R. Graham

Quantum theory of light propagation in a fluctuating laser-active medium

The basic equations are derived which describe the propagation of an electromagnetic field in a fluctuating laser-active medium. The well-known methods of Langevinequations and master-equation for a few discrete modes are generalized to meet also the case of a radiation field with continuous spectrum. The medium is described by two-level atoms which are embedded in a merely passive solid matrix and homogeneously distributed over space. They have an inversion which is kept constant by an externally applied pump. The atomic line may be homogeneously or inhomogeneously broadened. We obtain a complete set of partial differential equations for the field operators with damping terms and fluctuating forces homogeneously distributed over the material. The telegraph equation with a fluctuating force occurs as a special case. After the exact elimination of the atomic variables we obtain a nonlinear field equation for the radiation field alone. By means of a pseudo-Hamiltonian and by a simple one-dimensional example we show that in a certain sense there exists a close formal analogy between the present theory and the theory of an interacting Bose gas. The characteristic differences between the two theories are also discussed. We find, that there occurs a phase transition of the radiation field because above a certain threshold of the pump the photons condense into a single mode and establish an “offdiagonal-long-range order”. The amplitude fluctuations and the phase fluctuations, which restore the broken phase symmetry, are calculated in detail. A new condition for the occurrence of undamped spiking (pulse formation) for a continuum of modes is derived.

Laserlight — first example of a second-order phase transition far away from thermal equilibrium

We solve the functional Fokker-Planck equation established in a previous paper in the vicinity of laser threshold. The stationary solution is obtained explicitly in the formP=N exp [−ϕ({ū, ū*})]. ϕ has exactly the same form as the Ginzburg-Landau expression for the free energy of a superconductor, if the pair wave function is replaced by the electromagnetic field amplitudeū. This gives us the key for a thermodynamic reinterpretation of all laser phenomena.In particular the laser threshold appears as a second-order phase transition in all details. It is indicated that our theory provides a new formalism also for the Ginzburg-Landau theory.

Generalized thermodynamic potential for Markoff systems in detailed balance and far from thermal equilibrium

We analyze the question why there exist systems far from thermal equilibrium, e.g. lasers, whose stationary state may be described by a potential function, which has all the properties of a thermodynamic potential. It is argued, that the physical property, common to thermal equilibrium states and to these more general stationary states is detailed balance.The proof of this argument is given explicitly for systems whose macroscopic, collective variables can be described as a continuous Markoffian random process.We show that in order to preserve detailed balance the coefficients of the governing Fokker-Planck equation have to fulfill a set of restrictive conditions, special cases of which are well known in the mathematical theory of Markoff processes under the name of “potential conditions”. These restrictive conditions are derived as a consequence of the fact that the condition of detailed balance has to be compatible with the simultaneous validity of the Fokker-Planck equation (forward equation) and its adjoint equation (backward equation).By means of the potential conditions it is always possible to calculate the generalized thermodynamic potential explicitly. We further show that the validity of the potential conditions in turn implies that detailed balance holds.As the validity of the potential conditions has already been demonstrated for important cases in laser theory and in nonlinear optics the existence of detailed balance can now be inferred for those systems. When we specialize our present treatment to closed systems, we obtain the general Fokker-Planck equation derived for the first time by M. S. Green. In that paper detailed balance was implicitly assumed.This shows that detailed balance can provide a basis for a unified description of both the thermal equilibrium systems and some more general stationary states far from thermal equilibrium. Our potential function appears as a natural generalization of thermodynamic potentials to more general stationary states with detailed balance.

