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Physical Review A

Nondegenerate parametric interactions and nonclassical effects

A. Bandilla, G. Drobný, Igor Jex

Published Jan 1, 1996• 29 citations
Publisher

Abstract

We consider the classical and quantum-mechanical processes of three-wave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the out-of-phase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly sub-Poissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sum-frequency generation. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.507

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We consider the classical and quantum-mechanical processes of three-wave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the out-of-phase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly sub-Poissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sum-frequency generation. \textcopyright{} 1996 The American Physical Society.
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.
In the framework of Hamiltonian formalism nonclassical effects of an optical field propagating inside a directional coupler containing nondegenerate parametric amplification have been studied. We investigate the effect of switching between the input modes and the out going fields from the coupler. Particular attention has been paid to two mode squeezing, second-order correlation function, quasiprobability distribution functions, and photon-number distribution. Incident number and coherent states are considered. It has been shown that when one of the modes enters the coupler in the Fock state and the other modes are in vacuum states, the coupler can serve as a generator for coherent state. Furthermore, regimes for generation and transmission of squeezed and/or sub-Poissonian light are found.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
Abstract We theoretically investigate the emergence of nonclassical features, two-mode squeezing, sub-Poissonian photon statistics, and bipartite entanglement, in the third harmonic generation (THG) process, employing a first-order interaction Hamiltonian within the short-time approximation. The study reveals that both first- and second-order squeezing exhibit strong dependence on the input field amplitudes, relative phase, and interaction strength. Asymmetric coherent inputs, where the harmonic mode carries greater intensity than the fundamental, enhance the degree of squeezing and lead to deeper nonclassical minima. The squeezing parameters display periodic phase-dependent oscillations, indicating phase-sensitive control of quantum correlations. Sub-Poissonian photon statistics in the fundamental mode further confirm the quantum nature of the generated light. Entanglement is analyzed through the Duan et al. inseparability criterion and the Hillery–Zubairy (HZ) criteria. While the Duan criterion fails to detect entanglement under purely coherent excitation in the short-time regime, both the HZ-1 and HZ-2 criteria confirm entanglement under specific phase conditions, with HZ-2 exhibiting greater sensitivity at elevated field strengths. These results underscore the potential of THG as a viable platform for the controllable generation of nonclassical light, with implications for continuous-variable quantum optics and integrated quantum photonics.
In this paper, we follow the model presented in our previous work [J. Opt. Soc. Am. B30, 1109 (2013)], in which the interaction between a Λ-type three-level atom and a quantized two-mode radiation field in a cavity in the presence of nonlinearities was studied. After giving a brief review of the procedure for obtaining the state vector of the atom-field system, we investigated some further interesting and important physical features of the whole system state, which are of particular interest to the quantum optics field of research, i.e., the number-phase entropic uncertainty relation (based on the two-mode Pegg–Barnett formalism) and a few of the nonclassicality signs that consist of sub-Poissonian statistics, Cauchy–Schwartz inequality, and two types of squeezing phenomena. During our presentation, the effects of intensity-dependent coupling, deformed Kerr medium, and the detuning parameters on the depth and domain of each of the mentioned nonclassical criteria of the considered quantum system are described in detail. The results show that each of the mentioned nonclassicality aspects can be obtained by appropriately choosing the related parameters.