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Physical Review A

Diagrammatic analysis of atom–squeezed-light interactions

S. Smart, S. Swain

Published May 1, 1992• 25 citations
Publisher

Abstract

We study an atomic system interacting with a broadband squeezed vacuum and several applied classical fields. Exact rate equations are derived from the Gardiner-Collett master equation. A diagrammatic analysis is presented, which enables one to write down the contributions in a relatively simple manner and to provide a physical interpretation of the various terms. The method is applied in the following paper [Smart and Swain, Phys. Rev. A 45, 6863 (1992)] to a study of resonance fluorescence.

DOI
10.1103/physreva.45.6857

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We study an atomic system interacting with a broadband squeezed vacuum and several applied classical fields. Exact rate equations are derived from the Gardiner-Collett master equation. A diagrammatic analysis is presented, which enables one to write down the contributions in a relatively simple manner and to provide a physical interpretation of the various terms. The method is applied in the following paper [Smart and Swain, Phys. Rev. A 45, 6863 (1992)] to a study of resonance fluorescence.
We study fluorescence from a three-level atom in a cascade configuration driven by two coherent fields and damped by a broadband squeezed vacuum. Under the conditions that one or both of these fields are intense, a secular approximation is invoked to derive an analytical solution of the relevant master equation. When the two fields are in exact resonance with their respective transitions, expressions for the fluorescent spectra show a narrowing of the central peak for some specific values of the parameters of the squeezed vacuum and the phases between the driving fields and the squeezed vacuum. We also present analytical expressions for second-order intensity correlations. The inhibition of fluorescence in a squeezed vacuum is reflected in the quantum-jump-like behavior shown by these correlations.
If a broadband squeezed vacuum is treated as a reservoir with respect to a two-level atom, the non-zero phase-dependent reservoir correlation functions characterizing the squeezed vacuum introduce `squeezing terms' to the master equation. These terms are responsible, for example, for the well known narrowing of the spectral lines in the resonance fluorescence spectrum of the atom. For a squeezed vacuum reservoir with finite bandwidth it is possible to derive the master equation that is consistent with the Born-Markov approximation by a two-step procedure which consists in dressing the atom first and next coupling it to the finite-bandwidth reservoir. The master equation is valid whenever the bandwidth of the reservoir is much broader than the atomic linewidth but not necessarily broader than the Rabi frequency. This procedure can be applied not only for the squeezed vacuum reservoir but also for the ordinary vacuum with the structure of the mode density that is not flat, e.g., modelled by a Lorentzian function as in a cavity. The master equation for this case, in the operator form, shows some similarities to the master equation for the squeezed vacuum reservoir due to the presence of the `squeezing-like terms'. The similarities and differences of the two master equations are discussed in this paper.
The interaction between matter and squeezed light has mostly been treated within the approximation that the field-correlation time is small. Methods for treating squeezed light with more general correlations currently involve explicitly modeling the systems producing the light. We develop a general purpose input-output theory for a particular form of narrowband squeezed light, a squeezed wave-packet mode, that only concerns the statistics of the squeezed field and the shape of the wave packet. This formalism allows us to derive the input-output relations and the master equation. We also consider detecting the scattered field using photon counting and homodyne measurements, which necessitates the derivation of the stochastic master equation. The non-Markovian nature of the field manifests itself in the master equation as a coupled hierarchy of equations. We illustrate these with consequences for the decay and resonance fluorescence of two-level atoms in the presence of such fields.
We review the use of operator reaction field theory in quantum optics by considering the problem of one atom in an isotropic three-dimensional broad-band squeezed vacuum. In these terms we give a quantum electrodynamical analysis of this system exact up to all contributions of order one in the ratio γω 0 -1 , the ratio of the A -coefficient to the atomic resonance frequency. These terms include all of the counter-rotating terms at this order. At zero order, namely in the usual rotating wave approximation, we find the usual Einstein rate equation while we confirm that the fluorescence spectrum consists of two Lorentzian peaks. However, at order one we find an additional oscillating term in the rate equation and additional resonant dispersive terms correct for each of the two Lorentzian peaks in the fluorescence spectrum. Similar results are expected for broad-band correlated squeezed vacua of arbitrary, rather than isotropic, geometry.
A complete treatment of a three-level ladder system interacting with a broadband squeezed vacuum field is presented. It is assumed that the ground state and the second excited state are decoupled in the electric-dipole approximation, and are tuned close to the sum frequency of the incident squeezed vacuum field. Using Zwanzig's projection-operator techniques, we derive the master equation, assuming that the system interacts with a broadband squeezed vacuum field in one or more dimensions. It is shown that, in the first case, the squeezed vacuum introduces new decay constants and frequency-shift parameters. These have the same dependence on the atom-radiation coupling parameter as the ordinary vacuum decay rate and frequency shift, the only major difference being that they are multiplied by the squeezing parameters M and N. In more dimensions, the decay constants and frequency-shift parameters depend on the solid angle \ensuremath{\Omega} over which the squeezing is propagated. For \ensuremath{\Omega}=0 these correspond to the usual Einstein A coefficients and Lamb shifts of the atomic levels, while for large \ensuremath{\Omega} they are similar to those for the one-dimensional squeezed vacuum.