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Journal of the Optical Society of America B

Spatial fluctuation measurement in the time region

Eiichi Hanamura

Published Jan 1, 1986• 0 citations
OpenAlex

Abstract

We develop a microscopic theory for the photon echo signal which is written in terms of a two-particle correlation function of the dipole operator. In the absence of long-range spatial correlations in the material system, this correlation function may be factorized into the product of two single-particle correlation functions in agreement with a conventional result.

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Physical Review A•Sep 26, 2008•2 citations
We develop a microscopic theory for the photon echo signal which is written in terms of a two-particle correlation function of the dipole operator. In the absence of long-range spatial correlations in the material system, this correlation function may be factorized into the product of two single-particle correlation functions in agreement with a conventional result.
A microscopic correlation-function expression for the photon-echo signal in systems with long-range spatial correlations is developed. In general, the signal is given by an eight-point correlation function of the dipole operator involving two particles. In the absence of long-range spatial correlations, this correlation function may be factorized into a product of two four-point, single-particle correlation functions. Only under these conditions is the signal proportional to the absolute square of the ensemble-averaged nonlinear polarization. We predict that photon echoes may be used to probe the long-range spatial correlations near critical points.
A microscopic correlation function expression for the signal in four-wave mixing is developed. Both stationary and transient (i.e., photon-echo) experiments with coherent or incoherent radiation fields are considered. In general, these observables are described by an eight-point correlation function of the dipole operator involving two particles, and the signal cannot be written as an amplitude squared. Conditions under which the correlation function may be factorized into a product of two four-point, single-particle correlation functions are specified. In this case the conventional approach, based on calculating a nonlinear polarization and substituting it as a source in Maxwell's equations, is justified. However, in the presence of long-range spatial correlations in the sample (e.g., near critical points) the conventional formulation breaks down and the present theory should be used.
A microscopic correlation-function theory for coherent, parametric, four-photon processes (four-wave mixing) and for single-absorber two-photon processes (two-photon absorption, Raman, and fluorescence spectra) is developed. Both types of experimental observables are expressed in terms of four-point correlation functions of the dipole operator. The latter are then evaluated semiclassically or within the factorization approximation in which they are expressed in terms of ordinary single-photon line-shape functions. The common results of the Bloch equations are shown to be the outcome of two successive approximations: the factorization approximation and the impact limit whereby the single-photon line-shape functions are taken to be Lorentzian. Effects of fluctuations in the radiation fields are easily incorporated within the present approach.
It is shown that any Bose-Einstein-condensed fluid in its ground state will exhibit macroscopic single-particle quantum behavior (MSPQB). That is, (1) the many-particle wave function $\ensuremath{\Psi}({\mathbf{r}}_{1},\dots{},{\mathbf{r}}_{n})$ factors into a single-particle product ${\ensuremath{\prod}}_{n}\ensuremath{\eta}({\mathbf{r}}_{n})$; (2) the function $\ensuremath{\eta}(\mathbf{r})$ extends over macroscopic length scales and obeys the usual quantum equations for particle flux in a single-particle system; and (3) $\ensuremath{\eta}(\mathbf{r})$ obeys a nonlinear single-particle Schr\"odinger equation. The latter equation reduces to the Gross-Pitaevskii equation when interactions are weak and determines the density distribution of the fluid and the time development of this distribution. The arguments used rely only on elementary concepts of probability theory and many-particle wave mechanics and are valid even in strongly interacting fluids such as superfluid 4He^{4}\mathrm{He}4He. It is shown that Bose-Einstein condensation implies that the NNN-particle wave function $\ensuremath{\Psi}$ is delocalized. That is, if one considers a single-particle coordinate r\mathbf{r}r, then for all values that occur of the other $N\ensuremath{-}1$ coordinates, $\ensuremath{\Psi}$ is a nonzero function of r\mathbf{r}r over a region of space proportional to VVV, where VVV is the total volume within which the fluid is contained. MSPQB is a consequence of this delocalization and the absence of long-range correlations between particle positions in fluids. The results are accurate provided that only averages over regions of space containing many particles are considered. For averages over volumes of space containing ${N}_{\ensuremath{\Omega}}$ particles, inaccuracies due to quantum fluctuations are $\ensuremath{\sim}1∕\sqrt{{N}_{\ensuremath{\Omega}}}$.