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Journal of Modern Optics

Squeezing in Correlated Quantum Systems

Stephen Mark Barnett, P. L. Knight

Published Jun 1, 1987• 103 citations
Publisher

Abstract

We discuss the connection between quantum correlations and squeezing in simple quantum optical systems. We illustrate this connection by a study of two-mode states of light produced by parametric down-conversion and similar two-photon processes. The intermode correlations in these systems are shown to be responsible for modifications in photon-number sum and difference operators, and for squeezing in the superpositions of the two modes. The disappearance of the diagonal coherent-state quasiprobability function P(α) when non-classical light properties are important is noted, and alternative and better-behaved Wigner functions and coherent-state expectation Q-functions for the two-mode system are developed.

DOI
10.1080/09500348714550781

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Physica Scripta•May 8, 2012•0 citations
We discuss the connection between quantum correlations and squeezing in simple quantum optical systems. We illustrate this connection by a study of two-mode states of light produced by parametric down-conversion and similar two-photon processes. The intermode correlations in these systems are shown to be responsible for modifications in photon-number sum and difference operators, and for squeezing in the superpositions of the two modes. The disappearance of the diagonal coherent-state quasiprobability function P(α) when non-classical light properties are important is noted, and alternative and better-behaved Wigner functions and coherent-state expectation Q-functions for the two-mode system are developed.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.
The properties of states generated by the application of the two-mode squeeze operator to the pair coherent states are studied. These states are the two-mode analogues of the single-mode squeezed states generated by the application of the single-mode squeeze operator to an ordinary coherent state. In the present case there are correlations between the modes and strong non-classical properties are to be expected. We study the statistical properties of the photon number distributions, squeezing, violations of the Cauchy-Schwartz inequality, quasiprobability distributions and the phase distributions.
Squeezing the quantum fluctuations of the two-mode light due to the nonstationary coupling between the quadrature components q1{q}_{1}q1​ and q2{q}_{2}q2​ is examined. The \ensuremath{\delta}- and step-function mode couplings are considered. The conditions of weak and strong step-function coupling are distinguished, the latter being the condition of the instability for the classical counterpart of the quantum system under study. Under the conditions of weak coupling the quadrature squeezing is established in both a two-mode electromagnetic noise in thermal equilibrium (the thermal state) and a two-mode correlated coherent state (CCS). Squeezing in the thermal state is suppressed at a high temperature. The photon distribution function (PDF) in the thermal state reveals oscillatory behavior for both high and low temperature, the oscillations decrease while the temperature increases. The PDF in the CCS can be either oscillatory or smooth, whereas the photon statistics is essentially non-Poissonian. The nonclassical intermode photon-number correlations in the CCS are briefly studied.
In this paper we introduce a new squeeze operator which is a combination of the two single photon squeezing operator and two mode correlated operator. We construct the general coherent-state wave function, and study the effects of this operator on the photon number sum and difference. We find in absence of the squeezing parameters λ1 and λ2 both the sum and the difference are conserved by the two mode correlated operator. The squeezing phenomenon is discussed, and the minimum uncertainty state is found for small value of the squeezing parameter λ. Besides the Glauber second order correlation function, the statistical investigations are carried out for the quasi-probability distribution functions (P-representation, Wigner function and Q-function). The discussion related to the phase space distribution function is given, and the matrix elements of the density operator are calculated. The photocount distribution is also considered.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
In this paper, we introduce a modified class of correlated coherent states. These states are realized during the consideration of a new model which represents the interaction between a two-level atom and two electromagnetic fields injected simultaneously within a cavity. By using the transition amplitude we have managed to establish the relationship between the usual coherent states and the modified states. The phenomenon of squeezing and the variances of the photon number sum and difference are discussed. It has been shown that the mean photon number controls the value of the correlation. Furthermore, our discussion on the quasi-probability distribution functions (the Wigner and Q-functions) shows that both functions are sensitive to the change in the ratio of the field modes as well as the coupling parameter λ1.