Generation of path-polarization hyperentanglement using quasi-phase-matching in quasi-periodic nonlinear photonic crystal

A compact scheme for the generation of path-polarization entangled photon pairs is proposed by using a quasi-periodic nonlinear photonic crystal to simultaneously accomplish four spontaneous parametric down-conversion processes. Moreover, we report experimental scheme to measure the polarization entanglement and path entanglement separately and theoretically get numerical results that verify some predictions about the hyperentanglement. This method can be expanded for the generation of multi-partite and two-photon path-polarization hyperentanglement in a single quasi-periodic nonlinear photonic crystal structure. This compact quantum light source can be used as a significant ingredient in quantum information science.

which verify our predictions about the hyperentanglement. In discussion, we discuss how the basic model can be expanded for the generation of multi-partite and two-photon path-polarization hyperentanglement. In method, we introduce the principle of designing the crystal.

Results
Generation of path-polarization hyperentangled photon pairs. The schematic for the generation of path-polarization hyperentangled photon pairs is displayed in Fig. 1. We have a pump photon with the frequency of ω p injected into the designed NPC-in which it will get through either of the 4 SPDC processes-and the signal and idler photons with frequency ω p /2 are assumed to be generated in our engineering. From an intuitive perspective, the signal and idler photons are firstly polarization entangled; and since they come out from either of the two spatial modes shown in Fig. 1(a), they are also path entangled.
The NPC displayed in Fig. 1(a) is designed to simultaneously accomplish QPM of the 4 different SPDC processes. The QPM condition can be depicted by Fig. 1 To illustrate our design method we have proposed, we are going to take an example with specific parameter values. It must be noted that, these specific parameter values are just used to justify our theory in calculation, maybe the values are not suitable for a realistic case. However, if necessary, we can design the lattice with appropriate parameter values in any realistic case, including wavelength, temperature, directions of the wave vectors and so on. Thus there is no loss of generality.
we consider a very typical laser, Nd:YAG laser, whose wavelength is 532 nm. Now we set the wavelength of the pump light as 532 nm and that of the signal and idler light is 1064 nm. are 0°, 58°, −58°, 74°, −74° respectively. Periodically poled lithium niobat (PPLN) is chosen as the NPC material and the working temperature is 21 °C. We adopt sellmeier equations under this condition 21  Through engineering of the PPLN NPC89 to accomplish QPM of the mismatch vectors, the structure of PPLN NPC is depicted by Fig. 2(a) and the tiling vectors shown in Fig. 2 (1) Each red dot with radius of 1 μm in Fig. 2(a) is called motif 9 . Figure 2(a) actually depicts the distribution of nonlinear coefficient χ (2) in the PPLN NPC, which is obtained by the convolution between the quasi-periodic lattice and motif. In the motif (red dot) χ (2) = 1 while χ (2) = −1 in other areas of the PPLN NPC. We can also ∫ ∫ where = k k , k x , k y indicate the x and y components of k, χ ∆ is the absolute difference between the positive and negative values used for χ (2) , J 1 is the first Bessel function, S is a circle of radius , A is a rectangle of sides L x × L y -which indicates the total area of PPLN NPC (L x = 0.5 mm, L y = 2.5 mm in our engineering), U(k) is the Fourier transform of lattice function u r ( ) and is the sum of delta functions. Figure 2(b) depicts the Fourier transform of the PPLN NPC. We can clearly distinguish Bragg peaks at the positions of the required mismatch vectors , , , ∆k x and ∆k y indicate the x and y components of Δk. Note here that if more spatial modes are introduced-which implies more SPDC processes to achieve-we can prevent the decrease of SPDC efficiency by promoting the size of our designed PPLN Under the first-order perturbation approximation 22 , through the QPM of 4 SPDC processes in the designed PPLN NPC, the two-photon state can be written as ). And the relationship between the detuning frequency υ and Δk is     , and . This is aptly the required path-polarization hyperentanglement. Note that it is significant to match the efficiency of SPDC processes of different spatial modes because it will achieve maximally entangled states. However to date, there is no general design methods available to achieve this condition which means that this condition can only be achieved in some selected cases of us.
We design an experimental scheme and the criterions 23 to verify the path and polariztion entanglement separately. The experiment setup is shown in Fig. 3.
In Fig. 3, Ê s k and Ê i k (k=1, 2) stand for the signal and idler light fields generated from SPDC processes in the PPLN NPC. They are expressed as The phase relation, β 1 between s 1 , s 2 and β 2 between i 1 , i 2 , can be set by tilting two prisms. And we have Generally, in Heisenberg picture, the evolution of operators in a BS is expressed as The evolution of operators in BS1 and BS2 can be expressed as where j = 1, 2. Without loss of generality, we show theoretically the detected result at D 1 , D 2 after classical interference at BS1 and D 3 , D 4 after classical interference at BS2. The polarizers are temporarily removed so that all polarization components are included. To verify the path entanglement, the coincidence count of detectors D 1 and D 3 is measured, which is proportional to the expected value of the opeartor ˆˆˆ † †

