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arXiv:1711.08017 · International Journal of Quantum Information

Continuous variable entanglement of counter-propagating twin beams

A. Gatti, E. Brambilla

Published Dec 1, 2017• 2 citations
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Abstract

This work describes the continuous variable entanglement of the counter-propagating twin beams generated in a Mirrorless Optical Parametric Oscillator (MOPO) below threshold, encompassing both their quadrature and photon-number correlation. In the first case, a comparison with the single-pass co-propagating geometry outlines a completely different stability of the two sources with respect to the phase-angle. In the second case, stimulated by the critical divergence of the correlation time evidenced by Corti et al., we address the issue of the temporal bandwidth of the intensity squeezing.

DOI
10.1142/s0219749917400172

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This work describes the continuous variable entanglement of the counter-propagating twin beams generated in a Mirrorless Optical Parametric Oscillator (MOPO) below threshold, encompassing both their quadrature and photon-number correlation. In the first case, a comparison with the single-pass co-propagating geometry outlines a completely different stability of the two sources with respect to the phase-angle. In the second case, stimulated by the critical divergence of the correlation time evidenced by Corti et al., we address the issue of the temporal bandwidth of the intensity squeezing.
This work analyzes the temporal coherence and correlation of twin beams generated in a quasi-phase-matched nonlinear crystal in a counterpropagating configuration, ranging from the low-gain regime, where counterpropagating photon pairs are generated spontaneously, to the regime of stimulated pair production, close to the mirrorless optical parametric oscillator threshold. Here, we show a critical divergence of the correlation time and slowing down of quantum fluctuations originating from the feedback mechanism responsible for the MOPO threshold.
Quantum entanglement and intensity correlations in continuous variable (CV) regimes have investigated in the time domain in addition to many analogous investigations performed in the frequency domain. The specific signatures of time-resolved CV quadrature entanglement and intensity difference squeezing have been analyzed for periodically-pulsed optical parametric oscillator (OPO). An application to quantum key distribution scheme based on intensity correlation of twin beams has also been considered.
Twin photon pairs generated through parametric downconversion (PDC) in a χ 2 medium is one of the most widely used source of entanglement. We focus here on a non-conventional geometry in which one of the twin photons propagates in the opposite direction with respect to the pump beam, exploiting quasi-phase-matching in a periodically poled crystal. Through predicted almost 50 years ago, this new PDC configuration has been realized experimentally only recently [1] thanks to new fabrication techniques achieving the required sub-micrometer poling period. Because of the presence of distributed feedback, the optical system has been shown to behave as a Mirrorless Optical Parametric Oscillator (MOPO) and exhibits peculiar spectral properties which strongly differ from those found in more common geometries involving co-propagating beams. In this work we provide a detailed analysis of the correlation and coherence properties of counter-propagating twin beams both in the purely spontaneous regime and in the neighborhood of the MOPO threshold. We consider on the on side the regime of spontaneous pair production where the characteristic narrow band of the counter-propagating twin beams offers the unique opportunity of generating heralded single photon states with a high degree of purity, a relevant property for applications in quantum communication. In this context, we investigate how the degree of separability of the twin photon state varies with the pump pulse duration τp. We find that two well separated time scales characterize the system dynamics: a short time scale τgvm, in the picoseconds range, corresponding to the typical temporal delay of co-propagating waves due to group-velocity mismatch, and a much longer time scale τgvs associated with the temporal separation of counter-propagating waves. Such a difference of time scales occurs naturally in the counterpropagating configuration, for basically any kind of material and tuning condition. Because of this same feature, counter-propagating twin photons in a pure state can in principle be heralded at any wavelength by choosing the appropriate poling period. We show that a high degree of separability can be achieved when the pump pulse duration satisfy the condition τgvm ≪ τp ≪ τgvs, as put in evidence from the evaluation of Schmidt number as a function of the pump pulse duration which reaches a minimum close to unity in this region. The separability is lost in the nearly monochromatic limit ( τp ≫ τgvs ) as well as for ultra-short pulses ( τp ≪ τgvm ), where the entanglement between the signal and idler frequencies can be inferred by the non factorable shape of the spectral biphoton amplitude. We offer a physical interpretation of such a behaviour, and a detailed analysis of the Schmidt number characterizing the entanglement of the state. We also considered a completely different regime of operation, close to the MOPO threshold, where the combined effect of stimulated PDC and distributed feedback affects dramatically the property of coherence of the field. Our analysis put in evidence a progressive narrowing of both the spectral twin beam correlation and the intensity spectra as the pump field intensity approaches its threshold value. This translates into a drastic increase of the correlation and coherence times in the temporal domain, a feature which can be attributed to the critical slowing down of the fluctuation dynamics characterizing the transition toward coherent emission occurring at the MOPO threshold. Furthermore, we investigate the potentiality of the source to generate squeezing and EPR type correlations in the threshold vicinity. In this regards, the obtained results shows that the system displays a behaviour which is very similar to that found in standard optical parametric oscillators enclosed in a resonant cavity. In the last part of the work, we present some preliminary results from numerical simulations illustrating the transition above the MOPO thresholds. We also take into account non collinear PDC emission, showing explicitely that the spatial and the temporal degrees of freedoms of the emitted twin photons are almost uncoupled. This feature strongly distinguish the counter-propagating configuration from standard co-propagating geometries where the phase-matching mechanism usually leads to strong angular dispersion.
We report the realization of a classically phase-locked source of quantum twin beams. Theoretical work by Reid and Drummond, and more recently by van Loock and Braunstein, predicts that such a source is adequate for creating bipartite and multipartite continuous-variable entanglement. The source is a type-II optical parametric oscillator (OPO) above threshold. Its exceptional frequency and intensity stability is derived from careful design and from three servo loops. The OPO can stably emit on the frequency-degenerate mode, pumped a few percent above the threshold. We observe a preliminary number-difference squeezing of 5.5 dB (6.4 dB inferred) at 200 kHz. In order to enable future EPR measurements, the signal and idler fields must also be classically phase-locked: we measure a signal-idler-frequency-difference linewidth smaller than 300 Hz for our OPO.
We present the first measurement of squeezed-state entanglement between the twin beams produced in an optical parametric oscillator operating above threshold. In addition to the usual squeezing in the intensity difference between the twin beams, we have measured squeezing in the sum of phase quadratures. Our scheme enables us to measure such phase anticorrelations between fields of different frequencies. In the present measurements, wavelengths differ by approximately 1 nm. Entanglement is demonstrated according to the Duan et al. criterion [Phys. Rev. Lett. 84, 2722 (2000)] Delta2p- + Delta2q+ = 1.41(2) < 2. This experiment opens the way for new potential applications such as the transfer of quantum information between different parts of the electromagnetic spectrum.