Abstract
1. INTRODUCTION
Quantum states of light are central resources for quantum information technologies. Indeed, besides their easy transmission and robustness to decoherence, photons provide a large variety of degrees of freedom (DOFs) to encode information, which can be either two-dimensional (such as polarization) or higher-dimensional (such as frequency, orbital angular momentum, or spatial modes) [1,2]. In addition, photonic information can be encoded either in individual photons or in the quadratures of the electromagnetic field, defining, respectively, the realms of discrete variable (DV) and continuous variable (CV) encoding.
Polarization is a paradigmatic two-dimensional photonic DOF that allowed for pioneering demonstrations in quantum information, ranging from fundamental tests of quantum mechanics [3] to quantum computing [4] and communication tasks [5,6]. Focusing on DV encoding, polarization Bell states, such as , where and stand for the horizontal and vertical polarizations of single photons, respectively, constitute a fundamental building block for many of these applications. They can be efficiently generated with parametric processes in nonlinear bulk crystals combined with external components (such as a walk-off compensator or a Sagnac interferometer) [7–10]. More recently, chip-based sources based on quantum dots [11–15] or parametric processes [16–20] allowed to generate polarization Bell states in a fully integrated manner, without requiring external optical elements.
Now turning to high-dimensional photonic DOFs, among the various candidates, frequency is attracting growing interest due to its robustness to propagation in optical fibers and its capability to convey large-scale quantum information into a single spatial mode. Frequency is intrinsically a continuous DOF that can be used to encode information as such [21–23], but it can also be viewed as a discrete DOF when divided into frequency bins [24,25]. In the latter case, the simplest maximally entangled state of two photons is the so-called two-color Bell state, , where and are well-separated single-photon frequency bins. Several experimental schemes have been implemented to generate such two-color entangled states, which could be exploited, e.g., as a metrology resource for precise time measurements [26], or to interconnect stationary qubits with dissimilar energy levels [27] in a quantum network. The first realizations relied on filtering out frequency bins from a continuous spectrum [28,29]. More recently, brighter sources have been demonstrated by using periodically poled crystals in crossed configurations [26,30], Sagnac loops [31], double passage configurations [32], or by transferring entanglement from the polarization to the frequency domain [33]. All these demonstrations relied on bulk nonlinear crystals, and while integrated sources such as microring resonators are powerful to generate frequency combs (involving a high number of frequency bins) [25], the direct and versatile generation of two-color entangled states with chip-based sources is still scarce. In the latter domain, an important advance was achieved in Ref. [34], combining on the same silicon chip two four-wave mixing sources and an interferometer with a reconfigurable phase shifter. The resulting interference between two independent sources allowed generating two-color entangled states in an integrated and controlled manner, albeit with limited efficiency.
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Besides entanglement into a single DOF, combining several DOFs can provide increased flexibility for quantum information protocols. To this aim, we demonstrate here a single chip-integrated semiconductor source that combines frequency and polarization entanglement, leading to the generation of hybrid polarization–frequency entangled biphoton states without post-manipulation. Our AlGaAs device is based on type-II spontaneous parametric downconversion (SPDC) in a counterpropagating phase-matching scheme, where the modal birefringence lifts the spectral degeneracy between the two possible nonlinear interactions occurring in the device. This allows the direct generation of polarization–frequency entangled photons in two distinct spatial modes, at room temperature and telecom wavelength. Such combination of DOFs opens greater capabilities for quantum information applications, allowing to switch from one DOF to another and thus to adapt to different experimental conditions in a versatile manner.
