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Physical Review A

Up-conversion dynamics for temporally entangled two-photon pulses

Masatoshi Nakatani, Ryōsuke Shimizu, Kazuki Koshino

Published Jan 26, 2011• 5 citations
Publisher

Abstract

We analyze the up conversion of a two-photon pulse having temporal entanglement on the basis of a full quantum formalism that treats both photons and optical media quantum mechanically. We derive a formula of the up-converted photon wave function, which is applicable to arbitrary input two-photon states for a three-level system, as the simplest second-order nonlinear optical system. As the input, we employ three kinds of temporally entangled two-photon pulses: correlated, uncorrelated, and anticorrelated. We observe the up-conversion efficiency and the temporal profile of the up-converted photon. Our results reveal the crossover behavior of the up conversion from anticorrelation to correlation and show how the temporal correlation in the input is reflected in the up-conversion process.

DOI
10.1103/physreva.83.013824

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We analyze the up conversion of a two-photon pulse having temporal entanglement on the basis of a full quantum formalism that treats both photons and optical media quantum mechanically. We derive a formula of the up-converted photon wave function, which is applicable to arbitrary input two-photon states for a three-level system, as the simplest second-order nonlinear optical system. As the input, we employ three kinds of temporally entangled two-photon pulses: correlated, uncorrelated, and anticorrelated. We observe the up-conversion efficiency and the temporal profile of the up-converted photon. Our results reveal the crossover behavior of the up conversion from anticorrelation to correlation and show how the temporal correlation in the input is reflected in the up-conversion process.
The up-conversion dynamics of two photons in a three-level system is investigated theoretically. The analysis is based on a rigorous formalism that incorporates the multimode nature of the photon field, enabling quantitative handling of the spatial profiles of the photon pulses. We have derived an analytic formula that connects the wavefunction of two input photons to that of an up-converted output photon. The up-conversion efficiency and the output pulse profile are numerically calculated using this formula.
Multiphoton effects in down-conversion are investigated based on the full-quantum multimode formalism by considering a three-level system as a prototype nonlinear system. We analytically derive the three-photon output wave function for two input photons, where one of the two input photons is down-converted and the other one is not. Using this output wave function, we calculate the down-conversion probability, the purity, and the fidelity to evaluate the entanglement between a down-converted photon pair and a non-down-converted photon. It is shown that the saturation effect occurs by multiphoton input and that it affects both the down-conversion probability and the quantum correlation between the down-converted photon pair and the non-down-converted photon. We also reveal the necessary conditions for multiphoton effects to be strong.
The time-energy entanglement of down-converted photon pairs is particularly difficult to characterize because direct measurements of photon arrival times are limited by the temporal resolution of photon detection. Here, we explore an alternative possibility of characterizing the temporal coherence of two-photon wave functions using single-photon interference with weak coherent light. Specifically, this method makes use of the fact that down conversion generates a coherent superposition of vacuum and two-photon states, so that the coincidence count rates for photon pairs after interference with weak coherent light are given by a superposition of the two-photon wave function from the down conversion with the product wave function defined by the weak coherent references. By observing the frequency-dependent interference pattern, it is possible to reconstruct the amplitudes and the phases of the two-photon wave function within the bandwidth of the reference pulses used in the single-photon interferences. The correlated single-photon interferences therefore provide a direct map of the entangled two-photon wave function generated in the down-conversion process.
When two-photon interactions are induced by down-converted light with a bandwidth that exceeds the pump bandwidth, they can obtain a behavior that is pulselike temporally, yet spectrally narrow. At low photon fluxes this behavior reflects the time and energy entanglement between the down-converted photons. However, two-photon interactions such as two-photon absorption (TPA) and sum-frequency generation (SFG) can exhibit such a behavior even at high power levels, as long as the final state (i.e., the atomic level in TPA, or the generated light in SFG) is narrow-band enough. This behavior does not depend on the squeezing properties of the light, is insensitive to linear losses, and has potential applications. In this paper we describe analytically this behavior for traveling-wave down conversion with continuous or pulsed pumping, both for high- and low-power regimes. For this we derive a quantum-mechanical expression for the down-converted amplitude generated by an arbitrary pump, and formulate operators that represent various two-photon interactions induced by broadband light. This model is in excellent agreement with experimental results of TPA and SFG with high-power down-converted light and with entangled photons [Dayan et al., Phys. Rev. Lett. 93, 023005 (2004); 94, 043602 (2005); Pe'er et al., ibid. 94, 073601 (2005)].
Time- and frequency-gated two-photon counting is given by a four-time correlation function of the electric field. This reduces to two times with purely time gating. We calculate this function for entangled photon pulses generated by parametric down-conversion. At low intensity, the pulses consist of well-separated photon pairs, and crossover to squeezed light as the intensity is increased. This is illustrated by the two-photon absorption signal of a three-level model, which scales linearly for a weak pump intensity where both photons come from the same pair, and gradually becomes nonlinear as the intensity is increased. We find that the strong frequency correlations of entangled photon pairs persist even for higher photon numbers. This could help facilitate the application of these pulses to nonlinear spectroscopy, where these correlations can be used to manipulate congested signals.
We theoretically investigate the sequential two-step upconversion of correlated photon pairs with positive and negative energy correlations, in terms of how the up-conversion efficiency depends on the incident pulse delay. A three-level atomic system having a metastable state is used to evaluate the up-conversion efficiency. It is shown that a photon pair with a positive energy correlation can drastically enhance the up-conversion efficiency compared with uncorrelated photons and correlated photons with a negative energy correlation.