Programmable photon pair source

Quantum correlated photon pairs are essential resources that must be freely available for implementing many of the novel functions of quantum information science (QIS). Different quantum information processing tasks require photon pairs with different spectral properties. For QIS protocols involving quantum interference between different sources, such as quantum teleportation and quantum computing, photon pairs with no frequency correlation (or with a factorable spectrum) are desirable.^1–3 For quantum optical coherence tomography, broad bandwidth photon pairs with negative frequency correlation are more suitable,⁴ while for quantum enhanced positioning, photon pairs with positive frequency correlation are preferable.^4–6 Moreover, for practical applications of QIS, high-dimensional entangled states are desirable.^7,8 Therefore, a photon pair source that possesses the dynamic programmability or reconfigurability of its spectral property will be a powerful tool for QIS.

A popular approach to generate photon pairs is the spontaneous parametric process in χ⁽²⁾- or χ⁽³⁾-nonlinear media.^9,10 Because of the energy and momentum conservation, the photon pairs are highly correlated in frequency and time. When a single-frequency continuous-wave laser is used as the pump, the photon pairs have a perfect negative correlation in frequency. When an ultrashort pulse train is used as the pump, there exists uncertainty in the frequency correlation between signal and idler photons.^11–13 For example, for a signal photon in a single frequency, its correlated idler photon occupies a wide frequency band since each frequency component of the broadband pulsed pump field contributes to the parametric process. With great effort put on directly generating frequency uncorrelated photon pairs to achieve single-mode operation, significant progress has been made in tailoring the spectral property by engineering the dispersion of the nonlinear media.^14–18 However, the successes are limited to specific wavelength ranges, especially for the frequency uncorrelated and positively correlated cases.

Recently, engineering the spectral property of a quantum state by using nonlinear interferometers (NLIs) has attracted much attention.^19–23 In the NLIs, formed by a sequential array of nonlinear media with gaps in between filled by linear dispersive media, photon pair generation in the parametric processes is determined by the phase matching of nonlinear media, whereas the spectral shaping is achieved independently by the linear dispersive media. Such a scheme was investigated for quantum state engineering in χ⁽²⁾-crystals.^24,25 The first experiment generating the frequency uncorrelated photon pairs by the quantum interferometric method was realized in a two-stage NLI formed by two identical nonlinear fibers with a linear medium of single-mode fiber in between.²¹ The results indicated that the original frequency negatively correlated joint spectral function (JSF) from a single-piece nonlinear fiber can be modified to nearly uncorrelated without sacrificing the collection efficiency. However, the factorability and collection efficiency of the photon pairs are slightly deviated from the ideal case due to the overlapping between two adjacent interference fringes. It has been proposed and experimentally verified that finer spectral control can be realized if the stage number of the NLI (i.e., number of nonlinear media) is greater than two.^22,26–28 By properly selecting the linear dispersive media and the stage number (number of nonlinear media) of NLIs,^22,26,28 both the central wavelengths and spectral property of photon pairs can be adjusted.

At the current stage, however, photon pairs with specific spectral properties have to be realized by specifically designed components and the structure of the setup. For example, for the photon pairs generated from a single-piece nonlinear medium via the parametric process, the spectral property is mainly determined by the dispersion of the nonlinear medium. Even for the new aforementioned interferometric approach, the linear dispersive media need to be changed or extra stages need to be added. In any case, to alter the spectral property, one needs to change the hardware of optical components. The changing process, together with the accompanied re-aligning process, hinders the flexibility of the photon pair source.

In this paper, we demonstrate a photon pair source with programmable spectral properties for the first time. The source is based on a two-stage NLI formed by two pieces of dispersion-shifted fibers as the nonlinear waveguides and a 4f-configuration with a liquid crystal spatial light modulator as a phase-control device in between. The phase-control device can introduce arbitrary phase shifts for different wavelengths, providing more flexibility and precision in engineering the spectra of photon pairs. By loading the properly designed phase functions on the phase-control device, we can realize photon pairs with various spectral properties, including non-correlation, positive correlation, and negative correlation. Moreover, a multi-channel source of photon pairs with factorable spectra is generated as well.

