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Physical Review A

Statistics of field modes in the process ofk-photon down-conversion with a quantized pump

G. Drobný, Igor Jex

Published Apr 1, 1992• 15 citations
Publisher

Abstract

We solve numerically the dynamics of a system related to the problem of k-photon down-conversion with a quantized pump. We analyze in detail the statistical properties of the field modes. We show that the fields exhibit sub-Poissonian statistics and anticorrelation. The influence of different initial Fock states as well as their coherent superpositions on the field statistics and squeezing of the modes is analyzed in detail.

DOI
10.1103/physreva.45.4897

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We solve numerically the dynamics of a system related to the problem of k-photon down-conversion with a quantized pump. We analyze in detail the statistical properties of the field modes. We show that the fields exhibit sub-Poissonian statistics and anticorrelation. The influence of different initial Fock states as well as their coherent superpositions on the field statistics and squeezing of the modes is analyzed in detail.
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 characteristics of the photon statistical properties are studied in a soluble three-level plus two-mode model. The model consists of a single lambda-configuration three-level emitter interacting with two modes of the quantized radiation field, the emitter being initially in the lowest level. The normally ordered variances of the photon numbers and the cross-correlation function are calculated All the results are obtained without using perturbation theory and decorrelation approximations. It is shown that the states of the signal-mode field produced from its initial vacuum are sub-Poissonian for all times. For the case when both the signal-mode and pump-mode fields are initially in the Glauber coherent states the temporal behaviour of the photon statistical characteristics are described Under suitable conditions photon antibunching and anticorrelation between the modes are potentially possible in the system.
We investigate the effect of the cumulative phase on the photon statistics of the three-mode state whose evolution is described by the trilinear Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}=i\ensuremath{\hbar}\ensuremath{\kappa}\left(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{c}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}c\right)$, wherein the pump is taken to be quantized (and prepared in a coherent state) and the signal and idler modes are initially seeded with coherent states. We provide a brief review of the two-mode squeezed coherent states generated by nondegenerate coherently stimulated parametric down-conversion, whereby the nonlinear crystal is driven by a strong classical field. The statistics of the resulting two-mode state have been shown to depend greatly on the cumulative phase $\mathrm{\ensuremath{\Phi}}={\ensuremath{\theta}}_{s}+{\ensuremath{\theta}}_{i}\ensuremath{-}2\ensuremath{\phi}$ where ${\ensuremath{\theta}}_{s\left(i\right)}$ are the signal(idler) coherent state phases and $2\ensuremath{\phi}$ is the classical pump phase. Using perturbation theory, we analytically show for short times how the photon statistics and entanglement properties of the resultant state depends strictly on this phase combination. We also present numerical results of the relevant quantities to show the evolution of the three modes and provide a qualitative analysis of the near-thermal-like nature of the steady state valid for long times. Finally, we examine the effect of the cumulative phase $\mathrm{\ensuremath{\Phi}}$ on the correlations between the bipartite system of pump and signal/idler modes, and the effect of the quantized pump on the correlations between signal and idler modes.
We study the coupled system of the dressed atom and the quantized mode of the electromagnetic field (an open modification of the Jaynes-Cummings model). The mode is initially excited to the coherent quantum state and exhibits Poissonian photon statistics. When the quasienergy levels of the dressed atom cross, the mean number of photons in the mode oscillates at high amplitude. Numerical results show that the photon statistics in the mode regularizes, correlating as the value of falls. Near the local minima of , the statistics can become sub-Poissonian with Fano factor many times smaller than unity.