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Oxford University Press eBooks

The master equation

J. C. Garrison, R. Y. Chiao

Published Jun 5, 2008• 2 citations
Publisher

Abstract

This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample on the environment is negligible compared to the action of the environment on the sample. This leads to the master equation for the (reduced) sample density operator. Photons in a cavity and a two-level atom are presented as examples. The P-function representation of the sample density operator yields the Fokker-Planck equation. This is used to show the robustness of coherent states, and to describe a driven mode in a lossy cavity. The discussion next turns to quantum jumps and their experimental observation. Quantum jumps are related to the master equations by means of the Monte Carlo wavefunction algorithm, quantum trajectories, and quantum state diffusion.

DOI
10.1093/acprof:oso/9780198508861.003.0019

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This chapter examines the evolution of an open system — the sample — with the quantum Liouville equation for the world density operator. The fundamental approximation is that the action of the sample on the environment is negligible compared to the action of the environment on the sample. This leads to the master equation for the (reduced) sample density operator. Photons in a cavity and a two-level atom are presented as examples. The P-function representation of the sample density operator yields the Fokker-Planck equation. This is used to show the robustness of coherent states, and to describe a driven mode in a lossy cavity. The discussion next turns to quantum jumps and their experimental observation. Quantum jumps are related to the master equations by means of the Monte Carlo wavefunction algorithm, quantum trajectories, and quantum state diffusion.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
In Chapter 9, we derived the master equation for a reduced density operator of a two-level atom coupled to a multi-mode vacuum field. We have seen that the master equation is an operator type equation that, in general, is not easy to solve. We have illustrated the solution of the master equation by representing the density operator in the basis of the two states of the two-level atom. The purpose of the representation of the master equation was to obtain c-number differential equations that are equivalent to the operator equations, but are more readily soluble.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
We propose physical interpretations, also valid for temperatures different from zero, for stochastic methods which have been developed recently to describe the evolution of a quantum system interacting with a reservoir. As opposed to the usual reduced density operator approach, which refers to ensemble averages, these methods deal with the dynamics of single realizations, and involve the solution of stochastic Schrödinger equations. These procedures have been shown to be completely equivalent to the master equation approach when ensemble averages are taken over many realizations. We show that these techniques are not only convenient mathematical tools for dissipative systems, but may actually correspond to concrete physical processes, for any temperature of the reservoir. We consider a mode of the electromagnetic field in a cavity interacting with a beam of two- or three-level atoms, the field mode playing the role of a small system and the atomic beam standing for a reservoir at finite temperature, the interaction between them being given by the Jaynes-Cummings model. We show that the evolution of the field states, under continuous monitoring of the state of the atoms which leave the cavity, can be described in terms of either the Monte Carlo wavefunction (quantum jump) method or a stochastic Schrödinger equation, depending on the system configuration. We also show that the Monte Carlo wavefunction approach leads, for finite temperatures, to localization into jumping Fock states, while the diffusion equation method leads to localization into states with a diffusing average photon number, which for sufficiently small temperatures are close approximations to mildly squeezed states. We prove analytically that, in the quantum jump situation, the system evolves in the mean towards a Fock state, even if an infinite number of photon-number amplitudes is present in the initial state.
We discuss Monte Carlo simulation of the quantum master equation in terms of system wave functions undergoing a sequence of quantum jumps based on the delay function for the occurrence of the “next jump”.1-3 The formalism can be applied to calculate the density operator and correlation functions (spectra) in quantum systems with a large number of degrees of freedom. We present the formalism and applications to (i) a multilevel atom undergoing spontaneous emission including mechanical light effects (in particular, laser cooling in the quantum limit), (ii) coupling of an atomic system to a squeezed vacuum, and (iii) cavity QED problems with coupling to thermal and squeezed baths.