Our spectrally programmable photon pair source is based on a two-stage NLI scheme, as shown in Fig. 1(a). The scheme is pumped by a Gaussian pulse train. The two χ⁽³⁾-nonlinear waveguides are identical and support single-mode propagation, and photon pairs are generated via the spontaneous four-wave mixing (SFWM) process in the nonlinear waveguides. The programmable phase-control device can introduce different phase shifts at different optical frequencies, described by a phase function ϕ(ω) that can be arbitrarily defined. We assume that the phase-control device has a uniform transmission efficiency of η for all the optical fields involved in the SFWM process. We first consider the ideal case of η = 1. The two-photon term of the output state from the scheme can be written as $Ψ2∝G∫dωsdωiFNLIωs,ωi|1s,1i〉$⁠, where G ∝ γP_pL is the gain parameter with γ, P_p, and L, respectively, denoting the nonlinear coefficient, peak pump power, and length of each waveguide; $FNLIωs,ωi$ is the joint spectral function (JSF) of the photon pairs from the NLI, which describes the probability amplitude of a pair of signal and idler photons emerging at frequencies ω_s and ω_i, respectively. Due to the nonlinear interference of the parametric processes in the two waveguides,²² $FNLIωs,ωi$ can be expressed as

where

is the JSF of photon pairs produced from a single-piece nonlinear waveguide and

is the interference function. Here, ω_p0 and σ_p are the central frequency and bandwidth of the Gaussian pump, respectively; $Δk=2k(ωs+ωi2)−k(ωs)−k(ωi)−2γPp$ is the wave vector mismatch of the pump, signal, and idler fields in the nonlinear waveguide; and $Δϕ=2ϕ(ωs+ωi2)−ϕ(ωs)−ϕ(ωi)$ is the phase difference introduced by the phase-control device.

We suppose that in the waveguides, the phase matching condition of SFWM is perfectly satisfied at the pump, signal, and idler frequencies, ω_p0, ω_s0, and ω_i0, respectively. This means that Δk = 2k(ω_p0) − k(ω_s0) − k(ω_i0) − 2γP_p = 0 and 2ω_p0 = ω_s0 + ω_i0. Around these perfect phase matching frequencies, we have ΔkL → 0. In this case, the JSF of the single-piece nonlinear waveguide can be simplified to the Gaussian pump envelope,

and the interference function becomes

Note that we have made the substitutions Ω_s = ω_s − ω_s0 and Ω_i = ω_i − ω_i0 and dropped the imaginary phase terms for simplicity. From Eq. (3), one sees that I(Ω_s, Ω_i), the key for tailoring the final JSF F_NLI(Ω_s, Ω_i), is determined only by the phase difference Δϕ introduced by the phase-control device.

Considering that the contour of $FSPΩs,Ωi$ has a Gaussian profile with a direction of −45° from the horizontal axis [see Fig. 1(c)], we generally expect that the contour of I(Ω_s, Ω_i) is also a Gaussian function but has an orthogonal direction of 45° so that we can obtain F_NLI(ω_s, ω_i) with various spectral properties, especially the round-shaped factorable one.^13,18 In other words, we expect $I(Ωs,Ωi)∼exp−(Ωs−Ωi)24a2$⁠, where a is a bandwidth parameter describing the width of the contour. We notice that the arctan function is odd with the domain (−∞, ∞) and range (−π/2, π/2). Hence, the Gaussian function can be approximated with a function composition cos{arctan[g(x)]}, where g(x) is an odd polynomial. We compare the Taylor series of exp(−x²/a²) and cos{arctan[g(x)]} and obtain

