Photon-pair generation in arrays of cubic nonlinear waveguides

We study photon-pair generation in arrays of cubic nonlinear waveguides through spontaneous four-wave mixing. We analyze numerically the quantum statistics of photon pairs at the array output as a function of waveguide dispersion and pump beam power. We show flexible spatial quantum state control such as pump-power-controlled transition between bunching and anti-bunching correlations due to nonlinear self-focusing. © 2012 Optical Society of America OCIS codes: (270.0270) Quantum optics; (190.4380) Nonlinear optics; four-wave mixing; (080.1238) Array waveguide devices. References and links 1. J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photonics 3, 346–350 (2009). 2. A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science 325, 1221–1221 (2009). 3. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010). 4. A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010). 5. A. S. Solntsev, A. A. Sukhorukov, D. N. Neshev, and Y. S. Kivshar, “Spontaneous parametric down-conversion and quantum walks in arrays of quadratic nonlinear waveguides,” Phys. Rev. Lett. 108, 023601 (2012). 6. A. Rai and D. G. Angelakis, “Dynamics of nonclassical light in integrated nonlinear waveguide arrays and generation of robust continuous-variable entanglement,” Phys. Rev. A 85, 052330 (2012). 7. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express 14, 12388–12393 (2006). 8. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. ichi Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett. 91, 201108 (2007). 9. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). 10. L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am. B 29, 2199–2212 (2012). 11. M. Grafe, A. S. Solntsev, R. Keil, A. A. Sukhorukov, M. Heinrich, A. Tunnermann, S. Nolte, A. Szameit, and Y. S. Kivshar, “Biphoton generation in quadratic waveguide arrays: A classical optical simulation,” Sci. Rep. 2, 562 (2012). 12. J. C. F. Matthews, K. Poulios, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Ismail, K. Wrhoff, M. G. Thompson, and J. L. O’Brien, “Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions,” http://arxiv.org/abs/1106.1166 (2011). 13. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007). 14. A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010). #175540 $15.00 USD Received 5 Sep 2012; revised 19 Oct 2012; accepted 20 Oct 2012; published 19 Nov 2012 (C) 2012 OSA 19 November 2012 / Vol. 20, No. 24 / OPTICS EXPRESS 27441 15. C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009).


Introduction
With the development of quantum optics it is now feasible to realize complex quantum logic algorithms, however an increasing number of optical elements is required for multi-step quantum simulations.Integrated photonic circuits have been considered as the solution to this challenge, as they are intrinsically scalable and interferometrically stable [1].Integrated realization of multi-photon entanglement [1], quantum factoring algorithms [2], and polarization entanglement [3] have already been demonstrated experimentally.
One of particularly interesting devices in integrated photonics is a waveguide array (WGA).Recently WGAs have been shown to generate unusual and strongly non-classical correlations of photon pairs propagating in the regime of quantum walks [4].Combining quantum walks with photon pair generation in nonlinear waveguide arrays further opens the possibility for enhanced spatial quantum state control and improved clarity of spatial correlations [5].While WGAs with quadratic nonlinearity have been recently studied [5,6], we expect that WGAs with cubic nonlinearity can provide an entirely new realm of all-optical control of the quantum photon statistics.With on-chip photon-pair sources based on spontaneous four-wave mixing (SFWM) being readily available [7,8], it becomes of much interest to study the quantum state dynamics in WGAs with cubic nonlinear response.
In this work, we describe the generation of correlated photon pairs through SFWM in a cubic nonlinear WGA and analyze the interplay between SFWM phase-matching and WGA dispersion for the generation of complex entangled quantum states.We also demonstrate the potential of Kerr-based self-phase modulation (SPM) and cross-phase modulation (XPM) for quantum state control by investigating a case of stronger pump with special spectral filtering.

