Quantification of nonlocal dispersion cancellation for finite frequency entanglement

Xiao Xiang; Xiao Xiang; Ruifang Dong; Ruifang Dong; Ruifang Dong; Baihong Li; Baihong Li; Baihong Li; Feiyan Hou; Feiyan Hou; Runai Quan; Runai Quan; Tao Liu; Tao Liu; Tao Liu; Shougang Zhang; Shougang Zhang

doi:10.1364/OE.390149

1. Introduction

Nonlocal dispersion cancellation (NDC) [1] has been recognized as a unique quantum feature of frequency entangled biphoton sources and has played a major role in many fiber-based quantum information applications, such as quantum communications [2–5], quantum-enhanced clock synchronization [6–9], as well as quantum nonlocality test for continuous variables [10,11]. According to Franson [1], the NDC effect refers to the preservation of the temporal correlation between the frequency-entangled photon pairs after their propagation through dispersive media with opposite signs of group-velocity dispersion. To-this-date theoretical research only deals with maximal frequency entanglement that leads to the complete dispersion cancellation [12,13]. However, analogous to all the other types of entanglement, practically generated frequency entanglement is finite. Therefore, it is rational that the NDC will be incomplete. In the real case of finite frequency entanglement, no prior investigation has been given to discuss the optimal condition for the NDC and to clearly quantify to which extent the temporal correlation can be recovered from the NDC, which is vital to the assessment of the function of NDC for various quantum information applications.

In this study, we present a theoretical model to quantify the NDC characteristics for the photon pairs with finite frequency entanglement. According to the model, the two-photon spectral width determines the extent of the dispersion cancellation, while the optimum NDC condition is directly related to the frequency correlation coefficient. Extending it to the specific cases of frequency disentanglement and maximal frequency entanglement, the same conclusion as given in the previous research [1,12,13] is deduced. Therefore, our model gives a full quantification of the NDC characteristics which incorporates all kinds of frequency relation properties. To verify our quantification model, we further utilized two different spontaneous parametric down-conversion (SPDC) photon pair sources with frequency correlation and anticorrelation to experimentally investigate the observed temporal coincidence width varying the dispersions in the signal and idler paths. The very good agreement between the experimental results and theoretical simulations proves the validity of our characterization methodology that is of great importance in the utilization and evaluation of the NDC effect for various quantum information applications.

2. Theoretical model

The state function of the generated two-photon source in terms of the joint spectral relation can be written as [14–16]

(1)$$\begin{array}{c} {\; |\Psi \rangle \equiv {\int\!\!\!\int }\frac{{d{{\tilde{\omega }}_s}}}{{2\pi }}\frac{{d{{\tilde{\omega }}_i}}}{{2\pi }}\psi ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )\hat{a}_s^ + ({{\omega_{s,0}} + {{\tilde{\omega }}_s}} )\hat{a}_i^ + ({{\omega_{i,0}} + {{\tilde{\omega }}_i}} )|0 \rangle,} \end{array}$$

where $\hat{a}_{s(i )}^ + $ denotes the creation operator for the signal (s) or idler photon (i), and $|0\rangle $ represents the vacuum state. In addition, ${\tilde{\omega }_{s(i )}}$ is the angular frequency deviation of the signal (idler) photon from its center angular frequency ${\omega _{s(i ),0}}$ with ${\omega _{s,0}} + {\omega _{i,0}} = {\omega _{p,0}}$, where ${\omega _{p,0}}$ is the pump center angular frequency. $\psi ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )$ represents the joint spectral amplitude function of the signal and idler which can be expressed by the following bivariate normal distribution [17]

(2)$$\psi ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )\sim exp\left[ { - \frac{1}{{2({1 - {r^2}} )}}\left( {\frac{{\tilde{\omega }_s^2}}{{\sigma_s^2}} + \frac{{\tilde{\omega }_i^2}}{{\sigma_i^2}} - \frac{{2r{{\tilde{\omega }}_s}{{\tilde{\omega }}_i}}}{{{\sigma_s}{\sigma_i}}}} \right)} \right].$$

Herein, $ {\sigma _s}$ and ${\sigma _i}$ correspond to the single-photon spectral widths of the signal and idler photons. $ r$ is the spectral correlation coefficient of the generated photon pair state. Depending on its different value, the photon pairs can be classified as frequency anticorrelated $(0 > r > - 1)$ frequency uncorrelated $({r = 0} )$, and frequency correlated $(0 < r < 1)$. According to Ref. [18], the operational frequency entanglement quantifier $(R )$ is thus associated with the spectral correlation coefficient by $R = 1/\sqrt {1 - {r^2}} $.

