Propagation of squeezed radiation through amplifying or absorbing random media

Squeezed radiation is in a state in which one of the quadratures of the electric field fluctuates less than the other [1,2]. Such a nonclassical state is useful, because the fluctuations in the photon flux can be reduced below that of a Poisson process — at the expense of enhanced fluctuations in the phase. Sub-Poissonian noise is a delicate feature of the radiation, it is easily destroyed by the interaction with an absorbing or amplifying medium [3]. The noise from spontaneous emission events is responsible for the degradation of the squeezing. Because of the fundamental and practical importance, there exists a considerable literature on the propagation of squeezed and other nonclassical states of light through absorbing or amplifying media. We cite some of the most recent papers on this topic [4–10]. The main simplification of these investigations is the restriction to systems in which the scattering is one-dimensional, such as parallel dielectric layers. Each propagating mode can then be treated separately from any other mode. It is the purpose of the present paper to remove this restriction, by presenting a general theory for three-dimensional scattering, and to apply it to a medium with randomly located scattering centra. Our work builds on a previous paper [11], in which we considered the propagation of a coherent state through such a random medium. Physically, the problem considered here is different because a coherent state has Poisson noise, so that the specific nonclassical features of squeezed radiation do not arise in Ref. [11]. Technically, the difference is that a squeezed state, as most other nonclassical states, lacks a diagonal representation in terms of coherent states [1,2]. We cannot therefore directly extend the theory of Ref. [11] to the propagation of squeezed states. The basic idea of our approach remains the same: The photodetection statistics of the transmitted radiation is related to that of the incident radiation by means of the scattering matrix of the medium. The method of random-matrix theory [12] is then used to evaluate the noise properties of the transmitted radiation, averaged over an ensemble of random media with different positions of the scatterers. The outline of this paper is as follows. In Sec. II we first summarise the scattering formalism, and then show how the characteristic function of the state of the transmitted radiation can be obtained from that of the incident state. This allows us to compute the photocount statistics as measured in direct detection (Sec. III), and in homodyne photodetection measurements (Sec. IV). The expressions in Secs. II–IV are generally valid for any incident state. In Sec. V we specialise to the case that the incident radiation is in an ideal squeezed state (also known as a squeezed state of minimal uncertainty, or as a two-photon coherent state [1,2]). The statistics of direct and homodyne measurements are expressed in terms of the degree of squeezing of the incident state. The Fano factor, introduced in Sec. VI, quantifies the degree to which the squeezing has been destroyed by the propagation through an amplifying or absorbing medium. The ensemble average of the Fano factor is then computed using random-matrix theory in Sec. VII. We conclude in Sec. VIII.