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Cambridge University Press eBooks

Complement 5B: One-photon wave packet

G. Grynberg, Alain Aspect, Claude Fabre, Claude N. Cohen-Tannoudji

Published Sep 2, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511778261.026

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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.
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.
Interference effects produced by independent quantum sources are investigated, when the state of the field is not describable as a simple mixture of coherent states. The results of classical and quantum-mechanical calculations are compared. Whereas correlation effects are predicted both classically and quantum mechanically when the sources have random phases, there are important differences when the number of atoms is small. In particular, when each source consists of just one atom, the joint probability of detecting two photons at two different points in the receiving plane is found to vanish when the distance between them is an odd number of half fringes. Finally it is shown that when the number of atoms of each source is subject to Poisson fluctuations, one recovers the solution given by classical optics for thermal light, no matter how weak the sources may be on the average.
For around five decades, physicists have been experimenting with single quanta such as single photons. Insofar as the practised ensemble reasoning has become obsolete for the interpretation of these experiments, the non-classical intrinsic probabilistic nature of quantum theory has gained increased importance. One of the most important exclusive features of quantum physics is the undeniable existence of the superposition of states, even for single quantum objects. One known example of this effect is entanglement. In this paper, two classically contradictory phenomena are combined to one single experiment. This experiment incontestably shows that a single photon incident on an optical beam splitter can either be reflected or transmitted. The almost complete absence of coincident clicks of two photodetectors demonstrates that these two output states are incompatible. However, when combining these states using two mirrors, we can observe interference patterns in the counting rate of the single photon detector. The only explanation for this is that the two incompatible output states are prepared and kept simultaneously—a typical consequence of a quantum superposition of states. (Semi-)classical physical concepts fail here, and a full quantum concept is predestined to explain the complementary experimental outcomes for the quantum optical “non-waves” called single photons. In this paper, we intend to demonstrate that a true quantum physical key experiment (“true” in the sense that it cannot be explained by any classical physical concept), when combined with full quantum reasoning (probability, superposition and interference), influences students’ readiness to use quantum elements for interpretation.
How can we construct a picture of the photon from the wealth of observation material available to us? The photon appears to have a split personality: it is neither a wave nor a particle but something else which, depending on the experimental situation, exhibits a wave- or a particle-like behavior. In other words, in the photon (as in material particles such as the electron) the particle–wave dualism becomes manifest. Whereas classically the wave and the particle pictures are separate, quantum mechanics accomplishes a formal synthesis through a unified mathematical treatment. Let us look first at the wave aspect familiar from classical electrodynamics, which seems to be the most natural description. It makes all the different interference phenomena understandable, such as the “interference of the photon with itself” on the one hand and the appearance of spatial and temporal intensity correlations in a thermal radiation field on the other (which are obviously brought about by superposition of elementary waves emitted independently from different atoms). It might come as a surprise (at least for those having quantum mechanical preconceptions) that the classical theory is valid down to arbitrarily small intensities: the visibility of the interference pattern does not deteriorate even for very small intensities – the zero point fluctuations of the electromagnetic field advocated by quantum mechanics do not have a disturbing effect – and is valid not only for conventional interference experiments but also for interference between independently generated light beams (in the form of laser light).
Within the framework of a two-dimensional microscopic, purely quantum mechanical model, we analyse the dynamics of single-photon wave packets interacting with optical elements (beam splitters, mirrors), modelled as systems of two-level atoms. That is, we utilize a two-dimensional cavity to simulate the quantum behaviour of simple optical components and networks made thereof. The field is quantized using the canonical procedure, and only the basis states with one unit of excitation are included. This, however, covers linear optical phenomena. The field is taken to interact with localized atoms through a dipole interaction. Using different configurations of atoms, and choosing their frequencies to be resonant or off-resonance, we can model mirrors, beam splitters, focusing devices and multicomponent systems. Thus we can model arbitrary linear networks of optical components. We show the time evolution of a photon wave packet in an interferometer as an example. As the state of the field is known at each instant, spectral properties and spatial coherence can immediately be obtained from the simulations. We also know the states of the two-level atoms constituting the components, which allows us to consider their quantum behaviour. Here the decay of an excited atom into the vacuum state of the electromagnetic field in the two-dimensional cavity is studied.
The description of photon-matter interaction upon control, transmission, and detection of single-photon, two-photon, and multiphoton states, including the entangled ones, will play an ever-increasing role in many areas of photonics. An appropriate description requires taking into consideration various types of interference effects associated with these states. However, the relatively complex apparatus of second quantization of the electromagnetic field is used even in the simplest single-photon experiments equivalent to the Young’s one, e.g., the experiments with the Mach–Zehnder interferometer. In the present work, the Young’s single-photon mental experiment is explained using the coordinate photon wave function (PWF). The explanation is illustrated by specific examples of the single-photon states at certain wavelengths and different duration of radiation within the framework of two approaches: the quantum mechanical and the “quasi-classical.” In the first approach, a 6-component coordinate PWF is constructed based on the spherically symmetric momentum distribution in a wave packet, followed by approximate analytical calculations. In the second approach, a one-component quasi-classical PWF corresponding to the electric-dipole radiation is constructed. The same pronounced interference pattern was obtained in both cases, which makes is possible to draw the conclusion that the coordinate PWF allows explaining the one- and two-photon interference phenomena. This conclusion sheds the light on theoretical interpretation of the measurement of the coordinate PWF carried out in some of the recent experiments.