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Physical Review A

Angular correlations of photons

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Published Sep 1, 1990• 7 citations
Publisher

Abstract

We study angular correlations of the radiation emitted by spatially extended sets of atoms. Such correlations result from the fact that the photons are identical quantum particles. We show that the symmetry (or the antisymmetry) alone of the wave function for identical, indistinguishable particles always leads to significant angular correlations and that in full agreement with our intuition, bosons tend to travel in the same direction while fermions tend to travel in opposite directions. In order to quantify these intuitive notions, we introduce---as a measure of angular correlations---the average value of the cosine of the angle between the directions of the momenta of two particles. The angular correlation coefficient is calculated for simple, model wave functions and the results are compared with those obtained from an exactly soluble model describing radiating harmonic oscillators. We find that even though the interaction modifies the exact values of the angular correlation coefficient, the gross features can be obtained from quantum statistics, i.e., from the symmetry of the many-photon wave functions. Moreover, the angular correlations as measured by the average cosine exhibit an almost universal character, at least for a small number of emitting atoms. Finally, we compare angular correlations obtained in quantum electrodynamics with the correlations resulting from the phenomenologically introduced randomness of the phases in the classical description of the electromagnetic field.

DOI
10.1103/physreva.42.2829

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We study angular correlations of the radiation emitted by spatially extended sets of atoms. Such correlations result from the fact that the photons are identical quantum particles. We show that the symmetry (or the antisymmetry) alone of the wave function for identical, indistinguishable particles always leads to significant angular correlations and that in full agreement with our intuition, bosons tend to travel in the same direction while fermions tend to travel in opposite directions. In order to quantify these intuitive notions, we introduce---as a measure of angular correlations---the average value of the cosine of the angle between the directions of the momenta of two particles. The angular correlation coefficient is calculated for simple, model wave functions and the results are compared with those obtained from an exactly soluble model describing radiating harmonic oscillators. We find that even though the interaction modifies the exact values of the angular correlation coefficient, the gross features can be obtained from quantum statistics, i.e., from the symmetry of the many-photon wave functions. Moreover, the angular correlations as measured by the average cosine exhibit an almost universal character, at least for a small number of emitting atoms. Finally, we compare angular correlations obtained in quantum electrodynamics with the correlations resulting from the phenomenologically introduced randomness of the phases in the classical description of the electromagnetic field.
We study the angular correlations of photons emitted spontaneously by a set of sources placed at fixed positions with the use of a model of harmonic oscillators coupled to the electromagnetic field. In our previous study of angular correlations, in which the oscillators were treated as one dimensional and their Coulomb interactions were disregarded, we found only positive angular correlations over the whole range of interatomic separations for which the model was applicable. In this paper we use the model of three-dimensional oscillators interacting with the quantized electromagnetic field and also directly among themselves via the Coulomb forces. We solve this model analytically in the dipole approximation, and we calculate the angular correlation coefficient for different separations between the emitters. For separations less than half of the wavelength, we find negative angular correlations (spatial antibunching). This is a purely quantum effect that is not present in the corresponding classical stochastic model.
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
The connection between the intrinsic angular momentum (spin) of particles and quantum statistics is established by considering the response of identical particles to a common background radiation field. For this purpose, the Hamiltonian analysis previously performed in stochastic electrodynamics to derive the quantum description of a one-particle system is extended to a system of two identical bound particles subject to the same field. Depending on the relative phase of the response of the particles to a common field mode, two types of particles are distinguished by their symmetry or antisymmetry with respect to particle exchange. While any number of identical particles responding in phase can occupy the same energy state, there can only be two particles responding in antiphase. The calculation of bipartite correlations between the response functions reveals maximum entanglement as a consequence of the parallel response of the particles to the common field. The introduction of an internal rotation parameter leads to a direct link between spin and statistics and to a physical rationale for the Pauli exclusion principle.
