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IEEE Journal of Quantum Electronics

Interference in a Quantum Channel Due to Classical Four-Wave Mixing in Optical Fibers

Nuno A. Silva, Álvaro J. Almeida, Armando N. Pinto

Published Jan 31, 2012• 10 citations
Publisher

Abstract

The second-order coherence function for idler photons generated through stimulated four-wave mixing (FWM) is theoretically derived, considering both spontaneous and stimulated Raman contributions. Two distinct cases and two different power regimes are analyzed. First, we consider that the idler wave is fully generated inside the fiber. In a second scenario, we admit that at the fiber input there exist a quantum channel and idler photons will be created at that frequency through the combined processes of FWM and Raman scattering. Results show that for the first case, in a low power regime, the idler photons follow a thermal statistics. In a moderate power regime, the statistics of the generated idler wave presents a Poissonian distribution. Results also show that in the second case, in a low power regime, the statistics of the input quantum channel goes from a Poissonian statistics at the fiber input to a thermal statistics at the fiber output. Findings show that in a moderate power regime, the quantum channel maintains its Poissonian distribution through fiber propagation.

DOI
10.1109/jqe.2012.2185219

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The second-order coherence function for idler photons generated through stimulated four-wave mixing (FWM) is theoretically derived, considering both spontaneous and stimulated Raman contributions. Two distinct cases and two different power regimes are analyzed. First, we consider that the idler wave is fully generated inside the fiber. In a second scenario, we admit that at the fiber input there exist a quantum channel and idler photons will be created at that frequency through the combined processes of FWM and Raman scattering. Results show that for the first case, in a low power regime, the idler photons follow a thermal statistics. In a moderate power regime, the statistics of the generated idler wave presents a Poissonian distribution. Results also show that in the second case, in a low power regime, the statistics of the input quantum channel goes from a Poissonian statistics at the fiber input to a thermal statistics at the fiber output. Findings show that in a moderate power regime, the quantum channel maintains its Poissonian distribution through fiber propagation.
We report on the theoretical statistical characterization of a single-photon source based on the stimulated four-wave mixing process inside an optical fiber. We quantify the photon statistics by means of the expectation number of photons generated on the signal and idler waves, 〈Ňs〉 and 〈Ňi〉 respectively, and Mandel's Qsand Qiparameters. The obtained solutions take into account the spontaneous and stimulated Raman contributions for the generation of the idler wave. Results show that the idler wave generated inside the fiber follows a super-Poissonian statistics. Findings also show that the signal wave change their own statistics from a Poissonian to a super-Poissonian, due to the interaction with the pump field and the optical fiber.
The rising interest in quantum-level communication has resulted in proposals for coexistence schemes with classical signals within the same fiber optic channel, where the most recent proposals leverage novel fibers designed for space-division multiplexing (SDM) transmission. In all cases the large power difference between classical and quantum channels presents challenges for such schemes, as the classical signals generate interfering noise that corrupts the quantum signal. In this work, we discuss the main interference mechanisms in coexistence scenarios and provide a model to quantify their impact on the quantum signal quality. Analytical approximations in the model allow accurate and fast numerical solutions in the millisecond time-scale. The model accounts for out-of-band non-linear interference effects, namely spontaneous Raman scattering (SpRS) and four-wave-mixing (FWM) in both cases of single-mode and SDM fibers with weakly-coupled degenerate mode groups. Rayleigh and SpRS backscattering are considered in counter-propagating scenarios. Since broadband classical transmission is targeted, the model also accounts for the effect of stimulated Raman scattering (SRS)-induced power tilt. Use of the model in sample scenarios indicates that the interference noise power is minimized at the high end of the transmission band in both cases were the quantum is co- and counter-propagating with respect to the classical signals, with a preference of one or the other scheme depending on the link length and quantum signal center frequency. Our model reveals that FWM has negligible impact in counter-propagating schemes, but can be relevant in co-propagating schemes under certain scenarios. Nevertheless, the FWM interference can be mitigated by deallocating the classical signals adjacent to the quantum channel.
