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Physical Review A

Transitional behavior between self-Kerr and cross-Kerr effects by two photons

Kazuki Koshino

Published Jun 8, 2007• 4 citations
Publisher

Abstract

The transitional behavior of the two-photon Kerr effect between the self-Kerr and cross-Kerr cases is investigated. To this end, we have developed a semiclassical method for evaluating the two-photon Kerr effect that is applicable to any two-photon input state. It is revealed that the maximum Kerr effect is obtained when the second photon is input with a delay time that corresponds to the absorption time of the first photon by the optical material.

DOI
10.1103/physreva.75.063807

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The transitional behavior of the two-photon Kerr effect between the self-Kerr and cross-Kerr cases is investigated. To this end, we have developed a semiclassical method for evaluating the two-photon Kerr effect that is applicable to any two-photon input state. It is revealed that the maximum Kerr effect is obtained when the second photon is input with a delay time that corresponds to the absorption time of the first photon by the optical material.
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.
While the theoretical studies in waveguide quantum electrodynamics predominate with single-photon and two-photon Fock-state (photon number states) input, the experiments are primarily carried out using a faint coherent light. We create a theoretical toolbox to compare and contrast linear and nonlinear light scattering by a two-level or a three-level emitter embedded in an open waveguide carrying Fock-state or coherent-state inputs. We particularly investigate light transport properties and the Kerr and cross-Kerr nonlinearities of the medium for the two types of inputs. A generalized description of the Kerr and cross-Kerr effect for different types of inputs is formulated using the first-order correlation function to compare the Kerr and cross-Kerr nonlinearity between two photons in these models.
The self-Kerr interaction is an optical nonlinearity that produces a phase shift proportional to the square of the number of photons in the field. At present, many proposals use nonlinearities to generate photon-photon interactions. For propagating fields these interactions result in undesirable features such as spectral correlation between the photons. Here we engineer a discrete network composed of cross-Kerr interaction regions to simulate a self-Kerr medium. The medium has effective long-range interactions implemented in a physically local way. We compute the one- and two-photon SSS matrices for fields propagating in this medium. From these scattering matrices we show that our proposal leads to a high-fidelity photon-photon gate. In the limit where the number of nodes in the network tends to infinity, the medium approximates a perfect self-Kerr interaction in the one- and two-photon regime.
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)
For pt.I see ibid., vol.8, no.3, 339 (1975). The transient behaviour of two short optical pulses in two-photon coherent propagation is obtained by calculating numerically the coherent propagation equations which couple the two electromagnetic pulses and the atomic system in self-consistent fashion. The coherent interaction of the two different frequency pulses gives rise to lossless propagation in the absorptive medium. This description is characterized by a two-photon area theorem. Novel coherent propagation effects of peak amplification and multiple pulse breakup are found. These effects are ascribed to two-photon self-induced transparency analogous to that of a one-photon transition in a two-level system. A difference of input intensity between the two different frequency pulses yields curious modulation effects involving sharp peak amplification and marked pulse breakup for the low-intensity input. In contrast, the high-intensity input pulse displays a smooth variation with distance and time. Low-loss pulse transmission above threshold, and pulse advance and delay are found for the present case in contrast to that for one-photon self-induced transparency.
A cross-Kerr interaction produces a phase shift on two modes of light proportional to the number of photons in both modes and is sometimes called cross-phase modulation. Cross-Kerr nonlinearities have many applications in classical and quantum nonlinear optics, including the possibility of a deterministic and all-optical controlled-phase gate. We calculate the one- and two-photon SSS matrices for fields propagating in a medium where the cross-Kerr interaction is spatially distributed at discrete interaction sites comprised of atoms. For the interactions considered, we analyze the cases where the photons copropagate and counterpropagate through the medium and give a physical interpretation to the differences between the two cases. Finally, we obtain the SSS matrix in the limit of infinitely long chains, showing that it corresponds to a perfect controlled-phase operation.