Abstract It is shown by a quantum-mechanical treatment that the emission times of photoelectrons at different points illuminated by a plane wave of light are partially correlated, and identical results are obtained by a classical theory in which the photocathode is regarded as a square-law detector of suitable conversion efficiency. It is argued that the phenomenon exemplifies the wave rather than the particle aspect of light and that it may most easily be interpreted as a correlation between the intensity fluctuations at different points on a wavefront which arise because of interference between different frequency components of the light. From the point of view of the corpuscular picture the interpretation is much less straight­forward but it is shown that the correlation is directly related to the so-called bunching of photons which arises because light quanta are mutually indistinguishable and obey Bose-Einstein statistics. However, it is stressed that the use of the photon concept before the light energy is actually detected is highly misleading since, in an interference experiment, the electromagnetic field behaves in a manner which cannot be explained in terms of classical particles. The quantitative predictions of the theory have been confirmed by laboratory experi­ments and the phenomenon has been used, in an interferometer, to measure the apparent angular diameter of Sirius: these results, together with further applications to astronomy, will be discussed in detail in later papers. It is shown that the classical and quantum treatments give identical results when applied to find the fluctuations in the photoemission current produced by a single light beam, and the connexion between these fluctuations and the correlation between photons in coherent beams is pointed out. The results given here are in full agreement with those obtained by Kahn from an analysis based on quantum statistics: however, they differ from those derived on thermodynamical grounds by Fellgett and by Clark Jones and the reasons for this discrepancy are discussed.
Sub- and superradiant dynamics of spontaneously decaying atoms are manifestations of collective many-body systems. We study the internal dynamics and the radiation properties of two atoms in free space. Interesting results are obtained when the atoms are separated by less than half a wavelength of the atomic transition, where the dipole-dipole interaction gives rise to new coherent effects, such as (a) coherence between two intermediate collective states, (b) oscillations in the two-photon correlation G(2){G}^{(2)}G(2), (c) emission of two photons by one atom, and (d) the loss of directional correlation. We compare the population dynamics during the two-photon emission process with the dynamics of single-photon emission in the cases of a $\ensuremath{\Lambda}$ and a V scheme. We compute the temporal correlation and angular correlation of two successively emitted photons using the G(2){G}^{(2)}G(2) for different values of atomic separation. We find antibunching when the atomic separation is a quarter wavelength $\ensuremath{\lambda}∕4$. Oscillations in the temporal correlation provide a useful feature for measuring subwavelength atomic separation. Strong directional correlation between two emitted photons is found for atomic separation larger than a wavelength. We also compare the directionality of a photon spontaneously emitted by the two atoms prepared in phased-symmetric and phased-antisymmetric entangled states ${\ensuremath{\mid}\ifmmode\pm\else\textpm\fi{}⟩}_{{\mathbf{k}}_{0}}={e}^{i{\mathbf{k}}_{0}∙{\mathbf{r}}_{1}}\ensuremath{\mid}{a}_{1},{b}_{2}⟩\ifmmode\pm\else\textpm\fi{}{e}^{i{\mathbf{k}}_{0}∙{\mathbf{r}}_{2}}\ensuremath{\mid}{b}_{1},{a}_{2}⟩$ by a laser pulse with wave vector k0{\mathbf{k}}_{0}k0​. Photon emission is directionally suppressed along k0{\mathbf{k}}_{0}k0​ for the phased-antisymmetric state. The directionality ceases for interatomic distances less than $\ensuremath{\lambda}∕2$.
A general comparison is presented between random electrodynamics and quantum electrodynamics for the two systems which can be solved exactly in both theories, free electromagnetic fields and point dipole oscillators. The NNN-point correlation functions of the fields are computed in both theories and are found to differ in general because of the dependence upon the order of the quantum operators within products of operators. However, if all products of quantum operators are symmetrized by taking all permutations of the operator order, then the two theories give identical results for the correlation functions. Analogous results hold to all orders in the fine-structure constant for dipole oscllators in quantum and random electrodynamics. The theories agree only if the quantum operator products are symmetrized. In the limit that the oscillator couplings to the radiation fields vanish, th oscllators can be regarded as mechanical oscillators in quantum mechanics and in random mechanics. The theory of random mechanics is defined in terms of this limit which uncouples a mechanical oscillator from the radiation field. The average values of oscillator variables in random mechanics agree with those of symmetrized products in quantum mechanics. The question is then raised as to the physical significance of the many quantum operators which differ only in the order of their factors. It is pointed out that some operator products which are regarded as physically important, such as the square of the angular momentum, indeed involve unsymmetrized products of operators. On this account the average values of the angular momentum squared in the ground state of an isotropic three-dimensional harmonic oscillator differ between the random-mechanical and quantum-mechanical descriptions. However, there seems to be no case in which experiments have shown that the (unsymmetrized) quantum operator value is to be preferred to that provided by random mechanics. The presence of thermal radiation is next treated for free electromagnetic fields and for dipole-oscillator systems. Despite extraordinary differences in the points of view toward thermal radiation taken by the two theories, the conclusion is the same as that found for zero temperature; the two theories agree in their average values if all products of quantum operators are symmetrized. Finally, as a further example of the power of random electrodynamics to give an account of phenomena where Lorentz's classical electron theory failed, we investigate the diamagnetism of a charged three-dimensional isotropic oscillator. The mathematical descriptions at finite temperature are developed in full random electrodynamics and quantum electrodynamics and in second-order perturbation theory in quantum mechanics.