In this chapter we treat single-mode laser operation in which both the atoms and the field obey the laws of quantum mechanics. For simplicity, we suppose for most of the discussion that the field is tuned to atomic line center and that the active atoms are two-level systems which pass through the cavity in a time r, as in Sec. 16-1. To simulate a homogeneously broadened medium, we suppose that the transit times are distributed with the probability y exp(— yr). The more realistic (and complicated) calculation underlying the theory of Chap. 8 is given in Appendix I. We also suppose that excitation occurs only to the upper level; Probs. 17-13 and 17-14 consider excitation to the lower level as well. We describe cavity losses by an ensemble of nonresonant atoms injected into the lower level, as discussed in Sec. 16-1. The response of these “loss” atoms does not saturate (i.e., small coupling constant), as does that of the active atoms. The loss atoms here play the same role as the fictitious con ductivity in the semiclassical laser theory of Chap. 8. As shown in Sec. 16-5, an equivalent description is obtained by a reservoir of simple harmonic os cillators with a spread of frequencies. Our discussion depends on the reservoir theory of Sec. 16-1, in which the reduced density operator represents the ra diation field. In Chap. 20 we give a related development in which the Langevin (Heisenberg picture) approach is used.
In this chapter we treat single-mode laser operation in which both the atoms and the field obey the laws of quantum mechanics. For simplicity, we suppose for most of the discussion that the field is tuned to atomic line center and that the active atoms are two-level systems which pass through the cavity in a time r, as in Sec. 16-1. To simulate a homogeneously broadened medium, we suppose that the transit times are distributed with the probability y exp(— yr). The more realistic (and complicated) calculation underlying the theory of Chap. 8 is given in Appendix I. We also suppose that excitation occurs only to the upper level; Probs. 17-13 and 17-14 consider excitation to the lower level as well. We describe cavity losses by an ensemble of nonresonant atoms injected into the lower level, as discussed in Sec. 16-1. The response of these “loss” atoms does not saturate (i.e., small coupling constant), as does that of the active atoms. The loss atoms here play the same role as the fictitious con ductivity in the semiclassical laser theory of Chap. 8. As shown in Sec. 16-5, an equivalent description is obtained by a reservoir of simple harmonic os cillators with a spread of frequencies. Our discussion depends on the reservoir theory of Sec. 16-1, in which the reduced density operator represents the ra diation field. In Chap. 20 we give a related development in which the Langevin (Heisenberg picture) approach is used.
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.
Abstract This present paper is an attempt to describe the operation of a laser by a method patterned closely after the classical theory of dispersion. There, the electromagnetic field is treated as a classical system interacting with a collection of classical harmonic oscillators which are at rest. In the present case the radiation field is still treated classically, but, in accordance with the correspondence principle, the oscillators are now virtual oscillators associated with the up-and-down transitions in effective two-state atoms which are described quantum mechanically. The atoms carrying these virtual oscillators are not at rest but move with different velocities. We show how the Lorentz averaging can be performed for such systems, and derive a closed set of equations linking the average electric field E, with the average polarization density P„ and the average population inversion density Mp associated with atoms moving with the velocity v. Next, we investigate the conditions for the existence of a steady state if pumping is present, and if the electric field is represented by a standing wave, or a travelling wave of a single frequency (single-mode case). It turns out that, notwithstanding the nonlinear nature of the equations, the steady-state conditions give a simple complex dispersion relation; moreover, the real and imaginary part of this dispersion relation are equivalent to the usual heuristic expressions which specify the operating frequency in terms of the index of refraction and cavity length, and which balance the gain against the losses. The single-mode case is analysed in detail without any smallness assumptions for the resultant intensity. The small-intensity case gives the usual results exhibiting the tuning dip. For high intensities the relative depth of the dip tends to zero. At the end of the paper we discuss proposed extensions and additional applications.
A quantum theory of a two-level single-mode laser with injected atomic coherence is developed by generalizing the Scully-Lamb laser theory to a form appropriate for the analysis of a coherently pumped laser. We assume that the active atoms are prepared initially in a coherent superposition of the upper and lower levels, and we derive the master equation for the field density operator by treating the interaction of the laser field with many active atoms simultaneously. It is shown that the photon-number distribution can be exactly Poissonian. The laser operation is analyzed in terms of the Fokker-Planck equation for the laser field. Both the intensity and phase diffusion coefficients are phase sensitive and, for stable laser operation, become much smaller than those of an ordinary laser. Consequently, the injected atomic coherence reduces both the photon-number noise and phase noise simultaneously. The intensity diffusion coefficient can vanish exactly, and at the same time the phase diffusion coefficient can become very small. This leads to spontaneous-emission noise quenching in the photon-number distribution, and the laser field can become very close to a coherent state. A scheme to generate the proper form of the initial atomic coherence necessary for the quantum noise quenching is proposed and analyzed.
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.
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.
In Chap. 17 we developed the quantum theory of the laser obtaining in particular expressions for the laser photon statistics and linewidth. In this chapter we discuss and amplify those results in terms of measurement theory. Section 18-1 provides an overview of measurement theory as it applies to the quantum electric field. The discussion leads naturally into a specific discussion of the measurement of photon statistics and the laser linewidth. In Sec. 18-2 a heuristic model of a photodetector is presented with reveals how the photo electron statistics resulting from incident radiation is related to the photon statistics of that radiation, t In Sec. 18-3 we discuss the laser linewidth with the use of a simple two level atom beam much like that of Sec. 16-1. Our present discussion utilizes the density matrix technique throughout rather than the density matrix-state vector combination of Sec. 16-1. The laser spectrum “seen” by the atomic beam has, in fact, the same Lorentzian profile given in Sec. 17-3. In Sec. 18-4 the laser linewidth is discussed in terms of a quantum version of the classical correlation function <£(* -f t )E(t)y. In spite of the fact that two time correlations are apparently required, we show that a single time and solutions to the density operator equation of motion are sufficient. This reduction is an example of the so-called Onsager regression hypothesis, which shows how to proceed from single-time to multitime correlation functions.