When the signal and idler photons respectively travel through two dispersive media with dispersion coefficients $k_1^{\prime \prime}$ and $k_2^{\prime \prime}$, and with lengths ${l_1}$ and ${l_2}$, the second-order Glauber correlation function of the output photons can be written as [19]

(3)$$\begin{aligned} G^{\left( 2 \right)}\left( {t_1,t_2} \right) &\propto \left| {\rm \int\!\!\!\int }\displaystyle{{d{\tilde{\omega }}_s} \over {2\pi }}\displaystyle{{d{\tilde{\omega }}_i} \over {2\pi }}exp\left( -\displaystyle{1 \over {2\left( {1-r^2} \right)}}\left( {\displaystyle{{\tilde{\omega }_s^2 } \over {\sigma _s^2 }} + \displaystyle{{\tilde{\omega }_i^2 } \over {\sigma _i^2 }}-\displaystyle{{2r{\tilde{\omega }}_s{\tilde{\omega }}_i} \over {\sigma _s\sigma _i}}} \right)\right.\right.\\ & \quad \left.\left.+ i\left( {{\tilde{\omega }}_s\tau _1 + {\tilde{\omega }}_i\tau _2 + \displaystyle{{k_1^{{\prime \prime}} l_1\tilde{\omega }_s^2 + k_2^{{\prime \prime}} l_2\tilde{\omega }_i^2 } \over 2}} \right) \right) \right|^2 \end{aligned}$$

where ${\tau _1} = {t_1} - k_1^{\prime}{l_1}$ and ${\tau _2} = {t_2} - k_2^{\prime}{l_2}$. As the direct coincidence measurement for ${G^{(2 )}}$ can only yield the difference of the time arrivals between the signal and idler photons, we can replace ${\tau _ + } = {\tau _1} + {\tau _2}$ and ${\tau _ - } = {\tau _1} - {\tau _2}$, and apply integration over ${\tau _ + }$ on Eq. 3). The correlation function with respect to the difference of the time arrivals can then be expressed as

(4)$${G^{(2 )}}({{\tau_ - }} )\propto exp\left( { - \frac{{{{({{\tau_ - } - {{\bar{\tau }}_ - }} )}^2}}}{{2{\Delta^2}}}} \right),$$

where ${\bar{\tau }_ - } = k_1^{\prime}{l_1} - k_2^{\prime}{l_2}$ denotes the average time delay difference between the two propagation arms. The temporal coincidence width of ${G^{(2 )}}({{\tau_ - }} )$ is given by

(5)$${\Delta ^2} = \Delta _{0,\tau }^2 + \frac{{{{({k_1^{\prime\prime}{l_1}} )}^2}\sigma _s^2 - 2r({k_1^{\prime\prime}{l_1}} )({k_2^{\prime\prime}{l_2}} ){\sigma _s}{\sigma _i} + {{({k_2^{\prime\prime}{l_2}} )}^2}\sigma _i^2}}{2},$$

where ${\Delta _{0,\tau }} = \sqrt {\frac{1}{{2({1 - {r^2}} )}}\left( {\frac{1}{{\sigma_s^2}} + \frac{1}{{\sigma_i^2}} + \frac{{2r}}{{{\sigma_s}{\sigma_i}}}} \right)} $ denotes the initial time difference width determined by the photon pair generation process. Correspondingly, the temporal width at the full-width-at-half-maximum (FWHM) is given by $\Delta {\tau _ - } = 2\sqrt {2ln2} \Delta $. Defining the relation between the two dispersive media by the ratio $t = ({k_2^{^{\prime\prime}}{l_2}} )/({k_1^{^{\prime\prime}}{l_1}} )$, Eq. 5) is reduced to

(6)$${\Delta ^2} = \Delta _{0,\tau }^2 + {({k_1^{\prime\prime}{l_1}} )^2}\frac{{({\sigma_s^2 - 2rt{\sigma_s}{\sigma_i} + {t^2}\sigma_i^2} )}}{2}.$$

