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Physical Review

Quantum Theory of an Optical Maser. II. Spectral Profile

Marlan O. Scully, Willis E. Lamb

Published Feb 10, 1968• 60 citations
Publisher

Abstract

In Paper I of this series, we derived equations of motion for a quantum-laser field interacting with atomic reservoirs. In the usual region of sustained oscillation, the off-diagonal elements of the radiation density matrix ${\ensuremath{\rho}}_{n,n+k}(t)$ were found to have an expontential decay associated with phase diffusion. In I we found the spectrum of the laser radiation by calculating the single time-ensemble average electric field implied by ${\ensuremath{\rho}}_{n,n+1}(t)$. This electric field was then treated as a classical variable whose Fourier analysis gave the spectrum. In the present paper, we establish the validity of this procedure by analyzing a simple model for a spectrometer. It is also shown that the same spectrum can be obtained from a two-time correlation function derived from the equations of motion.

DOI
10.1103/physrev.166.246

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Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences•Feb 13, 1969•18 citations
In Paper I of this series, we derived equations of motion for a quantum-laser field interacting with atomic reservoirs. In the usual region of sustained oscillation, the off-diagonal elements of the radiation density matrix \ensuremathρn,n+k(t){\ensuremath{\rho}}_{n,n+k}(t)\ensuremathρn,n+k​(t) were found to have an expontential decay associated with phase diffusion. In I we found the spectrum of the laser radiation by calculating the single time-ensemble average electric field implied by \ensuremathρn,n+1(t){\ensuremath{\rho}}_{n,n+1}(t)\ensuremathρn,n+1​(t). This electric field was then treated as a classical variable whose Fourier analysis gave the spectrum. In the present paper, we establish the validity of this procedure by analyzing a simple model for a spectrometer. It is also shown that the same spectrum can be obtained from a two-time correlation function derived from the equations of motion.
A quantum statistical analysis of an optical maser is presented in generalization of the recent semiclassical theory of Lamb. Equations of motion for the density matrix of the quantized electromagnetic field are derived. These equations describe the irreversible dynamics of the laser radiation in all regions of operation (above, below, and at threshold). Nonlinearities play an essential role in this problem. The diagonal equations of motion for the radiation are found to have an apparent physical interpretation. At steady state, these equations may be solved via detailed-balance considerations to yield the photon statistical distribution ${\ensuremath{\rho}}_{n,n}$. The resulting distribution has a variance which is significantly larger than that for coherent light. The off-diagonal elements of the radiation density matrix describe the effects of phase diffusion in general and provide the spectral profile ${| E(\ensuremath{\omega}) |}^{2}$ as a special case. A detailed discussion of the physics involved in this paper is given in the concluding sections. The theory of the laser adds another example to the short list of solved problems in irreversible quantum statistical mechanics.
We discuss the quantum theory of a simple model of a maser oscillator, consisting of one radiation-field mode interacting with a large number of stationary three-level atoms. The field and the atoms also interact with separate heat reservoirs which represent dissipation mechanisms and an incoherent pumping mechanism. The model is sufficiently simple that some analytical progress can be made with the nonlinear quantal equations before further approximation is necessary. We start from the quantal equation of motion of the field-atom density operator. We make immediate use of a diagonal coherent-state expansion for the field part of the density operator and a somewhat similar expansion for the atom part. This yields an exact equation of the Fokker-Planck form for a ccc-number weight distribution, which retains all the significance of the original operator equation, and which has the semiclassical equation for the same model as a first, fluctuation-free approximation. We make use of our basic Fokker-Planck equation in a variety of ways. We discuss the reduction of the equation under conditions that the atomic decay constants are large (large atomic linewidth), arriving finally at an equation of motion for a field-only weight function which serves to demonstrate the basic coherence properties of a maser. We derive and discuss the equation of motion of the generalized Wigner density for the maser model. The generalized Wigner density is a smoothed version (a convolution) of our basic weight distribution, and from it we derive an equivalent classical model including noise sources. Finally, we discuss other useful weight distributions and the number representation for the field. The equations we derive in these discussions make contact with the rate equations of Shimoda, Takahasi, and Townes, as well as with the more recent work of Lax and Louisell, Lax, and of Scully and Lamb.
We generalize the method of deriving a kinetic equation for a gas laser developed in the first paper of this series to include radiation-matter correlations. As a consequence, we are able to show that the first Born approximation with asymptotic conditions which contain radiation-matter correlations is sufficient to explain the nonthermal photon distribution observed in photon-counting experiments. Our derivation includes the spatial and velocity effects of the motion of the gas atoms. After adiabatically eliminating the matter variables, we obtain a master equation for the radiation density matrix RRR with nonlinear coefficients. Our radiation master equation for stationary atoms without spatial effects is the same as the generalized Fokker-Planck equation of the Langevin-noise-equation treatment of lasers. We show that the generalized Fokker-Planck equation is the result of the first-Born-approximation treatment of the j\ifmmode\cdot\else\textperiodcentered\fi{}A interaction and is not dependent upon the microscopic structure of the reservoirs. In the first Born approximation, the diagonal and off-diagonal matrix elements of RRR do not interact with each other. Consequently, an initially diagonal radiation density matrix remains diagonal. However, we show that even though the stationary state is diagonal it is necessary to know the propagator of the off-diagonal part of RRR to answer time-dependent questions about quantities, such as the line shape, that depend on phase.
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.
A quantum statistical theory of resonant optical phenomena is developed for the case when the fluctuations of the laser, used for exciting transitions, are important. The general equations describing the dynamics of a relaxing two-level atom (TLA) are solved exactly for different types of mean values and the two-time amplitude and intensity correlations. The phase-diffusion model is adopted for laser fluctuations. The exact results are used to analyze the effect of laser fluctuations on a number of optical effects-optical free-induction decay, adiabatic following, the spectrum of the scattered light from a relaxing TLA, the energy-absorption spectrum from a weak field, Hanle resonances, etc. In each case, the laser fluctuations are found to affect in an important way the characteristics of the above optical-resonance phenomena. For example, the spectrum of scattered radiation from a TLA has the usual three-peak structure for fields at resonance and with strengths $\ensuremath{\alpha}$ above the threshold and for $\ensuremath{\alpha}\ensuremath{\gg}{\ensuremath{\gamma}}_{c}$; however, now the peak heights (widths) are in the ratio 3x3x3x (23x\frac{2}{3x}3x2​), $x=\frac{[{\ensuremath{\gamma}}_{c}+\frac{({T}_{1}^{\ensuremath{-}1}+{T}_{2}^{\ensuremath{-}1})}{3}]}{[2{\ensuremath{\gamma}}_{c}+{T}_{2}^{\ensuremath{-}1}]}$, where ${\ensuremath{\gamma}}_{c}^{\ensuremath{-}1}$ is the correlation time for laser phase fluctuations. For ${\ensuremath{\gamma}}_{c}\ensuremath{\gg}\ensuremath{\alpha}$, one gets a single-peak spectrum. The laser field is treated as a second-quantized field with excitation in either a coherent state or a Fock state. The results, obtained by straightforward perturbation theory, but valid for arbitrary values of the relaxation parameters, detuning, and the laser correlation time, are also presented.
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.