Master equation

About: Master equation is a research topic. Over the lifetime, 10541 publications have been published within this topic receiving 276095 citations.

Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps

Abstract: Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must have essentially different properties. Theorem 2 dictates a condition under which, in the ``semiclassical'' limit, LEPs and HEPs recover the same properties. In particular, we show the validity of Theorem 1 studying systems which have (1) an LEP but no HEPs and (2) both LEPs and HEPs but for shifted parameters. As for Theorem 2, (3) we show that these two types of EPs become essentially equivalent in the semiclassical limit. We introduce a series of mathematical techniques to unveil analogies and differences between the HEPs and LEPs. We analytically compare LEPs and HEPs for some quantum and semiclassical prototype models with loss and gain.

Quantum stochastic processes as models for state vector reduction

Abstract: An elementary introduction of quantum-state-valued Markovian stochastic processes (QSP) for N-state quantum systems is given. It is pointed out that a so-called master constraint must be fulfilled. For a given master equation a continuous and, as a new alternative possibility, a discontinuous QSP are derived. Both are discussed as possible models for state reduction during measurement.

Markovian Master Equations: A Critical Study

Abstract: We derive Markovian master equations of single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding Markovian master equations with exact numerical simulations of the evolution of the global system, we precisely delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced dynamics. The proposed method is sufficiently general to suggest that the conclusions made here are widely applicable to a large class of settings involving interacting chains subject to a weak interaction with an environment.

Variational Neural-Network Ansatz for Steady States in Open Quantum Systems

Abstract: We present a general variational approach to determine the steady state of open quantum lattice systems via a neural-network approach. The steady-state density matrix of the lattice system is constructed via a purified neural-network Ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof of principle, we apply the method to the dissipative quantum transverse Ising model.

Global weak solutions to compressible navier-stokes equations for quantum fluids

Abstract: The global-in-time existence of weak solutions to the barotropic compressible quantum Navier–Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Mehats [Derivation of viscous correction terms for the isothermal quantum Euler model, 2009, submitted] from a Wigner equation using a moment method and a Chapman–Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier–Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak s...