2. THEORETICAL FRAMEWORK
Our semiconductor integrated source of photons pairs is sketched in Fig. 1(a). It is a Bragg ridge microcavity made of a stack of AlGaAs layers with alternating aluminum concentrations [19,35,36]. The source is based on a transverse pump scheme, where a pulsed laser beam impinging on top of the waveguide (with incidence angle with respect to the axis) generates pairs of counterpropagating photons (signal and idler) through SPDC [19,37]. As a consequence of the opposite propagation directions for the photons, two type-II SPDC processes occur simultaneously in the device [19]: one that generates a TE-polarized signal photon [propagating along ; see Fig. 1(a)] and a TM-polarized idler photon (propagating along ), and a second one that generates a TM-polarized signal and a TE-polarized idler. We later refer to these two generation processes as and , respectively, using the shorter notation (horizontal) for TE and (vertical) for TM. The central frequencies and of the signal and idler photons obey energy conservation (, where is the pump frequency) and momentum conservation along the waveguide direction, which reads for the two interactions
Figure 1.(a) Schematic of an AlGaAs ridge microcavity emitting counterpropagating twin photons by SPDC in a transverse pump geometry. Two type-II interactions occur, generating either an
Figure 1(b) shows the resulting SPDC tunability curve, i.e., the calculated central wavelengths of signal and idler photons as a function of the pump incidence angle , for both interactions, by taking into account our sample properties and the used pump central wavelength . The two interactions are not degenerate because of the small modal birefringence of the waveguide ( at the working temperature 293 K). In the low-pumping regime, the generated two-photon state resulting from both interactions reads
To produce polarization–frequency entangled states, we consider the situation where the pump beam impinges at normal incidence on the waveguide (). The two interactions give rise to two distinct peaks in the biphoton spectrum, as seen in the simulation of Fig. 2(a) showing the joint spectral intensity (JSI), which is the modulus squared of the JSA (plotted in the wavelength space). The two peaks, centered at wavelengths and [see Fig. 1(b)], are symmetric with respect to the degeneracy wavelength . Since the separation between the peaks is much larger than their spectral widths, a reasonable approximation is to discretize the frequency DOF and replace the JSAs by Dirac deltas, (with ). Inserting these expressions into Eq. (3), the emitted biphoton state can be rewritten as a hybrid polarization-frequency (HPF) state, , with
Figure 2.(a) Simulated joint spectral intensity (JSI) of the hybrid polarization–frequency biphoton state of Eq. (
To reveal and quantify the entanglement level of the HPF state, two-photon interference in a Hong–Ou–Mandel (HOM) experiment provides a powerful tool, as it directly probes the quantum coherence between the two components of the state. When the signal and idler photons are delayed by a time and sent into the two input ports of a balanced beam splitter, the coincidence probability between the beam splitter outputs can be calculated from the JSAs and of the two SPDC processes:
3. EXPERIMENTS
The epitaxial structure of the sample is made of a 4.5-period core, surrounded by two distributed Bragg mirrors made of a 36- and 14-period stacks for bottom and top mirrors, respectively. The Bragg mirrors provide both a vertical microcavity to enhance the pump field and a cladding for the twin-photon modes. From this planar structure, a ridge waveguide (length 2.6 mm, width 5 μm, and height 7 μm) is fabricated by UV photolithography followed by wet etching (see Appendix A for more details). The waveguide facets are then coated with a thin film (target thickness 270 nm) deposited by plasma-enhanced chemical vapor deposition (PECVD), resulting in a modal reflectivity for the SPDC modes.
The experimental setup is shown in Fig. 1(c). The sample is pumped with a pulsed Ti:Sa laser of central wavelength , pulse duration 4.5 ps, repetition rate 76 MHz, and average power 30 mW on the sample. A cylindrical lens focuses the pump beam into a Gaussian elliptical spot on top of the waveguide (waist along the waveguide direction) at perpendicular incidence (), and the generated infrared photons are collected with two microscope objectives and collimated into single-mode optical fibers (with a total chip-to-fiber coupling efficiency ). The measured propagation losses are of for both TE and TM fundamental modes of the waveguide. The sample temperature is stabilized at 293 K using a Peltier controller.
We first characterize the generated quantum state by measuring the JSI using a fiber spectrograph [42]. For this, the signal and idler photons are separately sent into a spool of highly dispersive fiber, converting the frequency information into time-of-arrival information. The latter is recorded using superconducting nanowire single-photon detectors (SNSPDs, of detection efficiency 90%) connected to a time-to-digital converter (TDC); long-pass filters are used to remove slight luminescence noise from the sample. The measured JSI, reported in Fig. 3(a), shows two well-defined frequency peaks, symmetric with respect to the degeneracy wavelength, in good qualitative agreement with the numerical simulation of Fig. 2(a). This can be better seen in Fig. 3(b), showing the marginal spectrum of signal (red) and idler (blue) photons as extracted from the experimental JSI. The frequency peaks have an FWHM of and separation , i.e., about 10 times higher than their linewidths. The measured is smaller than in the simulation [6.2 nm in Fig. 2(a)], pointing to a discrepancy between the experimental and simulated modal birefringence . This could be due to a slight deviation of the epitaxial structure from the nominal one and/or imperfections of the simulation (used material refractive indices and exact etching shape of the waveguide); for large waveguides as considered here, numerical simulations show that the epitaxial structure (slight uncertainty in the aluminum concentration and height of the layers) is the dominant factor.