Thus, the minimum temporal coincidence width can be deduced and formulated as

(7)$${\Delta _{min}} = \sqrt {\Delta _{0,\tau }^2 + \frac{{{{({k_1^{^{\prime\prime}}{l_1}\sigma_s^c} )}^2}}}{2}} , $$

at

(8)$${t_{opt}} = r{\sigma _s}/{\sigma _i}. $$

Here $\sigma _s^c = {\sigma _s}\sqrt {1 - {r^2}} $ corresponds to the two-photon spectral width. As shown by Eq. 7)–(8), the narrowing of the dispersion broadened temporal coincidence term through NDC is ultimately limited by the term ${({k_1^{^{\prime\prime}}{l_1}\sigma_s^c} )^2}/2$, while the optimum NDC condition is directly related to the frequency correlation coefficient r. When the paired photons are frequency disentangled, the two-photon spectral width becomes equal to the single-photon spectral width, and the dispersion experienced by the signal photons can no longer be cancelled by the other. If the two-photon spectral width can be infinitesimally small such that the maximal frequency entanglement is asymptotically reached, a complete dispersion cancellation will be approached. To further give a clearer description on the condition for optimum NDC and the minimum achievable temporal correlation width as a function of the two-photon spectral width $\sigma _s^c$, a schematic diagram on NDC-based temporal coincidence measurement where a two-photon source with maximal and finite frequency anticorrelation are exampled in Fig. 1.

Fig. 1. Schematic diagram on NDC-based temporal coincidence measurement where two-photon sources with maximal and finite frequency anticorrelation are exampled. (a) joint-spectral intensity plots of frequency anticorrelated entangled biphoton state with finite entanglement (upper) and maximal entanglement (lower); (b) layout for the NDC-based temporal coincidence measurement, $k_1^{^{\prime\prime}}{l_1}$ and $k_2^{^{\prime\prime}}{l_2}$ denote the dispersion experienced by the signal and idler photons, respectively; (c) temporal coincidence distributions when the idler photons bypass (I) and propagate through (II) the NDC setup. When the two-photon spectral width $\sigma _s^c = 0$, the temporal coincidence width after NDC will be suppressed to the initial time difference width determined by the photon pair generation process. While for $\sigma _s^c \ne 0$, the temporal coincidence width after the optimum NDC is stretched by $\frac{{({k_1^{^{\prime\prime}}{l_1}\sigma_s^c} )}}{{\sqrt 2 }}$.

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Therefore, our model gives a full quantification of the NDC characteristics that incorporates all types of frequency relation properties. It can be directly exploited to assess the capability of the NDC effect in various quantum information applications, such as the quantum nonlocality t.st based on the NDC effect, whereby the temporal correlation width after dispersive propagation will set a critical limitation for the violation of the Bell-like inequality [11] (see details in the Appendix).

As the process of SPDC is the most accessible and controllable way to generate the photon pairs, the NDC characteristics are further investigated as a function of the pump spectrum and phase matching properties of the SPDC. Assume that the pump is Gaussian-shaped with a spectral bandwidth ${\sigma _p}$, which can be expressed by [19]

(9)$$\alpha ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )\propto exp\left[ { - \frac{{{{({{{\tilde{\omega }}_s} + {{\tilde{\omega }}_i}} )}^2}}}{{2{\sigma_p}^2}}} \right]. $$

For a type-II nonlinear crystal with a length of L, the phase matching function can be approximated by a Gaussian function which can be expressed as [20],

(10)$$\phi ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )\propto exp[{ - a{L^2}{{({{\gamma_s}{{\tilde{\omega }}_s} + {\gamma_i}{{\tilde{\omega }}_i}} )}^2}} ],$$

where $a = 0.04822$, ${\gamma _{s(i )}} = k_p^{\prime}({{\omega_{p,0}}} )- k_{s(i )}^{\prime}({{\omega_{s(i ),0}}} )$. $k_j^{\prime},\; \; j = p,\; s,\; i$, represent the first-order derivative of the wave vector around center frequency of the pump, signal, or idler photons. Accordingly, the joint spectral amplitude function $ \psi ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )$ is given by the product of $\alpha ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )$ and $\phi ({{{\tilde{\omega }}_s},{{\tilde{\omega }}_i}} )$. Combination of Eq. 9) and (10) with Eq. (2) yields the expressions for the spectral correlation coefficient r, the single-photon spectral width of the signal (idler) photon as a function of the pump bandwidth and the phase matching function according to