E t E t E t E t ( ) ( ) ( ) ( ) c i c s c s c i
c  Here we introduce a phase difference ∆ = − t t t s s s

i c s c s c i c s c i c s c i
After submitting the numerical two-photon mode function to Eq. (18), approximately we have We can get coincidence counts of (D 1 , D 4 ), (D 2 , D 3 ) and (D 2 , D 4 ) by the same method. The expected coincidence count can be written as The entangled state ψ can be adoped to test the violation of a Bell inequality 24   Two photon coincidence can be measured by using the phase setting The expected value, = .
> S 2 357 2 k , verifies the path entanglement between s 1 , i 1 and s 2 , i 2 . Then we discuss the measurement of polarization entanglement 25 . The evolution of operators in Polarizer1 and Polarizer2 can be written as To verify the polarization entanglement, the Coincidence Counting of detectors D 1 and D 3 is measured, which is proportional to the expected value of the operator . We have Note here that the effect of Prism1 and Prism2 (β 1 and β 2 ) can be used to compensate for the phase difference between the signal photons (Δt s ) and that between idler photons (Δt i ). Specifically, we need to adjust Prism1(Prism2) until the classical interference at BS1(BS2) results in a peak detected intensity at D 1 (D 2 ). After phase difference compensating, when θ θ = ( ) 0 2 1 , the relation between Coincidence Counting of D 1 , D 2 and θ 1 (θ 2 ) can be depicted by Fig. 4(a)

Discussion
Multi-partite and two-photon path-polarization hyperentanglement. Equation (8) illustrates the proposed path-polarization hyperentanglement. This is a basic 4-SPDC model and some adjustments to the NPC engineering will give new forms to the hyperentanglement. For example, if the engineering of NPC incorparates 8 SPDC processes instead of 4, the QPM condition can be illustrated by Fig. 5(a). And the result is that the number of spatial modes becomes 4 instead of 2, which implicates that the path entanglement part in the hyperentanglement changes from 4-partite to 8-partite. It can be described as  Theoretically, if multiple paths are established, multi-partite path-polarization entanglement can be generated. Moreover, if there are still 4 designed SPDC processes but the signal and idler are designed to be emitted from the same path, the QPM condition can be illustrated by Fig. 5(b).

Method
Designing a proper crystal is a key point for phase matching 4 SPDC processes. A general method to design frequency converters that will phase match any set of interacting waves is provided by the so-called generalized dual grid method (DGM) 9 . In this method, a dual structure, called the dual grid, which contains all the topological information required to built the quasi-crystal is first constructed. Then, using a simple transformation, this dual grid is transformed to a quasi-crystal. The Dual Grid Method can be adapted to match different processes. For different processes, the only thing you need to do is changing the mismatch vectors. Moreover, the Dual Grid Method could be implemented by a computer program, which is convenient to design a crystal.