## Abstract

High-dimensional entangled states of light provide novel possibilities for quantum information, from fundamental tests of quantum mechanics to enhanced computation and communication protocols. In this context, the frequency degree of freedom combines the assets of robustness to propagation and easy handling with standard telecommunication components. Here, we use an integrated semiconductor chip to engineer the wavefunction and exchange statistics of frequency-entangled photon pairs directly at the generation stage, without post-manipulation. Tailoring the spatial properties of the pump beam allows generating frequency-anticorrelated, correlated and separable states, and to control the symmetry of the spectral wavefunction to induce either bosonic or fermionic behaviors. These results, obtained at room temperature and telecom wavelength, open promising perspectives for the quantum simulation of fermionic problems with photons on an integrated platform, as well as for communication and computation protocols exploiting antisymmetric high-dimensional quantum states.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Nonclassical states of light are key resources for quantum information technologies thanks to their easy transmission, robustness to decoherence, and variety of degrees of freedom to encode information [1]. In recent years, great efforts have been directed towards entanglement in high-dimensional degrees of freedom of photons as a means to strengthen the violation of Bell inequalities [2,3], increase the density and security of quantum communication [4,5], and enhance flexibility in quantum computing [6]. In addition, high-dimensional degrees of freedom of photons display a perfect analogy with the continuous variables (CV) of a multiphoton mode of the electromagnetic field [7], which make them a promising platform to realize CV quantum information protocols in the few-photon regime [8,9]. Photonic high-dimensional entanglement has been recently demonstrated in orbital angular momentum [3,10], transverse spatial [11] and path [12,13] modes, and frequency (or frequency–time) [14,15] degrees of freedom.

Among these different degrees of freedom, frequency is particularly attractive thanks to its robustness to propagation in optical fibers and its capability to convey large-scale quantum information into a single spatial mode. This provides a strong incentive for the development of efficient and scalable methods for the generation and manipulation of frequency-encoded quantum states [16–18]. Nonlinear parametric processes such as parametric down-conversion (PDC) and four-wave mixing offer a high versatility for the generation of frequency-entangled photon pairs [19,20]. However, under CW pumping, energy conservation naturally leads to the emission of frequency-anticorrelated states, whereas other types of correlations are needed for certain applications: for instance, non-correlated states are required for heralded single-photon sources [21,22] and correlated states are key resources for clock synchronization [23] or dispersion cancellation in long-distance communication [24]. At a deeper level, it is desirable to gain a higher control over the frequency degree of freedom by manipulating the biphoton joint spectrum both in amplitude and phase. Such shaping can be performed by post-manipulation using time lenses [25], spatial light modulators (SLM) [26,27], dispersive elements [28], or programmable phase filters [14], but this inevitably reduces the brightness of the source and its integrability into chip-based photonic circuits.

Direct shaping of quantum frequency states at the generation stage is therefore preferable. Using parametric processes in solid-state systems, this has been recently realized by engineering the spectral [15,21,29,30] and spatial [31] properties of the pump beam, by temperature tuning [32], or by tailoring the material nonlinearity in domain-engineered crystals [33]. Among these different approaches, the spatial tuning of the pump combines the advantages of reconfigurability and extended possibilities of frequency-state engineering [34]. However, to our knowledge, no previous work has demonstrated a complete toolbox for frequency-state engineering through pump spatial tuning, including a control over the symmetry of the joint spectrum and thus the exchange statistics of the photon pairs—an important feature of quantum state engineering though, particularly in view of quantum simulation [35–37].

In this work, we exploit the high flexibility offered by PDC in a semiconductor AlGaAs microcavity under a transverse pump geometry [38–40] to engineer the spectral wavefunction and exchange statistics of photon pairs without post-manipulation. Tuning the pump spatial intensity allows producing frequency-anticorrelated, correlated, and separable states, while tuning the spatial phase enables switching between symmetric and antisymmetric spectral wavefunctions, leading respectively to bosonic and fermionic behaviors in a quantum interference experiment [11,41]. We also demonstrate the generation of non-Gaussian entanglement [42,43] in the continuous variables formed by the frequency and time degrees of freedom of the photon pairs. We thus demonstrate a general method providing a complete toolbox for frequency-state engineering at the generation stage, and using a chip-based source: these characteristics are crucial in the perspective of the real-world deployment of photonic quantum technologies based on the frequency degree of freedom. Our results, obtained at room temperature and telecom wavelength, open promising perspectives for quantum simulation with particles of various statistics on a monolithic platform without requiring external sources of quantum light [35–37], and to serve as a compact and flexible source for communication and computation protocols based on antisymmetric high-dimensional quantum states [44,45].

