Quantum optical synthesis in 2D time–frequency space

The invention of mode-locked lasers opened the door for ultrafast optical science and technology in the femtosecond region. This field has been continuously developed by manipulating the temporal waveform of optical pulses through the one-dimensional (1D) Fourier transform relationship of electric field distributions between the time and frequency domains.^1–3 Today, these technologies are known as “optical synthesis” (OS), which can precisely control and generate arbitrary temporal shapes of optical pulses by manipulating the phase and amplitude of spectral components.^4–6 Such OS technologies are expected to be applied to optical measurements, sensing, and spectroscopy.^2,7–9

From the viewpoint of the particle nature of light, the 1D Fourier transform treatment is valid only for an ensemble of photons without quantum correlation. On the other hand, recent developments in quantum optical technologies allow us to observe the time–frequency behavior of quantum-mechanically correlated photon pairs, e.g., biphotons.¹⁰ It has been revealed that biphoton distributions in the time and frequency domains are connected to Fourier optical phenomena in two-dimensional (2D) time–frequency space.^11–17 Such a quantum optical aspect of light has the potential to expand the conventional optical synthesis to its quantum counterpart, i.e., optical synthesis in high-dimensional time–frequency space. This treatment allows us to incorporate the quantum correlation into the OS. Hereafter, we refer to OS in high-dimensional space in a quantum manner as “quantum optical synthesis” (QOS).

In this work, we present a proof-of-concept demonstration of QOS by manipulating the amplitude and phase of the spectral distribution of biphotons in 2D frequency space and by directly observing the temporal distribution in 2D time space. Furthermore, as a potential application of QOS, we show the shaping of heralded single photons via manipulation in 2D time–frequency space.

The scheme of our QOS experiment is shown in Fig. 1. Laser pulses with a center wavelength of 792 nm and a bandwidth of 7.4 nm were used to pump a 30-mm-long PPKTP crystal in a bidirectional pumping configuration,^18,19 as shown in Fig. 1(a). The PPKTP crystal with type-II phase matching satisfies the group-velocity-matching (GVM) condition at the telecom wavelength.^20–22 The polarization of the constituent photons is aligned along either the crystallographic y or z axis. Thanks to the GVM condition with femtosecond pulse pumping,²³ the down-converted biphotons have an elliptical distribution along the diagonal direction in the frequency domain and along the anti-diagonal direction in the time domain, as shown on the left of Fig. 1(b). After passing twice through the dual wave plate (DWP) (quarter-wave plate for the biphotons and half-wave plate for the pump), the y(z)-polarized photon is interchanged with a z(y)-polarized photon, but the pump pulse is unchanged. As a result, the two-photon spectral (temporal) distribution is inverted with respect to Δν_z = Δν_y(Δτ_z = Δτ_y), as shown in the center of Fig. 1(b). Here, Δν is defined as the shifted frequency from the center frequency of 189.4 THz (1584 nm), and Δτ is the shifted time from 0 defined by the temporal position of the pump pulse. Then, only the temporal distribution is shifted along Δτ_z = −Δτ_y after passing through the PPKTP crystal again, as shown on the right of Fig. 1(b). By combining the biphoton wave packets produced by the first and second pumpings, a separated two-mode frequency distribution s₁ and s₂ was prepared, as shown in Fig. 1(c). Since the separation and relative phase between the two frequency modes would directly affect the two-photon temporal distribution, we could expect to synthesize two-photon temporal distributions by means of two-photon spectral manipulation.

Based on the scheme described above, we performed proof-of-principle experiments for QOS from two aspects: (1) the multiple mode effect in 2D frequency space and (2) the phase manipulation effect. These effects were confirmed by directly measuring two-photon spectral and temporal distributions in 2D space, where the 2D spectral and temporal distributions are known as the joint spectral intensity (JSI) and the joint temporal intensity (JTI) of biphotons, respectively. The JSI was measured using a fiber spectrometer,^24,25 and the JTI was measured by using a time-resolved up-conversion system.^12,13,26 See the supplementary material for the detailed experimental setup. To identify the multiple mode effect, we begin by presenting the JSI and JTI of the biphotons with a single pumping, realized by placing a filter that blocks biphotons but passes the pump pulse instead of the DWP, as shown in Figs. 2(a) and 2(b). Here, we set the crystal temperature to 65 °C. The frequency (time) of the biphotons in the JSI (JTI) was positively (negatively) correlated, and no modulations were observed in the JTI. In contrast, changing the experimental configuration to a bidirectional pumping scheme, we can clearly see two-mode spectral distributions with a positive frequency correlation [Fig. 2(c)], where the resultant peak separation between the two modes is 4.9 nm (0.58 THz). Here, the peak separation is defined as the distance along the line of Δν_z = −Δν_y. On the other hand, the observed JTI in Fig. 2(d) shows a negative correlation with almost the same full length as that in Fig. 2(b), but it has a distinct four-mode structure along the anti-diagonal direction with a peak separation of 1.6 ps between modes. The experimental peak separation of 0.58 THz in the JSI and that of 1.6 ps in the JTI satisfy an almost inverse relationship: 0.58 × 1.6 = 0.93; this suggests that the time and frequency distributions could have a conjugate relationship and partially validates our concept of QOS. In order to confirm the Fourier transform relationship between the data in Fig. 2(c) and those in (d), we theoretically constructed the two-photon spectral distribution [Fig. 2(e)] by adopting the experimental parameters, viz., pump bandwidth and crystal length. We then obtained the simulated JTI [Fig. 2(f)] by performing a two-dimensional Fourier transformation on the amplitude of the JSI in Fig. 2(e). The peak separation in Fig. 2(f) is 1.7 ps, which is consistent with the value of 1.6 ps in Fig. 2(d). Therefore, this theoretical simulation can well reproduce the experimental data, meaning that it can successfully demonstrate our concept of QOS. The slightly “fatter” distribution in Fig. 2(d) is caused mainly by the limited temporal resolution of our systems.

