Effects of polarization mode dispersion on polarization-entangled photons generated via broadband pumped spontaneous parametric down-conversion

An inexpensive and compact frequency multi-mode diode laser enables a compact two-photon polarization entanglement source via the continuous wave broadband pumped spontaneous parametric down-conversion (SPDC) process. Entanglement degradation caused by polarization mode dispersion (PMD) is one of the critical issues in optical fiber-based polarization entanglement distribution. We theoretically and experimentally investigate how the initial entanglement is degraded when the two-photon polarization entangled state undergoes PMD. We report an effect of PMD unique to broadband pumped SPDC, equally applicable to pulsed pumping as well as cw broadband pumping, which is that the amount of the entanglement degradation is asymmetrical to the PMD introduced to each quantum channel. We believe that our results have important applications in long-distance distribution of polarization entanglement via optical fiber channels.

polarization entangled states prepared via the pulsed SPDC are considered in refs 20 and 21. Furthermore, it has recently been experimentally demonstrated that narrowing frequency bandwidths of the photons is helpful for preserving more entanglement when each photon suffers from the same amounts of PMDs 22 . However, although ref. 22 provides insights about finding a way to suppress entanglement degradation due to PMD, the detailed analysis on how the photon's frequency bandwidth affects the behavior of the entanglement degradation caused by PMD has not been studied. Moreover, there still has been no experimental demonstration that reveals the overall features of entanglement degradation when each photon undergoes different amount of PMD.
In this paper, we consider broadband pumped SPDC for generating polarization entangled photons and a realistic quantum communication scenario in which the initially prepared polarization entangled photons are distributed through optical fiber channels. We theoretically calculate the concurrence of the polarization entanglement after the initial entangled photon pairs undergo independent PMDs. Then, we experimentally demonstrate that the polarization entanglement from the broadband CW pumped SPDC experiences unreported PMD effects in that the amount of the entanglement degradation is asymmetrical to the PMD introduced to each quantum channel. Here, we emphasize that this result is equally applicable to pulsed pumping as well as CW broadband pumping. We believe that our results have important applications in long-distance distribution of polarization entanglement via optical fiber channels.

