Precise manipulation of the time–frequency modes of entangled photons is crucial for future quantum science and technologies. Recently, the frequency-domain-quantum-optical-synthesis (FD-QOS) method was demonstrated by creating a superposition of different joint temporal amplitudes: those temporal distributions can be controlled by manipulating the joint spectral amplitude in 2D frequency space via a Fourier optical relation. This FD-QOS method provides an efficient, flexible, and easy-to-control way to precisely modulate the temporal distributions of the entangled photon in an ultrafast region. However, manipulation of only the temporal modes is not sufficient for various applications in quantum information, since spectral modulations are also needed on many occasions. Here, we present a proof-of-concept experiment of two-photon spectral modulation via temporal manipulation of a biphoton wave packet. This protocol, called time-domain-quantum-optical-synthesis (TD-QOS), is achieved by adjusting the relative phases between two joint temporal distributions. In addition, the two-photon joint spectral distributions are characterized by measuring the joint spectral intensities and Hong–Ou–Mandel interferences. The combination of FD-QOS and TD-QOS enables complete control over the biphoton states. Our work would further develop quantum technologies that rely on the time–frequency modes of entangled photons.

Time and frequency are essential degrees of freedom not only for classical laser technologies but also for quantum technologies using entangled photons. In practice, time–frequency (or energy-time) entangled photons can be encoded in a variety of ways for the implementation of quantum information protocols, such as frequency-bins, time-bins, and time–frequency modes (TFM).^{1–3} Compared with the widely used polarization encoding, time–frequency encoding has two significant advantages. First, the Hilbert space available in the time–frequency domain is theoretically infinite; hence, this high-dimensional feature makes it a promising candidate for large-capacity communication. Second, the time–frequency encodings are robust against environmental noise, and this noise-resistant feature makes it suitable for long-distance optical fiber transmission. Additionally, the TFM can provide a complete set of building blocks for implementing diverse quantum information protocols.^{4–7} For example, in a quantum communication scenario, the orthogonal TFMs can be used as independent channels for information multiplexing and demultiplexing;^{5} in a linear optical quantum computation algorithm, the TFM qubits can propagate through a linear-optical system and realize single- or two-qubit operations;^{6} in a large-scale quantum network, a temporal mode can be utilized and reshaped for efficient interfacing between distinct nodes.^{7} Notably, a broader spectrum of entangled photons is crucial for high-speed quantum communications, and neat spectral manipulation is indispensable for practical multiplexed quantum networks. Although there are several ways to generate entangled photons, spontaneous parametric downconversion can provide the most sophisticated approach to generating and precisely manipulating the TFM of broadband entangled photons.^{8–10}

TFM manipulation methods developed, thus, far can be classified into two categories. The first is internal modulation; that is, by engineering a pump laser or the phase-matching conditions of nonlinear crystal, the joint spectral distributions (JSDs) of biphotons from spontaneous parametric down conversion (SPDC) can be modulated. For example, by shaping the pump pulse, the JSDs can be manipulated along the diagonal axis;^{11} by tailoring the dispersion of the crystal [e.g., the group-velocity-matching (GVM) condition], the joint spectral of biphotons can be positively correlated, negatively correlated, or non-correlated;^{12,13} by engineering the poling period of the KTP crystal, the phase matching function (PMF) can have a Gaussian, square, or triangular shape;^{14} by choosing a different PMF, the joint spectral intensity (JSI) can be prepared in different orders of Hermite distribution;^{2,15} by modulating the nonlinear material in a nonlinear interferometer, the spectral distribution of the biphoton can be engineered.^{16} Such internal modulation is effective but usually has low flexibility because once the scheme is fixed, the controllable parameters are limited. The second category is external modulation, i.e., after being generated by an SPDC process, the biphotons can be further modulated by adding external devices. For example, by adding a spatial light modulator (SLM), the temporal shape can be modulated;^{17} by adopting a double-pump scheme, the relative phase between the biphotons can be arbitrary adjusted;^{18} different Hermitian modes can be selected by passing through a quantum pulse gate based on sum-frequency generation;^{19,20} by filtering the biphotons in a cavity, quantum optical microcombs can be prepared.^{21} Such external modulation schemes are effective and flexible, but the setup is usually complex, since it involves more components.