To strengthen the QOS, we verified how the phase manipulation of the biphoton in the frequency domain affects the two-photon temporal distribution. First, we decreased the crystal temperature from 65 to 45 °C and obtained the mode peak separation of 2.2 nm (0.26 THz) in Fig. 3(a). The crystal temperature impacts not only the separation of spectral modes but also the relative phase (see the supplementary material for details). The resultant JTI in Fig. 3(b) shows a distinct two-mode distribution with a peak separation of 3.4 ps. The disappearance of the coincidence counts around the 0-delay position, i.e., Δτ_y = Δτ_z = 0, suggests the relative phase ϕ between the two spectral modes in Fig. 3(a) to be ∼π. Indeed, a precise phase value of ϕ = (1.01 ± 0.01)π is obtained by fitting the JTI in Fig. 3(b). In order to ensure the relative phase value, we also carried out the Hong–Ou–Mandel (HOM) interference experiment²⁷ and obtained the interference pattern in Fig. 3(c). From the fitting, we extracted the relative phase value as ϕ = (0.93 ± 0.03)π, which approximately agrees with the phase value estimated in the JTI. Next, we controlled the relative phase by inserting and tilting a silica glass plate with a thickness of 1.5 mm between the DWP and the concave mirror in the setup and then obtained the JTI in Fig. 3(d). The maximum coincidence counts of this JTI are around Δτ_y = Δτ_z = 0, implying the relative phase value of ϕ = 0. By fitting the JTI in Fig. 3(d), we achieve a precise phase value of ϕ = (0.00 ± 0.01)π. In contrast, we obtained the HOM pattern reflecting the relative phase value of ϕ = (0.00 ± 0.05)π in Fig. 3(e). On the other hand, the modulation intervals are almost the same between Figs. 3(b) and 3(d), meaning that the relative phase ϕ only affects the phase of the modulation in the JTI. Comparing the phase value of Figs. 3(b) and 3(c) and comparing the phase value of Figs. 3(d) and 3(e), we can conclude that the 2D Fourier transformation well explains all the phenomena between the JSI and JTI.

In addition to the biphoton manipulation, our scheme can offer a new approach to shape a heralded single-photon wave packet via manipulation of a high-dimensional time–frequency space. To clarify this point, we show the manipulation of a single-photon temporal waveform in a heralding scheme. In this experiment, z-polarized photons were detected by the superconducting nanowire single photon detector (SNSPD) without the time-resolved measurement, while the y-polarized photons were sent to the time-resolved up-conversion system. With this setup, we measured the temporal shapes of the heralded single photons at the crystal temperatures of 35, 45, 55, and 65 °C. Figures 4(a)–4(d) show the counts of the y-polarized photons triggered by the heralding signal from the SNSPD. We can clearly see the distinct modulation and observe the increase in the numbers of peaks with the increase in the crystal temperature, manifesting the advantage of our scheme in comparison with the earlier works on biphoton temporal manipulations with the spatial light modulator (SLM). However, the corresponding spectra in Figs. 4(e)–4(h) are not changed at different temperatures. This implies that there is no direct connection in the Fourier transform operation in 1D space and the 2D treatment is essentially required for understanding the time–frequency behavior even for the heralded single-photon wave packet.

