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Appendix a TWO-LEVEL SYSTEMS and RATE EQUATIONS

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Appendix a TWO-LEVEL SYSTEMS and RATE EQUATIONS Appendix A TWO-LEVEL SYSTEMS AND RATE EQUATIONS Chapters 2 and 3 show that for semiconductor laser purposes, the in­ teraction of light with a semiconductor medium can often be modeled in terms of electronic transitions between a valence and a conduction band. A spread of transition energies occur that depend on the value of the car­ rier momentum k. Elsewhere in physics such a range of transitions is known to occur, namely in ensembles of inhomogeneously broadened two­ level systems. These appear to good approximation in the interaction of light with atoms and with the magnetic dipoles of nuclear magnetic reso­ nance. Hence people have been led to model semiconductor laser media using the theory of two-level systems. In appropriate limits, the two-level approaches reduce to rate equation theory, also very popular in modeling aspects of semiconductor laser operation. On the other hand, Sec. 3-1 rev­ eals that the semiconductor medium is at the very least an inhomogene­ ously broadened four-level medium, all of whose levels have appreciable probability in a gain medium. Two of the four levels correspond to the levels in a two-level medium, but the other two are absent in the two-level medium. Hence the degree to which a two-level model can describe semi­ conductor response is disturbingly uncertain. One is really better off using a real semiconductor model, for which the approximations are well de­ fined. Nevertheless much of the physics of the two-level model has coun­ terparts in the semiconductor models and one can study this physics in a relatively simple context. This appendix seeks to present the two-level model in a way well suited to people primarily interested in semiconductor media. More complete discussions can be found, for example, in Meystre and Sargent (1991). Section A-I reviews the physics of two-level systems, starting with the wave function and proceeding to the density matrix. The corresponding equations of motion are often called the optical Bloch equations. Section A-2 derives simple rate equations and solves for the gain and index of homogeneously and inhomogeneously broadened two-level media. As Sec. 1-9 shows, the simple semiconductor formalism is very similar to the hom­ ogeneously broadened two-level formalism. At first this might be surpris­ ing, since the semiconductor obviously involves a wide range of transition frequencies, i.e., is inhomogeneously broadened. The resolution of this §A-I TWO-LEVEL PHYSICS 439 apparent contradiction lies in rapid carrier-carrier scattering, which for sufficiently small laser fields allows the fields to experience gain from a large part of the carrier distribution, rather than from just in the vicinity of the laser frequency. The material in this appendix is not necessary for an understanding of semiconductor lasers. However, it helps both in defining many common variables and nomenclature, as well as in talking with people about semi­ conductor lasers, since they may tend to think in terms of two-level media. After reading this appendix, the reader should read the end of Sec. 3-2, which discusses the relationship between two-level systems and semicon­ ductor media in greater detail. A-I. Two-Level Physics In this section, we start with the wave function and its equation of motion, the SchrOdinger equation. We expand this function in terms of energy eigenfunctions, and simplify the treatment to two levels. We solve for the time evolution of the level probability amplitudes under the influ­ ence of light. We then define the corresponding density matrix, whose diagonal elements are level probabilities and off-diagonal elements are pro­ portional to induced electric-dipole moments. The latter give the polariza­ tion of the medium that acts as a source in Maxwell's equations to amplify or absorb light. The density matrix elements obey equations of motion that reduce to rate equations in appropriate limits. Think of this section as a crash course in the basics of two-level theory. More detailed discussions are available in many textbooks, such as Sargent, Scully, and Lamb (1977) and Meystre and Sargent (1991). According to the postulates of quantum mechanics, the best possible knowledge about a quantum mechanical system is given by its wave func­ tion tj;(r, t). Although .,p(r, t) itself has no direct physical meaning, it allows us to calculate the expectation values of all observables of interest. This is due to the fact that the quantity tj;(r,t) *.,p(r,t)d 3 r is the probability of finding the system in the volume element d 3r. Since the system described by .,p(r, t) is assumed to exist, its probability of being somewhere has to equal 1. This gives the normalization condition 440 TWO-LEVEL SYSTEMS AND RATE EQUATIONS App. A (1) where the integration is taken over all space. An observable is represented by a Hermitian operator & and its expec­ tation value is given in terms of t/J(r,t) by (2) Experimentally this expectation value is given by the average value of the results of many measurements of the observable & acting on identically prepared systems. The accuracy of the experimental value for (&) typi­ cally depends on the number of measurements performed. Hence enough measurements should be made so that the value obtained for (&) doesn't change significantly when still more measurements are performed. The reason observables, such as position, momentum, energy, and dipole moment, are represented by Hermitian operators is that the expectation values (2) must be real. Denoting by (</J, 1/J) the inner or scalar product of two vectors </J and 1/J, we say that a linear operator & is Hermitian if the equality (</J, &1/J) = (&rjJ, 1/J) *. (3) holds for all </J and 1/J. An important observable in the interaction of light with electrons is the electric dipole er. This operator provides the bridge between the quantum mechanical description of a system and the polarization of the medium P used as a source in Maxwell's equations for the electromagnetic field. According to Eq. (2), the expectation value of er is (4) where we can move er to the left of 1/J(r,t)* since the two commute (an operator like V cannot be so moved). Here we see that the dipole-moment expectation value has the same form as the classical value if we identify p = elt/J(r,t)12 as the charge density. §A-l TWO-LEVEL PHYSICS 441 In nonrelativistic quantum mechanics, the evolution of ..p(r, t) is gov­ erned by the Schrodinger equation ill :t1/l(r,t)=%1/I(r,t) , (5) where % is the Hamiltonian for the system and Il = l.054xl0-34 joule-sec­ onds is Planck's constant divided by 271". The Hamiltonian of an unper­ turbed system, for instance an atom not interacting with light, is the sum of its potential and kinetic energies %= E...+ ~(r) (6) 2m ' where p is the system momentum, m is the system mass, and V(r) the potential energy. We suppose that the time and space dependencies in Eq. (5) separate as (7) for which the un (r) satisfy the energy eigenvalue equation (8) The eigenfunctions un (r) can be shown to be orthonormal and complete so that any function can be written as a superposition of the un (r). In partic­ ular the wave function ..p(r, t) itself can be written as the superposition 1/I(r,t) = L Cn(t) un(r) e-iwnt (9) n Here the expansion coefficients Cn (t) are constants for problems described by a Hamiltonian satisfying the eigenvalue equation (8). We include the time dependence in anticipation of adding an interaction energy to the Hamiltonian. Such a modified Hamiltonian wouldn't quite satisfy Eq. (8), thereby causing the Cn (t) to change in time. Substituting Eq. (9) into the normalization condition (I) and using the orthonormality of the un (r), we find 442 TWO-LEVEL SYSTEMS AND RATE EQUATIONS App. A (10) n The 1C n 12 can be interpreted as the probability that the system is in the nth energy state. The Cn are complex probability amplitudes and completely determine the wave function. In terms of the Cn (t), the expectation value (2) of the operator &- is given by (&-) = L L Cn (t)Cm (t)e-iwnm t &-mn , (11) n m where &-mn is the matrix element fd 3r um *(r)&-un (r). We are primarily interested in the interaction of a medium with light. To treat such interactions, we add the appropriate interaction energy to the Hamiltonian, that is (12) If we expand the wave function in terms of the eigenfunctions of the "unperturbed Hamiltonian" %0' rather than those of the total Hamiltonian %, the probability amplitudes Cn (t) change in time. To find out just how, we substitute the wave function (9) and the Hamiltonian (12) into SchrOd­ inger's equation (5) to find en (t) = - ~ L (nIVlm) eiwnm t Cm (t) (14) m where the matrix element (15) and the frequency difference wnm = wn - wm. We can solve Eq. (14) approximately by using first-order perturbation theory. Since we are interested in interactions with light, we suppose that §A-I TWO-LEVEL PHYSICS 443 the interaction energy matrix element has the time-dependent form (nlVlm) = Vnm (O)cosvt , (16) where v is an optical frequency. Starting at time t = 0 with the system in the initial level i (Ci(O) = I, Cm#(O) = 0», we integrate Eq. (14) with Cm (t) ~ Cm (0) to find (17) Here the superscript (I) says that the interaction energy has been applied I time, i.e., to first order.
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