1. Introduction

Entanglement is a prominent feature of quantumly indistinguishable states and a key resource for a variety of quantum devices and processes. An entangled state can be constructed from photon pairs by using photon degrees of freedom such as polarization, wave-vector, or frequency [1,2]. To realize the potential of quantum applications, the ways of generating entangled photon pairs need to be coherent and versatile. A typical method is based on spontaneous parametric down conversion (SPDC) [3], in which one pump photon is coherently split into a pair of correlated photons, a signal and an idler. A remarkable feature of SPDC light sources is that the polarization, momentum (wave-vector) and frequency are uniquely determined by energy and momentum conservation rules, so that one can have a quantum-correlated photon pair with strong correlation of polarization, momentum and/or frequency state.

One of the most popular schemes to generate the entangled state is to use the two symmetry non-collinear processes which create the polarization-entangled photon pairs with strong momentum correlation [1]. Though this is a simple scheme using a single pump and crystal, there is a limitation to using spatially separated photon pairs for quantum applications. For this reason, it is essential to develop a generation method of entangled photon pairs using the collinear SPDC process. In many cases, the collinear SPDC process has been realized with additional operations such as filtering post-selection [4], interferometry [2,5,6], and double-pass configuration [7,8]. These complicated schemes, however, typically have problems with the tunability of the output wavelength, the purity of the quantum entanglement [9,10] or compatibility with integrated applications [11]. A simple but promising scheme has been explored in many proof-of-principle experiments.

Recently, in order to overcome such technical challenges, it has been demonstrated that two nonlinear optical processes take place simultaneously in a cascaded-grating nonlinear crystal with two different polling periods, which can be used to generate frequency-entangled states with strong polarization correlation in the collinear geometry [12–15]. One significant point is that the photon pairs generated in the two-period crystal exhibit entanglement in the polarization and frequency domain by spatially separating the photon pairs via a polarization or frequency beam splitter. In particular, Kaneda et al. [15] demonstrated that two non-degenerate SPDC pairs produced in the two-period PPLN exhibit the non-degenerate HOM interference. They confirmed that the cascaded-grating nonlinear crystal can directly form the frequency-bin entangled photon pairs. On the other hand, the output wavelengths generated by these techniques are in a limited spectral range due to the phase-matching conditions. For the fabricated crystal, only a set of wavelengths can excite the non-degenerate symmetric SPDC process. To realize the full potential of these techniques, tuning of the SPDC wavelengths over a wide spectral range including the infrared region remains a pressing issue for quantum sensing and communication implementations.

Here, we report a novel way to generate entangled photon pairs in the frequency and polarization domains by using simultaneous collinear spontaneous parametric down conversion (s-SPDC) [16,17]. Periodically poled stoichiometric lithium tantalate (PPSLT) crystals have a relatively small difference in refractive index between extra-ordinary and ordinary light rays [18,19]. It allows two type-II phase-matching conditions to be satisfied with the identical pump frequency, crystal period, and temperature. Two symmetric non-degenerate type-II SPDC pairs are spectrally indistinguishable after separated by a polarization beam splitter [15]. Therefore, the s-SPDC pairs become directly entangled without any complicated procedure as mentioned above. Furthermore, a key feature of the s-SPDC method is that the signal and idler can be set to a wide range of arbitrary frequencies by tuning the pump frequency and crystal temperature. Thus, it can readily generate entangled photon pairs consisting of signals and idlers with wavelengths not only in the telecom region but also far apart in the visible and infrared regions. Our light source will bridge the gap between the demands and technical limitations on applicability for quantum sensing and communication.

2. Theory

In this section, we discuss the theory of simultaneous phase-matching pair generation using the special birefringence properties of PPSLT [18]. Suppose that two independent collinear type II quasi-phase matching conditions are simultaneously satisfied in the same crystal with one polling period. In this case, the phase matching condition and law of the energy conservation can be written as follows:

 

Fig. 1. (a) Calculated curves of $y = {F_T}(x )$ for PPSLT with different temperatures 75-105 °C based on the Sellmeier equations from Ref. [18]. The number and position of the solutions that satisfy Eq. (2) vary as changing the temperatures. (b) Enlarged plot of the curve at 81.8 °C. The output wavelengths and corresponding polling period and pump wavelength are arranged by changing the value of $1/{\lambda _C}$.

