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Physical Review A

Electronic-field correlation functions

Tomáš Tyc

Published Dec 1, 1998• 7 citations
Publisher

Abstract

In this paper the correlation function of an arbitrary order is derived for an electron field in which the components corresponding to different particle numbers and momenta are mutually uncorrelated. The result is expressed in terms of the complex degree of coherence. The correlation functions clearly demonstrate the antibunching effect. All the computational steps are shown in detail.

DOI
10.1103/physreva.58.4967

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In this paper the correlation function of an arbitrary order is derived for an electron field in which the components corresponding to different particle numbers and momenta are mutually uncorrelated. The result is expressed in terms of the complex degree of coherence. The correlation functions clearly demonstrate the antibunching effect. All the computational steps are shown in detail.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
We theoretically study the high-order correlation functions of chaotic bosons and fermions. Based on the different parity of the Stirling number, the products of the first-order correlation functions are well classified and employed to represent the high-order correlation function. The correlation of bosons conduces a bunching effect, which will be enhanced as order NNN increases. Different from bosons, the anticommutation relation of fermions leads to the parity of the Stirling number, which thereby results in a mixture of bunching and antibunching behaviors in high-order correlation. By further investigating third-order ghost diffraction and ghost imaging, the differences between the high-order correlations of bosons and fermions are discussed in detail. A larger NNN will dramatically improve the ghost image quality for bosons, but a good strategy should be carefully chosen for the fermionic ghost imaging process due to its complex correlation components.
The physical origin of the feature of strong correlation between electrons is mainly ascribed to doubly occupied atomic orbitals. Two-particle wave functions are believed necessary to describe strongly correlated systems. We emphasize that the on-site Coulomb interaction, which is important for strong correlation, should appear in equations as a natural term. Hartree-Fock-like self-consistent equations are derived with wave functions including two-particle wave functions for such systems. In the self-consistent equations, a strong correlation term is separated from the mean field. To our knowledge, this is the first attempt to formulate self-consistent equations in which the strong correlation effect is directly embodied. Some features of the equations are discussed. Koopmans' theorem and Brillouin's theorem are applied to the equations. The former is valid for singly occupied orbitals, and the latter is entirely available. By separating an angular correlation factor which is believed important for a strong correlation effect, an approximate form of a two-particle wave function is suggested. This kind of wave function causes the six multiple integrals to degrade to triple integrals. Under such forms of two-particle wave functions, corresponding Roothaan equations are established. Through these steps, it is believed that the equations become solvable within present computational limits.
Interference experiments characterized by the degree of first-order coherence (Young’s fringes, Fresnel, and Fraunhofer diffraction, etc.) have been performed with massive particles, in particular electrons and neutrons, as well as with light. Optical intensity interference or correlation experiments characterized by the degree of second-order coherence have not yet been performed with particles. The theory for such experiments shows striking differences in the behavior of electrons compared with that of light deriving from the differences in quantum statistics and charge. Photons are neutral bosons. They can be prepared in states that exhibit bunching, no bunching, or antibunching; they are unaffected by the presence of electromagnetic potentials. Electrons are charged fermions. Theory indicates that they exhibit antibunching regardless of the coherence of the source, and that their degree of second-order coherence can be influenced by electromagnetic fields through which they do not pass. A remarkable example is the Hanbury Brown-Twiss experiment with electrons that have interacted locally with the vector potential of an inaccessible magnetic field. The electron intensity correlation at two detectors is shown to be sensitive to the confined magnetic flux; experimental conditions differ considerably from those of the well-known Aharonov-Bohm effect.