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The quantum mechanical density operator and its time evolution

Abraham Nitzan

Published Jun 14, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1093/9780191947971.003.0010

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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.
Typically, we are interested in a small system (a few particles, or some region in space), entangled with its surroundings. Exact equations (Dyson or Nakajima–Zwanzig) for the reduced system dynamics are readily derived using suitable projection operators. They are rarely solvable, however, since full account is taken for mutual influences between the system and its environment. Nevertheless, for weak mutual coupling, environment-induced dynamics in the (small) system can be much slower than system-induced dynamics in the (large) environment. This justifies the Born–Markov approximation, leading to closed equations for the system. In Hilbert space, we demonstrate the emergence of irreversible dynamics and exponential decay for pure sates. In Liouville space, we derive the Redfield equation. Invoking the secular (or, rotating-wave) approximation, we derive Pauli’s master equations, which properly account for relaxation to equilibrium. Rates of spontaneous emission, coherence transfer (Bloch equations), and pure dephasing are derived and analyzed for a dissipative qubit.
Abstract In this chapter we discuss statistical mixtures of pure quantum states, resulting in so-called mixed states, and their representation using density operators. The properties of density operators for pure and mixed states are analysed and we examine their time evolution governed by the Liouville-von Neumann equation. As particular examples of interest we consider the density operators for thermal states, focusing on harmonic oscillators and the application of this model to calculate the average photon number for the modes of the electromagnetic field at thermal equilibrium. We then turn to two-level systems, or qubits, which are of particular relevance in quantum information theory and we inspect the geometry of the state space of density operators as a convex subspace of the Hilbert-Schmidt space. Finally, we take a closer look at density operators for bipartite systems which forms the mathematical basis for considering entanglement in later chapters.
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
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
Abstract This chapter is dedicated to open quantum systems. We explain the origin of non-unitary time evolution by studying different interactions between a qubit and the environment. The first part of the chapter introduces the time evolution equation of an open quantum system, also called the Born-Markov master equation and the Redfield equation. We then study the interaction between a qubit and a thermal bath of harmonic oscillators and introduce the qubit relaxation and the dephasing times. After discussing the relationship between relaxation and dephasing times of a qubit and the power spectral density of the noise source, we present the Hahn echo and the Carr-Purcell-Meiboom-Gill experiments, two protocols that provide information about the noise spectrum. The chapter ends with a discussion of the driven evolution of a qubit subject to spontaneous emission and dephasing
We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.