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- Photonics Research
- Vol. 6, Issue 4, A6 (2018)

Abstract

Keywords

1. INTRODUCTION

The development of a new generation of devices for efficient generation of entangled photons is underpinning a range of quantum technologies, including computations, communications, and simulations ^{[1,2]}. Integrated sources of photon pairs in nonlinear waveguide circuits enable the increase of complexity and enhancement of stability compared to bulk optical implementations (see reviews ^{[3,4]} and references therein).

A powerful approach for the control of light propagation in waveguide-based circuits stems from the concept of parity-time (PT) symmetry in optical systems ^{[5,6]}, which can be implemented with coupled waveguides incorporating symmetrically arranged regions of gain and loss. One of the most interesting effects is the emergence of a phase transition behavior arising from a spontaneous breakdown of PT symmetry, which offers many novel possibilities for shaping optical beams. A broad range of effects qualitatively different from those usually observed in conservative systems has been identified, including power oscillations, nonmonotonous dependence of the transmission on absorption, unidirectional invisibility, conical diffraction, unusual switching regimes, and formation of spatial and temporal solitons in nonlinear structures (see recent reviews ^{[7,8]}).

Based on the success of PT symmetry in classical photonics, this approach is also likely to be useful for quantum photonic chips, in particular for the generation of spatially entangled photon pairs ^{[4]}. Loss is always present in realistic photonic structures, and it has recently been shown that it can enable new possibilities for shaping quantum entangled states in plasmonic circuits ^{[9]}. Using the inevitable loss for control of nonlinear photon state generation via PT-symmetry breaking can be a very promising solution. We note that quantum effects have been extensively studied in PT structures with gain and loss, where amplification can lead to both the photon generation and noise ^{[10–13]}. In this paper, we explore photon-pair generation in a quadratically nonlinear directional coupler, a structure that has been extensively studied in the non-PT regime with weak homogeneous losses ^{[14]}. We consider a regime when material gain is absent, and PT symmetry is introduced purely through loss in one of the waveguides ^{[5,15]}, which avoids the quantum noise amplification, in contrast to previous studies ^{[10–13]}. We show that use of losses and PT-symmetry breaking opens new opportunities for the control of spatially entangled quantum states of the generated photon pairs.

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2. METHODS AND RESULTS

We study the generation of photon pairs through spontaneous parametric down-conversion (SPDC) in a quadratically nonlinear coupler, as schematically shown in Fig. ^{[5,15]}. The effect of linear losses on SPDC was previously considered in various contexts ^{[16–18]}. We follow the approach of Ref. ^{[19]} and formulate Schrödinger-type equations for the photon-pair amplitudes ^{[19]}. Nevertheless, in the regime of a weak pump when four-photon generation probability is small, the two-photon state remains pure even in the presence of loss ^{[18]}.

Figure 1.(a) Scheme of generation of photon pairs through the spontaneous parametric down-conversion in a nonlinear PT-symmetric coupler with linear absorption in the second waveguide. (b) Graphical representation of biphoton correlation function

In the regime of photon-pair generation, the classical pump remains essentially undepleted, and its evolution is governed by linear coupled-mode equations: where

We derive the analytical solutions of Eqs. (^{[18,20]}, which we define as a solution of linear coupled-mode equations: ^{[18]}.

In the following, we analyze in detail the regime of near-degenerate signal and idler photon frequencies, when accordingly ^{[5,6]}. Compared to signal and idler, the fundamental pump mode is much stronger localized in a waveguide, resulting in very weak coupling ^{[14]}. Additionally, while tailored signal and idler losses can be introduced by depositing metal on top of dielectric waveguides, this will not affect the pump mode. Therefore, we consider

Below the critical gain/loss value (^{[5]}.

We now analyze the effect of PT-symmetry breaking on the generation of quantum photon-pair states. For this purpose, we study the signal and idler photon correlations between different waveguides, which are proportional to the squared modulus of the corresponding wave-function elements and can be measured by detecting coincident clicks with single-photon detectors, as schematically illustrated in Fig. ^{[14]}, the maximum correlations between the first and second waveguides are anticipated around phase matching with signal and idler supermodes of the same symmetry, which condition is found using linear supermode dispersion ^{[5]} as

Figure 2.Evolution of spatial signal and idler photon correlations between the two waveguide modes along the propagation direction (

Next, we explore the effect of varying loss in the second waveguide for a fixed propagation distance and present the normalized photon correlations in Fig. ^{[19]}. For the phase mismatch corresponding to stronger correlations between two waveguides,

Figure 3.Normalized photon correlations,

The experimental characterization of quantum states requires accumulation of statistics over multiple photon pairs measured with single-photon detectors, which can be time consuming. This motivated the development of approaches for fast classical characterization of nonlinear devices, which then enable one to predict their performance in the regime of quantum photon-pair generation ^{[21,22]}, including arbitrary multiwaveguide circuits as validated with high experimental fidelity ^{[23]}. In the following, we formulate this correspondence for PT nonlinear coupler, considering classical sum-frequency generation (SFG) by the input lasers with the same frequencies as for the signal and idler photons, which are generated in the quantum regime [see Fig.

Figure 4.(a) Scheme of SFG in passive PT-symmetric nonlinear coupler with linear absorption in one waveguide. (b) Mismatch

We first consider the regime of relatively weak input amplitudes, when

While the quantum-classical correspondence is exact in the limit of very weak SFG efficiency, we perform numerical simulations of full Eqs. (

3. CONCLUSION

In summary, we predict a possibility to perform flexible control of the quantum state of photon pairs generated in nonlinear coupled waveguides even in the presence of loss, because photon correlations between different waveguides can develop efficiently when operating below the PT-breaking threshold. However, correlations between waveguides get suppressed above the threshold loss value. Furthermore, we formulate a quantum-classical correspondence with SFG for fast evaluation of device performance. These results can facilitate the development of integrated photon sources based on nonlinear plasmonic structures.

References

[17] D. Klyshko**. Photons and Nonlinear Optics(1988)**.

[20] W. Vogel, D.-G. Welsch**. Quantum Optics(2006)**.

[21] M. Liscidini, J. E. Sipe. Stimulated emission tomography**. Phys. Rev. Lett., 111, 193602(2013)**.

[23] F. Lenzini, A. N. Poddubny, J. Titchener, P. Fisher, A. Boes, S. Kasture, B. Haylock, M. Villa, A. Mitchell, A. S. Solntsev, A. A. Sukhorukov, M. Lobino**. Direct characterization of a nonlinear photonic circuit’s wave function with laser light(2017)**.

Diana A. Antonosyan, Alexander S. Solntsev, Andrey A. Sukhorukov. Photon-pair generation in a quadratically nonlinear parity-time symmetric coupler[J]. Photonics Research, 2018, 6(4): A6

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