Recently, a frequency-domain quantum optical synthesis (FD-QOS) scheme was demonstrated to modulate the joint temporal intensity by manipulating the joint spectral amplitude, which provides a way to precisely synthesize the temporal modes of biphotons, as shown in Fig. 1(b).^{3} The foundation of QOS is a Fourier duality between the time and frequency domains, as shown in Fig. 1(a).^{10} The FD-QOS method provides a simple, flexible, and easy-to-control way to modulate temporal distributions. However, for various applications in quantum information, it is not sufficient to manipulate only the temporal modes, since spectral modulations are also needed on many occasions.^{22} In FD-QOS, it is technically hard to prepare a multi-peak spectral distribution, which is important for frequency-bin encoding or wavelength-multiplexing technology. To overcome this problem, in this work, we present two-photon spectral modulation via temporal manipulation of a biphoton wave packet, according to the concept shown in Fig. 1(c). This time-domain quantum-optical-synthesis (TD-QOS) of spectral modes is the inverse of the previous FD-QOS of temporal modes.^{3} The combination of temporal and spectral operation provides a full ability to control the biphoton state.

The scheme for TD-QOS of spectral modes is shown in Fig. 2. First, in Figs. 2(a1)–2(a3), we show the principle by which a rectangular joint temporal distribution (JTD) is prepared. The nonlinear crystal of PPKTP is type-II phase-matched (*y* $\u2192$ *y *+* z*), i.e., the pump and the signal are *y*-polarized, while the idler is *z*-polarized. At 1584 nm, the PPKTP crystal naturally satisfies the GVM condition of $Vi\u22121+Vs\u22121=2Vp\u22121$, where $Vp\u22121,\u2009Vs\u22121$, and $Vi\u22121$ are the inverses of the group velocities of the pump, the signal, and the idler, respectively.^{12,23} Thanks to the GVM condition, the signal and the idler are always distributed symmetrically with respect to the pump pulse position in the time domain. For example, in Fig. 2(a1), the signal and idler photons generated at the front end of the PPKTP are located at both ends of the biphoton pulse. In contrast, Fig. 2(a2) shows that the biphotons generated at the rear end of the PPKTP are located at the center of the biphoton pulse. Consequently, the Gaussian temporal distribution of the pump is transformed to a near-rectangular shape of the biphoton wave packet in the time domain, and the *y*-polarized idler is faster than the *z*-polarized signal as shown in Fig. 2(a3). In this scheme, the crystal length L is 30-mm-long, and $Vp=1.6583\xd7108$ m/s, $Vs=1.6997\xd7108$ m/s, and $Vi=1.6186\xd7108$ m/s. The biphoton pulse length can be calculated as $L/Vp\u2212L/Vs\u2248L/Vi\u2212L/Vp\u22484.4$ ps for signal and idler.^{24}

Second, we show how to prepare a phase-tunable two-mode square-shaped JTD using a double pump scheme. By passing through the PPKTP twice, the *y*-polarized and *z*-polarized wave packets are separated by temporal spacing of two-wave packets, as shown in Fig. 2(b1). This one-dimensional distribution can also be illustrated in the two-dimensional coordinate $\tau y\u22120\u2212\tau z$ in Fig. 2(b4). In this coordinate, the 0 temporal position is defined by the center of the pump pulse. Similarly, when the pump pulse passes through the PPKTP for the second time [Fig. 2(b2)], the second JTD can be prepared in Fig. 2(b5). By combining the cases in Figs. 2(b1) and 2(b2), we can obtain a superposition of two JTDs, as shown in Figs. 2(b3) and 2(b6). The relative phase $ei\phi $ in Fig. 2(c) between the first and second JTDs can be adjusted in the experiment using three methods, which are demonstrated later.

Finally, by adjusting the relative phase $\phi $, we can prepare different two-mode JTDs of the biphotons. Figures 2(d1)–2(d5) simulate the real portion of the joint temporal amplitudes (JTAs) at the relative phases of 0, $\pi /4,\u2009\pi /2,\u20093\pi /4$, and *π*. Figures 2(e1)–2(e5) simulate the corresponding imaginary portion of each JTA. Note that, by using different combinations of the real and imaginary portions in Figs. 2(d1)–2(d5) and 2(e1)–2(e5), different joint spectral intensities (JSI) can be synthesized using 2D Fourier transform, as shown in Figs. 2(f1)–2(f5). See the supplementary material for more details about the simulation conditions. The spectral and temporal differences can also be reflected in the Hong–Ou–Mandel (HOM) interference.^{18,25} Figures 2(g1)–2(g5) show the simulated HOM interference patterns at different relative phases.^{26} See the supplementary material for more details.