(11)$$r ={-} \frac{{1 + 2a{{({{\sigma_p}L} )}^2}{\gamma _s}{\gamma _i}}}{{\sqrt {1 + 2a{{({{\sigma_p}L{\gamma_s}} )}^2}} \sqrt {1 + 2a{{({{\sigma_p}L{\gamma_i}} )}^2}} }},$$

(12)$${\sigma _s} = \frac{{\sqrt {1 + 2a{{({{\sigma_p}L{\gamma_i}} )}^2}} }}{{\sqrt {2a} |{{\gamma_s} - {\gamma_i}} |L}},\; {\sigma _i} = \frac{{\sqrt {1 + 2a{{({{\sigma_p}L{\gamma_s}} )}^2}} }}{{\sqrt {2a} |{{\gamma_s} - {\gamma_i}} |L}}.$$

Substitution of Eq. 11) and (12) into Eq. 6) yields

(13)$$\begin{array}{c} {\Delta = \sqrt {\Delta _{0,\tau }^2 + {{({k_1^{\prime\prime}{l_1}} )}^2}\frac{{{{({1 + t} )}^2} + 2a{{({{\sigma_p}L} )}^2}{{({t\; {\gamma_s} + {\gamma_i}} )}^2}}}{{4a{L^2}{{({{\gamma_s} - {\gamma_i}} )}^2}}}} .} \end{array}$$

The initial time difference width is given by ${\Delta _{0,\tau }} = \sqrt a L|{{\gamma_s} - {\gamma_i}} |$. Knowledge of the properties of the pump spectrum and SPDC crystal yields the condition for optimum NDC according to ${t_{opt}} ={-} \frac{{1 + 2a{{({{\sigma_p}L} )}^2}{\gamma _s}{\gamma _i}}}{{1 + 2a{{({{\sigma_p}L} )}^2}{\gamma _s}^2}}$. Correspondingly, the achievable minimum temporal coincidence width ${\Delta _{min}}$ can be evaluated from Eq. 7) with $\sigma _s^c = \frac{{{\sigma _p}}}{{\sqrt {1 + 2a{{({{\sigma_p}L{\gamma_s}} )}^2}} }}$. As the phase matching bandwidth is expressed as ${\sigma _f} = \sqrt 2 /\left( {\sqrt a L|{{\gamma_s} - {\gamma_i}} |} \right)$ [21], ${\Delta _{0,\tau }} = \sqrt 2 /{\sigma _f}$ can be defined. Under the extended phase matching condition (EPM, leading to ${\gamma _s} \approx{-} {\gamma _i}$) [20], we can achieve

(14)$$\sigma _s^c = \sqrt {\frac{{\sigma _p^2\sigma _f^2}}{{({\sigma_f^2 + \sigma_p^2} )}}} ,$$

(15)$${t_{opt}} = r ={-} \frac{{\sigma _f^2 - \sigma _p^2}}{{\sigma _f^2 + \sigma _p^2}}.$$

It is clear to see that Eq. 15) turns out to be the same as the expression given in Ref. [12] when ${\sigma _p} = 0$. In real cases, the pump laser always has a finite bandwidth. Dependent on the relation between ${\sigma _p}$ and ${\sigma _f}$, there are two different regimes for corresponding ${t_{opt}}$, $\sigma _s^c$ and ${\Delta _{min}}$. If ${\sigma _p} \ll {\sigma _f}$, ${t_{opt}} \approx{-} 1$ and $\sigma _s^c \approx {\sigma _p}$ is approached. The minimum temporal coincidence width is given by the pump bandwidth, i.e.,

(16)$${\Delta _{min,p}} \approx \sqrt {\frac{2}{{\sigma _f^2}} + {{({k_1^{^{\prime\prime}}{l_1}} )}^2}\frac{{\sigma _p^2}}{2}} . $$

When ${\sigma _p} \gg {\sigma _f}$, .. and $\sigma _s^c \approx {\sigma _f}$. The minimum temporal coincidence width is determined by the phase matching bandwidth ${\sigma _f}$, i.e.,

