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Physical Review Letters

Evaluation of Two-Photon Nonlinearity by a Semiclassical Method

Kazuki Koshino, Hajime Ishihara

Published Oct 19, 2004• 28 citations
Publisher

Abstract

In order to discuss the two-photon nonlinearity theoretically, both photons and nonlinear materials should be treated quantum mechanically, which usually is a heavy theoretical task. Contrarily, nonlinear optics for classical light has been developed well and a detailed analysis is possible for realistic complex nonlinear systems. Here we show that the two-photon nonlinearity can be evaluated from the linear and third-order nonlinear output fields against a classical input pulse, which contains 2(-1/2) photons on average.

DOI
10.1103/physrevlett.93.173601

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In order to discuss the two-photon nonlinearity theoretically, both photons and nonlinear materials should be treated quantum mechanically, which usually is a heavy theoretical task. Contrarily, nonlinear optics for classical light has been developed well and a detailed analysis is possible for realistic complex nonlinear systems. Here we show that the two-photon nonlinearity can be evaluated from the linear and third-order nonlinear output fields against a classical input pulse, which contains 2(-1/2) photons on average.
When two distinguishable photons are simultaneously input into a nonlinear system, the output photons undergo nonlinear optical effects of the cross--Kerr type. Theoretical quantification of this two-photon cross--Kerr effect requires, in principle, a fully quantum-mechanical analysis involving heavy computation. In this paper, we propose a method for evaluating the two-photon cross--Kerr effect using a semiclassical optical response theory. The semiclassical method enables precise evaluation of the cross--Kerr effect with greatly reduced computation. The validity of the method is confirmed using a model nonlinear system.
In order to investigate the nonlinear dynamics of photons, one must in principle solve a quantum many-particle problem, which usually requires intensive computation. In this study, we show that the spatial wave function of photons after nonlinear interaction can be obtained with less computation by assuming a classical input pulse and calculating a correlation function in the output field. This method is particularly useful when nonlinear optical media have many mechanical degrees of freedom, where quantum many-particle problems become extremely difficult.
Exploiting the field-amplification effect of a cavity, the possibility of optical nonlinearity by only two photons was indicated experimentally. In the present article, we review our recent analysis of the two-photon dynamics in a cavity quantum electrodynamics (QED) system. Since a cavity-QED system is highly dispersive around its resonances, the shapes of photonic pulses are significantly deformed through interaction with the system. Thus, the present analysis is based on a formalism beyond single-mode approximations. The external photon field is treated rigorously as a continuum, which enables us to handle the two-photon wavefunction in the space representation. The degree of optical nonlinearity in a two-photon state is quantified by comparing the output wavefunction with the linear output wavefunction. It is revealed that the semiclassical optical response theory can be applied for evaluation of the two-photon optical nonlinearity. The two-photon nonlinearity appears not purely as a phase shift in the output wavefunction. The degradation of the fidelity between the output wavefunction and the linear output wavefunction always occurs, which hinders the application of this nonlinear effect as a quantum phase gate. The optimum condition for maximizing the two-photon nonlinearity is clarified, suggesting that pulse shape control is more essential than the Q-value control of the cavity QED system.
Abstract In order to analyze the dynamics of single photons, a full‐quantum analysis is indispensable in principle, where both the photon field and the material system are treated quantum‐mechanically. However, the semiclassical optical response theory can be utilized to infer the photon dynamics, bypassing the full‐quantum analyses. This fact is demonstrated for a concrete problem of evaluation of the Kerr effect appearing in two photons. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)