The TD-QOS scheme can be experimentally verified by measuring the JSI and HOM interference. We demonstrate the manipulation of the quantum state using three methods: tilting the glass plate, changing the thickness of the glass plate, and changing the propagation length of photons in air.

The experimental setup for generating biphotons is shown in Fig. 3(a1). The pump pulses from the Ti sapphire laser that we used have a center wavelength of 792 nm and a temporal width of around 130 fs (corresponding to a spectral width of 0.46 nm). The nonlinear crystal of PPKTP with a length of 30 mm and a polling period of 46.1 *μ*m is type-II phase-matched (y $\u2192$ y + z).^{27} The PPKTP crystal is pumped by the laser twice, as designed in the scheme in Fig. 2. The downconverted biphotons are separated by a fiber polarization beam splitter (FPBS) and then coupled into single-mode fibers. The relative phase between the two joint temporal intensities (JSIs) is adjusted by tilting the angles of the BK7 glass with a thickness of 1 mm. The JSIs of the biphotons are measured by a homemade dispersion-fiber spectrometer [Fig. 3(a2)] with a spectral resolution of 0.11 nm.^{3}

Figure 3(b1) shows the experimentally measured JSI, which has a single elliptical shape. By projecting the JSI onto the anti-diagonal direction, we can obtain its marginal distribution (the difference frequency spectrum), as shown in Fig. 3(c1). By contiguously tilting the angle of the glass plate, we can measure the JSI and marginal distributions at different relative phases, as shown in Figs. 3(b2)–3(b5) and 3(c2)–3(c5).

The relative phase between the two temporal modes can be precisely measured in a HOM interference with the setup shown in Fig. 3(a3). The measured results are shown in Figs. 3(d1)–3(d5). By fitting the patterns, we obtain the relative phase of $(0.17\u2009\xb1\u20090.02)\pi ,\u2009(0.33\u2009\xb1\u20090.01)\pi ,\u2009(0.48\u2009\xb1\u20090.00)\pi ,\u2009(0.62\u2009\xb1\u20090.00)\pi $, and $(0.94\u2009\xb1\u20090.05)\pi $. We then fit the difference frequency spectrum in Figs. 3(c1)–3(c5) using these phase values. The fitting curve matches well with the experimentally measured difference frequency spectrum. The fitted visibility in Fig. 3(d1) is 0.92, and the degradation of the HOM visibility is mainly caused by the asymmetry of the JSI resulting from the dispersion of the glass plate.

In addition to tilting the glass plate, the quantum state can also be manipulated by changing the plate thickness. In this case, we keep the glass plate standing vertically (i.e., the incident angle of the pump is 0) and change the plate thickness, as shown in Fig. 4(a). Specifically, we change the BK7 glass plate into three combinations: (1) a fused silica glass plate (1.25 mm); (2) a fused silica plate (2 mm) plus a BK7 glass plate (3 mm); and (3) a fused silica plate (2 mm) plus a fused silica plate (1.25 mm). As analyzed in the supplementary material, the time differences between the pump and biphotons under these conditions are 38, 99, and 195 fs, respectively. The measured JSIs are shown in Figs. 4(b)–4(d). Interestingly, with the increase in thickness, the amount of two-mode shift along the diagonal direction increases. This phenomenon is caused by the fact that the pump and the biphotons have different group velocities in the glass plate. We provide a solid theoretical analysis of this later in this work.

The relative phase between the two spectral modes can also be manipulated by changing the propagation length of the air. Specifically, we change the relative distance between the concave mirror and the PPKTP crystal. The manipulation results are confirmed by measuring the HOM interference. Figure 5(a) shows the experimental setup. Figures 5(b)–5(g) show the measured HOM interference patterns at different phases. The relative phase is increased from $0.28\pi $ to $0.78\pi $ when the distance between the concave mirror and PPKTP is increased from the reference point of 0 to 20.5 mm. The relationship between each phase and the relative distance is summarized in Fig. 5(h). It is noteworthy that the phases increased linearly as a function of relative distance. Considering a round trip, the phase difference is 0.5*π* over 41 mm. This result is also caused by the fact that the pump and the downconverted photons have different refractive indexes in the air. This is well explained later.

