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Physical Review A

Relationship between semiclassical and quantum-mechanical input-output theories of optical response

ARTUR K. EKERT, P. L. Knight

Published Apr 1, 1991• 66 citations
Publisher

Abstract

We discuss the relationship between fully quantized and semiclassical approaches to input-output theory that connect asymptotically free fields prior to and following interactions that we describe in terms of canonical transformations. We demonstrate the utility of the Wigner function in describing the connection between input and output fields, and show how it is particularly well suited to describe many quantum-optical systems in terms of positive quasiprobability permitting classical stochastic simulation.

DOI
10.1103/physreva.43.3934

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We discuss the relationship between fully quantized and semiclassical approaches to input-output theory that connect asymptotically free fields prior to and following interactions that we describe in terms of canonical transformations. We demonstrate the utility of the Wigner function in describing the connection between input and output fields, and show how it is particularly well suited to describe many quantum-optical systems in terms of positive quasiprobability permitting classical stochastic simulation.
The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.
This paper proves a new theorem on the relationship between optical field Wigner function's two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field Ψ( x ') propagates through an optical [ D (− B ) (− C ) A ] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D , B . It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A , C .
Based on the correspondence between the Collins diffraction formula (optical Fresnel transform) and the transform matrix element of a three-parameter two-mode squeezing operator in the entangled state representation 1 Fan, H-Y and Lu, H-L. 2006. Opt. Lett., 31: 2622–2624. [Crossref], [PubMed] , [Google Scholar] we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip, both in the spatial domain and the frequency domain. The entangled Wigner function is introduced for recapitulating the result.
The Wigner formalism in the Heisenberg picture constitutes a bridge that connects Quantum Optics to Stochastic Optics. The vacuum field appears explicitly in the formalism, and the wavelike aspects of light are emphasised. In addition, the zeropoint intensity as a threshold for detection is a common denominator in both theories. In this paper, after summarising the basic rules of the Wigner approach and its application to parametric down-conversion, some new results are presented that delve into the physical meaning of the zeropoint field in optical quantum communication. Specifically, the relationship between Bell-state distinguishability and the number of sets of zeropoint modes that take part in the experiment is analysed in terms of the coupling between the phases of the different fields involved and the subtraction of the zeropoint intensity at the detectors. Additionally, the connection between the compatibility theorem in quantum cryptography and zeropoint field is stressed.
The quantum-stochastic-differential-equation formulation of driven quantum-optical systems is carried out in the interaction picture, and quantum stochastic differential equations for wave functions are derived on the basis of physical principles. The Ito form is shown to be the most practical, since it already contains all the radiation reaction terms. The connection between this formulation and the master equation is shown to be very straightforward. In particular, a direct connection is made to the theory of continuous measurements, which leads directly to the method of quantum-jump simulations of solutions of the master equation. It is also shown that all conceivable spectral and correlation-function information in output fields is accessible by means of an augmentation of the simulation process. Finally, the question of the reality of the jumps used in the simulations is posed.