Results
Here we consider the scenario that the two-photon polarization entangled state generated via the broadband pumped SPDC process undergoes independent PMD as shown in Fig. 1. The broadband pumped SPDC which we consider in this manuscript deals with a cw pump laser having broadband but all modes are incoherent. Note that, in general, the longitudinal (frequency) modes of a free-running cw lasers are incoherent. In order to prepare the polarization entangled state, a pair of two photons is generated from type-I non-collinear parametric down conversion process in β-BaB 2 O 4 (BBO) crystal. When the multi-mode diode laser is used for pumping the nonlinear crystal, the two-photon state after transmitting through the interference filters (IFs) can be expressed in the following form 13,14 , where ω p is the frequency of the pump photon and the spectral power density function Φ (ω p ) is given by where ω p0 is the center frequency of the pump photon, δω p is the mode spacing, n is the mode number, and the spectral profile of the pump Φ 0 (ω p ) is Gaussian, where Δ ω p is the bandwidth of the pump photon. The monochromatic laser pumped SPDC two-photon state |ψ(ω p )〉 is written as 13,14 The pump photon is split into two down-converted photons (one is called the signal and the other is called the idler) via the broadband pumped SPDC process. Then, the down-converted photons are prepared as a two-photon polarization entangled state, and each photon undergoes independent PMD. Since the group velocities of the horizontal and vertical photons become different in birefringent medium, the distinguishabilities between two polarization components in each photon result in degradation of the polarization entanglement. Here the pump and the down-converted photons are correlated in a frequency mode due to the phase matching condition of the SPDC process. Since PMD is closely related to the frequency of photons, the polarization entanglement degradation due to PMD is influenced by frequency bandwidths of the pump (Δ ω p ), the signal (Δ ω s ), and the idler (Δ ω i ) photons.  , and ω k0 and Δ ω k are the center wavelength and the bandwidth of the IF, respectively. Since the IFs in the two optical paths are identical to each other, the frequency centers and bandwidths of the signal and the idler photons are the same; that is, ω s0 = ω i0 and Δ ω s = Δ ω i . The subscript s and i refer to the states of the signal and the idler photons, respectively.
The polarization entangled photon pair can be prepared using the Hong-Ou-Mandel (HOM) 9,23 interferometer and post-selection. We send the two photons into the HOM interferometer and post-select the case that each photon comes out from different output ports a and b of the HOM interferometer as depicted in Fig. 1 9,15 . The post-selected state is the entangled state having the following form, where H and V refer to the horizontal and vertical polarizations, respectively. The subscripts indicate the output modes of the interferometer. In each optical path, PMD may come from birefringent optical elements such as PM fibers and quartz crystals. The origin of PMD is the difference of the refractive indices between two orthogonal polarization modes, which causes the difference in the amounts of phase shift between the two polarization modes. After each photon undergoes PMD in the mode a and b, the state evolves to where L 1 and L 2 are the lengths of the birefringent material in the optical path a and b, and n H (ω) and n V (ω) refer to the refractive indices for horizontally and vertically polarized photons at frequency ω, respectively. The refractive indices of materials are dependent on the frequency of photons. Here we take an approximation on the refractive indices up to the first order in a frequency, so that ( 1) 0 , where n H (0) and n V (0) are the refractive index values at the frequency ω s0 and ω ω ≡ are the first order derivatives of the refractive indices at ω s0 . c is the speed of light.
Since single photon detector cannot distinguish photon's frequency, the final state is obtained by tracing out the frequency degree of freedom in Eq. (6). The detailed information on how to evaluate concurrence 24 of the final state is provided in the Methods section.
The experimental setup is schematically shown in Fig. 2 and is composed of four parts: SPDC, Bell state preparation, PMD, and quantum state tomography (QST). Detailed information about the experimental setup is described in the Methods section.   Fig. 3(b) shows the concurrence of the case that the amount of PMD on one path is fixed (with L 1 = 13.6 mm), while the PMD on the other optical path is varying. In the former case the concurrence decreases gradually as the length of the quartz plates increases. In the latter case, however, the concurrence does not monotonically decreases as the amount of PMD on one photon increases. Moreover, the concurrence maximum value is obtained when L 2 is around 10 mm, which is clearly smaller than L 1 = 13.6 mm, i.e., L 2 < L 1 . We conclude from this result that when only one of the polarization entangled photon undergoes PMD, making the other photon undergo the smaller amount of the PMD can effectively preserve more entanglement. For instance, when only one photon undergoes PMD (L 1 = 13.6 mm), the concurrence C(0L) is more than two times smaller than its maximum value in Fig. 3(b). In addition, the maximum concurrence value is about 10% larger than the case that each photon undergoes the same amount of the PMD effect.
To confirm the validity of our theoretical calculation, we estimate the frequency bandwidths of the pump and the down-convert photons from the experimental data. (See the Method section for the concurrence of the final state.). We used the mode spacing of the pump photon as δω p = 3.24 × 10 11 Hz, which corresponds to the mode spacing of the wavelength Δ λ = 0.0282 nm 13 . The fitting values are evaluated to be Δ ω p = (3.95 ± 0.10) × 10 12 Hz and Δ ω s = (7.37 ± 0.28) × 10 12 Hz. The red lines in Fig. 3 are the fitting curves with these values. From the HOM interference signal, we obtained the bandwidth of the signal photons as ω ∆ = . ± . × (6 74 0 03) 10 Hz s HOM 1 2 , which is close to the estimated values from the result of Fig. 3.
Theoretical simulation results of the pump and the down-converted photon's frequency bandwidths effects on the concurrence are graphically shown in Fig. 4. The concurrence degradation behavior is depicted by changing either Δ ω p or Δ ω s values. Figure 4(a,b) show the effect of the pump photon's frequency bandwidth, while Fig. 4(c,d) describe the down-converted photon's frequency bandwidth effect. Δ ω p = 4 × 10 12 Hz and Δ ω s = 7 × 10 12 Hz are chosen based on the realistic values from our experiment. Figure 4(a,c) represent the case that the amounts of PMD applied on both photons are the same, showing that concurrence degradation increases as the bandwidths of the pump, the signal, the idler photons increase. Interestingly, the concurrence appears to be more sensitive to the increments of Δ ω p than the increments of Δ ω s . Figure 4(b,d) show the concurrence tendency where the amount of the PMD on one optical path is varied while the amount of the PMD on the other optical path is fixed as L 1 = 13.6 mm. For a fixed bandwidth Δ ω s , the concurrence has its maximum value near at L 2 = 13.6 mm when Δ ω p is small, while its maximum point asymptotically approaches L 2 = 0 as Δ ω p increases. In contrast, for a fixed Δ ω p value, the quartz length giving the concurrence maximum approaches to L 2 = L 1 as Δ ω s increases.

Discussion
A broadband CW pumped SPDC using an inexpensive and compact frequency multi-mode diode laser is an attractive practical resource for entangled state generation. In this work, we have considered a realistic quantum communication scenario in which the initial two-photon polarization entangled states are distributed through optical fiber channels. Since PMD in the birefringent optical element leads to the degradation of the polarization entanglement, understanding PMD effects on polarization entangled state is one of the important issues in practical quantum information such as long distance communication. We have theoretically and experimentally investigated how the frequency bandwidths of the pump and the down-converted photons affect the two-photon polarization entanglement degradation caused by PMD. From the results, we find that the amount of the entanglement degradation is asymmetrical to the PMD introduced to each quantum channel, which is an unique effect of the broadband pumping, equally applicable to pulsed pumping as well as CW broadband puming. This is an intriguing result since the same amount of the PMD is introduced on each photon for preserving entanglement against PMD effects in ref. 22. Therefore, we believe that our results can give an insight on the degradation of entanglement caused by PMD in long-distance communications via optical fiber channels and are helpful for overcoming decoherence effect on the polarization entanglement. where A*(ω p ) is the complex conjugate of A(ω p ), and D(ω p ) and A(ω p ) have the following form, . After integrating over the pump photon spectrum, the concurrence of the total multi-mode final state can be calculated. The concurrence of the two-qubit density matrix is found to be C(ρ) = max[0, λ 1 − λ 2 − λ 3 − λ 4 ] where λ n 's are the eigenvalues of the matrix ρ ρ ρ  with ρ σ σ ρ σ σ = ⊗ ⊗  ⁎ ( ) ( ) y y y y in decreasing order and ρ* is the complex conjugate of ρ 24 . In our case, the final density matrix has only a few non-vanishing terms, the concurrence of the final state can be calculated to be as follows: