Rate equations for photons and for electrons in systems of two groups of energy levels (energy separation kTηG) are set up allowing for absorption, stimulated and spontaneous emission, nonradiative transitions, photon loss or dissipation as well as photon gain from the ambient, and an externally applied pumping rate. The rate equations are discussed qualitatively, and the stability of the solutions is obtained by the use of phase portraits. The quantitative discussion includes steady-state photon distributions. Earlier work showing that these distributions can obey the Bose–Einstein law with a nonzero chemical potential kTF is taken into the laser regime, which was not covered by recent relevant work. The second-order phase transition that occurs at threshold is established explicitly, and it is shown that the transition requires the neglect of certain generation terms in the rate equations. Some time-dependent solutions of the rate equations are also discussed, using the slaving principle of adiabatic elimination for the electrons. The application of this work to semiconductors is discussed.
The paper analyses some time-dependent and steady-state properties of coupled photons and electrons in two-level systems, by taking into consideration the radiative transitions and photon dissipation. The main assumption is the slaving principle for adiabatic elimination of the electrons. A master equation is derived for the photon number. The stability of the photon population is studied and the laser threshold concentration of the upper-level electrons determined. A first-order phase transition that occurs at threshold is established explicitly, and it is shown that the transition requires the neglect of the spontaneous emission and of the ambient influence. The steady-state photon distribution is not Poissonian and is characterized by large fluctuations. Fluctuations become negligible when the stimulated emission is zero. The steady-state distribution can obey the Bose-Einstein law with a nonzero chemical potential. The equation of state of the photon population is derived. When dissipation is weak and dilution increases, the equation of state is similar to that of a degenerate Bose gas in equilibrium.
We arrive at transition probabilities for a laserlike system by neglecting wave properties of matter and radiation, and treating the problem, in an analogy to chemical systems, entirely in a number representation. Thus, for example, stimulated emission is idealized as a collision between a photon and an excited atom and the transition rate is proportional to the amount of each species present in the cavity. In all, transition probabilities for stimulated emission, absorption of radiation, cavity loss, atomic deexcitation (other than stimulated emission), and pumping of atoms are included in the model. The resulting stochastic master equation analogous to that employed for the discussion of fluctuations in chemical systems is then used to obtain the deterministic behavior of atoms and field. This agrees with that of the quantum mechanical laser model of Casagrande and Lugiato; thus the rate parameters of the stochastic model may be related to the parameters of their model. Returning to the master equation, we project out the atomic variable and use adiabatic elimination to arrive at a closed, single-variable equation for P(n), the probability of having n photons in the cavity. The stationary solution of this equation is obtained and graphs of (n) and log[(n2)/(n)2] versus the pumping parameter show close agreement with the microscopic results of the Casa-grande-Lugiato model. A similar discussion of the fluctuations can also be given for a laser with saturable absorption. It shows a first-order phase transition. A comparison with the microscopic results of Lugiato et al. is outlined. Again a remarkable agreement of the stochastic approach via the master equation and the microscopic theory is obtained.
The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems. In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function. In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.
The basic results of theoretical nonperturbative analysis of spontaneous and stimulated emission from two-band excited intrinsic semiconductor interacting with a single mode of radiation field in a lossless micro-cavity are presented. The self-consistent dynamics of the coupled system ''semiconductor + single mode radiation field", accompanied with spontaneous and stimulated emission was investigated as dependent of the initial level of the excitation of semiconductor in the presence initially of the physical field vacuum. Calculations have been carried out with the help of the commutation version of the evolution method without introducing explicitly the phonon mechanisms. The initial non-equilibrium electron and hole populations in the bands have been supposed to be created by ultra-short laser pulses and described by the Fermi-Dirac quasi-equilibrium distributions with the given chemical potentials. The interaction between the electron-hole subsystem and the mode field has been adopted to be switched on suddenly. The calculations have shown that depending on the initial excitation level of electron-hole subsystem three states of the mode field are realized: the equilibrium Bose-Einstein photon distribution with nonzero chemical potential; the quasi-equilibrium distribution with the average number of photons equal to unit and accompanied with their fluctuations, and finally, the state of photon avalanche. The first regime takes the place when the coefficient of the light amplification is negative. Then, the equilibrium between the electron-hole subsystem and arising photons exists until the average number reaches the unit. For negative and =1 the equilibrium still exists but is accompanied with field fluctuations. Finally, when the steady growth arises the lower but stable values are established
We derive the kinetic equations for the coupled single-particle density matrix $\ensuremath{\rho}$ and the electromagnetic density matrix
R to lowest order in the dimensionless coupling constant ${\ensuremath{\beta}}^{2}\ensuremath{\equiv}{(\frac{{\ensuremath{\omega}}_{L}}{{\ensuremath{\omega}}_{D}})}^{2}$. The laser frequency ${\ensuremath{\omega}}_{L}$ is ${(4\ensuremath{\pi})}^{\ensuremath{-}\frac{1}{2}}{(\mathfrak{N}{r}_{0}{\ensuremath{\lambda}}^{2})}^{\frac{1}{2}}{\ensuremath{\omega}}_{0}$ where
N is the number of two-level systems per unit volume,
r0 is the classical electron radius, $\ensuremath{\lambda}$ is the wavelength of the radiation, and $k{\ensuremath{\omega}}_{0}$ is the two-level energy difference. The Doppler frequency ${\ensuremath{\omega}}_{D}$ characterizes the center-of-mass motion. For gas lasers ${\ensuremath{\beta}}^{2}$ is much less than 1 and, consequently, we generalize and use the Bogoliubov derivation of kinetic equations for weak interactions. We find solutions when the average field vanishes and which include spontaneous emission correctly. The single-particle density matrix and the radiation density matrix are coupled through their second moments. When we substitute the solution of the second-moment equations into the density-matrix equations, we find that each density matrix satisfies an uncoupled linear equation with known time-dependent coefficients. We introduce and discuss dissipation from the density-matrix point of view. With the use of the density-matrix formalism we indicate that the correct expansion parameter for higher order kinetic equations is ${\ensuremath{\beta}}^{2}$.
The quantum theory of the three-energy-level two-mode laser in which all the levels are being pumped is investigated and the master equation of the laser operation is obtained. It contains two variables,
n1, the photon number of the first mode, and
n2, the photon number of the second mode. There appear four terms in the master equation which represent two-photon processes between the two modes, the absorption of a photon from one mode, and emission of a photon to the other. They are absent in the single-mode case and in the two-mode case with pumping only to the upper level. The master equation can be represented by a diagram of probability in two dimensions which can be extended to infinity, and each arrow of it represents a term on the right-hand side of the master equation. By summing each variable (
n1 or
n2) of the master equation, two equations can be obtained. Each equation contains only one variable, and can be expressed by a diagram of probability flow in one dimension. By considering the correspondence between macroscopic equilibrium and microscopic detailed balance the equations of motion in the steady state are obtained. By introducing a parameter
H, the equation of motion is simplified and a formal solution can be deduced which is dependent on
H. It appears that the parameter
H cannot be fully determined, but some properties of it can be deduced which prove that the introduction of
H is reasonable. The operation characteristics of the laser under different conditions are discussed on the basis of the formal solution: among them are the threshold condition, the condition for one-mode operation, and the change of the photon-statistical distribution. We especially discuss the laser output power curves under two-mode operation above threshold. The curve of one mode, which oscillates first, shows a bending-down phenomenon as the excitation increases when the other mode goes beyond its threshold as demonstrated experimentally by Otsuka. This phenomenon is explained qualitatively and is attributed to the two-photon processes between the two modes.