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Photon pair generation and pump filtering in nonlinear adiabatic waveguiding structures Che Wen Wu,* Alexander S. Solntsev, Dragomir N. Neshev, and Andrey A. Sukhorukov Nonlinear Physics Centre, CUDOS, Research School of Physics and Engineering, Australian National University, Canberra, ACT 0200, Australia *Corresponding author:
[email protected] Received October 10, 2013; revised January 9, 2014; accepted January 10, 2014; posted January 10, 2014 (Doc. ID 197985); published February 11, 2014 We propose a novel integrated scheme for generation of Bell states, which allows simultaneous spatial filtering of pump photons. It is achieved through spontaneous parametric down-conversion in the system of nonlinear adiabatically coupled waveguides. We perform detailed analytic study of photon-pair generation in coupled waveguides and reveal the optimal conditions for the generation of each particular Bell state. Furthermore, we simulate the performance of the device under realistic assumptions and show that adiabatic coupling allows us to spatially filter the pump from modal-entangled photon pairs. Finally, we demonstrate that adiabatic couplers open the possibility of maintaining the purity of generated Bell states in a relatively fabrication-fault-tolerant way. © 2014 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (190.4390) Nonlinear optics, integrated optics; (270.0270) Quantum optics; (130.7405) Wavelength conversion devices. http://dx.doi.org/10.1364/OL.39.000953
Obtaining highly efficient generation of entangled photon pairs is extremely important in the field of quantum information for applications, such as quantum cryptography, quantum metrology, quantum teleportation, quantum information processing, and quantum computation [1–3]. Photons can be entangled in different degrees of freedom, thus realizing spatial, temporal, or polarization entanglement. The entanglement is vital for the realization of so-called Bell states, which are essential in quantum computation [4]. Bulk optics already demonstrates the ability to generate bright polarization-entangled photons; however, it still faces drawbacks, such as the instability, low precision, and large physical size of the devices. Integrated photonic circuits [3] provide a solution to the abovementioned challenges, as they enable integration of the source of entangled photons with quantum logic as well as offer scalability, miniaturization, and interferometric stability. A feasible way to perform such integration is to explore the functionalities of nonlinear waveguides. Nonlinear waveguides are advantageous for a vast variety of applications, including all-optical modulators [5], logic gates [6], high-brightness sources of entangled photons through spontaneous four-wave mixing [7], and spontaneous parametric down-conversion (SPDC) [8]. In the latter case, the coupling between two or more waveguides can result in the generation of strongly correlated biphoton states at the coupler output [8]. For example, the SPDC in an array of coupled nonlinear waveguides can be used to generate photon-pair quantum states with bunching or antibunching spatial correlations through the quantum walks of photons [9]. Importantly, the incorporation of waveguide coupling in the regime of SPDC can also allow for the simultaneous filtering of the pump beam from the generated photon pairs. If the coupling between the waveguides is chosen appropriately, the pump beam can stay in the input waveguide, whereas the generated photon pairs having approximately two times larger wavelengths can couple to the neighboring waveguides [9,10]. Such 0146-9592/14/040953-04$15.00/0
spatial filtering may in certain cases be preferable to spectral Bragg-grating-based filtering [11], because spatial filtering does not necessarily incur pump backreflection. Having access to spatial filtering may open broader opportunities for designing integrated photonics circuits with enhanced pump routing flexibility. However, when using this kind of spatial pump filtering, it is important to ensure that the photon-pair state remaining in the pumped waveguide is separable from the photon-pair state in the other waveguides; then the generated photon-pair state in waveguides other than the pumped waveguide will be a pure state. This effect can theoretically be achieved by careful waveguide design [10]; however, it may prove challenging for practical implementation. Furthermore, using simple nonadiabatic waveguide coupling makes pump filtering a complicated task that requires high fabrication precision