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Physical Review A

Dynamics of processes with a trilinear boson Hamiltonian

Salvatore Carusotto

Published Aug 1, 1989• 31 citations
Publisher

Abstract

We consider a Hamiltonian trilinear in boson operators, which describes several physical processes, such as the interaction of N two-level atoms with a single-mode resonant radiation field, parametric amplification, Raman and Brillouin scattering, and frequency conversion. The dynamics of systems obeying this trilinear Hamiltonian is analyzed and the equations of motion are solved exactly. So the time development of the boson operators is described by means of Laplace transforms of suitable functionals. The integration method is based on iteration techniques and makes use of the Manley-Rowe relations. Finally, the results are applied to describe the time evolution of the boson operators when one of the three modes is initially in a high state of excitation.

DOI
10.1103/physreva.40.1848

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We consider a Hamiltonian trilinear in boson operators, which describes several physical processes, such as the interaction of N two-level atoms with a single-mode resonant radiation field, parametric amplification, Raman and Brillouin scattering, and frequency conversion. The dynamics of systems obeying this trilinear Hamiltonian is analyzed and the equations of motion are solved exactly. So the time development of the boson operators is described by means of Laplace transforms of suitable functionals. The integration method is based on iteration techniques and makes use of the Manley-Rowe relations. Finally, the results are applied to describe the time evolution of the boson operators when one of the three modes is initially in a high state of excitation.
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.
The interaction of a resonant radiation field with two-level atoms (in the long-wavelength limit) is described by a Hamiltonian trilinear in the boson operators. It is shown that the dynamics of the system can be determined nearly exactly if either at least one of the three modes is strongly populated for all times or any two modes are separately strong in two complementary time domains. Explicit results are given in two important specific cases: (i) pumping of a ground-state atomic system by a coherent radiation field and (ii) coherent spontaneous emission. The results are compared with previously known approximate results. Our results are also applicable to certain parametric processes such as parametric amplification, frequency conversion, Raman and Brillouin scattering, etc., where an identical trilinear Hamiltonian is used to describe these processes.
The trilinear Hamiltonian of Tucker and Walls, governing the parametric amplification, leads to three coupled nonlinear Heisenberg equations in terms of creation and annihilation operators. Using the Manley-Rowe relations the equations are reduced to a single nonlinear equation in terms of one of the number operators. Thereafter, it is solved by means of a perturbation method around the diagonal, allowing calculation of the mean number of photons and the variance of the amplified field. The short-time approximation, the parametric approximation, and the classical limit are reobtained by suitable limiting processes.
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.