The problem of the formulation and simulation of quantized optical systems using the positive-P representation is considered. We show that feasible computational strategies are determined mainly by the eigenstructure of the system of stochastic differential equations that correspond to the positive-P Fokker-Planck equation. We demonstrate analytically that the stability and accuracy of numerical solutions are generally much improved by adopting time-centered implicit schemes. These findings are then related to the specific problem of second-harmonic generation. It is shown that the eigenstructure of the stochastic system possesses interrelationships that can lead to severe difficulties in numerical simulations. In particular we show how the problem of ``spiking'' encountered at high levels of quantum noise can be related to the numerical ``stiffness'' of the problem. We also demonstrate the advantages of time-centered implicit methods by highlighting the distortions that can occur in the Euler-type simulation of limit cycle solutions. Of physical interest is the fact that the phase-space morphology of the quantized second-harmonic generator admits families of limit cycles. This is in marked contrast to the purely classical picture in which, for a given parameter set, there exists only a single limit cycle. Finally we demonstrate by direct analytic solution that, under the conditions of adiabatic elimination, the positive-P representation for the second-harmonic generator with zero driving field fails, in the sense that it produces nonphysical results. This failure is in all respects analogous to the breakdown encountered in the problem of nonlinear optical damping.
The quantum statistical equations for the field produced in the process of second-harmonic generation (SHG) are expressed, based on a method proposed for four-wave mixing.1 We start from the following assumptions: 1) the field absorption in the nonlinear medium may be neglected; 2) the amplitude of the fundamental field is much greater than that of the SH field; 3) the Wigner function associated with the density operator of the field depends significantly only on the variables of the SH field. The coefficients of the Fokker–Planck equation thus obtained depend on the Laplace transforms of the correlation functions of the atomic polarization operator. The moments of the Wigner function2 are related to the moments of the SH field photon number; consequently, we impose the condition for sub-Poissonian photon-statistics and derive the corresponding relationships satisfied by the atomic correlation functions. A numerical interpretation is given.
With the help of second-order nonlinear interactions and hence Hamiltonian, we construct the equations of motion corresponding to pump and signal (second harmonic) fields in a second harmonic generation. The corresponding coupled differential equations involving the non-commuting field operators are not solvable in closed analytical forms. With the help of a perturbation method, we obtain analytical solutions of these field operators up to cubic orders in the interaction constant. In an appropriate limit (truncating the solution up to the second order in the interaction constant), these solutions lead to the existing solutions under the short-time approximation. The present analytical solutions are exploited to investigate the squeezing and the antibunching of photons of the input coherent light coupled to the said second-order nonlinear medium. We report the squeezing and antibunching effects of the pump field even for solutions up to the second order and hence the present results are consistent with earlier results. However, for the second harmonic mode, we report the squeezing involving the leading cubic term in the interaction constant.
In this paper the equations for quantum optical pulse propagation in nonlinear and dispersive single-mode fibers are presented in terms of two phase-space formulations based on the positive-P and the Wigner distributions. Included are the effects due to the coupling of the electromagnetic modes to the vibrational states of a vitreous silica fiber. By making use of the well-known equivalence of Fokker-Planck and Ito stochastic equations, we demonstrate two alternative methods for formulating the equations of motion as coupled stochastic c-number equations for the propagating field. The first method involves a representation of the density operator in terms of the positive P distribution function. This leads to exact stochastic equations of motion. The second method makes use of the Wigner distribution function. This method, which requires truncation of third-order derivative terms in the corresponding Fokker-Planck equation, is necessarily approximate. However, we discuss certain advantages to the Wigner approach that have made it the preferable method for exploratory work.
In this paper we present a quantum statistical analysis of a non-linear optical system with a mode coupled to its second harmonic, studied semiclassically in a previous paper. Quantum fluctuations are included via a Fokker-Planck equation for a generalized P representation. The spectrum of the light is shown to split into two peaks for critical values of the input fields. It is found that under certain conditions the non-classical effect of photon antibunching is possible in the steady state. In certain limits exact solutions for the generalized P representation are possible which allows an estimate of lifetimes in the metastable region to be calculated.
A fully quantum-mechanical treatment is given of field and population fluctuations in a homogeneously broadened laser. The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources. The atomic polarization variables are adiabatically eliminated. The mean motion of an arbitrary operator function $M(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2})$ of the field variables
b, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population variables
N1 and
N2 is calculated. The equation of motion for the density matrix $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2}, t)$ is deduced, as well as the Fokker-Planck equation for the associated classical distribution function $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, {N}_{1}, {N}_{2}, t)$. The diffusion coefficients in the latter equation demonstrate the shot-noise origin of the fluctuations. The Fokker-Planck equation for
P is replaced by Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$,
N1, and
N2 to facilitate adiabatic elimination of the population variables (when valid). The resulting Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$ are converted to a Fokker-Planck equation for $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$, to an operator equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, and to an equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$, the field density matrix in the photon-number representation. The transformation $\ensuremath{\beta}={I}^{\frac{1}{2}}{e}^{\ensuremath{-}i\ensuremath{\phi}}$ to an ($I, \ensuremath{\phi}$) representation leads (a) to the steady-state solution
P(I) valid at all operating levels, and (b) to the phase linewidth. The transformation to new variables
N=21(m+n), $k=m\ensuremath{-}n$, permits the equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$ to be replaced by a Fokker-Planck equation for $\ensuremath{\rho}(N, k, t)$. The latter permits a verification of the previously obtained phase linewidth and a solution for the steady-state $\ensuremath{\rho}(N)$.