Biphoton manipulation in the time–frequency degree of freedom has been mainly studied with SPDC pumped by a continuous-wave (CW) laser.^3,11,28 In those works, biphoton temporal distributions are described as a function of relative delay between the constituent photons. Therefore, only the time–frequency distribution along the anti-diagonal direction in 2D time–frequency space plays a crucial role, and no temporal modulation would appear in the diagonal direction. Eventually, a 1D treatment for the time and frequency difference between the photons is sufficient for the CW-pumped SPDC. In contrast, the temporal structure of the pump pulse in SPDC directly affects the distribution along the diagonal direction in 2D time–frequency space. Therefore, it is crucial to consider the time–frequency distribution in 2D space, providing an intuitive and comprehensive understanding of biphoton manipulation.

Here, we discuss the advantages of our manipulation methods. In our bidirectional pumping scheme, the whole two-photon spectral distribution S is expressed by the superposition of the two-photon spectral modes s₁ and s₂; S(ω₁, ω₂) = s₁(ω₁, ω₂) + e^iϕs₂(ω₁, ω₂), where s₂(ω₁, ω₂) = s₁(ω₂, ω₁) in our case. The spectrum of the constituent photon with ω₁₍₂₎ is given by f₁₍₂₎(ω₁₍₂₎) = ∫S(ω₁, ω₂)dω₂₍₁₎. In order to control the two-photon spectral distribution S(ω₁, ω₂), we need to independently control the spectral modes s₁ and s₂. So far, the SLM has been frequently used and enables us to arbitrarily shape an ultrafast optical pulse. In previous reports, this technique is also applied to quantum optical experiments to control a biphoton temporal waveform.^10,13,28,29 In those experiments, the SLM manipulates the spectral amplitude and phase distribution of the constituent photons f₁ or f₂. However, the SLM cannot be utilized in our experimental situation because a certain frequency component in the one-photon spectrum f₁ may contain both the biphoton spectral modes s₁ and s₂. In contrast, our bidirectional pumping scheme is based on manipulation along a difference–frequency axis, allowing the independent control of s₁ and s₂ even in the spectral condition in Figs. 2 and 3.

Other approaches have been demonstrated for engineering biphoton spectral distributions with the Schmidt decomposition of the joint spectral amplitude. In those studies, the manipulation of the JSI is achieved either by shaping the pump-pulse amplitude^14,15 or by tailoring the phase matching of a nonlinear crystal.¹⁷ Shaping the pump spectrum allows the JSI manipulation just along a sum–frequency axis, while our scheme is based on manipulation along the difference–frequency axis. These two axes are orthogonal to each other; thus, these two methods have a complimentary relationship for the arbitrary shaping of the biphoton wave packet in 2D time–frequency space. In Ref. 17, a similar double-peak joint spectrum can be generated by carefully designing the domain structure. This approach may allow one to create a more complicated spectral structure in the JSI. On the other hand, our scheme can provide better flexibility on the spectral manipulation, retaining a symmetric or antisymmetric state, especially for controlling the frequency spacing between the modes. Combining these techniques, we may control a complex temporal waveform of biphotons and could approach an arbitrary shaping of the biphoton wave packet toward the realization of ultrafast quantum optical measurements.

In our scheme, the number of spectral modes of the biphotons is determined by the number of crystals; one crystal creates two modes. Thereby, we could increase the number of spectral modes by putting additional crystals with different phase-matching conditions. However, we may face technical difficulties in collecting biphotons generated from all the crystals. Another approach would fabricate a complex domain structure in one crystal, but the phase-controllability problem would remain.

Finally, it is worthwhile to discuss the plausible future direction of the QOS technique. The main motivation of our work is to bridge the research fields between quantum optics and ultrafast optical measurements, especially for spectroscopy with ultrafast lasers, which has been utilized for the investigation of the dynamical process in physical, chemical, and biological materials.^30,31 Indeed, spectroscopy with quantum light is attracting much attention as emerging quantum technology.^32–34 For instance, a recent theoretical work predicts that heralded single photons produced by SPDC could emulate sunlight conditions and that such photons could be used to investigate the dynamical process in complex molecules.³⁵ Although the arbitrary shaping of temporal modes is crucial for investigating the light–matter interactions in such complex systems, it is hard to control the temporal characteristics of thermal light, such as sunlight, because of its incoherent properties. Quantum optical synthesis in high-dimensional time–frequency space could provide thermal light with high controllability and may contribute to deeper insights into complex molecular systems, such as photosynthetic materials.

See the supplementary material for more details on the experimental setup, temperature dependence in HOM interference, and theoretical simulation of the JSI and JTI.

We thank Akihito Ishizaki and Zheshen Zhang for helpful discussions. This work was supported by the MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) (Grant No. JPMXS0118069242), JSPS KAKENHI (Grant Nos. JP18H05245 and JP17H01281), JST CREST (Grant No. JPMJCR1671), and the National Natural Science Foundation of China (Grant Nos. 91836102, 12074299, and 11704290).

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