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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum fields in disequilibrium: Squeezed states

Mark Burgess

Published Jan 15, 1997• 5 citations
Publisher

Abstract

A spacetime description of squeezed states in quantum fields is presented, revealing the connection between squeezing and nonequilibrium dynamics. Squeezings in configuration space, occupation number space, and phase space are distinguished; generating transformations and criteria for their physical realization are discussed. The results have an immediate applicability to atoms and ions in traps, as well as to quantum optics in relativistic and nonequilibrium systems. Squeezing, sub-Poissonian statistics, and antibunching are all shown to be a direct consequence of spacetime inhomogeneities in the quantum field. The finite speed of communication between separate regions of the field (finite speed of light) places a lower limit on the attainable spectral width of squeezed states. The squeezing parameter for field quadratures has the appearance of a chemical potential in an inhomogeneous field, and through a renormalization may be generated by a Chern-Simons-like term.

DOI
10.1103/physrevd.55.951

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A spacetime description of squeezed states in quantum fields is presented, revealing the connection between squeezing and nonequilibrium dynamics. Squeezings in configuration space, occupation number space, and phase space are distinguished; generating transformations and criteria for their physical realization are discussed. The results have an immediate applicability to atoms and ions in traps, as well as to quantum optics in relativistic and nonequilibrium systems. Squeezing, sub-Poissonian statistics, and antibunching are all shown to be a direct consequence of spacetime inhomogeneities in the quantum field. The finite speed of communication between separate regions of the field (finite speed of light) places a lower limit on the attainable spectral width of squeezed states. The squeezing parameter for field quadratures has the appearance of a chemical potential in an inhomogeneous field, and through a renormalization may be generated by a Chern-Simons-like term.
The statistical fluctuations of light can, under certain circumstances, exhibit a number of interesting nonclassical features. These include photon antibunching, sub-Poissonian statistics, and squeezing, all of which exist only in a quantum field. The three effects are distinct, and do not necessarily go together, although the first two are sometimes mistakenly treated as interchangeable. Both photon antibunching and sub-Poissonian statistics have been observed in experiments on resonance fluorescence from single atoms, which is described. Squeezing has not yet been seen experimentally, although many nonlinear processes are expected to give rise to it. The squeezed state can be recognized in heterodyne or homodyne experiments, where the squeezing becomes manifest as sub-Poissonian photon fluctuations.
Relative entropy serves as a fundamental measure of state distinguishability in both quantum information theory and relativistic quantum field theory. Despite its conceptual importance, however, explicit computations of relative entropy remain notoriously difficult. Thus far, results in closed form have only been obtained for ground states, coherent states, and, more recently, single-mode squeezed states. In this work, we extend the analysis to multi-mode squeezed states, imposing that the squeezing generators be local either in space or in spacetime, which results in a continuum of squeezed modes. We provide a detailed and self-contained analysis of such states for a free scalar quantum field on Minkowski spacetime, connecting also with older results on the essential self-adjointness of the Wick square, and showing that they lie in the folium of the Minkowski vacuum representation. We uncover that local-in-space squeezing leads to physically unreasonable states which have a divergent energy density and relative entropy, while local-in-spacetime squeezing provides a physically reasonable class of states with finite relative entropy.
In this paper we discuss the nonclassical properties of quantum superpositions of coherent states of light. Using general expressions for the Wigner functions of superposition states we analyze the consequences of quantum interference between coherent states. We describe in detail nonclassical properties of a superposition of two coherent states. In particular, we study the oscillatory behavior of the photon number distribution of the even and odd coherent states. We show under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics. We examine the sensitivity of nonclassical effects such as oscillations in the photon number distribution or second-order squeezing to dissipation. We demonstrate that quantities such as the photon number distribution and interferences in phase space are highly sensitive to even a quite small dissipative coupling, because they depend on all moments of the field observables, and higher moments decay more rapidly than lower moments. Quantities such as quadrature squeezing, on the other hand, are more robust against dissipation because they involve only lower moments. Finally, we find a remarkable effect whereby fourth-order squeezing is generated by damping.
We study the statistical properties of squeezed Kerr states generated by squeezing, with a two-photon parametric process, the output of a Kerr medium whose input is coherent light. The squeezing has the effect of allowing the photon statistics to depend on the Kerr nonlinearity. Without the squeezing the statistics are Poissonian, but after squeezing they may be either sub-Poissonian or super-Poissonian. We also study the Q-function quasiprobability distribution for these states and related distributions associated with the quantum-mechanical phase. Finally, we show that quadrature squeezing can be enhanced and that squeezing and sub-Poissonian statistics can coexist.
Recently a number of laboratories have demonstrated the generation of squeezed light, a new kind of light which in some of its aspects exhibits less quantum noise then coherent light. Here a description will be given of how such light is generated and detected. With the successful generation and detection of squeezed light by a number of laboratories1-5 a new degree of control over the quantum statistics of light has been achieved. Light can now be generated which, in some of its aspects, has less quantum noise than coherent light. Such light may find applications in optical communications and in sensitive and precision measurement. Here a description will be given of how squeezed light is generated and detected.6-8 The discussion will be restricted to what is called "quadrature squeezing". The electric field operator for a single mode of the electromagnetic field can be written in the form
A squeezed state of light is usually taken to mean a state in which the fluctuations of one of the quadratures are below the level associated with a coherent or vacuum state. These squeezed fluctuations are phase sensitive in the sense that they vary periodically as the field evolves, but the relationship between squeezing and phase fluctuations is not trivial. While squeezed states can exhibit an arbitrarily low level of phase fluctuations, the phase states, which are eigenstates of the Hermitian optical phase operator are not themselves quadrature squeezed. A phasesqueezed state is one with a lower level of phase fluctuations than a coherent state with the same mean photon number. Phase squeezing is fundamentally different from quadrature squeezing in much the same way that sub-Poissonian photon statistics is different. For this reason optimally squeezed states and states with minimum phase noise will be distinct. We derive the form of the states which minimum phase noise for a given intensity. These have an Airy function behavior for the photon number probability distribution. These phase squeezed states have a far better phase resolution than coherent states of similar intensity, but do not seem to offer a significant reduction of phase fluctuations compared with conventional squeezed states.