In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
Our treatment of the quantum theory of radiation so far has predominantly involved entire systems, such as the electromagnetic field in a cavity. In most areas of quantum optics, however, we are interested in only part of the entire system. For example, in the laser problem, we want to know the field but are not particularly interested in what happens to the atoms. We find it convenient to separate the system of primary interest from that (or those) of secondary interest and to call the former simply the “system” and the latter the “res ervoir.” We can eliminate the reservoir by using the reduced density operator method in the Schrodinger (or interaction) picture or the noise operator method in the Heisenberg picture. In this chapter we discuss the density operator method, which provides computational convenience while stressing the statistical aspects of the problem. In Chap. 19 we use the noise operator approach, which often requires more calculation but offers a direct physical appeal in its resemblance to the classical Brownian motion problem.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.
A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.
A fully quantum-mechanical treatment is given of field and population fluctuations in a homogeneously broadened laser. The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources. The atomic polarization variables are adiabatically eliminated. The mean motion of an arbitrary operator function $M(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2})$ of the field variables
b, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population variables
N1 and
N2 is calculated. The equation of motion for the density matrix $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2}, t)$ is deduced, as well as the Fokker-Planck equation for the associated classical distribution function $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, {N}_{1}, {N}_{2}, t)$. The diffusion coefficients in the latter equation demonstrate the shot-noise origin of the fluctuations. The Fokker-Planck equation for
P is replaced by Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$,
N1, and
N2 to facilitate adiabatic elimination of the population variables (when valid). The resulting Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$ are converted to a Fokker-Planck equation for $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$, to an operator equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, and to an equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$, the field density matrix in the photon-number representation. The transformation $\ensuremath{\beta}={I}^{\frac{1}{2}}{e}^{\ensuremath{-}i\ensuremath{\phi}}$ to an ($I, \ensuremath{\phi}$) representation leads (a) to the steady-state solution
P(I) valid at all operating levels, and (b) to the phase linewidth. The transformation to new variables
N=21(m+n), $k=m\ensuremath{-}n$, permits the equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$ to be replaced by a Fokker-Planck equation for $\ensuremath{\rho}(N, k, t)$. The latter permits a verification of the previously obtained phase linewidth and a solution for the steady-state $\ensuremath{\rho}(N)$.
In this chapter we use the Langevin (Brownian motion) method of Chap. 19 to once again treat quantum mechanically both the field and the active atoms in a laser. The present treatment comprises a Heisenberg picture approach to the problem, as compared to the approach in Chap. 17, which represents the Schrodinger picture. The two methods are in complete agreement (as they should be!), a point discussed in some detail by Lax (1968). As before, the atoms are considered to comprise systems which interact with independent pumping and dissipation reservoirs, and the laser field is regarded as another system with reservoirs. The atom and field systems are then coupled by an electric-dipole perturbation energy, as depicted schematically in Fig. 20-1.