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arXiv:quant-ph/0304072 · Physical Review A

Nonperturbative quantum solutions to resonant four-wave mixing of two single-photon wave packets

Mattias Johnsson, Michael Fleischhauer

Published Aug 13, 2003• 10 citations
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Abstract

We analyze both analytically and numerically the resonant four-wave mixing of two co-propagating single-photon wave packets. We present analytic expressions for the two-photon wave function, and show that quantum solutions exist which display a shape-preserving oscillatory exchange of excitations between the modes. Potential applications including quantum-information processing are discussed.

DOI
10.1103/physreva.68.023804

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We analyze both analytically and numerically the resonant four-wave mixing of two co-propagating single-photon wave packets. We present analytic expressions for the two-photon wave function, and show that quantum solutions exist which display a shape-preserving oscillatory exchange of excitations between the modes. Potential applications including quantum-information processing are discussed.
This thesis deals with applications of nonlinear four-wave mixing in the context of optical quantum information science. By enabling diverse functionalities such as single-photon generation, temporal shaping, and frequency conversion, four-wave mixing is a highly versatile tool for processing optical quantum information. The first part addresses the use of spontaneous four-wave mixing for generating entangled photon pairs with the particular property that the photons are uncorrelated in time and frequency. Two schemes are proposed to achieve this. The first proposal relies on a gradually varying interaction between two orthogonally polarized pump pulses in a birefringent waveguide. It is shown that, if the two pump pulses undergo a complete walk-through, the two generated photons emerge in orthogonal polarizations with uncorrelated- and tailorable time-frequency distributions. The second proposal deals with photon-pair generation in a microring resonator. In this setting, it is shown that a dual-pulse configuration can be used to excite- and subsequently de-excite the resonator on a sub-lifetime timescale. This turns out to strongly diminish time-frequency correlations. The second part considers the four-wave-mixing process Bragg scattering, which is known to enable noise-free frequency conversion. The Bragg-scattering process treated here, which occurs in a birefringent waveguide, is shown to allow for shape-preserving and strongly uni-directional conversion with a large acceptance bandwidth.
Three-level atoms in lambda configuration find diverse applications in quantum information processing, and a promising way to manipulate their quantum states is with single-photon pulses propagating in a waveguide (which can be theoretically regarded as a highly broadband regime of the Jaynes–Cummings model). Here, we analytically find the nonperturbative dynamics of a lambda atom driven by a two-photon wave packet, propagating in a 1D electromagnetic environment. We study the dynamics of a quantum state purification as an application. By comparing our exact model with an approximated model of two cascaded single-photon wave packets, we show how two-photon nonlinearities and stimulated emission affect the purification.
We present a full quantum analysis of resonant forward four-wave mixing based on electromagnetically induced transparency (EIT). In particular, we study the regime of efficient nonlinear conversion with low-intensity fields that has been predicted from a semiclassical analysis. We derive an effective nonlinear interaction Hamiltonian in the adiabatic limit. In contrast to conventional nonlinear optics, this Hamiltonian does not have a power expansion in the fields and the conversion length decreases with decreasing input power. We analyze the stationary wave-mixing process in the forward scattering configuration using an exact numerical analysis for up to 103{10}^{3}103 input photons and compare the results with a mean-field approach. Due to quantum effects, complete conversion from the two pump fields into the signal and idler modes is achieved only asymptotically for large coherent pump intensities or for pump fields in few-photon Fock states. The signal and idler fields are perfectly quantum correlated which has potential applications in quantum communication schemes. We also discuss the implementation of a single-photon phase gate for continuous quantum computation.
A microscopic correlation-function theory for coherent, parametric, four-photon processes (four-wave mixing) and for single-absorber two-photon processes (two-photon absorption, Raman, and fluorescence spectra) is developed. Both types of experimental observables are expressed in terms of four-point correlation functions of the dipole operator. The latter are then evaluated semiclassically or within the factorization approximation in which they are expressed in terms of ordinary single-photon line-shape functions. The common results of the Bloch equations are shown to be the outcome of two successive approximations: the factorization approximation and the impact limit whereby the single-photon line-shape functions are taken to be Lorentzian. Effects of fluctuations in the radiation fields are easily incorporated within the present approach.
We propose a method to achieve quantum entanglement of two Fock states with perfectly efficient, ultraslow propagation enhanced four-wave mixing. A cold atomic medium is illuminated with a two-mode cw control laser to produce coherent mixtures of excited states. An ultraslowly propagating, single-photon quantum probe field completes the four-wave mixing with 100% photon flux conversion efficiency, creating a depth dependent entanglement of two Fock states. We show that at a suitable propagation distance, a maximum entangled state is created with a single-photon wave-packet state that has 50% probability of being in each of two product-type Fock states.