SFWM in WGAs at low pump powers
We consider lossless near-degenerate SFWM with signal and idler frequencies being close to the pump frequency, with the rest of photon-pairs being filtered out.In this case the mode profiles and coupling between waveguides remain approximately the same for pump, signal and idler photons.It has been demonstrated that even with less than 0.5% difference between these frequencies, the pump can be effectively filtered at the output of the system, and photon pair correlations can be measured with high signal-to-noise ratio [7,8].The main difference between spontaneous four-wave mixing in bulk or single waveguides in comparison to WGAs [Fig.1(a)] is a different spatial dispersion leading to modified phase-matching.We begin the analysis of SFWM in WGA by studying the four-wave-mixing phase-matching for plane waves, Δβ = 2β p − β s − β i .Here β p,s,i are the propagation constants for pump, signal and idler.In a WGA they depend on normalized transverse momenta k p,s,i as β p,s,i = β (0) p,s,i + 2C cos(πk p,s,i ), where C is the coupling coefficient between the waveguides [9].Meanwhile the overlap between interacting Bloch waves [5,9] can be written as where n is a waveguide number.Since , where N ∈ Z, then the transverse momenta k ⊥ p,s,i for periodical solutions will satisfy the condition 2k Therefore we can write an analytical expression for plane-wave four-wave mixing phase-mismatch in a WGA: The spatial dispersion and therefore the phase-matching conditions for SFWM in WGAs are qualitatively different from that in bulk, opening new possibilities for generating photon pairs with unusual quantum statistics.If pump is no longer a plane wave, but instead it is initially coupled to a finite number of waveguides, it then propagates in the regime of discrete diffraction [9] as illustrated in Fig. 1(b).For developing a model describing SFWM based photon-pair generation with a negligible number of higher multiphoton events in a cubic WGA, we first consider low pump powers that do not lead to pump spatial reshaping or nonlinear phase modulation.In the absence of SPM, XPM and losses, the system Hamiltonian consists of linear [4] and nonlinear [10] Here n s and n i are the waveguide numbers describing the positions of the signal, and the idler photons, and E (p) n p (z) is the pump amplitude in waveguide number n p .Δβ (0) is the linear four-wave mixing phase-mismatch in a single waveguide, γ is a nonlinear coefficient.The normalized pump field profile evolution along the propagation distance z is defined through the classical coupled-mode equations [9]: Generation of photon pairs in cubic nonlinear WGAs through SFWM in the absence of multiple photon pairs can be characterized by the evolution of a bi-photon wave function ψ n s ,n i (z) in a Schrödinger-type equation.The equation is obtained from the Hamiltonian, and it has a form similar to that of quadratic media [11]: ( We notice that the spatial dispersion described by Eqs. ( 2) and (3) exactly agrees with Eq. (1).
After calculating the wave function, we obtain two-photon correlations in real-space as where L is the propagation length.In order to find correlations for the signal and idler photons in k-space we apply the two-dimensional Fourier-transform, For the examples presented below, we normalize all parameters to the WGA length L = 1 and nonlinearity γ = 1.The physical value of the nonlinear coefficient can be determined following the approach of Ref. [10].We use the coupling coefficient C = 5, and consider a pump beam coupled only to the central waveguide (n = 0).
We study the case of low pump amplitude E (p) n p =0 (0) = 10 −5 when the input beam exhibits linear discrete diffraction [9], see Fig. 1(b).We analyze three different types of group velocity dispersion (GVD): anomalous Δβ (0) = −18, zero Δβ (0) = 0 and normal Δβ (0) = 18.In the case of anomalous GVD, photons in a pair tend to end up mostly away from the central waveguide, with higher probability to be at either the same or the opposite waveguides, see photon-pair probability correlation in Fig. 1(c).This behavior corresponds to weakly pronounced simultaneous spatial bunching and antibunching.The quantum statistics in this case is quasi-anyonic, which can be interesting for quantum simulations [12].The k-space correlations show an elliptical shape centered at k ⊥ s = k ⊥ i = 0 [Fig.1(d)].This shape corresponds to the wavenumbers with the most efficient phase-matched interactions, i.e.Δβ = 0.For zero GVD the signal and idler photons mostly leave the structure from the same waveguides, thus demonstrating strong spatial bunching behavior [Fig.1(e)].Figure 1(f) shows that the transverse wavenumbers for photon pairs satisfy the relations k ⊥ s + k ⊥ i ±1.We note that for zero dispersion photon pairs have much higher probability to arrive to the center of the WGA, because phase matching can now be achieved for a broader range of transverse momenta.In the case of normal GVD, the real-space correlations [Fig.1(g)] are the same as for anomalous GVD [Fig.1(c)].Indeed for low pump powers with negligible SPM and XPM, the system is symmetrical with respect to the GVD sign.In k-space the correlations form an elliptical shape centered around k ⊥ s = k ⊥ i = ±1 [Fig.1(h)].We note that there is a gradual transition from ellipses centered at 0 [Fig.1(d)] through linear shapes [Fig. 1 when tuning the GVD from anomalous to normal.GVD tuning can achieved by changing the pump wavelength [13], however such tuning can be complicated due to the required corresponding spectral shift of output filters for signal and idler photons.