Figure 3.(a) Measured joint spectral intensity (JSI) of the hybrid polarization–frequency state, and (b) corresponding marginal spectrum of signal (red line) and idler (blue line) photons. (c) Measured HOM interferogram (black symbols with error bars) fitted with Eq. (
We now perform two-photon interference in an HOM setup [see Fig. 1(c)] to reveal the entanglement properties of the generated HPF state. A half-wave plate (HWP) in the signal arm and a fibered polarization controller (FPC) in the idler arm are used to compensate for polarization rotation on the optical path, hence maintaining signal and idler photons in the same state as at the chip output so that they enter the beam splitter with crossed polarizations. The resulting interferogram, shown in Fig. 3(c) (black points with error bars), displays a clear sinusoidal modulation with a Gaussian envelope. Each point is obtained by a 20 s integration time, and error bars are calculated assuming a Poissonian statistics. The data are fitted (blue line) using a modified version of Eq. (7) accounting for experimental imperfections:
The experimental raw visibility is . We attribute the main part of the visibility reduction to the non-zero reflectivity of the facets. Indeed, as a consequence of the latter, the two photons of each pair have a non-zero probability to exit through the same facet, instead of opposite ones. This results in quantum interference between the situation where both photons exit from one facet and the situation where both photons exit from the other facet, leading to a modulation at a frequency equal to the sum of signal and idler frequencies, i.e., the pump frequency . This interference occurs for a time delay shorter than the temporal width of the SPDC photons, and therefore, it is superimposed to the HOM interference [43]. The corresponding period is , which is beyond the resolution of our HOM setup. The measured interferogram thus averages out over these rapid oscillations, reducing the effective visibility of the fringes. For our sample with 10% reflectivity for SPDC modes, numerical simulations including this averaging effect predict a visibility of 82% (see Appendix A). We attribute the remaining visibility drop to slight imperfections of the pump spatial profile and incidence angle (see Appendix A for a detailed discussion on this effect and on other possible causes of visibility reduction in the experiment).
Using the joint spectrum and HOM measurements, we can now quantify the entanglement of the generated HPF state by estimating a restricted density matrix [33] in the polarization-frequency discrete space. The full basis would include all combinations of the {, } polarizations and {} frequencies, resulting in a density matrix. However, physical considerations allow for restricting the relevant basis dimension, projecting it onto the relevant subspace. Indeed, the employed type-II SPDC process does not allow the production of photons of the same polarization, while energy conservation forbids the production of photons of the same frequency; in addition, the phase matching requires that photons of frequency (resp. ) are always (resp. ) polarized. This leads to the following 4 × 4 restricted density matrix:
The JSI measurement [Fig. 3(a)] yields the population term (obtained by integrating coincidence counts in a square window of 2 nm width centered on each of the two spectral modes) while the HOM interferogram [Fig. 3(c)] gives the coherence modulus from the visibility deduced above; the phase between the two interactions is deduced from the fact that the source is pumped by a single pump beam. The resulting density matrix is shown in Fig. 4. It allows to extract the purity of the generated state, , its fidelity to the ideal state of Eq. (4), , and its concurrence, (all raw values). This confirms the direct generation of HPF entanglement by our chip-integrated source. The generation rate, estimated from single and coincidence counts data [44], is pairs/s at the chip output for the used pump power incident on the sample.
Figure 4.Experimental reconstruction of the restricted density matrix [Eq. (
4. DISCUSSION AND CONCLUSION
In summary, we have demonstrated a chip-based semiconductor source that combines polarization and frequency entanglement, enabling the generation of HPF entangled photon pairs directly at the generation stage, in two distinct spatial modes. Such combination of DOFs provides an increased flexibility for quantum information protocols, allowing to adapt the source to different applications in a versatile manner. The demonstrated device operates at room temperature and telecom wavelength and has a strong potential for integration within photonic circuits [45–47]. Due to the direct bandgap of AlGaAs, it is also compliant with electrical pumping, either by monolithic integration with a laser diode [48] or through adhesive bonding of a VCSEL with a large rectangular aperture, as demonstrated, e.g., in Ref. [49], on top of the structure.
