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The Density Matrix: Bloch Equations

Duncan G. Steel

Published Apr 30, 2021• 0 citations
Publisher

Abstract

Abstract One of the most powerful tools for calculating quantum device performance is based on the density matrix operator. The operator is unique because it is time dependent in the Schrödinger picture. The approach is quite general, but in the systems of interest here, the Hilbert space of the operator includes both the quantum system such as a nano-vibrator or two-level system and the quantized vacuum radiation field. The equation of motion follows from the time dependent Schrödinger equation. It is possible, as we show, to include the generation of spontaneous emission in this system and then, because observables of interest do not depend on the vacuum field, trace over the vacuum field to create a new density matrix called the reduced density matrix. The resulting equations of motion are the Bloch equations. We use these equations to analyze several problems involving two- and three-level systems.

DOI
10.1093/oso/9780192895073.003.0018

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Abstract One of the most powerful tools for calculating quantum device performance is based on the density matrix operator. The operator is unique because it is time dependent in the Schrödinger picture. The approach is quite general, but in the systems of interest here, the Hilbert space of the operator includes both the quantum system such as a nano-vibrator or two-level system and the quantized vacuum radiation field. The equation of motion follows from the time dependent Schrödinger equation. It is possible, as we show, to include the generation of spontaneous emission in this system and then, because observables of interest do not depend on the vacuum field, trace over the vacuum field to create a new density matrix called the reduced density matrix. The resulting equations of motion are the Bloch equations. We use these equations to analyze several problems involving two- and three-level systems.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.
The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector , which will allow us to give simple geometrical pictures of the evolution of the system.
Master equations are derived for the time evolution of density-matrix elements characterizing atoms interacting with classical radiation fields that drive transitions between electronic-state manifolds of levels. The atomic center-of-mass motion is quantized and recoil accompanying absorption, stimulated emission, and spontaneous emission is included. An adiabatic elimination of the optical coherences is used to obtain quantum rate equations for ground- and excited-state density-matrix elements. Subsequent adiabatic elimination of excited-state density-matrix elements results in rate equations for the slow evolution of ground-state density-matrix elements. It is shown that recoil during spontaneous emission mixes spatial and internal atomic degrees of freedom. \textcopyright{} 1996 The American Physical Society.
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the n=3n=3n=3 states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.
Our treatment of the quantum theory of radiation so far has predominantly involved entire systems, such as the electromagnetic field in a cavity. In most areas of quantum optics, however, we are interested in only part of the entire system. For example, in the laser problem, we want to know the field but are not particularly interested in what happens to the atoms. We find it convenient to separate the system of primary interest from that (or those) of secondary interest and to call the former simply the “system” and the latter the “res­ ervoir.” We can eliminate the reservoir by using the reduced density operator method in the Schrodinger (or interaction) picture or the noise operator method in the Heisenberg picture. In this chapter we discuss the density operator method, which provides computational convenience while stressing the statistical aspects of the problem. In Chap. 19 we use the noise operator approach, which often requires more calculation but offers a direct physical appeal in its resemblance to the classical Brownian motion problem.
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.