SFWM and pump self-focusing at high pump powers
Next we investigate the potential of SPM and XPM at high pump powers for flexible quantum state control.When the pump power is increased, the beam self-focusing results in a sharp transition from discrete diffraction to the formation of a spatial soliton, as shown in Fig. 2(a).However, the powers required for soliton formation are at least an order of magnitude higher than those needed to remain in the regime with small number of multiphoton events.For example, in a 3 mm long Si WGA (waveguides 200 − 300 nm high and 400 − 500 nm wide) the pump peak power required for noticeable nonlinear phase modulation is of the order of 10−20 W [14].The characteristic pump peak power P for photon-pair generation is much smaller: in a 10 mm long Si waveguide P ≈ 0.1 − 0.2 W [7,8] (corresponding to 1 − 2 W for a 3 mm long waveguide).
To realize the influence of SPM and XPM on the photon-pair generation in WGAs in the ab-  sence of multiphoton events, we suggest asymmetric filtering approach for a pulsed pump beam of transform-limited pulses having a spectrum shown in Fig. 2(b) with a green solid line.In this approach we choose two narrowband spectral filters for measuring the signal (red dashed line) and the idler (blue dotted line) photons, such that they are located asymmetrically with respect to the central pump frequency.Then only a narrow window of pump frequencies with small peak power (indicated by gray shading) would be responsible for the detected photon-pairs, due to the energy conservation ω .With such filtering the pump peak power can be strong enough to induce pump beam self-focusing, while multiphoton events are mostly excluded from the measurement.The complete modeling of this system should account for the spatio-temporal pump dynamics.Here we investigate a simplified steady-state model that is valid if the pump pulse does not experience any dispersion-related reshaping.Signal and idler filters in this case should be far enough from the pump in frequency domain, so that the phase-matching is affected by XPM and SPM, but also close enough, so that the coupling coefficients are still similar for signal, idler and pump waves.Since coupling dispersion in WGAs is usually smaller than the temporal dispersion (unless specifically engineered otherwise [15]), these assumptions should be valid for a large variety of systems.
Since high pump peak powers lead to the pump beam focusing [Fig.2(a)], generated photon pairs will have different spatial distributions depending on the pump power.The pump power will also effectively change the SFWM phase-matching conditions due to XPM.We incorporate these effects by adding the terms responsible for XPM and SPM into the Eqs.
We acknowledge that this model is only the first-order approximation and that simulations designed to give precise quantitative results should incorporate full dispersion curves both for propagation and coupling constants, as well as spatio-temporal dynamics.However, we believe that the simplified model that we present here is useful to obtain a qualitative insight into the quantum statistics control that can be achieved in cubic WGAs.

2 .
(a) Output pump power profile vs. pump amplitude (single waveguide excitation).(b) Normalized pump spectrum (green solid line), normalized signal (red dashed line) and idler (blue dotted line) spectral filters.Gray shading marks a part of the pump spectrum contributing to the filtered photon-pair generation.