with state-dependent transitions. Focusing on the dynamics emerging in the limit of infinitely large systems, we build on the exactness of the mean-field equations for the dynamics of average operators. In this framework, we derive the dynamics of quantum fluctuation operators, that can be used in turn to understand the fate of quantum correlations in the system. We then apply our results to quantum generalized Hopfield associative memories. Here we show that, asymptotically and at the description level of quantum fluctuations, only a very weak amount of quantum correlations, in the form of quantum discord, emerges beyond classical correlations.
Abstract We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.
We consider quantum particles coupled to local and collective thermal quantum environments. The coupling is energy conserving, and the collective coupling is scaled in the mean-field way. There is no direct interaction between the particles. We show that an initially factorized state of the particles remains factorized at all times, in the limit of large particle number. Each single-particle factor evolves according to an explicit, nonlinear, dissipative and time-dependent Hartree–Lindblad equation. The model is exactly solvable; we do not make any weak coupling or any Markovian approximations, and our results are mathematically rigorous.
This work is about a quantum-mechanical generalisation of transition rate theory that accounts for temperature-dependent quantum-dynamical effects. The starting point of this investigation is a formulation of the classical transmission coefficient where the activated dynamics is needed to obtain the time evolution of the classical state function and the rate is expressed as an average flux in momentum space. Traditional approaches to quantum transition rate theory are briefly discussed, with emphasis on the fact that these do not generally require knowledge of the open quantum dynamics. The latter can be studied by means of the Lindblad equation for mixed states, which is shown to be analogous to an average over realisations of a certain type of stochastic Schrödinger dynamics. The stochastic evolution in Hilbert space is used in combination with an ansatz for the wave function as a Gaussian wave packet to show that under appropriate assumptions the stochastic Schrödinger equation converges to the classical Langevin equation. A novel formulation of the quantum transmission coefficient is presented which features a quantum-mechanical version of the state function obtained via the density operator, so that the activated quantum dynamics can be taken into account by solving the Lindblad equation. A numerical investigation of the quantum transmission coefficient thus obtained for different temperatures is carried out, along with a comparison to the classical results. The quantum-mechanical transmission coefficient is found to display some expected features, such as vanishing in the limit as t → 0, displaying a “plateau” value for certain values of the temperature, and decaying to zero exponentially fast when the fundamental assumptions of TRT do not hold. Additionally the plateau value of the quantum transmission coefficient is found to be smaller than the classical counterpart in all regimes, and to approach the corresponding classical value in the limit of large temperatures.
The dynamics of a quantum system coupled to a classical environment and subject to constraints that drive it out of equilibrium are described. The evolution of the system is governed by the quantum-classical Liouville equation. Rather than evaluating the evolution of the mixed quantum-classical density operator, we derive exact equations of motion for the nonequilibrium average values of a set of operators or variables, along with correlation function expressions for the dissipative coefficients that enter these equations. These equations are obtained by requiring that the exact nonequilibrium averages are equal to local nonequilibrium averages that depend on auxiliary fields whose values satisfy evolution equations obtained using projection operator methods. The results are illustrated by deriving reaction-diffusion equations coupled to fluid hydrodynamic equations for a solution of quantum particles that can exist in two metastable states. Nonequilibrium steady states are discussed along with the reaction rate and diffusion correlation functions that characterize such states.