The inclusion of damping in the quantized radiation field in pure and mixed states is accomplished in a natural way, without using traditional loss reservoirs, through the application of appropriate boundary conditions in a simple model of an optical cavity coupled to the outside world. By using collective operators (which are suitable weighted superpositions of the usual creation and annihilation operators for modes of the continuous spectrum, extended over each Fox–Li band) and convenient initial conditions, we are able to describe the radiation loss from an optical cavity to the outside world. The physical situations corresponding to initial conditions leading to the exponential damping are also investigated.
The inclusion of damping in the quantized radiation field in pure and mixed states is accomplished in a natural way, without using traditional loss reservoirs, through the application of appropriate boundary conditions in a simple model of an optical cavity coupled to the outside world. By using collective operators (which are suitable weighted superpositions of the usual creation and annihilation operators for modes of the continuous spectrum, extended over each Fox–Li band) and convenient initial conditions, we are able to describe the radiation loss from an optical cavity to the outside world. The physical situations corresponding to initial conditions leading to the exponential damping are also investigated.
A semiclassical treatment of the radiation loss in a laser, previously investigated by Baseia and Nussenzveig [Opt. Acta 31, 39 (1984)], is extended to the quantum-theory level. The model employs an optical cavity having an output coupling that includes the radiation loss due to the beam extraction through appropriate boundary conditions, instead of the unrealistic loss reservoir of the conventional treatment. This allows one to work entirely within the continuous spectrum and obtain the equation for the radiation field both inside and outside the cavity. Inside the cavity the results agree, in the coherent representation and absence of quantum noise, with those of the previous semiclassical theory. Also, the results of both treatments for the field inside the cavity coincide with those of the conventional theory, since we reinterpret its ``mode amplitude'' as a collective variable. Outside the cavity (an inaccessible region for a standard approach) the explicit solution for the laser field exhibits a natural relationship between the intensities inside and outside the cavity. Further considerations, such as those concerning the role played by the noise term in the field buildup, are also discussed.
We use Thomson's classical concept of mode of a leaky cavity to develop a quantum theory of cavity damping. This theory generalizes the conventional system-reservoir theory of high-Q cavity damping to arbitrary Q. The small system now consists of damped oscillators corresponding to the natural modes of the leaky cavity rather than undamped oscillators associated with the normal modes of a fictitious perfect cavity. The formalism unifies semiclassical Fox-Li modes and the normal modes traditionally used for quantization. It also lays the foundations for a full quantum description of excess noise. The connection with Siegman's semiclassical work is straightforward. In a wider context, this theory constitutes a radical departure from present models of dissipation in quantum mechanics: unlike conventional models, system and reservoir operators no longer commute with each other. This noncommutability is an unavoidable consequence of having to use natural cavity modes rather than normal modes of a fictitious perfect cavity.
In this chapter, a one-dimensional optical cavity with output coupling is con-sidered. The optical cavity has transmission loss at one or both of the end surfaces. The classical, natural cavity mode is defined, and decaying or growing mode functions are derived using the cavity boundary conditions. A series of resonant modes appears. But these modes are not orthogonal to each other and are not suitable for quantum-mechanical analysis of the optical field inside or outside of the cavity. Hypothetical boundaries are added at infinity in order to obtain orthogonal wave mode functions that satisfy the cavity and infinity boundary conditions. These new mode functions are suitable for field quantization, where each mode is quantized separately and the electric field of an optical wave is made up of contributions from each mode. Some results of quantization are described in the next chapter. Chapter 3 deals with the usual quasimode model: a perfect cavity with distributed internal loss or with a fictitious loss reservoir.
A recent result of a previous paper describing the field loss in an optical cavity in a natural way is applied to laser theory. The field loss, due to the transmitted beam light, and the field gain, now attributed to the presence of active atoms but previously attributed to two noninteracting mechanisms, are correctly added to provide the equation of motion for the density operator of the radiation field. The replacement of individual operators, which appear in the conventional treatment, with collective operators is a conceptual difference that emerges from the present approach.
Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.
An approximate solution is given for the Jaynes-Cummings model with cavity losses, i.e., the problem of a two-level atom interacting with a single mode of the quantized radiation field, in the rotating-wave approximation, when the field is damped by a reservoir at zero temperature. The approximate solution is derived for initial coherent field states with moderately large numbers of photons. It is simpler in form than earlier results derived by other authors and, over the appropriate parameter range, substantially more accurate than some of them, as shown by direct numerical integration of the master equation. In particular, it is found that an earlier treatment of this problem based on a secular approximation is seriously flawed, in that the conditions for its validity are much more restrictive than was previously believed. Among the results derived it is shown that, just as for the lossless case, when the atom is initially prepared in one of the semiclassical eigenstates the evolution is very simple, with the field and the atomic dipole drifting together in phase. For moderate losses this leads, as in the lossless case, to a ``state preparation''; i.e., to a good approximation, the state of the atom at a specific time can be made independent of its initial state. The effect of losses on the recently discovered ``Schr\"odinger cat'' state of the field is also analyzed. It is found that, although the dissipation destroys the coherence of the macroscopic superposition very rapidly, preparation and observation of the ``cat'' should be possible with the cavity quality factors reported in recent micromaser experiments.