For convenience, we introduce u(x, a) = arctan[g(x)] and have cos[u(x, a)] ≈ exp(−x²/a²). Then, we construct a piece-wise phase function ϕ(ω), as shown in Fig. 1(b), to introduce different phase shifts in the pump, signal, and idler bands. The central frequencies of the pump, signal, and idler bands are ω_p0, ω_s0, and ω_i0, respectively. The width for the three bands is not necessarily restricted to a particular value, provided that the pump spectrum is effectively covered and the overlap between different bands is avoided. For the Gaussian pump used in our model, a half spectral width of 3σ_p covers more than 99.99% of the pump intensity. For convenience, we set the half-width of all three bands to be 3σ_p. For the pump band, we add a fixed phase shift of π/2; for the signal band, the introduced phase shift is described by function ϕ(Ω_s) = −u(Ω_s, a) + π/2; and for the idler band, the phase shift is ϕ(Ω_i) = u(Ω_i, a) + π/2. Besides the three bands, we fill the undefined intervals with 0 or π to keep the continuity of ϕ(ω). From Fig. 1(b), one sees that the constructed ϕ(ω) is symmetric about ω_p0. After applying the constructed ϕ(ω), the interference function becomes

whose contour plot is shown in Fig. 1(d). One sees that the ridge of the contour is described by u(Ω_s, a) − u(Ω_i, a) = 0, solving which we get Ω_i = Ω_s. Thus, the direction of the contour is 45°. Along the cross section line of Ω_i = −Ω_s, the interference function can be expressed as

which is just what we expected.

We can control the final JSF F_NLI(ω_s, ω_i) by adjusting the bandwidth parameter a. For example, we can obtain the round-shaped factorable JSF by setting a = σ_p to make the contour of I(Ω_s, Ω_i) having the same width as that of $FSPΩs,Ωi$ [this is exactly the case shown in Fig. 1(d)]. The obtained factorable JSF is shown in Fig. 1(e). To check its factorability, we perform a singular mode decomposition²⁹ and find the mode number K = 1.01, which is very close to the ideal single-mode case. Besides the factorable JSF, we can also create a positively correlated JSF by setting a < σ_p or a negatively correlated one by setting a > σ_p. As examples, Figs. 2(a1) and 2(b1), respectively, show the contours of I(Ω_s, Ω_i) and F_NLI(ω_s, ω_i) when a = 0.5σ_p, while Figs. 2(a2) and 2(b2) show the results when a = 1.7σ_p.

The above examples show that we can obtain JSFs with a customized “island” structure centering at (Ω_s = 0, Ω_i = 0). In fact, we can create such an island structure centering at any arbitrary frequencies/wavelengths within the gain bandwidth of SFWM, just like painting on a blank canvas. We can also realize JSFs with a multi-island structure by repeating the phase shift patterns in the signal and idler bands shown in Fig. 1, and the central frequencies/wavelengths and spectral correlation property of each island can be controlled independently. As an example, we create a JSF with three well-separated factorable islands, and the intensity contours of I(Ω_s, Ω_i) and F_NLI(ω_s, ω_i) are shown in Figs. 2(a3) and 2(b3), respectively. From Fig. 2(b3), one sees that the JSF can be decomposed as $FNLI(ωs,ωi)=∑k=13rkFNLI(k)ωs,ωi$⁠, where $∑k=13|rk|2=1$⁠, and $FNLI(k)ωs,ωi=ψ(k)ωsϕ(k)ωi$ is the partial JSF for the kth factorable island. Since each island can be seen as an independent spectral-temporal mode, these modes are coherently superposed and form a high-dimensional entangled state,

where $ks(i)$ represents the kth mode in the signal (idler) field. Note that the state in Eq. (5) can also be generated from a cavity, and the frequency difference between different wave-packets is determined by the size of the cavity.⁷ However, using our source, the frequency difference between adjacent factorable islands can be flexibily tuned by loading properly designed phase functions on the phase-control device.

Compared with the NLI using fixed linear dispersive media for phase-control,^21,22,26 our scheme has the following merits. (1) Hardware-free reconfigurability: the output of our scheme can be reconfigured without changing any optical hardware. (2) Better flexibility: our scheme can be used to create spectrally tailored photon pairs centering at any arbitrary wavelengths, as long as the wavelengths are within the gain range of SFWM. (3) More accurate phase control: different from the cosine interference function of the fixed NLI, the interference function of our scheme can be a nearly perfect Gaussian function, from which a near perfect factorable state with the mode number K = 1.01 is achievable.