when dealing with different signal and idler photons, such as in cases of polarization or spectral entanglement. In this Letter, we show that, by employing adiabatic waveguide coupling [12,13], we can generate SPDC photon pairs entangled in the degree of modal entanglement, while simultaneously keeping the filtered photon states separable from the photons in the pumped waveguide. Our proposed scheme reveals the possibility of combining flexible reconfigurable quantum state generation and spatial filtering of the pump beam in an integrated photonic chip. Importantly, adiabatic coupling is also tolerable to fabrication inaccuracies, and does not require careful waveguide manufacturing even when used to couple photons of different polarizations and frequencies. We perform careful systematic analysis of photon-pair generation in quadratic nonlinear coupled waveguides and determine the optimal conditions for the generation of all four possible maximally entangled Bell states. We model the spatial biphoton wave function dynamics by extending the coupled-mode approach developed to simulate degenerate SPDC in homogeneous waveguide arrays [14]. We incorporate variable adiabatic coupling for nondegenerate biphotons as follows: © 2014 Optical Society of America
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i
dΨns ;ni s −C s ns Ψns 1;ni C ns −1 Ψns −1;ni dz i C i ni Ψns ;ni 1 C ni −1 Ψns ;ni −1 ideff Ans δns ;ni expiΔβ0 z:
(1)
Here, Ψns ;ni is the biphoton wavefunction, z is the propagation distance, and ns and ni denote the waveguide numbers of the signal (s) and idler (i) photons, respectively. Δβ0 ωp nωp ∕c − ωs nωs ∕c − ωi nωi ∕ c − π∕Λ is the phase mismatch in a single waveguide, and ωp;s;i are the angular frequencies of the pump, signal, and idler photons. Λ is the poling period, nωs;i;p are the effective waveguide refractive indices, c is the speed of light, deff is the effective nonlinear coefficient, and δ is a Kroncecker delta function. Anp is the complex pump amplitude in the waveguide np . C s;i n z is the coupling between the waveguides n; n 1 for signal and idler photons accordingly; for example, C s ns indicates the coupling of signal photons between the original waveguide and its right-hand neighbor, while C s ns−1 indicates the coupling of signal photons between the original waveguide and its left-hand neighbor. If the structure contains N waveguides, then the generated photon pairs can exit from any combination of waveguides ns;i 1; 2; …; N, depending on the pump and the effective coupling between the waveguides. Our aim is that at the output of this system, the generated photons form Bell states, which are maximally entangled biphotons. Our formalism is applicable to the analysis of modal entanglement for both type-I and type-II SPDC. In the case of type-I SPDC, the idler and signal photons have the same coupling [C s C i ]. In the case of type-II SPDC, the idler and signal photons have unequal coupling [C s ≠ C i ]. Both cases are useful for quantum optics applications, but are difficult to realize in an integrated system in combination with spatial filtering to remove the pump. Adiabatic coupling allows for spatially pump-filtered generation of modal-entangled photon pairs, while maintaining the purity of generated Bell states in a way that does not require high fabrication precision. To study the generation of Bell states on chip, we start with the structure incorporating two coupled quadratic nonlinear waveguides [15]. Pump beams are coupled to both waveguides 1 and 2, with the corresponding amplitudes A1 Aa and A2 Ab [see Fig. 1(a)]. Idler and signal photon pairs can couple to both waveguides, and form the following Bell state wavefunctions at the output: 1 0 1 1 1 0 jΦ i ∼ p ; jΨ i ∼ p : (2) 2 1 0 2 0 1 We identify the optimal conditions for the generation of all Bell states and summarize them in Table 1. In the Pump column, we use “Even” to denote in-phase pumping of both waveguides (Aa Ab ), and “Odd” denotes out-of-phase pump with Aa −Ab . For optimal generation, the single-waveguide mismatch should be zero (Δβ0 0); however, the generation efficiencies of
Fig. 1. (a) Scheme of the structure with two coupled waveguides: pump beams (green) are coupled to both waveguides, and generated Bell states (red) exit the structure from the same waveguides at the output. (b)–(e) Generated Bell states (b), (c) jΦ− i for C s C i 0 and (d), (e) jΨ− i for C s 1, C i 2. Shown are the (b), (d) biphoton correlations, Γns ;ni , and (c), (e) phases of the biphoton wavefunctions, θns ;ni . For all the plots, Aa 1, Ab −1, deff 0.1, Δβ0 0, and Lt π.