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Now, we focus on the special case of 81.8 °C (solid curve in Fig. 1(a)) which corresponds to the experimental condition as described below. Figure 1(b) shows the enlarged plot around the minimum value of the solid curve. The dashed lines represent several $1/{\lambda _c}$s to determine the solutions. For example, for $1/{\lambda _c} = 1.919 \times {10^{ - 4}}$, which is close to the minimum, the signal and idler are generated at 0.955 and 0.998 µm. This directly gives the pump wavelength ${\lambda _p}\; $ as 488 nm from Eq. (1c). The polling period $\mathrm{\Lambda }$ is obtained as 6.04 µm from Eq. (1a) or (1b). As increasing $1/{\lambda _c}$, the output wavelengths are arranged to the widely-separated combinations as shown in Fig. 1(b). One essential feature is that the crystal temperature determines the tunability of the output wavelength. According to our calculation, the widest tunability can be obtained with the temperature at 84.7 °C, in which the signal and idler wavelength can be arranged from the degenerate pair of 1.14 µm and 1.14 µm ($\mathrm{\Lambda } = 9.69$ µm, ${\lambda _p} = 0.57$ µm) to the non-degenerate pair of 0.55 µm and 3.78 µm ($\mathrm{\Lambda } = 9.80$ µm, ${\lambda _p} = 0.47$ µm). It should be noticed that one could not set the crystal temperature, pump wavelength or polling period independently. We will discuss the case of the fixed crystal polling period in the next section and the case of fixed pump wavelength in the Supplement 1.

Note that other nonlinear crystals such as periodically poled lithium niobate (PPLN) have no solution for Eq. (2) due to the larger birefringence. The difference between ${n_e}$ and ${n_o}$ of PPLN crystals monotonically changes and is more than 0.05 in the transparent spectral region [20]. In such case, ${F_T}(x )$ also monotonically changes, which implies that there are no combinations of ${\lambda _p}$, T, and $\mathrm{\Lambda }$ to satisfy Eq. (2). However, this is not the case for some crystals such as PPSLT crystal. The difference between ${n_e}$ and ${n_o}$ of PPSLT crystals is not only less than 0.004 but also does not monotonically change [18]. These features allow ${F_T}(x )$ to have several extremum points in the visible and infrared region, which means that there is a set of ${\lambda _p}$, T, and $\mathrm{\Lambda }$ that satisfies both equations above. The detail calculations for PPLN crystals are shown in Supplement 1.

The output quantum state $\left| \varphi \right\rangle $ of the simultaneous SPDC pairs can be described as follows:

Here, the quantum states are described using horizontal (H) or vertical (V) polarization modes, and the angular frequency ${\omega _{1(2 )}}$ in place of the wavelength ${\lambda _{s(i )}}$. $\theta $ is the relative phase between two quantum states, which is derived from the temporal gap caused in the generation process between signals and idlers and is arranged by the external delay line after separating the photons by a polarization beam splitter (PBS) or dichroic mirror (DM). By using PBS or DM, $\left| \varphi \right\rangle$ is spatially separated and reduced to a frequency-bin or polarization-bin state [15]. In our experiment, we separated the photon pairs by using a PBS and obtained a frequency-bin state as follows:

3. Wavelength tuning

3.1 Phase matching condition

To investigate the potential tunability of the type-II s-SPDC processes, we calculated the possible combinations of the pump wavelengths and crystal temperatures while using the fixed polling period. Figure 2(a)-(c) show the output wavelengths and pump wavelengths as function of the crystal temperature for several polling periods. The red, blue and black solid curves indicate the signal, idler and pump wavelengths, respectively. As shown in Fig. 2(a), the s-SPDC process with the polling period of Λ = 6.04 µm has a potential coverage in the visible and near infrared region. Especially, the degenerate pairs are generated around 1 µm with the temperature at 81.8 °C. In the case of Λ = 10.6 µm, the idler wavelengths can be arranged even into the infrared region. Moreover, by setting Λ = 20.6 µm, the degenerate pairs are generated in the telecom band around 1.5 µm, and the non-degenerate pairs can be widely separated in the visible and infrared region. It should be noticed that the tuning range of the pump wavelength and crystal temperature is quite narrow, which means that the pump wavelength and the crystal temperature should be finely tuned for generating a certain combination of signal and idler. Additionally, this tuning capability can be also achieved by changing the polling period and crystal temperatures while using the fixed pump wavelengths. In this case, the tunability of the output wavelength is realized by setting the temperature to satisfy Eq. (1a) and (1b) with the polling period in a certain range. The calculated results of these tuning arrangement are shown in Supplement 1.