## 2. THEORETICAL FRAMEWORK

The working principle of our semiconductor integrated source is sketched in Fig. 1(a). It is a Bragg ridge microcavity made of a stack of AlGaAs layers with alternating aluminum contents [39,40,46]. The device is based on a transverse pump geometry, in which a pulsed pump laser beam impinging on top of the ridge (with an incidence angle $ \theta $) generates pairs of counterpropagating, orthogonally polarized telecom-band photons (signal and idler) through PDC [40,47]. The Bragg mirrors provide both a vertical microcavity to enhance the pump field and a cladding for the twin-photon modes. Of the two possible nonlinear interactions occurring in the device, in the following we consider the one that generates a TM-polarized signal photon [propagating along $ z \gt 0 $, see Fig. 1(a)] and a TE-polarized idler photon (propagating along $ z \lt 0 $). The corresponding biphoton state reads $|\psi \rangle =\iint {\rm d}{{\omega }_{s}}{\rm d}{{\omega }_{i}}\text{JSA}({{\omega }_{s}},{{\omega }_{i}})\hat{a}_{s}^{\dagger }({{\omega }_{s}})\hat{a}_{i}^{\dagger }({{\omega }_{i}})|0,0{{\rangle }_{s,i}}$, where the operator $ \hat a_{s(i)}^\dagger (\omega ) $ creates a signal (idler) photon of frequency $ \omega $. The joint spectral amplitude JSA gives the probability amplitude of measuring a signal photon at frequency $ {\omega _s} $ and an idler photon at frequency $ {\omega _i} $. Neglecting group velocity dispersion (which is justified by the narrow spectral range of the generated photon pairs), and in the limit of narrow pump bandwidth, the JSA can be expressed as [34,48]

Equation (1) indicates that the shape of the JSA along the diagonal direction of the biphoton frequency space ($ {\omega _ + } = {\omega _s} + {\omega _i} $) and that along the antidiagonal direction ($ {\omega _ - } = {\omega _s} - {\omega _i} $) can be tuned independently by varying respectively the spectral or spatial properties of the pump beam, providing a simple and versatile means to engineer the frequency–time correlations of the photon pairs [34]. In addition, in contrast to the co-propagative regime of guided-wave PDC [15,49], the signal and idler photons are here produced in two distinct spatial modes, facilitating their further utilization in protocols. Here, we will exploit the spatial control of the pump beam in intensity and phase by using a spatial light modulator (SLM).

## 3. EXPERIMENTAL SETUP

The experimental setup is shown in Fig. 1(b). The AlGaAs source (ridge length $ L = 2\,\,{\rm mm} $, width $ 6\,\,\unicode{x00B5} {\rm m} $, height $ 7\,\,\unicode{x00B5} {\rm m} $) is pumped with a pulsed Ti:Sa laser with wavelength $ {\lambda _p} \simeq 773\,\,{\rm nm} $, pulse duration $ \simeq 6\,\,{\rm ps} $, repetition rate 76 MHz, and average pump power 50 mW incident on the sample. The pump beam is shaped in intensity and phase using a reflective phase-only SLM (Holoeye Leto) in a 4f configuration, and analyzed with a wavefront analyzer (WFA) to verify the obtained modulation. Finally, a cylindrical lens focuses the beam on the top of the waveguide, and the PDC photons are collected with two microscope objectives and collimated into telecom optical fibers. To characterize the emitted quantum states, we measure the joint spectral intensity (JSI), which is the modulus squared of the JSA, by using a stimulated emission tomography (SET) technique [50] as sketched in Fig. 1(c). In this technique, in addition to the transverse pump beam, a TM polarized CW telecom laser (seed beam), injected through one facet of the waveguide, stimulates the generation of the (TE polarized) idler field by difference frequency generation, and its spectrum is recorded with an optical spectrum analyzer (OSA). The wavelength of the seed laser is swept so as to iteratively reconstruct the whole JSI.

