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Journal of Physics Condensed Matter

Time-dependent and steady-state statistics of photons at nonzero chemical potential

Viorel Bădescu

Published Aug 19, 1991• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/3/33/025

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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.
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 behavior of a single mode of radiation field coupled to an excited two band intrinsic semiconductor has been investigated with the help of the commutator version of evolution operator method. As dependent on the excitation level three states of the field has been shown to exist: the equilibrium with the number of photons determined by the Bose–Einstein distribution with nonzero chemical potential; the quasi-equilibrium with the average number of photons equal to unity and accompanied by fluctuations; the state of photon avalanche.
We derive the equation of motion for the field density matrix of the degenerate two-photon laser under conditions of two-photon resonance starting from the full microscopic Hamiltonian. Our results are compared with the corresponding quantities obtained from the standard effective Hamiltonian. As we have shown for the nondegenerate case, the full diagonal density-matrix equations tend to the effective Hamiltonian density-matrix equations in an appropriate limit, but the equations of motion for the off-diagonal elements do not coincide. The equations obtained using a more accurate form of the effective Hamiltonian, in which Stark shifts are included, do agree. In this paper we concentrate on the photon-number distribution and the nature of the phase transition that takes place in the neighborhood of the two-photon lasing transition threshold. We determine the steady-state mean photon numbers and consider the stability of the solutions. A solution is found where the mean photon number is zero, but this is found to be a stable solution only for sufficiently weak pumping, whereas in the standard effective Hamiltonian approach it is always stable. As the pumping strength is increased, a range of detunings is reached in which there are two stable solutions. The amplitude of the zero-photon peak diminishes very rapidly as the pumping strength increases. Finally, for sufficiently large detunings, a single, stable, steady-state solution is obtained. The nature of the phase transition is illustrated by presenting plots of the photon-number distribution against detuning and pumping rate. The photon-number fluctuations about the mean are also discussed.
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 fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function p(n,m)p(n,m)p(n,m) to find nnn photons in the light field and mmm atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.
This paper presents new results on aspects of the two-photon laser. A semiclassical investigation of threshold behaviour reveals that the atom-field coupling is already so strong at threshold that a non-perturbative treatment of the dynamics is necessary. A master equation is derived on the basis of an effective Hamiltonian generalized in that it includes the effect of an overall two-photon detuning, the effect of ionization of the upper atomic level and the effect of dynamic Stark shifts. From the diagonal elements of the master equation, a Fokker-Planck equation is obtained and shown to yield a general analytical expression for the photon distribution which is found to simplify considerably in special cases. The agreement with the result obtained through numerical solution of the master equation itself is very good. It is further shown that a detuning from two-photon resonance and a significant Stark shift lead to sub-Poissonian statistics. Finally, time-dependent solutions for the photon distribution under an injected coherent state are presented.