E. WOLF, PROGRESS IN OPTICS XI 0 NORTH-HOLLAND 1973 I MASTER EQUATION METHODS IN QUANTUM OPTICS * BY G.S. AGARWAL Department of Physics and Astronomy, University of Rochester, Rochester, N. Y., 14621, U.S.A. * Work supported by the U.S. Air Force Office of Scientific Research and by the U.S. Army Research Office (Durham). Present address: Institut fur Theoretische Physik, Universitat Stuttgart, Stuttgart, Germany. CONTENTS PAGE 1. INTRODUCTION . 3 0 2. PHASE SPACE METHODS. 8 3 3. MASTER EQUATION FOR A GENERAL SYSTEM. 13 5 4. MASTER EQUATIONS FOR SYSTEMS INTERACTING WITH STOCHASTIC PERTURBATIONS . 18 5 5. MASTER EQUATIONS FOR OPEN SYSTEMS . 23 0 6. RELAXATION OF A HARMONIC OSCILLATOR. 27 5 7. BROWNIAN MOTION OF A QUANTUM OSCILLATOR 37 9 8. RELAXATION OF AN ATOM. 40 5 9. INCOHERENT AND COHERENT (SUPERRADIANCE) SPONTANEOUS EMISSION . 44 0 10. LASER MASTER EQUATION . 52 0 11. MASTER EQUATIONS FOR STRONGLY INTERACTING QUANTUM SYSTEMS IN CONTACT WITH HEAT BATHS 58 0 12. AN APPLICATION OF PHASE SPACE TECHNIQUES TO A PROBLEM IN SOLID STATE PHYSICS . 65 APPENDIX.. , . 67 REFERENCES. , . , . 73 8 1. Introduction In recent years increasing use has been made of methods of quantum statistical mechanics and stochastic processes in treatments of various problems in quantum optics. This is seen from a large number of publica- tions dealing with the theory of lasers (GORDON[1967], HAKEN[1970], LAX [1966c, 1968a1, SCULLYand LAMB [1967]), with superradiance (AGARWAL[1970; 1971b, c, el, BONIFAC~Oet al. [1971a, b]), with problems in nonlinear optics such as parametric oscillators (GRAHAM[ 19681). More- over some of the methods were specifically developed to treat the problems in quantum optics. These include the well known phase space methods (GLAUBER[ 1963, 19651, SUDARSHAN[ 19631, CAHILL and GLAUBER[ 1969a, b], AGARWALand WOLF [1968, 1970a, b, c], LAX [1968b]). In phase space methods the c-number distribution functions for quantum systems are introduced, which in many physical situations are found to obey equations of the Fokker-Planck type. One may then use the language of stochastic processes to study various quantum systems. In quantum optics, one is usually concerned with the study of a subsystem which is a part of a large system, for example, in case of the laser one is mainly interested in the statistical properties of the emitted radiation. In this context master equation methods have played a very important role. Master equation methods have found applications in many branches of physics such as in the theory of relaxation processes (BLOCH[1956, 19571, REDFIELD[1957, 19651, ABRAGAM[1962], ARGYRESand KELLEY[1964]), anharmonic interaction in solids (BROUTand PRIGOGINE[ 19561, PRIGOGINE [ 19621, CARRUTHERSand DY [ I966]), superfluids (LANCER[ 1969]), transport phenomenon (CHESTERand THELLUNG[1959], VANHOVE[1959], KOHNand LUTTINGER[ 19571, ARGYRES[ 1966]), optical pumping (WILLIS[ 1970]), superradiance (ACARWAL[1970, 1971b, c, el, BONIFACIOet al. [1971a, b]), Brownian motion of a quantum oscillator (AGARWAL[1971d]), in the quantum theory of damping (LOUISELLand MARBURGER[ 19671, AGARWAL [ 1969]), in the kinetic theory of gases (see, for example, PRIGOGINE [1962]), in the theory of lasers (LAX [ 1966~1,HAAKE [ 1969b1, HAKEN[ 1970]), etc. 3 4 MASTER EQUATION METHODS IN QUANTUM OPTICS [I, 0 1 We start by giving a brief history of the subject. A master equation was first obtained by PAULI[1928]. He obtained an equation of motion for the diagonal elements pnn of the density operator by making the statistical hypothesis of random phases at all times. Pauli’s master equation has the form where ynm is the transition probability per unit time for the system to make a transition from the state Irn) to the state In). It should be noted that pnn is the probability that the system be found in the state In) and, therefore, (1.1) is of the form of a rate equation. An equation of the type (1.1) for P.,~is also expected from the principle of detailed balance. We will now outline Pauli’s derivation. Our presentation follows that of VAN HOVE[1962]. We write the Hamiltonian of the system as H = Ho+gH,, (1.2) where H,, is the unperturbed Hamiltonian and gH, is a small perturbation. We work in a representation in which Ho is diagonal, with eigenfunctions I$,,) and eigenvalues En. The state of the system at time t+At is related io the state at time t by the unitary transformation I$(t+At)) = exp {-i(Ho+gH,)At}l$(t)), (1.3) where h has been put equal to unity and this we do throughout this article. On expanding I$(t+ At)) and I$(t)) in a complete set of states Pauli made the statistical assumption of random phase at all times. This assumption enables us to ignore