 

Fig. 2. Phase matching conditions of the type-II s-SPDC processes in PPSLT calculated using the Sellmeier equations from Ref. [18]. Two symmetric non-degenerate type-II SPDC pairs are generated in a single-pass configuration. Output wavelengths (signal: red line, idler: blue line) and pump wavelengths (dotted line) are shown as function of the crystal temperature for several polling periods: (a) Λ=6.04 µm, (b) Λ=10.6 µm, (c) Λ=20.6 µm.

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3.2 Spectral filtering system

As described in the next subsection, we have experimentally demonstrated the type-II s-SPDC processes in a PPSLT crystal excited by a 488 nm pump laser with a broad linewidth of 1 nm. To show the affection of the 1-nm linewidth to the spectral distribution of the s-SPDC process, we have calculated phase matching conditions for three representative pump wavelengths in the 1-nm spectral window, 0.4872, 0.4877, and 0.4882 µm. In each case, we calculated the phase-matching curves under the ideal condition with a narrow-linewidth pump laser (0.1 nm) and a single-grating PPSLT crystal.

The upper panel of Fig. 3 shows the phase-matched signal (blue and green dots) and idler (red and orange dots) wavelengths as function of the crystal period. It is noteworthy that the phase-matched crystal temperature is almost unchanged as shown in the lower panel of Fig. 3. In this case, the vertical dashed line (6.04 µm) has intersections with the phase-matching curves among the 0.92 µm and 1.04 µm. These results suggest that under the conditions of a given crystal temperature (81.8°C) and period (6.04 µm), the distribution of signal and idler wavelengths continuously covers around 0.98 µm and extends over a 120-nm range from 0.92 µm to 1.04 µm. Note that the right-side phase-matching curve (0.4882 µm) has no intersection with the vertical dashed line as long as the corresponded period is larger than 6.058 µm. In this condition, only a part of 1-nm pump spectral window, 0.4872-0.488 µm, contributes to the distribution of the s-SPDC output wavelength.

 

Fig. 3. The phase matching conditions for the simultaneous type-II SPDC process in PPSLT when the pump wavelength is fixed around 0.488 µm calculated using Sellmeier equations from Ref. [18]. Upper panel represents the wavelengths of generated signals (blue) and idlers (red) as function of the polling period of the PPSLT crystal. Lower panel represents the phase matched temperatures.

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Figure 4(a) shows the simulated s-SPDC spectra for two pump-linewidth cases, 1 nm (blue dashed curve) and 0.1 nm (blue solid curve) under Λ = 6.04 µm, $T$ = 81.8 °C. In the case of 1 nm pump linewidth, one can see only a broad spectrum with a single peak as expected from Fig. 3. On the other hand, in the case of 0.1-nm pump linewidth, signal and idler are clearly separated with much narrower spectrum width than that of 1-nm pump linewidth.

 

Fig. 4. (a) Numerical simulations of the s-SPDC spectra for PPSLT without spectral broadening by single grating and monochromatic pump (blue solid curve) and with spectral broadening (blue dashed curve) with the poling period ranges from 6.038 µm to 6.043 µm and the bandwidth of the pump around 1 nm. (b) Red curve shows the transmission spectrum of the combination of the filters we used for extracting the corresponding signals and idlers. Refractive indexes are taken from the Sellmeier equations from Ref. [18].

