Quantum-theoretical treatments of three-photon processes M. K. Olsen 1 , L. I. Plimak, 2 and M. Fleischhauer 2 1 Instituto de Fı ´sica da Universidade Federal Fluminense, Boa Viagem 24210-340, Nitero ´i, Rio de Janeiro, Brazil 2 Fachbereich Physik, Universita ¨t Kaiserslautern, 67663 Kaiserslautern, Germany Received 18 November 2001; published 22 April 2002 We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation GFPE for the positive-P function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predic- tions for the quantum statistics. DOI: 10.1103/PhysRevA.65.053806 PACS numbers: 42.65.Ky, 42.50.-p, 02.50.Ey I. INTRODUCTION The theoretical study of the interaction of electromagnetic waves via a nonlinear medium has a long history, going back at least to the groundbreaking paper of Armstrong et al. 1, wherein a classical treatment was performed for the pro- cesses of second- and third-harmonic generation, degenerate and nondegenerate down-conversion, and four-wave mixing. Experimentally, second-harmonic generation SHG and parametric down-conversion are well-known sources of quantum states of the electromagnetic field. Third-harmonic generation THG, wherein input fields at frequency  pro- duce output fields at frequency 3  is a process that has been observed experimentally in a number of different situations. An early experiment 2 produced both third- and fifth- harmonic light at the interface of glass and liquids and it was suggested that odd-multipole generation may be a wide- spread phenomenon. THG has been observed in a number of other situations, for example, in the interaction of laser light with a nematic liquid-crystal cell 3, and in the interaction of pulsed light from an Nd:YAG yttrium aluminum garnet laser with organic vapors 4 and with polyimide films 5. As all the processes mentioned above are highly nonlin- ear, a full quantum-theoretical treatment is often difficult without resorting to the phase-space representations of quan- tum optics, generally the positive-P 6 or Wigner represen- tations 7. In the usual approach, the system Hamiltonian is mapped onto a Fokker-Planck equation for the particular pseudoprobability distribution being used, which may either be solved directly or further mapped onto a set of stochastic differential equations 8,9. As the usual methods only allow for the mapping of genuine Fokker-Planck equations, that is, equations with derivatives of no higher than second order this is the content of Pawula’s theorem 10, onto stochastic differential equations, the systems that can be investigated using this approach are limited. There is at least one other known method for the derivation of stochastic differential equations for interacting bosonic systems 11. Although this method gave enhanced stability in the numerics over the nor- mal positive P, it was assumed at the beginning of the deri- vation that the time development could be modeled by Ito ˆ stochastic differential equations, so that it is subject to the limitations of Pawula’s theorem when it comes to the repre- sentation of third-order noises. The present situation of triply degenerate four-wave mix- ing results in generalized Fokker-Planck equations of third or higher order and hence Langevin equations may not be de- rived. A common procedure in these cases is to use a Wigner equation truncated at second order, which is equivalent to the semiclassical theory of stochastic electrodynamics 12. This procedure necessarily discards the deeper quantum aspects of the problem and gives answers at odds with quantum me- chanics for several systems 13,14. There are also situations where even a P-representation Fokker-Planck equation must also be written in generalized form 15–18, and more are likely to be investigated in the future. The present problem is one of these situations. Using the techniques of quantum field theory, we have previously developed methods to represent this class of prob- lems using stochastic difference equations 18–20. These equations may be numerically integrated using computers, so that the fact that stochastic differential equations cannot be defined for these systems is not a problem. As the problem of triply degenerate four-wave mixing has a Hamiltonian cubic in creation and annihilation operators, it results in a general- ized Fokker-Planck equation with third-order derivatives for the positive-P pseudoprobability distribution. We note here that a related problem, namely, that of intracavity third- harmonic generation, has previously been dealt with 15,16. In Ref. 15, the quantum properties of the fields are ana- lyzed in a linearized approximation, Ref. 16 looks at mac- roscopic quantum interference. In both cases the steady states, which are natural for a resonator, rather than tran- sients, which are natural in the traveling-wave case, are in- vestigated. It should also be noted that there are certain tech- nical problems with Refs. 15,16. It appears the authors were unaware that the stochastic equations used could not, in fact, be considered Langevin equations in the formal sense, and that these should be considered stochastic difference equations. Furthermore, the third-order noise contribution to PHYSICAL REVIEW A, VOLUME 65, 053806 1050-2947/2002/655/0538069/$20.00 ©2002 The American Physical Society 65 053806-1