Quantum correlations, where two or more parties exhibit strong correlations in a particular degree of freedom are important resources for fundamental science [1], such as the exploration of non-locality and entanglement, as well as non-classical technologies including quantum computation, quantum key distribution and quantum metrology. One typical way to generate correlated photon pairs is to use the spontaneous four-wave mixing process, with demonstrations on various platforms [2]. So far, single-photon stimulated four-wave mixing (FWM) has been demonstrated in optical fibers [3]. Here we show induced correlation by single photon seeded FWM through exploiting the orthogonal polarization modes in an integrated ring resonator [4]. We allow two different four-wave mixing processes to occur simultaneously with the same signal frequency mode but different idler frequency modes (see Fig.1(a)). The single-photon induced FWM process, named here seeded FWM, was created through the link of type-0 SFWM and type-2 SFWM. In the type-0 process, two excitation photons from the same field are converted into two new photons, signal and idler, with the same polarization as the pump field. While in the type-2 FWM, two annihilated photons, each one from a different polarized pump field, namely TE and TM, generate an orthogonal-polarized photon pair. More importantly, spatial and temporal overlapping of two processes and their sharing of one common resonance lead to correlation between the other two signal photons, one from each process. For example, an idler photon created in type-0 FWM seeds the type-2 FWM process or vice versa, i.e. the TM idler photon in Fig. 1, induces correlation between the signal TM and TE photons. This is proven by the observation of a clear coincidence peak for both signal photons in Fig.1(b). Furthermore, we confirmed our observation via power-dependent correlation measurements (see Fig.1 (c)). We summarized the true coincidence count (C) and its coincidence to accidental ratio (CAR) when only the TE pump power is increased while maintaining the TM mode pump power (black squares), and both pump powers are increased with a constant ratio (blue dots). We found that the seeded FWM scales with the square of both pump powers (C ∝ PjePtm) and that the CAR reach its maxima of 1.5 at the available pump power, indicating thermal statistics. Our findings are relevant to the fundamental understanding of spontaneous parametric effects as well as single-photon induced processes and single photon cloning.
The impact of fiber absorption on the generation rate of quantum-correlated photon pairs through spontaneous four-wave mixing in optical fibers is studied theoretically. We quantify the impact of the loss through the analysis of the Cauchy-Schwarz correlation parameter and the Clauser, Horne, Shimony, and Holt inequality. Results show that fiber loss can improve the quantum-correlation between the photon pairs. This is a consequence of the decrease of the individual signal and idler photon-fluxes, which reduces the rate of uncorrelated photons generated through Raman scattering inside the optical fiber. Findings show that a high degree of polarization entanglement is obtained when the ratio of signal to idler photon-fluxes lies between 0.5 and 0.8, in a regime with high losses on the signal wave.
We present a general quantum theory capable of describing photon statistics under the combined effects of four-wave mixing and Raman scattering inside optical fibers. Our theory is vectorial in nature and includes all polarization effects. Our analysis shows that spontaneous Raman scattering degrades the pair correlation in all cases but the extent of degradation depends on the pumping configuration employed. In a single-pump configuration, photon pairs can be created with polarization either parallel or orthogonal to the pump. Our results show that the orthogonal configuration can improve the extent of quantum correlation considerably over a broad bandwidth. In the case of a dual-pump configuration, we show that imbalance of two pump powers can be used to improve the quality of photon pairs. We show that orthogonally polarized pumps can generate photon pairs automatically in a polarization-entangled state. In particular, orthogonal pumping with circular polarizations produces such an entangled state with relatively high quality. We also quantify the quality of polarization entanglement as well as energy-time entanglement constructed using correlated photon pairs.
We investigate the impact of fiber birefringence and spontaneous Raman scattering on the properties of photon pairs that are generated by the spontaneous four-wave mixing process in birefringent fibers. Starting from the formulation of the theory of four-wave mixing, we show a theoretical model for a generated optical field with the consideration of the Raman scattering and a Gaussian-distributed pump. The theoretical model is then applied for deriving the closed expressions of the photon-pair spectral properties as a function of the fiber birefringence. Also, with the modeled Raman gain, we evaluate the reduction of the pair production rate due to the presence of the Raman effect as well as the contributions of the Raman-scattered photons over a broad wavelength range. The predictions are experimentally verified with a commercial polarization-maintaining fiber.