The above analyses have shown the effectiveness of our scheme, but there are several points that need to be elaborated further. First, in our model, we have omitted the term ΔkL. However, even in a case that ΔkL cannot be omitted, e.g., the frequencies of the signal and idler photons are not near the perfect phase-matched frequencies, we can always compensate ΔkL by adding an opposite compensation term while constructing the phase function ϕ(ω) (see Sec. 1 in the supplementary material for details) so that the final interference function I(Ω_s, Ω_i) can always follow our expectation. Second, in our model, we have made the contour of I(Ω_s, Ω_i) along 45° by constructing a symmetric ϕ(ω). However, the phase shift introduced in the signal and idler bands can be asymmetric as well. We can freely control the contour direction of I(Ω_s, Ω_i) by setting different bandwidth parameters in the signal and idler bands (see Sec. 2 in the supplementary material for details). This is very useful in tailoring the JSF when the contour of the single-piece JSF is along an arbitrary angle. Third, in practice, the ideal case of η = 1 is hard to achieve. We usually have η < 1 due to the existence of transmission loss. In the case of η < 1, the two-photon term of the output state from the scheme can be written as (see Sec. 3 in the supplementary material for details)

On the right-hand side of Eq. (6), the first three terms arise from the transmission loss and the last term is the two-photon state we expect. The first term is the vacuum state, which means both the signal and idler photons of a pair are lost due to the transmission loss, and the second (third) term is the one-photon state, which originates from the only surviving signal (idler) photon of a pair. The one-photon states can be seen as a background noise of the photon pairs, which can significantly influence the modal purity and heralding efficiency of the source (see the supplementary material for details).

Our analysis of the NLI with χ⁽³⁾-nonlinear media can be extended to other platforms, such as χ⁽²⁾-nonlinear media.^30,31 However, in the case of non-ideal transmission efficiency η < 1, additional attention should be paid in optimizing the parameters of the NLI. The key to achieve a complete two-photon interference in the NLI is that the gain parameters of waveguides 1 and 2, G and G′, respectively, should have the relation G′ = ηG (see the supplementary material for details). For χ⁽³⁾-nonlinear media, this relation is automatically satisfied when the lengths of the two nonlinear media are equal because the pump power of the second fiber is also reduced by a factor of η. However, for χ⁽²⁾-nonlinear waveguides, the length of each medium should be properly adjusted to meet the requirement of G′ = ηG.

As shown in Fig. 3, we experimentally implement the nonlinear interferometer scheme by employing two identical 30-m-long dispersion-shifted fibers (DSFs) as the nonlinear waveguide. The phase-control device in our nonlinear interferometer is realized by a 4f-configuration consisting of two diffraction gratings, two cylindrical lenses, and a spatial light modulator (SLM). We build the device on a sealed invar optical table to avoid the phase drifting due to the temperature fluctuation and air flow. The SLM (HOLOEYE Pluto-2) is based on a reflective liquid crystal micro-display with a resolution of 1920 × 1080 pixels. Note that in Fig. 3, the SLM is depicted as a transmissive device for clarity. The light of different wavelengths from the output of the first DSF is spatially separated by the first grating and focused on different columns of pixels of the SLM by the first cylindrical lens. The SLM can introduce programmed phase shifts for different wavelengths when a properly designed gray-level pattern is loaded. Then, the light of different wavelengths is recombined by using a second group of cylindrical lens and grating and then sent into the second DSF. The transmission loss in each DSF is negligible. The total transmission efficiency of the phase-control device between the two DSFs is 60%.

We employ a telecom-band mode-locked fiber laser with a repetition rate of 36.9 MHz as the pump source and a tunable filter (TF) realized by diffraction gratings to control the central wavelength and bandwidth of the pump. The fiber polarization controller (FPC) and polarization beam splitter (FPBS) are used for polarization purification and power control. The photon pairs generated via SFWM in the DSFs are co-polarized with the pump.³² Therefore, we use another set of FPC and FPBS at the output of the nonlinear interferometer to select the SFWM photons and suppress the photons from spontaneous Raman scattering (SRS) that are cross-polarized with the pump.³³ Then, we use a notch filter (NF) to reject the residual pump. To characterize the spectral profile of the photon pairs, the signal and idler photons are separated and selected by a dual-band TF. Two superconducting nanowire single-photon detectors (SPDs) are utilized to detect the signal and idler photons, respectively. The total detection efficiencies (including the filters and SPDs) for the signal and idler photons are both ∼15%. We use a computer-controlled data acquisition system to process the detection signals. The single-channel counting rates and twofold coincidence counting rates for photons that originated from the same pulse and adjacent pulses are recorded.

