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arXiv:1905.05756 · Physical Review A

Effect of an incoherent pump on two-mode entanglement in optical parametric generation

Stephen Vintskevich, D. A. Grigoriev, Sergey N. Filippov

Published Nov 6, 2019• 9 citations
arXivHTMLPDF

Abstract

Pumping a nonlinear crystal by an intense radiation results in the optical parametric generation of photons in two modes (the signal and the idler). The quantized electromagnetic field in these modes is described by a continuous-variable quantum state, which is entangled if the pump is a coherent state produced by a laser. The signal and the idler modes remain populated by photons even if the pump becomes incoherent (dephased by a medium, superposed with a thermal state, or produced by an alternative source such as the superluminescent diode). However, the incoherent pump does effect the entanglement and purity of the signal and the idler modes, which is of vital importance for quantum information applications and interferometry. Here we develop an approach to infer the signal-idler entanglement and purity for a general quantum incoherent pump with the given Glauber-Sudarshan function. We show that the signal-idler entanglement is extremely sensitive to the phase distribution of the pump and illustrate our findings by physically relevant examples of the incoherent pump: the noisy coherent state, slightly dephased and phase-averaged coherent states, the thermal state, and states modulated by the Kerr medium. The effect of an incoherent pump on the combined quadratures is discussed as well.

DOI
10.1103/physreva.100.053811

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Pumping a nonlinear crystal by an intense radiation results in the optical parametric generation of photons in two modes (the signal and the idler). The quantized electromagnetic field in these modes is described by a continuous-variable quantum state, which is entangled if the pump is a coherent state produced by a laser. The signal and the idler modes remain populated by photons even if the pump becomes incoherent (dephased by a medium, superposed with a thermal state, or produced by an alternative source such as the superluminescent diode). However, the incoherent pump does effect the entanglement and purity of the signal and the idler modes, which is of vital importance for quantum information applications and interferometry. Here we develop an approach to infer the signal-idler entanglement and purity for a general quantum incoherent pump with the given Glauber-Sudarshan function. We show that the signal-idler entanglement is extremely sensitive to the phase distribution of the pump and illustrate our findings by physically relevant examples of the incoherent pump: the noisy coherent state, slightly dephased and phase-averaged coherent states, the thermal state, and states modulated by the Kerr medium. The effect of an incoherent pump on the combined quadratures is discussed as well.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
The correlation properties of the pump field in spontaneous parametric down-conversion are crucial in determining the degree of entanglement of generated signal and idler photons. We find theoretically that continuous-variable entanglement of the transverse positions and momenta of these photons can be achieved only if the coherence of the pump beam is sufficiently high. The positions of signal and idler photons are found to be correlated, even for an incoherent pump. However, the momenta of the signal and idler photons are not anti-correlated, even though transverse momentum is conserved.
Abstract The correlation properties of the pump field in spontaneous parametric down-conversion are crucial in determining the degree of entanglement of generated signal and idler photons. We find theoretically that continuous-variable entanglement of the transverse positions and momenta of these photons can be achieved only if the coherence of the pump beam is sufficiently high. The positions of signal and idler photons are found to be correlated, even for an incoherent pump. However, the momenta of the signal and idler photons are not anti-correlated, even though transverse momentum is conserved.
We investigate the effect of the cumulative phase on the photon statistics of the three-mode state whose evolution is described by the trilinear Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}=i\ensuremath{\hbar}\ensuremath{\kappa}\left(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{c}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}c\right)$, wherein the pump is taken to be quantized (and prepared in a coherent state) and the signal and idler modes are initially seeded with coherent states. We provide a brief review of the two-mode squeezed coherent states generated by nondegenerate coherently stimulated parametric down-conversion, whereby the nonlinear crystal is driven by a strong classical field. The statistics of the resulting two-mode state have been shown to depend greatly on the cumulative phase $\mathrm{\ensuremath{\Phi}}={\ensuremath{\theta}}_{s}+{\ensuremath{\theta}}_{i}\ensuremath{-}2\ensuremath{\phi}$ where ${\ensuremath{\theta}}_{s\left(i\right)}$ are the signal(idler) coherent state phases and $2\ensuremath{\phi}$ is the classical pump phase. Using perturbation theory, we analytically show for short times how the photon statistics and entanglement properties of the resultant state depends strictly on this phase combination. We also present numerical results of the relevant quantities to show the evolution of the three modes and provide a qualitative analysis of the near-thermal-like nature of the steady state valid for long times. Finally, we examine the effect of the cumulative phase $\mathrm{\ensuremath{\Phi}}$ on the correlations between the bipartite system of pump and signal/idler modes, and the effect of the quantized pump on the correlations between signal and idler modes.
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.
The quantum fluctuations and correlations of the output signal and idler beams from a type-II polarization nondegenerate optical parametric amplifier (NOPA) inside an optical cavity are investigated theoretically, which is driven by the different kinds of quantized fields such as the two-mode thermal field, the phase-conjugate state, and a pair of EPR beams at the signal and idler frequency. Spectral line shapes due to quantum interferences between the input quantum fields and the generated down-converted subharmonic fields of NOPA are studied by scanning the length of the NOPA cavity. The entanglement degree of the EPR entangled beams injected into the NOPA as signal and idler beams can be increased or decreased, which are realized by controlling the relative phase between the pump beam and the injected beams for the NOPA. The results demonstrate coherent phenomena of NOPA in the quantum regime and show phase-sensitive manipulations of quantum entanglement for quantum information processing.