(17)$${\Delta _{min,f}} \approx \sqrt {\frac{2}{{\sigma _f^2}} + {{({k_1^{^{\prime\prime}}{l_1}} )}^2}\frac{{\sigma _f^2}}{2}} \; .$$

In the examples presented below, the SPDC process based on a collinear type-II periodically poled KTiOPO₄ (PPKTP) crystal is considered. The pump wavelength is centered at 791 nm and the PPKTP crystal has a period of 46.146 µm, so that the EPM condition is satisfied. The dispersive media in the signal path is assumed as a 10 km-long single-mode fiber (SMF) with a dispersion coefficient of $k_1^{^{\prime\prime}}\sim{-} 2.35 \times {10^{ - 26}}{\textrm{s}^2}/\textrm{m}$, while the dispersion in the idler arm satisfies the optimum NDC condition given by Eq. 13). The dependencies of r and $\Delta {\tau _{ - ,min}} = 2\sqrt {2ln2} {\Delta _{min}}$ (the full-width-at-half-maximum (FWHM)) on ${\sigma _p}$ and L are depicted in Fig. 2.

Fig. 2. Contour plots of (a) the spectral coefficient of r and (b) the FWHM minimum temporal correlation width after nonlocally cancelling the dispersion experienced the signal photons propagating through a 10 km-long single-mode fiber as a function of the pump bandwidth and the crystal length.

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One can see from Fig. 2(a) that the spectral correlation of the SPDC photon pair is changed dramatically from anticorrelated ($r < 0$) to uncorrelated ($r = 0$) and correlated ($r > 0$) as a function of the pump bandwidth and the crystal length. Given that ${t_{opt}} = r$, it is also inferred that the corresponding dispersion applied into the idler path ($k_2^{^{\prime\prime}}{l_2})$ should vary from negative to positive to achieve the minimum temporal correlation width. As shown in Fig. 2(b), $\Delta {\tau _{ - ,min}}$ is considerably stretched by broadening the pump bandwidth, and concurrently narrowed by increasing the crystal length, in accordance with two different behaviors described by Eq. 16) and (17).

3. Experimental verification

3.1 Experimental setup

To verify the theoretical model shown above, two different SPDC photon pair sources with frequency correlation and anticorrelation were utilized for experimental examinations. The frequency correlated biphoton source was generated from a collinear type-II PPKTP with a length of 20 mm and a 46.146 µm poling period (Raicol Crystals), which was pumped by a femtosecond laser (Fusion 20-150, FemtoLasers) with a center wavelength of 787 nm and a FWHM spectral bandwidth of 22 nm. The details of the experimental system can be found in [22], and the spectral correlation coefficient was estimated to be equal to $r = 0.997$. The frequency anticorrelated biphoton source was generated with the use of a continuous wave laser (at 780 nm) which was produced by the cavity-based frequency doubling of a 10 kHz linewidth fiber laser at 1560 nm to pump a collinear type-II PPKTP crystal with a length of 10 mm [23]. The FWHM of the 780 nm pump laser was indirectly simulated to be 0.025 nm [24], and the corresponding spectral correlation coefficient was deduced to $ r ={-} 0.9998$.

After filtering out the residual pump, the orthogonally polarized signal and idler photons were then coupled into the fiber polarization beam splitter (FPBS). With the help of a half-wave plate (HWP2) before the FPBS, the signal and idler photons were spatially separated and transmitted into two fiber paths. These paths are respectively marked with the symbols s and i in Fig. 3. The single-mode fiber (SMF) was placed in the signal path, while either an auxiliary dispersion-compensation fiber (DCF) or a SMF were placed in the idler path for the implementation of the NDC. The after-propagation signal and idler photons were then detected by superconducting nanowire single-photon detectors (SNSPD1 & SNSPD2) with an efficiency of approximately 50% [25]. The arrival times of the signal and idler photons to the detectors were recorded independently by two commercial event timers, ET A and ET B (Eventech Ltd, A033-ET), as time tag sequences $\{{t_A^{(j )}} \}$ and $\{{t_B^{(j )}} \}$. Application of a cross-correlation algorithm on the acquired time sequences [26] allows the construction of the temporal coincidence distribution with respect to the time difference ${G^{(2 )}}({{\tau_ - }} )$.

