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Cambridge University Press eBooks

Open quantum systems

Howard M. Wiseman, Gerard J Milburn

Published Nov 12, 2009• 1 citation
Publisher

Abstract

Introduction As discussed in Chapter 1, to understand the general evolution, conditioned and unconditioned, of a quantum system, it is necessary to consider coupling it to a second quantum system. In the case in which the second system is much larger than the first, it is often referred to as a bath, reservoir or environment , and the first system is called an open system . The study of open quantum systems is important to quantum measurement for two reasons. First, all real systems are open to some extent, and the larger a system is, the more important its coupling to its environment will be. For a macroscopic system, such coupling leads to very rapid decoherence . Roughly, this term means the irreversible loss of quantum coherence, that is the conversion of a quantum superposition into a classical mixture. This process is central to understanding the emergence of classical behaviour and ameliorating, if not solving, the so-called quantum measurement problem . The second reason why open quantum systems are important is in the context of generalized quantum measurement theory as introduced in Chapter 1. Recall from there that, by coupling a quantum system to an ‘apparatus’ (a second quantum system) and then measuring the apparatus, a generalized measurement on the system is realized. For an open quantum system, the coupling to the environment is typically continuous (present at all times).

DOI
10.1017/cbo9780511813948.004

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Introduction As discussed in Chapter 1, to understand the general evolution, conditioned and unconditioned, of a quantum system, it is necessary to consider coupling it to a second quantum system. In the case in which the second system is much larger than the first, it is often referred to as a bath, reservoir or environment , and the first system is called an open system . The study of open quantum systems is important to quantum measurement for two reasons. First, all real systems are open to some extent, and the larger a system is, the more important its coupling to its environment will be. For a macroscopic system, such coupling leads to very rapid decoherence . Roughly, this term means the irreversible loss of quantum coherence, that is the conversion of a quantum superposition into a classical mixture. This process is central to understanding the emergence of classical behaviour and ameliorating, if not solving, the so-called quantum measurement problem . The second reason why open quantum systems are important is in the context of generalized quantum measurement theory as introduced in Chapter 1. Recall from there that, by coupling a quantum system to an ‘apparatus’ (a second quantum system) and then measuring the apparatus, a generalized measurement on the system is realized. For an open quantum system, the coupling to the environment is typically continuous (present at all times).
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).
We now turn to the subject of thermodynamics, and to describing the process of thermalization of quantum systems. A quantum system that interacts with another system large enough to obey the usual rules of thermodynamics, is referred to as an open system. The thermal system with which it interacts is often referred to as a thermal bath , or simply as the environment. We will use these terms interchangeably. Modeling the behavior of open quantum systems has many applications. One of these is to model the noise that they are subjected to by their surroundings, noise that we will wish to control using measurement and feedback loops in Chapter 5. There are also two important connections between quantum measurements, information theory, and thermodynamics. The first of these is that the thermodynamic entropy of a macroscopic system can be identified with the information-theoretic entropy that characterizes the lack of knowledge of an observer about the microstates of the system. In turn, if an observer is allowed to make microscopic measurements on the system, then this modifies the thermodynamic rules regarding the conversion of heat to work. Such is the subject of Maxwell’s demon. Curiously it took more than a century to understand how Maxwell’s demon, essentially a microscopic measurement and feedback process, fits within the second law of thermodynamics. In another twist, it turns out that the process of erasing information requires an increase in the thermodynamic entropy of the environment, and this increase is directly related to the amount of information erased.
The interaction of a quantum system with a many degree of freedom environment entails damping and the loss of quantum coherence. In these two lectures we focus on two aspects of open quantum system dynamics. First, we present decoherence as a dynamical phenomenon and determine the corresponding time scales, discuss an experiment and the nature of robust states, and mention universal properties of short-time decoherence of macroscopic superpositions. Secondly, in analogy to the classical description of open systems in terms of Langevin equations, we present a stochastic approach to open quantum system dynamics in terms of a stochastic Schrödinger equation. We give a microscopic derivation that allows us to generalize the stochastic approach to non-Markovian situations. We discuss a convolutionless formulation of our theory and finally, we focus on the quantised version of Brownian motion as an often-employed example.
The interaction of a quantum system with a many degree o ff reedom environment entails damping and the loss of quantum coherence. In these two lectures we focus on two aspects of open quantum system dynamics. First, we present decoherence as a dynamical phenomenon and determine the corresponding time scales, discuss an experiment and the nature of robust states, and mention universal properties of short- time decoherence of macroscopic superpositions. Secondly, in analogy to the classical description of open systems in terms of Langevin equations, we present a stochastic approach to open quantum system dynamics in terms of a stochastic Schrodinger equation. We give a microscopic ¨ derivation that allows us to generalize the stochastic approach to non- Markovian situations. We discuss a convolutionless formulation of our theory and finally, we focus on the quantised version of Brownian motion as an often-employed example.
The process of measurement plays an important role in quantum theory. Measurements can be direct, if the physical system S interacts directly with the measurement apparatus M (as we have assumed until now), or indirect. In the latter case, the physical system first interacts with an ancillary system B , which may have a space of states that is very different from that of S , for instance much larger; after this interaction has finished, M is used to perform a measurement on B , without any direct interaction with A . Because S is then “protected” from any direct interaction with the measurement apparatus, the state of S is not necessarily strongly modified, and may even be only weakly affected. In both cases, the process implies entanglement between several physical systems. In this chapter, we study how this entanglement is created and used for measurements as well as the notion of weak and continuous measurements. These questions play an important role in several of the interpretations of quantum mechanics that we discuss in Chapter 10. Direct measurements The Von Neumann model of quantum measurement [4] provides a general frame for describing the process in terms of correlations appearing (or disappearing) in the state vector associated with the whole system S + M . In this model, the two systems S and M are initially described by a product state ∣ψ 0 〈 and interact during the time of measurement, so that they become entangled; they then reach a final state ∣ψ′〈 and do not interact anymore.
Chapter 3 explores open quantum systems, emphasizing their interactions with environments, unlike isolated closed systems. It introduces the concept of generalized measurements and mixed quantum states, reflecting the complex scenarios arising from these interactions. The chapter utilizes Positive Operator Valued Measures (POVMs) to describe generalized measurements, broadening the conventional approach to quantum measurements. A significant focus is on the evolution of open systems through quantum channels, which illustrate the transfer or transformation of quantum information amid noise and external disturbances. This section underpins the dynamics open systems exhibit, critical for understanding quantum computing and information processing in realistic settings. Through practical examples, the chapter elucidates how environmental factors influence quantum information, vital for applications in quantum technologies. It aims to equip readers with foundational knowledge of open quantum systems, highlighting their importance in the broader context of quantum mechanics.