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Annalen der Physik

Statistical Behaviour of Three Boson Field Modes with Trilinear Interaction

T. V. Trung, F.‐J. Schütte

Published Jan 1, 1978• 11 citations
Publisher

Abstract

Abstract The time variation of the density matrix for a system of three interacting boson field modes is described in the Schrödinger picture with the help of the unitary time development operator in coherent state basis. The nonlinear c‐number equation for this operator is solved using a perturbation expansion in powers of the coupling constant. The explicit time dependence is obtained for the matrix elements of the total density matrix and of the particular density matrices of the individual modes. Photon number distributions and second order correlations are calculated for several initial situations.

DOI
10.1002/andp.19784900306

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Abstract The time variation of the density matrix for a system of three interacting boson field modes is described in the Schrödinger picture with the help of the unitary time development operator in coherent state basis. The nonlinear c‐number equation for this operator is solved using a perturbation expansion in powers of the coupling constant. The explicit time dependence is obtained for the matrix elements of the total density matrix and of the particular density matrices of the individual modes. Photon number distributions and second order correlations are calculated for several initial situations.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.
The quantum theory of the stimulated Raman and Brillouin effects, the parametric amplifier, and the frequency converter is developed, starting with a simple Hamiltonian for a system of three coupled field modes. The intense incident beam is treated classically. The evolution in time of the statistical properties of the quantized Stokes and anti-Stokes waves are analyzed by means of appropriate time-dependent phasespace distributions under various assumptions for the initial fields. Expansions of the field variables in terms of complete sets of orthogonal operators are discussed. Several regimes of operation are considered. It is shown that an initially coherent state develops, in general, into a superposition of a coherent state and a chaotic state. The amplitudes of the coherent components follow the same equations of motion as the mode operators. The chaotic components stem from the amplification of the vacuum fluctuations.
In an earlier paper, we considered a Hamiltonian trilinear in boson operators, which described several physical systems of interest such as interaction of NNN two-level atoms with a single-mode resonant radiation field, parametric amplification, frequency conversion, Raman and Brillouin scattering, etc. We found that the dynamics of the system can almost exactly be determined if either one of the three modes is strong for all times or any two modes are separately strong in two complementary time domains. In the present paper, we determine the statistical behavior of the system in each case. We give explicit results for a number of initial states (e.g., coherent, chaotic, a mixture of coherent and chaotic, number state, etc.) of the modes 1, 2, and 3.
We consider a trilinear Hamiltonian in boson operators describing various physical processes such as frequency conversion, Raman or Brillouin scattering, or the interaction of N two-level atoms with a single-mode radiation field. Due to the fact that two independent integrals of motion can be found, the solution of the dynamics of the system is reduced to the diagonalization of a finite matrix (as was already shown by Walls and Barakat [Phys. Rev. A 1, 446 (1970)]). Performing a numerical diagonalization, we analyze the statistical properties of the field modes (sub-Poissonian statistics, anticorrelation, squeezing). We also pay attention to the appearance of collapses and revivals in the mean photon number of the modes. The relation of this model to the model of two coupled modes with an intensity-dependent coupling constant is pointed out.