Fig. 3. Experimental setup. Two different photon pair sources are generated with frequency correlation and anticorrelation. The frequency correlated biphoton source used a femtosecond laser centered at 787 nm as the pump laser, while the frequency anticorrelated biphoton source was generated with the use of a continuous wave 780 nm laser for the pump laser. PPKTP: nonlinear crystal for collinear type-II SPDC, DM: dichroic mirrors, HWP1 & HWP2: half-wave plates, C: fiber coupler, FPBS: fiber polarizing beam splitter, SMF: single-mode fiber, DCF: dispersion compensation fiber, SNSPD1 & SNSPD2: superconductive nanowire single photon detectors, ET A & ET B: event timers.

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3.2 Results and discussions

In our experimental setup, the combined FWHM jitters of the employed SNSPDs and the event timers (ETs) were measured to be approximately 37.6 ps, which are mainly attributed to the SNSPDs. The above jitter characterizing process was accomplished with frequency anticorrelated or correlated biphotons before dispersive propagations, providing a similar result. It means that these two kinds of biphotons both have a narrow (∼1 ps in theory) temporal correlation width much less than 37.6 ps when they leave the SPDC crystal. The dispersion coefficients of the SMF and DCF at 1560 nm were simulated based on the linear fits of the measured temporal coincidence widths versus their single-arm lengths, and yielded $k_1^{^{\prime\prime}}\sim{-} 2.37 \times {10^{ - 26}}{\textrm{s}^2}/\textrm{m}$ for SMF and $k_2^{^{\prime\prime}}\sim 1.99 \times {10^{ - 25}}{\textrm{s}^2}/\textrm{m}$ for DCF, respectively [11].

Considering the frequency anticorrelated biphotons as the SPDC source, Fig. 4(a) shows the measured FWHM temporal coincidence width ($\Delta {\tau _ - }$) versus the DCF length in the idler path, with a 62 km SMF connected in the signal path (in black squares). It can be observed that without the use of the DCF in the idler path, the observed $\Delta {\tau _ - }$ is greatly broadened owing to the dispersion added in the single arm. When the DCF was inserted, the temporal coincidence width was narrowed continually until $k_2^{^{\prime\prime}}{l_2} \approx{-} k_1^{^{\prime\prime}}{l_1}$. With a 7.47 km DCF inserted, $\Delta {\tau _ - }$ was narrowed to approximately 90 ps. With respect to the 7.47 km DCF, $\Delta {\tau _ - }$ was measured as the SMF fiber length in the signal path varied from 60 km to 63 km (inset plot in Fig. 4(a)). As shown, $\Delta {\tau _{ - ,min}}$ was observed at ${l_1}\sim 62\; \textrm{km}$ and ${l_2}\sim 7.47\; \textrm{km}$. Based on the experimental parameters, the phase matching bandwidth is equal to ${\sigma _f} \approx 2.2 \times {10^{12}}\; \textrm{rad}/\textrm{s}$, while ${\sigma _p} \approx 3.3 \times {10^{10}}\; \textrm{rad}/\textrm{s}$. Therefore, ${\sigma _p} \ll {\sigma _f}$ is satisfied. According to Eq. 16), a value of $\Delta {\tau _{ - ,min}} \approx 81\; \textrm{ps}$ is estimated. Inclusion of the timing jitter contribution from the single photon detector setup, $\Delta \tau _{ - ,min}^{\prime} \approx 89\; \textrm{ps}$ is achieved, which agrees well with the measured result. The simulated $\Delta {\tau _ - }$ as a function of the DCF length (${l_2}$) is also plotted in Fig. 4(a) (red solid curve), which shows a very good agreement with the experimental outcomes. To further visualize the effect of the pump bandwidth on $\Delta {\tau _{ - ,min}}$, the curve of $\Delta {\tau _ - }({{l_2}} )$ based on the assumption that the pump has a FWHM of 0.001 nm is shown in Fig. 4(a) as well (black dashed line). It can be observed that there is an apparent gap between the measured results and this simulation. The simulated $\Delta \tau _{ - ,min}^{\prime}$ is approximately 42 ps, which is mainly attributed to the jitter of the single-photon detectors, and is much smaller than the measured value. Therefore, the non-negligible spectral width of the pump (0.025 nm in FWHM) plays a dominant role in the minimum achievable temporal coincidence width.

