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Physical Review A

Photodetection and causality in quantum optics

Peter W. Milonni, Daniel F. V. James, H. Fearn

Published Aug 1, 1995• 96 citations
Publisher

Abstract

The theory of photodetection and quantum-optical coherence is formulated in the Heisenberg picture and in such a way that the causal propagation of fields at the velocity of light is manifest. Objections to the standard theory, based on a putative violation of causality, are shown to be unfounded, as is the notion that normal ordering is essential for the elimination of an infinite contribution from the vacuum field. In a similar vein we revisit the Fermi two-atom problem and explicitly demonstrate the causal nature of the interaction without invoking simplifying approximations that have recently been called into question.

DOI
10.1103/physreva.52.1525

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The theory of photodetection and quantum-optical coherence is formulated in the Heisenberg picture and in such a way that the causal propagation of fields at the velocity of light is manifest. Objections to the standard theory, based on a putative violation of causality, are shown to be unfounded, as is the notion that normal ordering is essential for the elimination of an infinite contribution from the vacuum field. In a similar vein we revisit the Fermi two-atom problem and explicitly demonstrate the causal nature of the interaction without invoking simplifying approximations that have recently been called into question.
A formal implementation of the concepts of mesoscopic electromagnetic interaction and of the propagating wave in quantum electrodynamics beyond the rotating wave approximation is discussed. Used as a guide, these concepts lead to a natural resolution of a long-standing controversy: causality violations in the Glauber–Kelley–Kleiner photodetection theory. The Glauber–Kelley–Kleiner definition of the time-normal operator ordering must be amended without the rotating wave approximation, which eliminates all causality problems.
Light in media The backbone of theoretical quantum optics is the quantum field theory of light, the quantum theory of the electromagnetic field. Quantum electrodynamics becomes surprisingly complicated if one insists on formulating it in a relativistically and gauge-invariant form, as one surely should do in elementary particle physics. Moreover, in quantum optics, we are concerned with light quanta in materials such as glass, with quantum electromagnetism in media, which is even more complicated. To cut a long story short, here we follow a minimalistic approach where we develop the essentials of quantum electrodynamics with as little technical effort as possible, sailing around the cliffs of relativistic quantum field theories. We do not assume any prior knowledge of classical field theory; we only borrow a few ideas from classical electromagnetism (Jackson, 1998). Complications and problems still remain, but hopefully these are mostly no longer formal difficulties, but rather conceptual problems of the quantum nature of light. As Dr Samuel Johnson said on poetry “We all know what light is; but it is not easy to tell what it is. ” Maxwell's equations What is light? Light is a quantum object - we describe its properties by quantum observables, Hermitian operators, and its state by a state vector | ψ _ or density matrix ρ . We use the Heisenberg picture of quantum mechanics where the operators evolve in time, but the quantum state does not change.
Conventional definition of the time-normal operator ordering (Kelley P. L. and Kleiner W. H., Phys. Rev. , 136 (1964) A316 ) is prone to causality violations (de Haan M., Physica A , 132 (1985) 375 ; 397 ). We show that such violations disappear if this definition is amended outside the rotating wave approximation. Nonrelativistic causality of an arbitrary time-normal product turns out to be a property of quantum kinematics, while relativistic causality is demonstrated for a time-normal product of two operators under the most general assumptions about quantum dynamics (commutativity of operators at space-like intervals). This eliminates the key obstacle preventing phase-space techniques of quantum optics from being extended to arbitrary quantum fields including fermions.
Introduction We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems. In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.
The idea that events obey a definite causal order is deeply rooted in our understanding of the world and at the basis of the very notion of time. But where does causal order come from, and is it a necessary property of nature? Here, we address these questions from the standpoint of quantum mechanics in a new framework for multipartite correlations that does not assume a pre-defined global causal structure but only the validity of quantum mechanics locally. All known situations that respect causal order, including space-like and time-like separated experiments, are captured by this framework in a unified way. Surprisingly, we find correlations that cannot be understood in terms of definite causal order. These correlations violate a 'causal inequality' that is satisfied by all space-like and time-like correlations. We further show that in a classical limit causal order always arises, which suggests that space-time may emerge from a more fundamental structure in a quantum-to-classical transition. Causal order is a concept that is engrained in the standard understanding of time, both in classical and quantum mechanics. Oreshkovet al.generalize the standard formalism of quantum theory to a framework with no pre-existing causal order, and find a new class of correlations that have no analogue in the classical world.
The theory of relativity is essentially the culmination of classical physics. Although revisions in our concepts of space and time are necessitated by both the special and general theories of relativity, the notion of causality, in which a cause precedes its effect, remains intact in the relativistic formulations of electrodynamics and mechanics and of gravitation. An upper limit is set on the speed of bodies, and of the propagation of energy, and physical quantities are often defined operationally, but once we adjust ourselves to these new rules, our representations of physical processes proceed pretty much as they did in classical physics. On the other hand, quantum theory seems to require a much more profound philosophical revision of our thought patterns. Even the viability of the usual meaning of a causal connection between one event and another is there called into question. SOME HISTORICAL BACKGROUND We begin with a brief and highly selective review of the experimental and theoretical problems that formed the conceptual backdrop to the emergence of quantum theory. We focus attention first on an important phenomenon – blackbody radiation – that was instrumental for the formulation of quantum mechanics. A blackbody is defined as a surface that absorbs all of the electromagnetic radiation that falls onto it. It is picturesquely so named since any light incident on such a body would be absorbed, making it appear black (at least at typical temperatures in our environment). A blackbody also emits radiation.