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Journal of the Optical Society of America B

Space–time description of squeezing

Zofia Białynicka-Birula, Iwo Białynicki‐Birula

Published Oct 1, 1987• 44 citations
Publisher

Abstract

We present a field-theoretical description of the squeezed states of the electromagnetic field. A definition of the squeezed state is introduced that is a natural generalization of the definition for one or two modes. We show that the squeezing produced by the medium with space- and time-dependent material coefficients ∊ and μ is directly related to the photon-pair production. In Appendix A we describe the time evolution of the Gaussian squeezed states in terms of the field variables.

DOI
10.1364/josab.4.001621

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Communications in Theoretical Physics•Mar 1, 2017•5 citations
We present a field-theoretical description of the squeezed states of the electromagnetic field. A definition of the squeezed state is introduced that is a natural generalization of the definition for one or two modes. We show that the squeezing produced by the medium with space- and time-dependent material coefficients ∊ and μ is directly related to the photon-pair production. In Appendix A we describe the time evolution of the Gaussian squeezed states in terms of the field variables.
This article contains a review of an alternative theory of squeezing, based entirely on the wave function description of the squeezed states. Quantum field theoretic approach is used to describe the squeezing of the electromagnetic field in its most complete form that takes into account temporal and spatial characteristics of the squeezed state. An analog of the Wigner function for the full electromagnetic field is introduced and expressed in terms of second order correlation functions. The field-theoretic approach enables one to study the propagation of the ``squeezing wave'' in space-time. A simple example of weak squeezing, that allows for all calculations to be done analytically, is discussed in detail.
In Section 11.5 we showed that, although the complex amplitude of the electromagnetic field has a well-defined value in any coherent state, yet the real and imaginary (Hermitian and anti-Hermitian) parts of the field fluctuate with equal dispersions. The phenomenon of vacuum fluctuations is a manifestation of this effect, because the vacuum state is an example of a particular coherent state. This behavior is quite different from that of an ordinary, classical field. In a squeezed state, which is even more non-classical, as we shall see, one part of the field fluctuates less and another part fluctuates more than in the vacuum state. In general, a squeezed state is one in which the distribution of canonical variables over the phase space has been distorted or ‘squeezed’ in such a way that the dispersion of one variable is reduced at the cost of an increase in the dispersion of the other variable. In the following we shall examine the properties of squeezed states when the two canonical variables are two quadratures of the electromagnetic field. Although the squeezing terminology is sometimes applied to variables other than the two field quadratures, it is less meaningful in those cases. A number of review articles on squeezing have been published and can be consulted for more details (Walls, 1983; Schumaker, 1986; Loudon and Knight, 1987; Teich and Saleh, 1989, 1990; Kimble, 1992).
It is found that two-mode output quantum electromagnetic field in two-mode squeezed states exhibits higher-order squeezing to all even orders. And the generalized uncertainty relations are also presented for the first time. The concept of higher-order squeezing of the single-mode quantum electromagnetic field was first introduced and applied to several processes by Hong and Mandel in 1985. Lately Li Xizeng and Shan Ying have calculated the higher-order squeezing in the process of degenerate four-wave mixing and presented the higher-order uncertainty relations of the fields in single-mode squeezed states. In this paper we generalize the above work to the higher-order squeezing in two-mode squeezed states. The generalized uncertainty relations are also presented for the first time.
We showed in Section 5.2 of the present chapter that according to quantum theory the value of the electric field of an electromagnetic wave could not be predicted to arbitrarily high precision, this being a consequence of the uncertainty relation satisfied by the two quadrature components of the field (Equation 5.32), which imposed a finite limit on the product of their variances. This limit is more than an abstract theoretical limit; it is often the overriding factor determining the resolution of high-precision optical measurements. In such situations measurements by photodetectors on the field exhibit uncontrollable fluctuations of quantum origin known as quantum noise . Fortunately, as we shall show in this complement, it is possible to overcome this by using non-classical states of the radiation field, namely the squeezed states , provided measurements are made of a single one of a pair of conjugate field variables. A field mode prepared in such a state exhibits reduced quantum noise in one of the variables at the expense of increased noise in the other, so that the uncertainty relation involving their variances is still satisfied. The potential of such states for increasing the obtainable precision of optical measurements is self-evident. However, the practical realization of the potential benefits requires delicate experimental techniques that we describe only briefly here.
Abstract Recently, the quantum description of electromagnetic waves in conducting media has been performed. It has been demonstrated that in particular case, the Hamiltonian of the corresponding field can be expressed by Caldirola–Kanai Hamiltonian. In this paper, using the associated annihilation and creation operators of the above-mentioned quantized field, the time- and conductivity-dependent squeezed vacuum and one-photon squeezed states as well as their superpositions, and also the time- and conductivity-dependent excited even and odd coherent states are produced. Also, using a few well-known nonclassicality criteria, the time evolution of nonclassicality features of the above classes of obtained states, in addition to the influence of medium conductivity on them are demonstrated, numerically. It has been shown that the nonclassicality indicators may be adjusted by tuning the conductivity of media.