These results could be further expanded along several directions. First, the fidelity of the experimentally generated state to the ideal HPF state of Eq. (4) could be improved by implementing a multi-layer coating of the waveguide facets, allowing to reach modal reflectivities . This enhancement would lead to an expected fidelity larger than 0.93, all other factors (including experimental imperfections) kept unchanged. The fidelity could be further improved by correcting for the small imperfections of the pump spatial profile (deviations from the ideal Gaussian shape) using, e.g., a spatial light modulator.
In addition, the frequency entanglement of our HPF state can be varied by different means. In Eq. (4), frequency entanglement is described as a discrete two-color entanglement, which reflects the dominant frequency anticorrelation of the state, but neglects intra-mode frequency entanglement, i.e., the continuous entanglement associated with the internal structure of each frequency mode [as determined by JSAs and in Eq. (3)]. This intra-mode entanglement, which manifests in the envelope of HOM oscillations [Eq. (7)], can be controlled
Frequency entanglement can also be tailored at the inter-mode level, i.e., by varying the separation between the two central frequencies ( and ). This can be achieved by playing with the modal birefringence of the waveguide, either
Acknowledgment
Acknowledgment. We acknowledge support from European Union’s Horizon 2020 research and innovation programme under the HORIZON EUROPE Marie Sklodowska-Curie Actions grant agreement No. 665850, Paris Ile-de-France Région in the framework of DIM SIRTEQ (LION project), Ville de Paris Emergence program (LATTICE project), IdEx Université Paris Cité (ANR-18-IDEX-0001), Labex SEAM (Science and Engineering for Advanced Materials and Devices, ANR-10-LABX-0096), and the French RENATECHnetwork.
APPENDIX A
From the planar AlGaAs epitaxial structure described in the main text, ridge waveguides are patterned by photolithography followed by wet etching. We employed UV photolithography (MJB4 machine) with a positive photoresist S1805 (nominal thickness of 500 nm) and a custom Cr photomask from Delta Mask. The resist is developed using an MF-319 developer, and the waveguides are then etched using a solution composed of acetic acid, potassium dichromate (), and hydrobromic acid (HBr) in stoichiometric proportions. This wet etching technique yields curved sidewalls. The width value given in the paper (5 μm) corresponds to the width at the top of the waveguide, while the height value (7 μm) corresponds to the total height between the top and bottom (far from the waveguide) of the structure. These parameters can be measured either by SEM or a Dektak profilometer, yielding consistent results. As mentioned in the main text, for such relatively large waveguide widths, we verified by numerical simulations that the precise shape of the confinement has little effect on the modal birefringence, which is mainly determined by the epitaxial structure.
We state in the paper that the non-zero reflectivity of the waveguide facets is the main source of limitation of visibility in the measured HOM interferogram of the HPF state [Fig.
We model the HOM experiment as shown in Fig.
Figure 5.Hong–Ou–Mandel scheme for a counterpropagating parametric source emitting photons through both
Figure 6.Simulated HOM interferogram for the HPF entangled state, taking into account the Fabry–Pérot effect of the sample with facet reflectivity
Figure 7.Simulated HOM interferogram obtained from the data of Fig.
The experimentally measured visibility is . We attribute the additional visibility drop (with respect to the above simulated 82%) to slight imperfections of the pump spatial profile and incidence angle, not considered in the simulation above. These pump imperfections, when considered alone, typically lead to a visibility reduction as we observed in Ref. [
Other possible sources of visibility reduction could in principle be related to multi-pair emission or sample photoluminescence. The coincidence-to-accidental ratio (CAR) measured in the experiment is 42, where accidental coincidences are dominated by coincidences between an “orphan” photon (whose twin has been lost or undetected) and a noise count coming from photoluminescence, multi-pair emission (photon coming from another, uncorrelated pair), or detector dark count. The high measured CAR suggests that photoluminescence and multi-pair emission are maintained to a low level in the experiment, thus having a negligible impact on the HOM experiment. This is corroborated by the fact that no detectable increase of HOM visibility was observed when reducing the pump power in our experiments.
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