all the terms in (1.5) with rn # 1. In fact the assumption of random phases implies that the terms in (1.5) with rn # I oscillate very rapidly and so they average out to zero. Eq. (1.5) then reduces to Pn(t+Al) = C I($nI ~XP{-i(Ho+gH,)At}Ill/m)12pm(t). (1.6) m I, 0 11 INTRODUCTION 5 Since At is small enough the expression I($"[ exp { -i(Ho+gHl)At}l$,)lZ for n # m is easily evaluated by first order perturbation theory and one finds that I($nI ~XP{-i(H~+gH,)At)Icl/m)IZ = YnmAt, (a # m), (1.7) where Ynm = 2ng2a(En- Em)I <$nI I$m) I (1.8) Equation (1.6) in the limit At -+ 0 leads to the Pauli master equation (1.1) if use is also made of the following identity c 1($,1 exp {-i(H0+g~,)At)l$,)lZ = 1- (1.9) m In the derivation it has also been assumed that the spectrum is continuous. Pauli's equation is valid for times such that z, << t << irelax,where z, is the interaction time and trelaxis the relaxation time. The 'interaction time' for the problem of anharmonic interaction in solids, for example, is of the order of (coo)-', where wD is the Debye frequency (see e.g. PRIGOCINE [ 19621). In the past two decades, there has been revival of interest in deriving the master equations by making far less reaching assumptions than Pauli did. The work on the derivation of the master equations can mainly be divided into three groups: (i) VANHOVE [1955, 1957, 19621, (ii) Prigogine and his co-workers (see e.g. PRIGOCINE[1962]) and (iii) ZWANZIG[1961a, 19641. VanHove appears to have been the first to give a rigorous derivation of the Pauli equation. He also derived master equations to all orders in perturba- tion. Prigogine et al. made extensive use of diagrammatic methods to derive master equations. Zwanzig developed very elegent projection operator methods for deriving the master equations to all orders in perturbation and also established the identity of various other master equations (MONTROLL [196I], PRIGOGINEand RESIBOIS119611). In the present article we will not be concerned with rigorous derivation of master equations but rather in reviewing research that demonstrates the power of the master equation approach in the study of many problems in quantum optics (for rigorous derivation of master equations see for example, VANKAMPEN [1954], VANHOVE [1955, 19571, EMCHand SEWELL[1968]). We will show how these techniques may be employed in studies of the relaxation of oscillators and spin systems (two-level atomic systems), of lasers and of superradiance. We will make use of Zwanzig's projection operator techniques to obtain master equations for a wide variety of systems. The density operator p characterizing the state of a quantum mechanical system satisfies the equation of motion 6 MASTER EQUATION METHODS IN QUANTUM OPTICS [I, 5 1 aplat = -i2p, (1.10) where the Liouville operator 3 is given by 9= [H, 1. (1.11) Zwanzig noted that the part of the density operator which is of interest can be obtained from the total density operator by suitably projecting it. To obtain an equation of motion for the diagonal elements of p ZWANZIG [1961a, 19641 introduced the following projection operator prnnmfn, = am, amm?am, 4 (1.12) Zwanzig regarded 9 and B tetradics. However, there is no need for introducing the notion of a tetradic, one may very well work instead with the projection operator 9.. = c 9,,Tr {B,, . .}, (1.13) m where 8, is the projection operator onto the state Im), i.e. Pm= Im)(ml. (1.14) It is seen from (1.13) that B projects out the diagonal elements of p, since the off diagonal elements of the operator 8pare identically equal to zero. We can treat both classical and quantum systems by using the projection operator methods. In classical statistical mechanics the distribution function @N({q}, {p};t) satisfies the Liouville equation of motion (1.15) which can be rewritten in the form aaNjat= -imjN, (1.16) where the Liouville operator 9is given by (1.17) In many problems one is only interested in the momentum’ distribution function aN({p};t) which is obtained from @N((q}, {p};t) by integrating over the volume V of the system: rr (1.18) 1,9 11 INTRODUCTION 7 The relation (1.18) can be written as (1.19) where 2' is the projection operator 9... = ./d({q}). .. (1.20) '1..VN In other problems, such as anharmonic interaction in solids, one is interested in calculating the energy distribution. In this case one introduces the action- angle variables Ji and pi and regards the distribution as a function of {Ji} and (qi>.Then the energy distribution is given by (1.22) In case of a system interacting with stochastic perturbations, such as a randomly modulated harmonic oscillator, the density operator of the distribution function becomes a random function.