First, we demonstrate the generation of photon pairs with different kinds of spectral properties. The DSFs are immersed in liquid nitrogen (temperature: 77 K) to suppress SRS.³² The zero-dispersion wavelength of the DSFs is ∼1548.5 nm, so we set the central wavelength of the pump to be 1549.32 nm (193.5 THz in frequency) with a full width at half maximum (FWHM) of 0.9 nm. In this case, the phase matching condition ΔkL → 0 is nearly satisfied when the signal/idler wavelength is around the pump wavelength within tens of nanometers. Without loss of generality, we choose the central wavelengths (frequencies) of the signal and idler photons to be 1554.13 nm (192.9 THz) and 1544.53 nm (194.1 THz), respectively. We first generate photon pairs with a factorable JSF. Following the analysis represented by Fig. 1(b), we create a phase function with the parameter a = σ_p = 0.042 THz and construct the corresponding gray-level pattern, as shown in Fig. 4(a1). Note that the patterns are rotated about 0.2° counterclockwise to offset the small angle between the focused light spots and the y axis of the SLM. We load the gray-level pattern on the SLM and measure the JSF of the signal and idler photons. In the measurement, we set the transmission profiles of both the signal and idler channels of the dual-band TF (realized by using Finisar WaveShaper 4000) to be flattop with a fixed full-width of 0.2 nm. Then, we scan the central wavelength of the signal (idler) channel from 1552.6 to 1555.6 nm (1543.0–1546.0 nm) by a step of 0.2 nm. At each step, we record the coincidence rate between the signal and idler photons. During this process, the average pump power is fixed at 0.7 mW. After subtracting the accidental coincidence from the signal and idler photons originated from adjacent pump pulses, we obtain the true coincidence rate per second as a function of the signal and idler wavelengths. Then, we can plot the two-dimensional contours shown in Fig. 4, which reflect the intensity distribution of the JSF.³⁴ From the results shown in Fig. 4(b1), one sees that the measured JSF has a round shape, which agrees well with the theoretically predicted result in Fig. 1(e). Then, we change the bandwidth parameter a to 0.21 and 0.71 THz, respectively, and construct the two gray-level patterns shown in Figs. 4(a2) and 4(a3), respectively. For each case, we load the pattern on the SLM and repeat the joint spectrum measurement. The measured JSFs are shown in Figs. 4(b2) and 4(b3). As expected, their spectra exhibit positive and negative correlations, respectively.

We then demonstrate the multi-channel output feature of our source. As an example, we generate photon pairs with a JSF consisting of three separated factorable islands. The central wavelengths of the islands are specified to match the standard grid of the wavelength division multiplexing in the fiber optical communication system. For the signal (idler) band, the central wavelengths of the three islands are 1554.13 (1544.53), 1555.75 (1542.94), and 1557.36 (1541.35) nm, respectively, while the corresponding frequencies are 192.9 (194.1), 192.7 (194.3), and 192.5 (194.5) THz, respectively. To achieve this, we construct the gray-level pattern shown in Fig. 4(a4). One sees that the patterns of the signal and idler bands in Fig. 4(a1) are repeated for three times to create the three islands. Compared with that for the other islands, the patterns for the central island are reversed for continuity. The measured JSF after loading the pattern on the SLM is shown in Fig. 4(b4). One sees that the three round-shaped islands are perfectly sitting at the designed wavelengths/frequencies.

