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Physical Review A

Conditioned homodyne detection at the single-photon level: Intensity-field correlations for a two-level atom in an optical parametric oscillator

C. Elliot Strimbu, Joseph Leach, Perry R. Rice

Published Jan 10, 2005• 14 citations
Publisher

Abstract

We consider intensity-field correlation functions for a two-level atom in a degenerate optical parametric oscillator (OPO), which would result from a conditioned homodyne measurement. Analytic results are obtained in the limit of weak driving fields using quantum trajectory methods for both the transmitted and fluorescent fields. This system is unique in that after detection of a photon, it is known that one excitation is in the system, in either the atom or cavity mode. We find large violations of inequalities satisfied by classical fields, for both transmitted and fluorescent fields. This is in contrast to the usual cavity QED system of an atom in a driven cavity where we do not find nonclassical behavior in the intensity-field correlation function in fluorescence. This is understood in terms of a relationship between the intensity-field correlation function and the second-order intensity correlation function, as well as the different amount of field-atom entanglement in the two systems. We show that for weak-field cavity QED one must have photon bunching to have nonclassical behavior in the intensity-field correlation function. We compare our results to those of an ordinary OPO. Finally, we also consider cross correlations, where we examine the transmitted (fluorescent) field conditioned by detection of a fluorescent (transmitted) photon.

DOI
10.1103/physreva.71.013807

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We consider intensity-field correlation functions for a two-level atom in a degenerate optical parametric oscillator (OPO), which would result from a conditioned homodyne measurement. Analytic results are obtained in the limit of weak driving fields using quantum trajectory methods for both the transmitted and fluorescent fields. This system is unique in that after detection of a photon, it is known that one excitation is in the system, in either the atom or cavity mode. We find large violations of inequalities satisfied by classical fields, for both transmitted and fluorescent fields. This is in contrast to the usual cavity QED system of an atom in a driven cavity where we do not find nonclassical behavior in the intensity-field correlation function in fluorescence. This is understood in terms of a relationship between the intensity-field correlation function and the second-order intensity correlation function, as well as the different amount of field-atom entanglement in the two systems. We show that for weak-field cavity QED one must have photon bunching to have nonclassical behavior in the intensity-field correlation function. We compare our results to those of an ordinary OPO. Finally, we also consider cross correlations, where we examine the transmitted (fluorescent) field conditioned by detection of a fluorescent (transmitted) photon.
We consider intensity-field correlation functions for a two-level atom in a degenerate optical parametric oscillator (OPO). Analytic results are obtained in the limit of weak driving fields using quantum trajectory methods. We find large violations of inequalities satisfied by classical fields. We compare our results to those of an ordinary OPO. We also calculate the second-order intensity correlation function for transmitted and fluorescent light, as well as for cross correlations. Here we report large violations of inequalities for the intensity cross-correlation functions in the weak field limit.
We consider two systems in the weak driving field limit. The first is an optical parametric oscillator, the transmitted field of which is highly bunched. We discuss a conditioned measurement of the second order intensity correlation function that yields antibunching. The second system we consider is a single two level atom inside a weakly driven cavity. Here we consider cross correlations between the transmitted and fluorescent fields. A strong asymmetry in delay time has been noted by Denisov et. al., here we find strong violations of a classical inequality for the cross correlation.
We investigate intensity cross-correlation functions for two cavity QED systems. These are a driven optical cavity containing a single two-level atom interacting with a single mode of the cavity field with quantized centre of mass motion, and a two-level atom in an optical parametric oscillator. We find analytic results in the weak driving field limit using quantum trajectory theory. We find large violations of inequalities that must be satisfied by classical fields. One of these inequalities is well known, g (2) ij (τ) ≤ √g (2) jj (0)g (2) jj (0), where i and j denote two modes of the field. We also derive a new inequality that cross-correlations must satisfy, |g (2) ij (τ) - 1| 2 ≤ |(g (2) ii (0) - 1)(g (2) jj (0) - 1)|. Large violations of classical inequalities and asymmetrical behaviour in delay time τ are found in complimentary regimes for the cavity QED system with quantized centre of mass motion. They always exist for the two-level atom inside an optical parametric oscillator.
We investigate the quantum fluctuations of a single atom in a weakly driven cavity, with an intracavity optical lattice. The weak driving field is on resonance with the atoms and the cavity, and is the second-harmonic of the lattice beam. In this special case we can find eigenstates of the Hamiltonian in terms of Mathieu functions.We present analytic results for the second order intensity correlation function g(2)(τ)g^{(2)}(τ)g(2)(τ) and the intensity-field correlation function hθ(τ)h_θ(τ)hθ​(τ), for both transmitted and fluorescent light for weak driving fields. We find that the coupling of the center of mass motion to the intracavity field mode can be deleterious to nonclassical effects in photon statistics; less so for the intensity-field correlations, and compare the use of trapped atoms in a cavity to atomic beams.
We consider an optical cavity containing a single two-level atom driven on resonance by an external laser source.1 We focus on the limit of weak excitation where the dynamic response of this system is governed by just three parameters: the atom-field coupling constant g, spontaneous emission rate γ for the atom, and cavity decay rate κ. The statistics of the transmitted light are analyzed as a function of g, γ, and κ, in terms of the second-order correlation function g(2)(τ), and the quadrature variances measured in a homodyne detection scheme. Squeezing and photon antibunching exist over a wide range of parameters. For γ ~ κ, the system can exhibit an oscillatory response, even when the mean intracavity photon number is much less than unity.2 The oscillations can be understood in terms of the coupling between the free atom-field eigenstates |0,−>, |0,+>, and |1,−>. The oscillation frequency is determined by the energy level splitting produced between the degenerate one-quantum states |0,+> and |1,−> by the atom-field interaction. The second-order correlation function can exhibit a novel nonclassical effect, where, for g(2)(0) ≠ 0, at some finite delay τ0, the correlation function dips exactly to zero—g(2)(τ0) = 0.