different states will be slightly different due to the positive and negative phase mismatches associated with odd and even guided supermodes, respectively. For the generation of jΦ i states, the signal and idler photons need to stay at the same waveguide, and we call them “bunching” states. To obtain the Bell state jΦ i, the pump should have an even phase. The phase mismatch Δβ0 may be arbitrary; however, zero phase mismatch corresponds to the most efficient generation. Tuning the pump to odd phase leads to the generation of the jΦ− i state [Figs. 1(b) and 1(c)]. We show the Pcorresponding photon correlations Γns ;ni jΨns ;ni j2 ∕ ns ;ni jΨns ;ni j2 and the phases of biphoton wavefunction θns :ni argΨns ;ni − argΨn1 ;n1 in Figs. 1(b) and 1(c), respectively. These states are fairly trivial, since they do not Table 1. Optimal Conditions for Bell State Generation State
Pump
C s;i
L
Δβ0
jΦ i jΦ− i jΨ i jΨ− i
Even Odd Even Odd
0 0 C s ≠ −C i C s ≠ C i
Any Any πjC s C i j−1 πjC s − C i j−1
0 0 0 0
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involve coupling between the waveguides and correspond to bosonic statistics [16]. The generation of the other two Bell states (jΨ i) requires a particular coupling between the waveguides, as the idler and signal photons need to be in different waveguides, and we call them “antibunching” states. We can obtain the Bell state jΨ i with even pump and zero phase mismatch Δβ0 0, but it requires a specific structure length, L πjC s C i j−1 . The Bell state jΨ− i [Figs. 1(d) and 1(e)], corresponding to fermionic statistics [16], can be realized with an odd pump, zero phase mismatch (Δβ0 0) and a structure length L πjC s − C i j−1 . Note that the coupling coefficients for idler and signal photons have to be different (C s ≠ C i ), which implies that the signal and idler photons have either different polarizations or different frequencies. We find that two-waveguide couplers [15] can be easily designed to satisfy the parameter requirements for generation of all four Bell states. However, this simple configuration does not allow for spatial filtering of the pump beam. Next, we demonstrate how adiabatic coupling can be used for efficient pump filtering even in the case of nondegenerate photon pairs. For this purpose, we suggest a structure incorporating six waveguides as shown in Fig. 2, with the adiabatic coupling introduced between the waveguides 1 and 3 and between the waveguides 4 and 6. The pump beams stay in the central waveguides 3 (A3 Aa ) and 4 (A4 Ab ), while the idler and signal photons can couple to the edge waveguides 1 and 6, where they can be spatially separated from the pump. The adiabatic coupling of photons is achieved through a modulated coupling with the intermediate waveguides 2 and 5, which can be approximated as C s;i 1 z s;i s;i s;i z κ exp−α z − L and C z C s;i C s;i 5 4 2 z κ s;i expαs;i z − L. Here, the total sample length Lt 2L, κs;i is the coupling strength at z L for signal
Fig. 2. (a) Scheme of adiabatically coupled waveguiding structure: pump beams (green) are coupled to the central waveguides 3 and 4, and the Bell states (red) are formed in the edge waveguides 1 and 6 that are separate from the pumped waveguides. (b) Coupling of the adiabatically coupled waveguides C 1 (red line), C 2 (blue line), C 3 (black line) with the cous s i C s C i pling coefficients C s 1 C5 , 1 C5 , 2 C4 , i i s;i s;i s;i 0.2, κ 2, and α 0.4. C2 C4 , γ