## 4. CONTROL OF FREQUENCY CORRELATIONS

We first demonstrate the control over frequency correlations by varying the spatial extension of the pump beam. We pump the device with Gaussian pump profiles, $ {{\cal A}_p}(z) = {e^{ - {z^2}/{w^2}}}{e^{ikz}} $, where $ w $ is the beam waist on the waveguide and $ k = {\omega _p}\sin (\theta )/c $ is the projection of the pump wavevector along the $ z $ direction. In this situation, the phase-matching term $ {\phi _{{\rm PM}}}({\omega _s} - {\omega _i}) $ is real and corresponds, in the biphoton frequency space $ ({\omega _s},{\omega _i}) $, to a stripe aligned along the diagonal, with a width inversely proportional to the pump waist (in the limit where $ L \gg w $). The other term of the JSA, $ {\phi _{{\rm spectral}}}({\omega _s} + {\omega _i}) $, is given by the spectral distribution of the pump beam: since we use unchirped (Fourier-transform limited) pulses, it is also a real function and corresponds to a stripe aligned along the antidiagonal, with a width inversely proportional to the duration of the pump pulses. The JSA is the product of these two functions: it thus has the shape of an ellipse whose size and orientation are determined by the pump waist and pulse duration.

Figure 2(a) reports the JSI measured by the SET technique for a pump waist $ w = 0.25 $ mm and a pulse duration of 6 ps; the pump angle $ \theta $ is slightly offset from degeneracy as required for the SET measurement [50]. The spectrum is aligned along the antidiagonal, corresponding to a frequency-anticorrelated state. We note the presence of a grid-like pattern, which is related to the reflectivity of the waveguide facets: this creates a Fabry–Perot cavity along the $ z $ direction, whose transmission resonances modulate the joint spectrum [50]. This effect could be exploited to facilitate the manipulation of the frequency degree of freedom by discretizing it, as is the case for quantum frequency combs [20,51,52]; on the other hand, it could be removed if needed by depositing an anti-reflection coating, e.g., in silicon nitride [53]. Starting from the anticorrelated spectrum of Fig. 2(a), Figs. 2(b)–2(d) show the JSI measured for increasing values of the pump waist. We observe that the extension of the JSI along the antidiagonal direction progressively shrinks, transforming the initial state into a frequency-correlated state when $ w = 1\,\,{\rm mm} $ [Fig. 2(d)]. For the intermediate value $ w = 0.6\,\,{\rm mm} $ [Fig. 2(c)], the width of the phase-matching and spectral terms of the JSA are nearly equal, yielding a circular joint spectrum corresponding to a frequency-separable state. The numerical simulations in Figs. 2(e)–2(h), which take into account modal birefringence, chromatic dispersion, and cavity effects in the sample, are in excellent agreement with the experiment. We show also on each panel the calculated Schmidt number $ K $ (obtained from the JSA), which quantifies the effective number of orthogonal frequency modes spanned by the biphoton wavefunction [21]. For the experimental data [Figs. 2(a)–2(d)], the Schmidt number is determined by assuming a flat-phase JSA, a reasonable approximation here since we use unchirped pulses with flat spatial phase profiles. The Schmidt number initially decreases, reaches $ K \simeq 1 $ (corresponding to a separable state) when the JSI is circular, before increasing again when the state becomes frequency-correlated. Note that quantum states with higher Schmidt numbers (i.e., involving more time–frequency modes) could be obtained with the same source by tuning the pumping parameters (see Supplement 1 for a quantitative discussion).

Overall, the results presented in Fig. 2 demonstrate a flexible frequency engineering of biphoton quantum states, which can be exploited to adapt the AlGaAs integrated source to different quantum information applications requiring either anticorrelated [14], separable [21], or correlated frequency states [23,24]. In contrast to filtering approaches that decrease the source brightness by removing unwanted parts of the spectrum [20,54], here the full biphoton spectral intensity is entirely directed into the desired shape at the generation stage. The pair production rate is here $\, \simeq 10\,{\rm MHz} $ at the chip output, corresponding to a brightness of $\, \simeq 200\,{\rm kHz}/{\rm mW} $.

## 5. CONTROL OF WAVEFUNCTION SYMMETRY AND EXCHANGE STATISTICS

We now investigate further control of the quantum frequency state by engineering the phase profile of the pump beam. A first natural way is to impose a phase step $ \Delta \varphi $ between the two halves of the pump spot, as sketched in Fig. 3(a). Placing the pump spot at the center $ z = 0 $ of the waveguide, the pump amplitude profile reads $ {{\cal A}_p}(z) = F(z){e^{ - {z^2}/{w^2}}}{e^{ikz}} $, with $ F(z) = 1 $ for $ z \lt 0 $ and $ F(z) = {e^{i\Delta \varphi }} $ for $ z \gt 0 $. When pumping at the degeneracy angle $ {\theta _{{\rm deg}}} $, one can show that the phase-matching term [Eq. (2)] takes the form (see Supplement 1)