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Basically, spectra of non-degenerate signals and idlers need to be explicitly separated for constructing the frequency bin mode, which means that the spectral widths should be much narrower than the wavelength separation. Therefore, we used the spectral filters in order to extract only the corresponding signal and idler pairs from the broad s-SPDC emissions [21]. The typical transmission spectrum after filtering is shown by the red curve in Fig. 4(b) with the combination of the filters (Thorlabs Corporation: FELH950, FESH1000 and NF980-41). We confirmed that the filter set has almost the same transmission window as the simulated spectrum shown in Figs. 4(a) and 4(b) by blue solid curves. Therefore, the filtering process corresponds to the selection of the SPDC pairs generated with the pump-laser linewidth of 0.1 nm.

3.3 Demonstration of s-SPDC wavelength tuning

Figure 5(a) shows the experimental setup to measure signal spectra using a wire grid polarizer (WGP) for analyzing the distribution of the polarization modes. A PPSLT crystal was excited by a 488 nm continuous wave laser diode (linewidth: 1 nm) with an average power of 1 mW. We employed two 11-mm-long fan-out PPSLT crystals (Oxide Corporation) for setting a variety of the polling periods. The output signals were collected and introduced into a silicon CCD spectrometer after collimated by a 50-mm-focal lens. We used appropriate frequency filters, depending on the situation. The WGP extracted a certain linear polarization mode.

 

Fig. 5. (a) Experimental setup for measuring the SPDC signal spectra. PPSLT was mounted on the movable stage. A wire grid polarizer (WGP) was inserted for extracting a certain linear polarization mode before coupled into the spectrometer. (b) Experimental spectra (solid curves in the upper panel) under the degenerate condition with Λ = 6.04 µm, $T$ = 75 °C, and theoretical spectra (solid curves in the lower panel) under the condition with Λ = 6.04 µm, $T$ = 81.8 °C. The temperatures shift by 7-8 K from the experimental values systematically occurred possibly due to deviation of the Sellmeier equation, local heating or temperature gradients. Theoretical simulations of s-SPDC spectra for pump wavelength of 488 nm using Eq. (1a) based on Eq. (4) in Ref. [16] and Sellmeier equations from Ref. [18]. Two curves correspond to the horizontal (blue) and vertical (green) polarization mode, respectively. (c) Experimental spectra (dashed curves in the upper panel) under the degenerate condition with Λ = 7.83 µm, $T$ = 76 °C, and theoretical spectra (solid curves in the lower panel) under the condition with Λ = 7.83 µm, $T$ = 83.4 °C.

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In the upper panel of Fig. 5(b), measured signal and idler spectra after filtering in H- and V-polarization (blue and green solid curves, respectively) are shown for Λ = 6.04 µm and $T$ = 75 °C. This corresponds to the near degenerate condition with the frequencies of the signal and idler waves close to each other. We picked up only narrow-bandwidth signals (0.955 µm) and idlers (0.998 µm), whose wavelength separation was explicitly broader than the linewidths of the signal and idler. We confirmed that this pair should be the phase-matched pair under the s-SPDC process by comparing with the theoretically simulated spectra for Λ = 6.04 µm, $T$ = 81.8 °C shown in the lower panel (blue and green dashed curves, respectively). Note that the temperatures are found to shift systematically by 7-8 K from the experimental values. Possible causes of this shift include the crystal being strictly different from the one that determined the Sellmeier equation [18], local heating due to the pump laser light, and temperature gradients, but it has not been clarified yet which of these factors is responsible. Except for this temperature shift problem, the clear peaks were observed in both polarizations, which is consistent with the theoretical simulation of the s-SPDC processes.

Next, we investigated a non-degenerate case of s-SPDC to demonstrate the output wavelength tuning by arranging the crystal temperatures and polling periods with the fixed pump wavelength 488 nm. In the upper panel of Fig. 5(c) is shown the spectrum of the signal for the condition of Λ = 7.83 µm and $T$ = 76 °C in H- and V-polarization (blue and green curve, respectively). The center wavelengths have a good agreement with that of the calculated signal spectrum (Λ = 7.83 µm, $T$ = 83.4 °C). One remarkable point is that the corresponding idlers in such non-degenerate conditions exists at 2.08 µm in the mid-infrared region. In other word, one can cover wide spectral range even in the infrared region by arranging the crystal temperatures and polling periods with the fixed pump wavelength 488 nm. Thus, the type-II s-SPDC generator can be served as a widely tunable frequency-entangled photon source covering from the visible to the infrared region. It should be mentioned that in both cases of Figs. 5(b) and 5(c), the intensity of the H-polarization is stronger than that of the V- polarization. This is because the first-order type-0 process (all photon is in the H-polarization) is also spontaneously caused in the PPSLT and contributes to the counts of the H-polarization. The discussion of the polarization analysis for the signals is shown in Supplement 1.

