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Proceedings of the Physical Society

On a representation in quantum optics

Saul M. Bergmann

Published Aug 1, 1967• 0 citations
Publisher

Abstract

A dynamical problem in quantum optics is considered to be that of solving Schrodinger's equation with the initial state of the radiation field as a coherent state By cairying out a particular unitary transformation it is shown how one can arrive at a representation of the radiation field operators which is particulaily convenient for Proc.esses in which the interaction with the material system only involves one photon or a few photons. It is shown in which sense this representation is a counterpart of the Fock representation. In the derivation of the representation, which is exact throughout, some features are brought out, which may be interpreted as being semi-classical.

DOI
10.1088/0370-1328/91/4/304

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A dynamical problem in quantum optics is considered to be that of solving Schrodinger's equation with the initial state of the radiation field as a coherent state By cairying out a particular unitary transformation it is shown how one can arrive at a representation of the radiation field operators which is particulaily convenient for Proc.esses in which the interaction with the material system only involves one photon or a few photons. It is shown in which sense this representation is a counterpart of the Fock representation. In the derivation of the representation, which is exact throughout, some features are brought out, which may be interpreted as being semi-classical.
In this chapter, we will introduce different representations of the electromagnetic (EM) field. One may ask, why do we need different representations for the EMfield? The answer is that usually we do not know the state of the EM field which in quantum optics is represented by the annihilation and creation operators. However, results of measurements of the field are given in terms of real variables, for example, intensity. The representations allow us to recognize the state of the field from values of the measured quantities such as average amplitude, intensity, and correlation functions. The nature of the state is present in terms of the interpretation of the apparently classical (measured) variables. We will discuss two basic types of representations often used in quantum optics, Fock state (photon number) representation and coherent states representation, the later one introduced by Glauber.a We also discuss properties of fields with thermal and Poisson distribution of photons. The photon number states are very often used as a basis for quantum optics problems, and despite of many difficulties have recently been generated experimentally [11-13].
We often hear at seminars and presentations ‘Show me the Hamiltonian of your problem and I will tell you what problem you are talking about’. Therefore, to understand what quantum optics is about, we start from the description of a standard Hamiltonian of the problems considered in quantum optics. We illustrate the method of derivation of the explicit form of the Hamiltonian for a simple system that is composed of two subsystems that interact (communicate) with each other. As we shall see, the explicit form of the Hamiltonian is essential to explicitly calculate energy levels of a given combined system and the temporal evolution of an arbitrary operator representing the system.
The interaction between matter and fields is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena, which includes transport properties. We propose a radically different approach based on the extension of the dynamics of integrable systems. We consider a simple model, a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of the particle and the actions of the field. This makes the system an example of Poincar\'e nonintegrable systems. We extend the unitary operator to a new star-unitary operator $\ensuremath{\Lambda}.$ This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can of course write the Hamilton equations. But we have the possibility to go to new descriptions. The invertible $\ensuremath{\Lambda}$ transformation gives many new aspects, which are hidden in the initial description. For example, we show that there are fluctuations, and that there exists an irreducible probability description. The transformation to $\ensuremath{\Lambda}$ representation corresponds to a transformation to Markovian probability equations. We can always come back to the initial representation by the inverse transformation. We have verified this remarkable prediction by detailed numerical calculations. We need the $\ensuremath{\Lambda}$ transformation to obtain the definition of a dressed unstable mode, which has a well-defined lifetime. In the initial representation there are various time scales and there is no strictly exponential lifetime. The situation is the same as the one we have studied in the quantum case in recent papers.
We have already shown that the coherent states are very close to the classical state of the well-defined amplitude and phase. A question not answered yet is: Does there exist an analogous quantum mechanical observable for the phase that is given by some Hermitian operator? We know from classical optics that phase of an EM field is an observable quantity. Hence, in quantum physics it should be associated with a Hermitian operator. In order to set up a quantitative description of the phase operator, we will introduce some of the ideas presented in the literature which led to the formulation of the Hermitian phase operator.
We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order ααα influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $α= 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.
A model for interacting single-mode electromagnetic radiation and matter consisting of NNN two-level particles is solved exactly using a representation in which the creation and destruction operators for the field act by multiplication and differentiation, respectively. The model includes off-resonant interactions. Closed-form expressions can be given for some special cases. The solutions are transformed to the more familiar Fock and Schr\"odinger representations. Numerical results are given to indicate the behavior of the solutions. An equation for a generating function is derived which would give solutions of the (algebraic) matrix equation to which the problem has been reduced.