We experimentally implement this concept and show in Figs. 3(b)–3(f) the measured JSI for increasing values of the phase step $ \Delta \varphi $, at fixed pump waist [1 mm, as in Fig. 2(d)] and pulse duration (4 ps). Starting from a frequency-correlated state at $ \Delta \varphi = 0 $, we observe the progressive appearance of a second lobe in the joint spectrum as $ \Delta \varphi $ increases. These results are in good agreement with the numerical simulations [Figs. 3(g)–3(k)], where we show also the calculated Schmidt numbers (here, the non-flat phase structure of the JSA does not allow determining $ K $ experimentally). When $ \Delta \varphi = \pi $ [Figs. 3(f) and 3(k)], the spectrum is split into two lobes of equal intensity, and vanishes along the diagonal axis between the two lobes. According to the previous theoretical analysis, there is a $ \pi $ offset between the spectral phase of points that are mirror-symmetric with respect to this diagonal axis. However, the JSI measurement is not sensitive to such phase information: to retrieve this information and probe the biphoton spectral wavefunction parity, we will exploit two-photon interference in a Hong–Ou–Mandel (HOM) experiment.

The experimental HOM setup is shown in Fig. 1(e). The polarization of the signal photon is rotated and aligned with that of the idler, then the signal photon enters a fibered delay line before recombining with the idler on a fibered 50/50 beamsplitter. Coincidence counts at the outputs (after a long-wave pass filter to remove luminescence noise) are monitored while scanning the delay $ \tau $ of the interferometer. This HOM experiment has in principle four possible outcomes: the two photons can either leave the beamsplitter through the same output port (bunching) or through different ports (antibunching), with two possibilities in each case. When the entangled state is symmetric, antibunching probability amplitudes cancel each other, leaving only bunching events; when the biphoton state is antisymmetric, the reverse scenario occurs, leaving only antibunching events as would be the case for (independent) fermions [11,41,55,56].

We first consider the quantum frequency state obtained when pumping the waveguide with a Gaussian of flat phase profile ($ \Delta \varphi = 0 $). Figure 4(a) shows the corresponding JSI measured at degeneracy with a fiber spectrograph [50] [see Fig. 1(d)]: each photon of the pairs is sent into a spool of highly dispersive fiber so as to convert the frequency information into a time-of-arrival information, which is recorded with single-photon avalanche photodiodes (SPAD, of detection efficiency 25%) connected to a time-to-digital converter (TDC). This technique has here a lower resolution ($ \Delta \lambda \sim 200\,{\rm pm} $) than the SET technique but, contrary to the latter, it can be implemented at frequency degeneracy. The result of the HOM experiment performed with this quantum state is shown in Fig. 4(b), with the corresponding simulation in Fig. 4(c). We observe a coincidence dip (i.e., two-photon bunching), confirming the symmetric nature of the frequency state. The experimental dip visibility, defined as $ V = ({N_\infty } - {N_0})/{N_\infty } $ with $ {N_\infty } $ ($ {N_0} $) the mean coincidence counts at long (zero) time delay, is 88% (using raw coincidence counts); our simulations indicate that this value is mainly limited by slight imperfections of the pump spatial profile and incidence angle (see Supplement 1).

We next consider the biphoton state obtained when imposing a phase step $ \Delta \varphi = \pi $ at the center of the pump spot, resulting in a split JSI as seen in the spectrum of Fig. 4(e), measured at frequency degeneracy. Here, the HOM interferogram [Figs. 4(f)–4(g)] show a coincidence peak (antibunching), demonstrating the antisymmetric nature of the frequency state and the effectively fermionic behavior of the photons. Here, the raw experimental visibility is 77%, again mainly limited by pump imperfections; the side dips at $ \pm 12\,{\rm ps} $ delay are due to the specific shape of the joint spectrum.

