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Physical Review A

Signal-pump entanglement in quantumk-photon down-conversion

Vladimír Bužek, G. Drobný

Published Feb 1, 1993• 24 citations
Publisher

Abstract

We study the quantum dynamics of the k-photon down-conversion process. We concentrate our attention on the entanglement between the pump and the signal modes. We show that the degree of the entanglement between the pump and the signal depends on the initial statistics of the light field in both modes. Moreover, the higher the nonlinear process the stronger the entanglement is. The entanglement between the modes is related to the marginal entropy in the pump (signal) mode; i.e., the larger the entanglement, the larger the entropy is. In the quantum k-photon down-conversion the signal mode at t>0 is not generally in a pure state which restricts applicability of the parametric approximation.

DOI
10.1103/physreva.47.1237

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We study the quantum dynamics of the k-photon down-conversion process. We concentrate our attention on the entanglement between the pump and the signal modes. We show that the degree of the entanglement between the pump and the signal depends on the initial statistics of the light field in both modes. Moreover, the higher the nonlinear process the stronger the entanglement is. The entanglement between the modes is related to the marginal entropy in the pump (signal) mode; i.e., the larger the entanglement, the larger the entropy is. In the quantum k-photon down-conversion the signal mode at t>0 is not generally in a pure state which restricts applicability of the parametric approximation.
We study entanglement between field modes in the process of nondegenerate two-photon down-conversion with quantized pump. We show that due to the quantum nature of the dynamics, strong entanglement between the pump and the signal-idler subsystems can be observed. We find that the higher the initial intensity of the pump mode the stronger the entanglement between the pump and the signal-idler subsystem is established during the first instants of the time evolution. We also show that the signal and the idler modes are strongly entangled (correlated). This entanglement is much stronger than the entanglement between the pump and the signal-idler subsystem. Correlation between the signal and the idler modes leads to a high degree of two-mode squeezing, which can be observed during the first instants of the time evolution when the pump mode is still approximately in a pure state. On the other hand, the back action of the signal-idler subsystem on the pump mode leads to a strong single-mode squeezing of the pump mode. At the time interval during which squeezing of the pump mode can be observed the pump mode is far from being in the minimum uncertainty state. We also analyze the long-time behavior of the quantum-optical system under consideration and we show that the interesting collapse-revival effect in the time evolution of the mean photon number and of the purity parameters of field modes can be observed. Finally, we show that the degree of entanglement between modes in the nondegenerate quantum-optical down-conversion strongly depends on the initial state of the system.
We show that in the k-photon down-conversion a fundamental limit on the energy transfer does exist providing the signal mode is initially in the vacuum state and the pump mode is excited. In particular, in the two-photon down-conversion less than 2/3 of the pump energy can be transferred from the pump to the signal mode. With the increase of the order of the nonlinear process under consideration the efficiency is even smaller. On the contrary, we show that no restriction on the efficiency of the energy transfer in the kth harmonic generation does exist, i.e., in this process the total energy from the initially excited mode can be transferred into the mode which initially was in the vacuum state. We study restrictions implied by the fundamental limit on the energy transfer in the k-photon down-conversion on nonclassical effects which can be observed in the signal mode in the process.
A quantum theory of parametric amplification and frequency conversion of an optical image in coupled nonlinear optical processes that include one parametric amplification process at high-frequency pumping and two up-conversion processes in the same pump field is developed. The field momentum operator that takes into account the diffraction and group velocities of the waves is used to derive the quantum equations related to the spatial dynamics of the images during the interaction. An optical scheme for the amplification and conversion of a close image is considered. The mean photon number density and signal-to-noise ratio are calculated in the fixed-pump-field approximation for images at various frequencies. It has been established that the signal-to-noise ratio decreases with increasing interaction length in the amplified image and increases in the images at the generated frequencies, tending to asymptotic values for all interacting waves. The variance of the difference of the numbers of photons is calculated for various pairs of frequencies. The quantum entanglement of the optical images formed in a high-frequency pump field is shown to be converted to higher frequencies during the generation of sum frequencies. Thus, two pairs of entangled optical images are produced in the process considered.
We use a nonperturbative theoretical approach to the parametric down-conversion (PDC) process, which generates an entangled-photon field for an arbitrarily strong pump-pulse. This approach can be used to evaluate multipoint field correlation functions to compute nonlinear spectroscopic signals induced by a strong pump. The entangled-photon statistics is studied using Glauber’s g (2) function, which helps understand the significance of the photon entanglement-time and the pump-pulse intensity on spectroscopic signals. Under the nonperturbative treatment of the entangled field, the two-photon absorption (TPA) signal shows linear to strongly nonlinear growth with the pump intensity, rather than the linear to quadratic scaling reported previously. An increase in the range of pump intensity for the linear scaling is observed as the pump bandwidth is increased. We propose an experimental scheme that can select contributions to the TPA signal that arise solely from interactions with the entangled photons, and filter out unentangled-photon contributions, which are dominant at higher pump intensities, paving a way to explore the entanglement effects at higher intensities.
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.