The quantum-fluctuations of the optical parametric oscillator. I

Our treatment is based on a microscopically correct Hamiltonian which contains the Bose-operators of the light modes and the Fermi-operators of the optically active electrons in the medium. The coupling between modes and atoms is taken from quantum-electrodynamics. Besides that, the light modes may interact with external “heat baths” like the mirrors, scattering centers etc., while the atoms interact with lattice vibrations, incoherent light fields etc. Using recently developed methods the effect of these heatbaths is taken into account in a quantum mechanically consistent fashion. In the present paper we apply quantum mechanical Langevin equations for the field and electron operators which contain dissipation and fluctuation terms. The elimination of the electron operators by an iteration procedure finally leaves us with a set of coupled nonlinear field equations which are shown to be quantum mechanically consistent. They are solved in the Heisenberg picture below threshold by linearization and well above threshold by quantum mechanical quasi-linearization. The solutions show that the line width of the signal mode below threshold is due to the vacuum fluctuations in the idler and vice versa, whereas the thermal noise of the resonator and the spontaneous emission noise of the medium may be neglected. Above threshold the linewidth is caused by the undamped diffusion of the phase difference between signal and idler, to which the vacuum fluctuations of both modes contribute in equal parts. The phase sum of both modes adiabatically follows the slow phase diffusion of the external pump light, produced by a laser, and therefore contributes to the linewidth too. Well above threshold the amplitudes are stable. Correlation and cross-correlation functions of their small residual fluctuations are calculated.

Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}.A functional of the time dependent probability density of the system is defined in close analogy to the Gibbs definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.

Photon statistics of the optical parametric oscillator including the threshold region

Starting from an effective Hamiltonian the derivation of a set of classical Langevin equations for the amplitudes of signal, idler, and pump is briefly reconsidered. From these equations all variables except those describing the signal mode are eliminated with the help of an adiabatic approximation and certain others, which are valid in the threshold region and somewhat above (i.e. photonumbers ≪ 1014). The signal mode amplitude then satisfies a van der Pol equation in the rotating wave approximation and is driven by a fluctuating force.With the exception of a slight difference due to the undamped phase diffusion of the pumping laser, the same Langevin equation has been derived earlier for the amplitude of a laser mode near threshold. We present the stochastically equivalent Fokker-Planck equation, whose solution is reduced to the known solution of the laser Fokker-Planck equation. Thus the complete photon statistics of the signal mode is revealed at once. In particular we obtain the stationary distribution and the amplitude and intensity correlation functions as well as the transient solution.

Quantum mechanical correlation functions for the electromagnetic field and quasi-probability distribution functions

We give new formulae permitting the calculation of time ordered but otherwise arbitrary correlation functionsK of electromagnetic field operators in terms of a class of quantum mechanical quasi-probability distribution functions. This class contains among others the socalledQ- andP-functions as well as the Wigner function.

Functional quantum statistics of light propagation in a two-level system

The functional Fokker-Planck formalism developed in a preceding paper is applied to the problem of a radiation field propagating in a medium, which contains resonant two-level atoms. Besides the electromagnetic field also the medium is described by continuous space dependent fields. We give the masterequation and transform it into ac-number functional differential equation for a characteristic functional. This equation is reduced considerably by the projection onto one dimension and the introduction of the diffusion approximation. It forms a solid basis for the study of all types of light propagation in resonant media including classical and quantum noise.We give an approximate solution of this equation by considering the problem of an externally pumped optical transmission line, in the case that saturation effects are absent. The spectral function of the electric field strength is obtained which describes a statistical mixture of photons with the quasiparticles of the polarization field. It shows the onset of a condensation of the quasiparticles into a single state. Self excitation of the transmission line is obtained at a certain threshold of the atomic inversion. This threshold is characterized by a finite occupation number of one single quasiparticle state. The influence of a finite length of the transmission line is briefly considered.

Functional Fokker-Planck treatment of electromagnetic field propagation in a thermal medium

The quantum statistics of continuous space time dependent electromagnetic fields is analyzed by means of functionals. The case of a field propagating in a thermal reservoir serves as a simple example to illustrate the succeeding steps: a masterequation is derived for the density operator which is a functional of the field operators. By means of the coherent state representation for continuous fields the masterequation is transformed into a functional differential equation in the function space, spanned by the coherent state amplitudes. This equation is of the Fokker-Planck type and determines a Gaussian process for a continuum of variables or a field. It is solved by determining the characteristics in function space of the associated equation of motion for the characteristic functional and subsequent functional integration. The solution is used to calculate some correlation functions and the spectral function of the field.

Theory of cross-correlation of signal and idler in parametric oscillators

Abstract The stationary joint Wigner distribution of signal and idler amplitudes is found. It is used to compare the cross-correlation of signal and idler with the auto-correlation of signal alone.