Interestingly, the anti-bunching behavior evidenced for the antisymmetric frequency state [Fig. 4(f)] is a direct proof of entanglement [41,57], and more precisely, of entanglement with non-Gaussian statistics [42,43] in the continuous variables formed by the time–frequency degrees of freedom of the biphotons. This non-Gaussian entanglement is associated with the negativity of the chronocyclic Wigner function (CWF) [58], $ W({\omega _s},{\omega _i},{t_s},{t_i}) $, which gives the quasi-probability amplitude of measuring a signal photon at frequency $ {\omega _s} $ and time $ {t_s} $, and an idler photon at frequency $ {\omega _i} $ and time $ {t_i} $. Similar to the JSA [Eq. (1)], in our case the CWF can be factorized into a spectral and phase-matching contributions, $ W = {W_ + }({\omega _ + },{t_ + }) {W_ - }({\omega _ - },{t_ - }) $, with $ {\omega _ \pm } = {\omega _s} \pm {\omega _i} $ and $ {t_ \pm } = ({t_s} \pm {t_i})/2 $. The coincidence probability $ P(\tau ) $ in the HOM experiment is determined by the cut of the $ {W_ - } $ function along $ {\omega _ - } = 0 $ [see dotted lines in Figs. 4(d) and 4(h)], $ P(\tau ) = \frac{1}{2}( {1 - {W_ - }(0,\tau )} ) $ [32,43]. Figures 4(d)–4(h) show the $ {W_ - } $ function calculated for our symmetric and antisymmetric frequency states, respectively. In the latter case, the CWF takes negative values (reaching the theoretical minimum of $ - 1 $) at $ {\omega _ - } = 0 $ (i.e., $ {\lambda _s} - {\lambda _i} = 0 $), evidencing non-Gaussian entanglement. Note that in Fig. 4(d), a small negativity ($ \sim - 0.05 $) also appears at non-zero values of $ {\omega _ - } $ due to the finite length of the device. A full experimental determination of the CWF could be performed by using a generalized HOM experiment, where a frequency shift is added between the two photons (using, e.g., an electro-optic modulator) in addition to the usual temporal delay. Measuring the HOM trace for various frequency shifts would then allow moving along the vertical axis of the CWF shown in Figs. 4(d)–4(h) and reconstruct the $ {W_ - } $ function slice by slice [32,43]; this provides an alternative and promising route to the characterization of a quantum frequency state that does not require a direct measurement of the phase of the JSA. Interestingly, in the particular case of our counter-propagative source, it has been shown that instead of using an electro-optic modulator, a simple change of the pump incidence angle can be used to scan the Wigner function along the frequency difference axis [34].

## 6. CONCLUSION

In summary, we have demonstrated a flexible control over the spectral wavefunction and particle statistics of photon pairs, with a chip-integrated source and directly at the generation stage. The symmetry control of high-dimensional entangled states has been demonstrated previously in the spatial degree of freedom [11,56], but using bulk sources only. In the frequency degree of freedom, displaying strong potential for applications thanks to its robustness to propagation and capability to convey large-scale quantum information into a single spatial mode, a recent work demonstrated the integrated and post-manipulation-free control of the spectrum of biphotons by engineering the spectrum of the pump field, leading in particular to the production of time–frequency Bell states and the implementation of high-dimensional operations in the time–frequency domain [15]. Another work developed a method to control two-color entanglement and gain control over the biphoton spectrum [28], but this approach requires two passages in a bulk source and post-manipulation with a dispersive element, and is limited to the production of two-color entangled states. By contrast, here we experimentally demonstrate a general method providing a complete toolbox to engineer quantum frequency states at the generation stage and using a chip-based source: these features are essential in view of practical and scalable applications for quantum information technologies. The demonstrated device operates at room temperature and telecom wavelength, is amenable to electrical pumping [59] thanks to the direct bandgap of AlGaAs, and has a high potential of integration within photonic circuits [60]: the monolithical integration with on-chip beamsplitters has been demonstrated [22], and the integration of electro-optic phase shifters [60,61] for further manipulation of the state and superconducting nanowires to achieve on-chip detection [62] can be envisaged. The used transverse pump configuration circumvents the usual issue of pump filtering and allows a direct spatial separation of the photons of each pair, facilitating their use in protocols. In particular, these results could be harnessed to study the effect of exchange statistics in various quantum simulation problems [35–37] with a chip-integrated platform, and for communication and computation protocols making use of antisymmetric high-dimensional quantum states [44,45]. Other non-Gaussian high-dimensional photonic states such as time–frequency Schrödinger cat or compass states could also be realized in the used device by a further engineering of the pump beam [34]. In addition, direct generation of polarization entanglement has already been demonstrated with this source design [40] and similar chip-integrated structures [63], opening the perspective to combine such discrete-variable entanglement with the continuous-variable-like entanglement demonstrated here in the time–frequency degrees of freedom of the photon pairs.

## Funding

Agence Nationale de la Recherche (SEMIQUANTROOM); Région Ile-de-France DIM NanoK (SPATIAL); Horizon 2020 Framework Programme (Marie Skłodowska-Curie grant agreement No 665850); Labex SEAM (ANR-10-LABX-0096); RENATECH network; IdEx Université de Paris (ANR-18-IDEX-0001).

## Acknowledgment

The authors thank M. Apfel and F. Bouchard for technical support.

See Supplement 1 for supporting content.

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