Finally, we characterize the modal purity of the round-shaped factorable JSF shown in Fig. 4(b1) by measuring the second-order correlation function g⁽²⁾ of the individual signal (or idler) field. The individual signal or idler field generated by SFWM is in the thermal state, and its mode number K is related to g⁽²⁾ through g⁽²⁾ = 1 + 1/K.³⁵ Using the equations derived in Sec. 3 of the supplementary material, we calculate g⁽²⁾ as a function of the transmission efficiency η. From the results shown by the curve in Fig. 5, one sees that the value of g⁽²⁾ can reach 1.99 in the ideal case of η = 100% but decreases significantly with the decrease in η. The relation between g⁽²⁾ and η can be comprehended as follows: From Eq. (6), one sees that the two-photon state is in the single mode with a tailored spectrum $FNLIωs,ωi$⁠, but the one-photon states (the background noise of photon pairs) have a large mode number since their spectra are described by the untailored $FSPωs,ωi$⁠. With the decrease in η, the intensity ratio between the one-photon states and two-photon states increases, leading to the decrease²¹ in g⁽²⁾. In the experiment, the round-shaped island is carved out by setting the bandwidths of both channels of the dual-band TF to 1.6 nm (200 GHz in frequency). We then conduct the g⁽²⁾ measurement for the individual signal field by using the HBT interferometer setup. In the measurement, the signal photons are sent into a 50/50 fiber coupler whose two output ports are fed into two SPDs. We record the coincidence count rates between the two SPDs, including the coincidence produced by the signal photons from the same pump pulses as well as the accidental coincidence produced by the signal photons from adjacent pump pulses. Then, we obtain g⁽²⁾ by calculating the ratio between the measured coincidence and accidental coincidence count rates. From the results shown by the solid circles in Fig. 5, one sees that the measured g⁽²⁾ has the same trend as the theoretical curve but is always lower than the curve. We think that the main factor accounted for this deviation is the existence of the SRS photons, which have a different modal structure from the SFWM photons,^36,37 and another factor is the imperfection of the components (such as the limited pixel resolution of the SLM). However, the photons from SRS can be almost entirely eliminated by further cooling the fiber³⁸ while the performance of the components can also be improved further. We believe that the measured g⁽²⁾ could approach the theoretical prediction after addressing the two factors. The only crucial issue is the transmission efficiency η. Currently, the efficiency η is limited by the efficiencies of the components in the phase-control device used in our experiment. It is possible to achieve η > 85% by using high efficiency components (e.g., transmission grating with efficiency $>94%$⁠) and optimizing the configuration. From the theoretical curve in Fig. 5, one sees that we can get g⁽²⁾ > 1.89 when η > 85%.

To conclude, we demonstrate a spectrally programmable photon pair source by using a pulse-pumped two-stage NLI scheme in which two pieces of DSFs are utilized as the nonlinear media for SFWM and a 4f-configuration with a liquid crystal micro-display SLM is used as the programmable phase-control device. We can customize the spectral properties of the photon pairs by loading a properly designed phase function on the phase-control device. To change the output property of the source, we only need to re-design the loaded phase function without replacing any components in the scheme. The change can be very fast since the typical frame rate of the micro-display SLM can be greater than 60 Hz. Many novel functions can be exploited using our source, such as a multi-channel output that can be used to generate high-dimensional entanglements. Our investigation also shows that the transmission loss between the two stages of the NLI is a crucial factor influencing the performance of the source. We believe that our scheme can be further transplanted to other platforms of quantum light generation. Specifically, by employing a compact integrated phase-control device,³⁹ our scheme can be realized on the photonic integrated circuit platform, providing a flexible and powerful tool for quantum information processing.

Moreover, although our analysis and experiment are focused on the temporal/spectral profile in the pulse-pumped NLI, this quantum interferometric method can be extended to the engineering of the spatial profile of the photon pairs by directly placing a programmable spatial dependent phase shifter in between the two nonlinear processes.^19,40 In this case, it should work for both continuous wave and pulsed pumping.²³ 

See the supplementary material for controlling the contour direction of the interference function, the influence of the transmission loss between the stages on the output state from the NLI, and compensating for the phase mismatch term in a nonlinear medium by defining the phase function.

The authors acknowledge financial support from the Science and Technology Program of Tianjin (Grant No. 18ZXZNGX00210) and the National Natural Science Foundation of China (Grant Nos. 11874279, 12074283, and 11527808).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.