Fig. 4. Measured (black squares) and simulated (lines) variations of the FWHM temporal coincidence width ($\Delta {\tau _ - }$) versus the DCF/SMF length in the idler path. (a) Experiments with frequency anticorrelated biphotons as the SPDC source whereby a 62 km SMF connected in the signal path (black squares). The red line corresponds to the simulation when the pump bandwidth (FWHM) is set to 0.025 nm, while the black dashed line denotes the simulation with the pump bandwidth (FWHM) set as 0.001 nm. The inset shows the measured (black squares) and simulated (red solid line) $\Delta {\tau _ - }$ value with the single-mode fiber (SMF) length as the signal path which varied from 60 km to 63 km, while the DCF length was fixed at 7.47 km. (b) Experiments with frequency correlated biphotons as the SPDC source, whereby a 10 km SMF connected in signal path (black squares). The corresponding simulation outcomes are shown in red line with the dispersion parameter chosen as $k_1^{\prime\prime}\sim 2.40 \times {10^{ - 26}}{\textrm{s}^2}/\textrm{m}$.

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Using the frequency correlated biphoton source, the NDC effect on the narrowing of the temporal coincidence width ($\Delta {\tau _ - }$) was also investigated. Based on the experimental parameters for the pulse pumped SPDC process, the phase matching and pump bandwidths are calculated to be ${\sigma _f} \approx 1.1 \times {10^{12}}\; \textrm{rad}/\textrm{s}$ and ${\sigma _p} \approx 2.8 \times {10^{13}}\; \textrm{rad}/\textrm{s}$ respectively. Therefore, ${\sigma _p} \gg {\sigma _f}$ is satisfied. By fixing the SMF length in the signal path at ${l_1} = $10 km, the FWHM temporal coincidence width versus the SMF length in the idler path was experimentally measured, and the results are plotted in Fig. 4(b) (black squares). For this case, the observed $\Delta {\tau _{ - ,min}}$ was achieved at $k_1^{^{\prime\prime}}{l_1} \approx k_2^{^{\prime\prime}}{l_2}$ with a value of about $560\; \textrm{ps}$. Simulations were also conducted according to Eq. 13). Results are shown in Fig. 4(b) (red solid curve). By choosing the dispersion parameter of the SMF at 1574 nm as a reasonable value of $k_1^{^{\prime\prime}}\sim{-} 2.40 \times {10^{ - 26}}{\textrm{s}^2}/\textrm{m}$, a very good agreement between the simulation and the experimental results has been achieved. According to Eq. 17), the value of $\Delta {\tau _{ - ,min}} \approx 440\; \textrm{ps}$ was estimated at ${l_2}\sim 9.5$ km rather than at ${l_2} \approx 10\; $km. This minor offset can be attributed to the non-negligible difference between the signal and idler single photon spectrum (${\sigma _s} \ne {\sigma _i})$, which is induced by the deviation of the center wavelength (787 nm) of the pulsed pump from the demanded wavelength (791 nm) for EPM.

4. Conclusions

In summary, we have quantified the NDC characteristics for the photon pairs with various types of frequency entanglement. It shows that, as long as the paired photons are frequency entangled, irrespective of whether the frequency is anticorrelated or correlated, the NDC may partially take effect. The extent of achievable dispersion cancellation is fundamentally limited by the nonzero two-photon spectral width of the photon pairs, which introduces an inevitable broadening by interaction with the dispersion in the signal path. To verify our quantification model, we further utilized two different types of SPDC photon pair sources with frequency correlation and anticorrelation properties to experimentally investigate the observed temporal coincidence width as a function of the dispersion relations in the signal and idler paths. The very good agreement between the experimental results and theoretical simulations proves the validity of our characterization methodology that is of great importance in the utilization and evaluation of the NDC effect for various quantum information applications. For instance, for the quantum nonlocality test based on the NDC effect, the necessary condition that a frequency entangled biphoton source should satisfy can be deduced by exploiting our characterization methodology [11] (see details in the Appendix).