4. Hong-Ou-Mandel interference

To demonstrate entanglement between two SPDC pairs, we constructed a HOM interferometer in the frequency-bin configuration for the near degenerate pairs shown in Fig. 5(b). The experimental layout of the HOM interference measurement is shown in Fig. 6. We used the same pump wavelength and crystal parameters as is for in Fig. 5(b). After the residual pump was removed by 700-nm-long-pass filters (LP700s), the SPDC pairs from the PPSLT were introduced into a Michelson interferometer with a polarization beam splitter (PBS1) and quarter wave plates (QWPs) [15]. PBS1 spatially separated signals and idlers into two branches, respectively, and the QWP was served for SPDC pairs to pass into another PBS1 port by rotating the polarization after reflected by the mirror. One of the two branches in the interferometer has a delay line for arranging the relative phase between two SPDC pairs by changing the optical length. After the interferometer, the polarizations of two pairs were changed to a diagonal mode (45 degrees) by a half wave plate (HWP) set at 22.5 degrees in order to equally divide both pairs into two output ports of PBS2, which functioned as a non-degenerate HOM interferometer. The photons were focused into single-mode-fiber coupling (SMF) systems and detected by silicon-based single photon-counting avalanche diodes (Si-SPADs). Two Si-SPADs were connected to a time-correlated single photon counter (TCSPC) module to measure coincidental events in the two PBS2 branches. The integration time was 50 seconds and the temporal window of the entire coincidence measurement system was set at 4 ns. The typical count rate of each port was around $3500 \pm 59$ cps after the filtering system to remove the excess spectral broadening and extracted only discrete-color photon pairs as described in Section 3.2. For measuring the HOM interference, the coincidence events ${C_O}$ between two detecting ports were recorded in the TCSPC as changing the delay line with step size of 50 nm. To convert the coincidental counts into normalized counts, we also measured the coincidence counts ${C_F}$ at the position of the delay far from the temporal origin, where no interference occurred. Note that we measured the coincidence counts with the center wavelengths of two peaks that does not satisfy energy conservation with the pump by arranging the transmission window of the filtering system, in which we found there were little coincidence counts. It follows that the filtering system was properly placed for extracting the narrow s-SPDC components. We also estimated the conversion efficiency was $1.4 \times {10^{ - 12}}$ at the pump power of 1 mW using two single counts and coincidence counts [22], which was sufficient for the entangled photon-pair generation applications even though it was one order of magnitude smaller than to the generation schemes using other nonlinear crystals such as PPLNs [15].

 

Fig. 6. Schematic layout of the experiment. LP700s: long pass filters. PBS: polarization beam splitter. QWP: quarter wave plate. HWP: half wave plate. SMF: single mode fiber coupling. APD: avalanche photo diode. TCSPC: time correlated single photon counter.

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Figure 7(a) shows the measured HOM interferogram of the normalized coincidental events ${C_N} = {C_o}/2{C_F}$. In Fig. 7(a), the counts oscillated in a certain range, with a frequency corresponding to the spectral difference (13.5 THz) between the two peaks as shown in Fig. 5(b). This interferogram is theoretically described by the non-degenerate interference equation for spatial beating [2] as follows:

Here, V is interference visibility and $\delta \omega $ is beating frequency determined by the wavelength difference between the signal and idler. ${t_c}$ is coherence time determined by the spectral width of the SPDC signals and idlers. By fitting the fringe to Eq. (5), we estimated that V, $\delta \omega $, $\theta $, and ${t_c}$ were 0.82 ${\pm} $ 0.01, 13.5 ${\pm} $ 0.1 THz, 0.205 ${\pm} $ 0.001 rad, and 0.69 ${\pm} $ 0.01 ps, respectively. The estimated value of $\delta \omega $ is in a good agreement with the one predicted from the measured spectral difference. The value of ${t_c}$ matched the bandwidth of the signals. The visibility indicates indistinguishability of the s-SPDC pairs, which is limited by a broad phase-matching window due to the spectral width of the SPDC pump, and excess photons from the type-0 SPDC process. The spectral filtering manipulation used in the measurement decreased the total throughput of the system, and the type-0 SPDC pairs increased the accidental coincidence counts, which limited the visibility. Note that we could obtain the accidental counts of the type-0 pairs by measuring the coincidence as blocking the V blanch of the PBS1 in Fig. 6, and subtract it from the raw coincidence counts data. Moreover, we estimated the density matrix ${\rho _F}$ of the s-SPDC pairs in the frequency domain. According to the method used in Ref. [2], the matrix is constructed by parametrization as follows:

Here, the balance parameter p is the probability of $\left| {\omega _1}{\omega _2} \right\rangle_{HV}$, which is obtained from the ratio of the coincidence events of $\left| {\omega _1}{\omega _2} \right\rangle_{HV}$ and $\left| {\omega _2}{\omega _1} \right\rangle_{HV}$. In our setup, we estimated $p = 0.538 \pm 0.002$ by comparing the number of single counts in each port.

 

Fig. 7. (a) Measured Hong-Ou-Mandel (HOM) interferogram for type-II s-SPDC frequency-entangled photon pairs in PPSLT (pump wavelength: 488 nm, crystal temperature: 75 °C, polling period: Λ = 6.04 µm). The estimated beating frequency and visibility of the fringe are 13.5 THz and 0. 82. (b) (c) Real and imaginary parts of the reconstructed density matrix elements in the frequency domain.

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Figures 5(b) and 5(c) show real and imaginary parts of the reconstructed density matrix elements using the parameter V, $\theta $, and p estimated by the fitting curve shown in Fig. 7(a). As Eq. (6) indicates, the non-diagonal components in the density matrix correspond to the purity and relative phase of entanglement between photon pairs. In Fig. 7(b) and 7(c), we obtained uneven but explicit non-diagonal components in the real part, and a finite but small value of non-diagonal components in the imaginary part. Theoretically, ideal entangled states have $V/2 = p = 0.5$, which means that the deviation from the ideal states may come from the fact that $V/2$ and p is not 0.5. The fidelity to the ideal entangled states was $F = 0.90 \pm 0.001$, even more than the classical limit 0.5 [23]. From these results, we conclude that our s-SPDC process successfully generated entangled photon pairs. Note that the visibility V is maximized at $p = 0.5$ for $V/2 \le \sqrt {p({1 - p} )} $. Therefore, the indistinguishability can be improved by making p come close to 0.5 if we employed adequate optical components such as the PBS perfectly splitting the polarization modes over the broadband range. Moreover, the filters we used also decreased the visibility because the extracted spectral coverage cannot be perfectly proper. Using a narrow pump would improve indistinguishability.

5. Conclusion

We proposed a way to generate widely-tunable frequency entangled photon pairs with strong polarization correlation in a single-pass configuration by using the simultaneous phase-matching conditions of a non-degenerate type-II SPDC. In this technique, the specific birefringence in the nonlinear medium is the key that allows the simultaneous processes. We investigated the capability of a PPSLT crystal in generating entangled photon pairs over a wide range of frequencies including the infrared domain. One significant aspect of the method is that the pump can be of wide wavelength range of 0.28-2.75 µm as described in Supplement 1, which allows the entangled photon pairs to be produced with arbitrary spectral coverage, e.g. on telecom wavelengths or within the molecular fingerprint range.

Moreover, we experimentally demonstrated entanglement between two SPDC pairs in the frequency domain. Our experimental setup used a filtering system for extracting narrow bandwidths, which lowered the purity of the entangled state, though we obtained a clear spatial beating of the coincidental events. Using a narrower-band pump laser source, long crystal or more accurate beam splitter would allow us to have more discrete spectrum and increase the purity of entanglement. Additionally, multi-wavelengths filtering or multi-grating crystal can readily multiply the entanglement dimensions by increasing the frequency modes [24,25]. Therefore, our method may propel the development of not only multi-dimensional quantum protocols using entangled states for computing and communication, but also techniques of quantum sensing in a wide spectral region.