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arXiv:1611.01003

The Commutation Relation for Cavity Mode Operators

Fesseha Kassahun

Published Nov 3, 2016• 6 citations
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Abstract

We have obtained the commutation relation for a cavity mode driven by coherent light and interacting with a two-level atom. We have found that the commutation relation for a free cavity mode with or without photons is just the usual commutation relation. However, the commutation relation for a cavity mode, which is interacting with a two-level atom, turns out to be different from the usual one.

DOI
10.48550/arxiv.1611.01003

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We have obtained the commutation relation for a cavity mode driven by coherent light and interacting with a two-level atom. We have found that the commutation relation for a free cavity mode with or without photons is just the usual commutation relation. However, the commutation relation for a cavity mode, which is interacting with a two-level atom, turns out to be different from the usual one.
We propose an experimental scheme to prove the photon commutation relation, [ a , a † ]=1. The scheme exploits interaction between a single-mode cavity field and three two-level atoms, two of which are prepared in an entangled state. The success of the scheme is subject to observation of the atoms in predetermined states after the interaction. We show, in particular, that a reasonably high success probability can be obtained with our scheme by preparing the cavity field in a superposition of two Fock states and choosing the interaction times appropriately.
The dynamics of few two-level atoms interacting with one cavity mode is described in master equation formalism. Two different configurations are considered: a transient one with all atoms initially in the excited state and a stationary one where a coherent pump acts on either the atoms or the cavity mode. Superradiance, resonant scattering and switching between a low and a high emission state are considered. Cooperation and decoherence as well as the statistics of the emitted radiation field in both cases are discussed.
The interaction between a squeezed vacuum and a two-level atom coupled to a cavity mode is studied via its influence on the intracavity photon number, the simplest measurable quantity. When the system is driven by a coherent field a remarkable sensitivity to the relative phase between the driving field and the squeezed vacuum appears. The effect is very sensible to the excess of two-photon correlation between a squeezed vacuum and a maximally correlated classical field. The possibility of an experimental demonstration of the effect is established by means of numerical calculations corresponding to a slight modification of an experiment previously reported in the literature.
We investigate a cavity mode that interacts with a three-level atom in an open cavity that is coupled to a vacuum reservoir and is driven by coherent light. In our analysis, the interaction of the three-level atom with the external vacuum reservoir is taken into account and the noise operators of the vacuum reservoir are placed in normal order. While the spontaneous emission rate fluctuates as [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], the cavity damping constant and stimulated emission decay constant, both set to 0.8, govern the system dynamics. Both the plus and negative components of the cavity light display quadrature squeezing. For coherent driving strengths [Formula: see text] and [Formula: see text] at [Formula: see text], and [Formula: see text] at [Formula: see text], the plus quadrature exhibits [Formula: see text] squeezing below the vacuum level, indicating that increasing the coherent drive can compensate for the decoherence introduced by spontaneous emission. Moreover, for [Formula: see text], the minus quadrature achieves [Formula: see text] squeezing. At high driving strengths, the squeezing in both quadratures becomes independent of spontaneous emission. In the absence of spontaneous emission, the mean photon number increases, and the photon number distribution shows a higher probability for even photon numbers than for odd ones, with the overall distribution decreasing as the photon number increases.