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and idler photons respectively, αs;i is the parameter related to the slope of adiabatic coupling, and the adiabatic condition here is that γ s;i ≡ αs;i ∕κ s;i ≪ 1. Under such conditions, both photons are generated and propagate in the two central waveguides until a distance of approximately z L, after which they couple out to the edge waveguides. The optimal conditions for the Bell state generation in the 6-waveguide structure remain the same as for the two-waveguide coupler discussed previously. We present simulations of the Bell state generation combined with pump filtering in Fig. 3. Figures 3(a)– 3(d) show the biphoton intensity distribution along P jΨ j2 the waveguides, calculated as I n N n ns 1 s ;n PN 2 − ni 1 jΨn;ni j , for the generation of states jΦ i and jΨ− i, respectively. We observe the adiabatic coupling between the waveguides 1, 3, 5, and 6 mediated by the waveguides 2 and 5. The corresponding photon states at the output are presented in Figs. 3(b), 3(c) and 3(e), 3(f). We see that the biphotons at the edge waveguides have the amplitudes and phases corresponding to the target Bell states. Additionally, these states have a high degree of purity, as there are almost no correlations between the edge and the inner waveguides. We have also confirmed that the same approach works with the other two Bell states, and that all four Bell states can be generated with a range of different coupling parameters corresponding to Table 1. Importantly, the adiabatic coupling approach is tolerant to the effects of the fabrication inaccuracies.
Fig. 3. Generation of Bell states combined with pump filtering: i s − (a)–(c) jΦ− i for C s 3 C 3 0 and (d)–(f) jΨ i for C 3 1, i C 3 2. Shown are (a), (d) the biphoton intensity distribution along the waveguiding structure, (b), (e) biphoton correlations, and (c), (f) phases of the biphoton wavefunctions θ; we only show the relative phases for the waveguide combinations with nonzero biphoton correlations. For all the plots, κs 2, κi 4, γ s;i 0.1, Aa 1, Ab −1, deff 0.1, Δβ0 0, αs;i 0.4, and Lt 2π.
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Finally, we estimate the performance of the spatial pump filtering. We numerically calculate coupling strength depending on the wavelength and the separation distance between the waveguides by simulating coupled waveguide supermodes in COMSOL Multiphysics. We utilize realistic parameters based on the previous experimental measurements in the nonlinear coupled waveguides [17]. We confirm that the coupling for the proposed 6-waveguide structure can be achieved for biphotons with the wavelengths around 1550 nm. To achieve the required coupling in our simulation, we consider that the structure length is equal to 6.3 cm, and the distance between the adiabatically coupled waveguides changes from 5.7 to 8.5 μm8.5 m. Then, to estimate the pump filtering order, we change the wavelength to 775 nm and perform the numerical simulation with the CW pump being initially coupled to the central waveguides 3 and 4. The filtering order is defined as FO −10 × logI WG1 ∕ I WG3 −10 × logI WG6 ∕I WG4 , where I WG1 I WG6 is the pump intensity at the output of the waveguide 1 or 6 and I WG3 I WG4 is the pump intensity at the input of the waveguide 3 or 4. We find that in the adiabatic coupling configuration, the structure provides the pumpfiltering order of 72 dB. The filtering can be improved to reach the values exceeding 100 dB by adding an extra set of adiabatic passages in a 10-waveguide configuration or adding a spectral filter. In conclusion, our Letter shows that adiabatic waveguide coupling is a promising solution for efficient and simple spatial pump filtering during the generation of modal-entangled photon pairs in integrated photonic circuits. In the future, this scheme may prove useful for on-chip generation of strongly entangled biphoton states with controllable correlations, when combined with more sophisticated waveguiding structures. We acknowledge support from the Australian Research Council, including Centre of Excellence CE110001018, Future Fellowship FT100100160, and Discovery Project DP130100135, and the Australian NCI National Facility.
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