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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

The photon wave function

Ashok Muthukrishnan, Marlan O. Scully, Muhammad Suhail Zubairy

Published Aug 1, 2005• 7 citations
Publisher

Abstract

We review and sharpen the concept of a photon wave function based on the quantum theory of light. We argue that a point-like atom serves as the archetype for both the creation and detection of photons. Spontaneous emission from atoms provides a spatially localized source of photon states that serves as a natural wave packet basis for quantum states of light. Photodetection theory allows us to give operational meaning to the photon wave function which, for single photons, is analogous to the electric field in classical wave optics. Entanglement between photons, and the uniquely quantum phenomena that result from it, are exemplified by two-photon wave functions.

DOI
10.1117/12.623373

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We review and sharpen the concept of a photon wave function based on the quantum theory of light. We argue that a point-like atom serves as the archetype for both the creation and detection of photons. Spontaneous emission from atoms provides a spatially localized source of photon states that serves as a natural wave packet basis for quantum states of light. Photodetection theory allows us to give operational meaning to the photon wave function which, for single photons, is analogous to the electric field in classical wave optics. Entanglement between photons, and the uniquely quantum phenomena that result from it, are exemplified by two-photon wave functions.
Spontaneous emission of a photon by an atom is described theoretically in three dimensions with the initial wave function of a finite-mass atom taken in the form of a finite-size wave packet. Recoil and wave-packet spreading are taken into account. The total atom-photon wave function is found in the momentum and coordinate representations as the solution of an initial-value problem. The atom-photon entanglement arising in such a process is shown to be closely related to the structure of atom and photon wave packets which can be measured in the coincidence and single-particle schemes of measurements. Two predicted effects, arising under the conditions of high entanglement, are anomalous narrowing of the coincidence wave packets and, under different conditions, anomalous broadening of the single-particle wave packets. Fundamental symmetry relations between the photon and atom single-particle and coincidence wave-packet widths are established. The relationship with the famous scenario of Einstein-Podolsky-Rosen is discussed.
One-photon sources are important elements in quantum optics. The archetypal example is an atom raised to an excited state at time t = 0, then de-exciting with emission of a single photon. The development of this kind of source depends on progress with experimental techniques, e.g. the possibility of isolating a single atom, molecule or quantum well. In this complement, we present the formalism for describing the corresponding radiation, and use it to discuss some spectacular experiments which bring out properties quite incompatible with a classical description of the electromagnetic field. We begin in Section 5B.2 by describing the anti-correlation between detections on either side of a semi-reflecting mirror, establishing the quantitative difference with a classical field. Section 5B.3 discusses a quantum optical effect that was only demonstrated at the beginning of the twenty-first century, namely the quantum coalescence of two one-photon wave packets on a semi-reflecting mirror, which occurs even when the two photons were emitted by independent atoms. An analogous effect, the Hong–Hou–Mandel effect, is discussed in Chapter 7. These effects exemplify quantum interference involving two photons. Finally, Section 5B.4 is concerned with quantum calculations involving quasi-classical states. As we now know, this leads to results that are identical to the predictions of semi-classical theory.
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).
Abstract The main provisions of the construction of the single-particle wave function of a photon in the coordinate representation are expounded and its application to the description of single-photon interference phenomena is discussed. The connection between the concepts of electromagnetic field strengths used in quantum electrodynamics and the characteristics of wave packets in photon quantum mechanics is substantiated. For illustration, we consider the relationship between the structure of the photon wave function in coordinate representation and the polarization of photon in the formalism of quantum transition amplitudes. The meaning of the concept of linear polarization of a photon in the state of a wave packet is explained. The wave function is constructed in the coordinate representation of the state of two entangled photons at the exit from a nonlinear crystal by modeling the polarization vectors and Gaussian momentum distribution for each of the photons.
The usefulness of the photon concept has been debated since its original proposal by Einstein in 1905. The photon picture has evolved via the “fuzzy-ball” and stochastic to the entangled, intrinsically quantum mechanical entity. However, in spite of the recent progress, questions about the nature of light still arise and often cause confusion from misinterpretations of various light-matter interaction phenomena. Questions are frequently asked about the existence of a `wave function for the photon’ and about the photon localization. Many light-matter interactions may be described using semiclassical theory and the question is when and where the quantum theory of radiation is needed to provide the best description? Here, we extend the discussion of these issues from several previous works on this subject. We use the definition of the photon wave function described by Scully and coworkers to discuss the photon detection in the context of single atoms and “giant atom” ensembles. We review the role of virtual photons, and discuss the effects of cooperative virtual processes on light harvesting efficiency of photodetectors and photosynthetic biological systems. Noise-induced quantum coherence may increase the performance of these quantum heat engines and the efficiency of photon harvesting.