The measured JSIs in Figs. 4(b1)–4(b5) are slightly different from the theoretically expected JSIs in Figs. 2(e1)–2(e5), since the two measured spectral modes are not symmetric but rather shift along the diagonal direction. This is mainly caused by the group velocity difference between the pump and the biphotons during propagation in the glass plates or in the air. As shown in Fig. 6(a1), the center position of the first pair and that in the second pair overlaps in the ideal condition without any dispersion effect. So, the JTIs of both pairs are distributed along the line of $\tau y=\u2212\tau z$. However, in the practical condition as shown in Fig. 6(a2), the second pair is slower than the first because the GVM at the pump wavelength (792 nm) is smaller than that at the biphoton wavelength (1584 nm) in the glass plates or in the air. As a result, the first pair is distributed on the line of $\tau y=\u2212\tau z$, while the second pair is distributed on the line of $\tau y=\u2212\tau z\u22122\xi $. *ξ* is the time delay between the center of the first pair and the center of the second. We also theoretically simulated the JSI and JTI under different time delays *ξ*. Figures 6(b1) and 6(b2) show the case of *ξ* = 38 fs, which corresponds to Fig. 4(b). Figures 6(c1) and 6(c2) show the case of *ξ* = 99 fs, which corresponds to Fig. 4(c). Figures 6(d1) and 6(d2) show the case of *ξ* = 195 fs, which corresponds to Fig. 4(d). The simulation results well reproduce the experiment results. In the case of the air propagation, the estimated delay is $\xi =0.77$ fs, which corresponds to a phase delay of 0.58*π* (see the supplementary material for details). This value is consistent with the experimentally measured value of 0.5*π* in Fig. 5.

Here, we compare the difference between our present and previous works on the FD-QOS of temporal modes.^{3} Essentially, the FD-QOS is the inverse process of the TD-QOS. In our FD-QOS scheme,^{3} by inserting a quarter-wave plate (QWP), the two JTIs overlap and JSIs are separated from each other. The QWP is removed in this TD-QOS, and as a result, the JTIs are separated from each other, and the JSIs completely overlap. The combination of the two methods provides full control over the biphoton spectral-temporal distributions.

It is meaningful to discuss the advantages and disadvantages of FD-QOS and TD-QOS. In FD-QOS, it is technically hard to prepare a multi-peak spectral distribution, but it is easier to prepare a multi-peak temporal distribution. In contrast, TD-QOS is the opposite case. FD-QOS would be useful for quantum communication technology with time-bin encoding. TD-QOS is compatible with frequency-bin encoding or wavelength-multiplexing technology.

This work is only a proof-of-principle demonstration; in the future, this scheme can be expanded. One future application of this work is to highly controllable frequency-bin qudits. Another possible application is to use HOM interference to test the refractive indices of samples, e.g., air or other gases. The HOM interference shows a phase sensitivity of 0.1 *π* over a relative delay of 165 fs. This sensitivity can be further improved by using an updated scheme. The third application is single-photon-level distance measurement. Compared to the classical methods (e.g., Lidar), the coincidence counting in this work may provide a higher signal-to-noise ratio. Furthermore, our experimental scheme is also a kind of nonlinear interferometer.

In summary, we have theoretically designed and experimentally demonstrated a time-domain QOS (TD-QOS) scheme. Specifically, we prepared a two-mode JTA in an SPDC process using a GVM-PPKTP crystal under a dual pump configuration. The relative phase between the two modes was manipulated using three methods. By setting different relative phases between the JTDs, different spectral distributions were synthesized. The JSIs are characterized by measuring the JSIs and HOM interferences. This TD-QOS is a complementary operation to the FD-QOS. The proof-of-principle demonstration of FD-QOS in this work might inspire the future development of frequency-bin qudits.

See the supplementary material for more theoretical analysis of the experiment.

This work was supported by the MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) (Grant No. JPMXS0118069242), the JSPS KAKENHI (Grant Nos. JP18H05245 and JP17H01281), the JST CREST (Grant No. JPMJCR1671), and by the National Natural Science Foundations of China (Grant Nos. 91836102, 12074299, and 11704290).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Rui-Bo Jin:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hiroki Oshima:** Data curation (equal). **Takumi Yagisawa:** Data curation (equal). **Masahiro Yabuno:** Data curation (equal). **Shigehito Miki:** Data curation (equal). **Fumihiro China:** Data curation (equal). **Hirotaka Terai:** Data curation (equal); Funding acquisition (equal). **Ryosuke Shimizu:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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

## References

_{4}with zero group-velocity mismatch