Appendix

As deduced by Wasak et al. [10], the quantum nonlocality based on the NDC effect can be tested by the violation of a Bell-like inequality, which is shown as follows,

(18)$$W = \frac{{\Delta _{min}^2}}{{\Delta _{0,\tau }^2 + {{(k_1^{\prime\prime}{l_1})}^2}/({2\Delta_{0,\tau }^2} )}} \ge 1. $$

As described in the body text, ${\Delta _{min}}$ is the minimum temporal coincidence width after dispersive propagation. ${\Delta _{0,\tau }}$ denotes the initial time difference width which is determined by parameters in photon pair generation process. Note that there is a minor difference in the expression of the denominator in comparison with the formulation given in Ref. [11]. This is due to the notation of the lower uncertainty bound in terms of time and frequency variables, which is determined by the limited time-bandwidth product and should be written as $\Delta _{0,\tau }^2\Delta \Delta {{\Omega }^2} \ge 1/2$ in our case (here $\Delta {\Omega }$ corresponds to the spectral bandwidth of the frequency difference term).

Substituting Eq. 7) into the above inequation, $\sigma _s^c\; < 1/{\Delta _{0,\tau }}$ is needed to ensure the violation. Consider the SPDC process in which the pump wavelength is centered at 791 nm and the PPKTP crystal has a period of 46.146 µm, the dependence of W on the pump bandwidth ${\sigma _p}$ and the crystal length L is depicted in Fig. 5(a). Compared with simulations in Fig. 2(a), the purple area ($W < 1$) is located in the region of $r < 0$. Therefore frequency anticorrelation property of utilized photon pairs is desired for quantum nonlocality test based on the NDC effect.

Fig. 5. Contour plots of the simulated W as a function of the pump bandwidth ${\sigma _p}$ and the crystal length L with jitter of 0 ps (a), 1 ps (b), 10 ps (c), 100 ps (d). In the simulation, the pump center wavelength is assumed at 791 nm and the PPKTP crystal has a period of 46.146 µm.

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Note should be taken that the jitter of single-photon detectors would limit the timing resolution that cannot be ignored in a practical situation. Further taking the jitter contribution from the single photon detectors (${\Delta _{jit}}$) into account, $\sigma _s^c\; < \sqrt {\frac{1}{{\Delta _{0,\tau }^2 + \Delta _{jit}^2}}} $ or $r < - \frac{{\Delta _{jit}^2}}{{\Delta _{0,\tau }^2 + \Delta _{jit}^2}}$ is demanded. It means that the presence of the jitter will raise a more stringent requirement on the two-photon spectral width of entangled photon pairs for violation of the Bell-like inequality which can be visualized with different jitters of 1 ps, 10 ps and 100 ps in Fig. 5(b)-(d), respectively. Meanwhile, given $\sigma _s^c$ and ${\Delta _{jit}}$, the value of W is also dependent on the dispersion broadening experienced by the signal photon. To visualize such effect, the simulated W as a function of SMF length in the signal path is given in Fig. 6 by assuming the pump bandwidth as 0.01 nm and the crystal length as 10 mm. It shows that, using single photon detectors with lower timing jitter and increasing the value of the dispersion term $k_1^{\prime\prime}{l_1}$ (here exampled with SMF) would be helpful to realize a significant violation of the above Bell-like inequality.

Fig. 6. Simulated W versus the SMF length in the signal path with different jitters (0 ps, 1 ps, 10 ps 100 ps) when the pump bandwidth is 0.01 nm and the crystal length is 10 mm.

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Funding

National Natural Science Foundation of China (61535012, 61801458, 61875205, 91836301); Chinese Academy of Sciences (QYZDB-SSW-SLH007, XAB2019B17, XAB2019B15); National Youth Talent Support Program of China ([2013]33); Strategic Priority Research Program of Chinese Academy of Sciences (XDC07020200); Chinese Academy of Sciences Key Project (ZDRW-KT-2019-1-0103); Natural Science Foundation of Shaanxi Province (2019JM-346); Xi’an University of Science and Technology (2019YQ2-13).

Disclosures

The authors declare no conflicts of interest.

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Quantification of nonlocal dispersion cancellation for finite frequency entanglement

Abstract

1. Introduction

2. Theoretical model

3. Experimental verification

3.1 Experimental setup

3.2 Results and discussions

4. Conclusions

Appendix

Funding

Disclosures

References

Cited By

Figures (6)

Equations (18)

Optics Express