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Physical Review A

Positive-Prepresentation for second-harmonic generation: Analytic and computational results

K. J. McNeil, I. J. D. Craig

Published Apr 1, 1990• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.41.4009

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This paper considers the ``positive-P'' description of nonlinear optical damping where one-photon and two-photon loss mechanisms are allowed. An exact solution of the corresponding stochastic differential equations is presented that graphically shows the breakdown of the positive-P representation found in earlier numerical-simulation work. The problem of ``spiking'' in simulations of nonlinear optical processes is then addressed, using the nonlinear damping process as an example. It is pointed out that although ``spiking'' can be exascerbated at high levels of quantum noise by numerical inaccuracy, the tendency to spike is in fact an analytic property of the stochastic variables.

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Nonclassical states of the electromagnetic field are states for which the Glauber-Sudarshan P function does not exist as a positive-definite nonsingular probability density. A classical stochastic description for systems that produce nonclassical states can be recovered, however, using the positive P representation. But difficulties can arise in numerical simulations of nonlinear stochastic models derived using this representation.1 This paper addresses the unresolved question of whether the difficulties are purely numerical or fundamental to the representation. Recently we used the positive P representation to provide an appealing picture of the nonclassical states produced by a degenerate parametric oscillator.2 The resolution of difficulties in that problem by a reflecting boundary condition seemed rather model dependent. We analyze the quantized anharmonic oscillator in the positive P representation. Correct operator averages are derived from the stochastic model analytically. However, an explicit construction of the positive P function shows that there are inherent difficulties for numerical simulations. Unconstrained diffusion in phase space occurs, and for long times the important trajectories for recovering correct moments are those that diffuse far from the origin. Even if numerical instability can be avoided, an enormous number of trajectories would be required to cover phase space sufficiently to obtain accurate averages.

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The positive Prepresentation has proved a useful tool for analyzing optical systems which exhibit squeezing. The Glauber-Sudarshan phase-space representation gives a Fokker-Planck equation without positive semidefinite diffusion for systems which generate nonclassical fields. The positive Prepresentation provides a method for guaranteeing positive semidefinite diffusion and enables a quantum- dynamical model to be formulated in terms of classical stochastic differential equations. In previous work we had some success in applying this approach to describe the decay of the unstable state above threshold in a degenerate parametric oscillator.1 Below threshold a linearized analysis of fluctuations recovers the predictions of squeezing obtained by alternative methods.

Annual Meeting Optical Society of America•Jan 1, 1987•0 citations

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We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations $(\mathrm{S}\ensuremath{\Delta}\mathrm{E}).$ Second, we show that introducing sources into the SDE's (or $\mathrm{S}\ensuremath{\Delta}\mathrm{E}\mathrm{'}\mathrm{s})$ generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.

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Nonequilibrium transitions in quantum optical systems

P. D. Drummond

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.

•Jan 1, 1979•0 citations
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.
This paper considers the ``positive-P'' description of nonlinear optical damping where one-photon and two-photon loss mechanisms are allowed. An exact solution of the corresponding stochastic differential equations is presented that graphically shows the breakdown of the positive-P representation found in earlier numerical-simulation work. The problem of ``spiking'' in simulations of nonlinear optical processes is then addressed, using the nonlinear damping process as an example. It is pointed out that although ``spiking'' can be exascerbated at high levels of quantum noise by numerical inaccuracy, the tendency to spike is in fact an analytic property of the stochastic variables.
Equation of motion for the density operator of the field in the process of second harmonic generation is converted into a Fokker-Planck equation by using the positive P-representation of the density operator. This Fokker-Planck equation is replaced by a set of four coupled stochastic differential equations using the Ito interpretation. The statistics of the light field are studied by numerically solving these stochastic equations. For long interaction times substantial antibunching and squeezing are possible in principle.
Nonclassical states of the electromagnetic field are states for which the Glauber-Sudarshan P function does not exist as a positive-definite nonsingular probability density. A classical stochastic description for systems that produce nonclassical states can be recovered, however, using the positive P representation. But difficulties can arise in numerical simulations of nonlinear stochastic models derived using this representation.1 This paper addresses the unresolved question of whether the difficulties are purely numerical or fundamental to the representation. Recently we used the positive P representation to provide an appealing picture of the nonclassical states produced by a degenerate parametric oscillator.2 The resolution of difficulties in that problem by a reflecting boundary condition seemed rather model dependent. We analyze the quantized anharmonic oscillator in the positive P representation. Correct operator averages are derived from the stochastic model analytically. However, an explicit construction of the positive P function shows that there are inherent difficulties for numerical simulations. Unconstrained diffusion in phase space occurs, and for long times the important trajectories for recovering correct moments are those that diffuse far from the origin. Even if numerical instability can be avoided, an enormous number of trajectories would be required to cover phase space sufficiently to obtain accurate averages.
The positive Prepresentation has proved a useful tool for analyzing optical systems which exhibit squeezing. The Glauber-Sudarshan phase-space representation gives a Fokker-Planck equation without positive semidefinite diffusion for systems which generate nonclassical fields. The positive Prepresentation provides a method for guaranteeing positive semidefinite diffusion and enables a quantum- dynamical model to be formulated in terms of classical stochastic differential equations. In previous work we had some success in applying this approach to describe the decay of the unstable state above threshold in a degenerate parametric oscillator.1 Below threshold a linearized analysis of fluctuations recovers the predictions of squeezing obtained by alternative methods.
We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations $(\mathrm{S}\ensuremath{\Delta}\mathrm{E}).$ Second, we show that introducing sources into the SDE's (or $\mathrm{S}\ensuremath{\Delta}\mathrm{E}\mathrm{'}\mathrm{s})$ generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.
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.