# Optimizing the quantum interference between single photons and local oscillator with photon correlations

Hubert Lam, Juan Álvarez, Petr Steindl, Ilse Maillette de Buy Wenniger, Stephen Wein, Anton Pishchagin, Thi Huong Au, Sebastien Boissier, Aristide Lemaître, Wolfgang Löffler, et al.

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## ► To cite this version:

Hubert Lam, Juan Álvarez, Petr Steindl, Ilse Maillette de Buy Wenniger, Stephen Wein, et al. Optimizing the quantum interference between single photons and local oscillator with photon correlations. Quantum Science and Technology, 2025, 10 (4), pp.045061. 10.1088/2058-9565/ae0a7a. hal-05370303

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# Optimizing the quantum interference between single photons and local oscillator with photon correlations

To cite this article: Hubert Lam et al 2025 *Quantum Sci. Technol.* 10 045061

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# Optimizing the quantum interference between single photons and local oscillator with photon correlations

Hubert Lam¹,*, Juan R Álvarez¹,², Petr Steindl¹,², Ilse Maillette de Buy Wenniger³, Stephen Wein⁴, Anton Pishchagin⁴, Thi Huong Au⁴, Sebastien Boissier⁴, Aristide Lemaître¹, Wolfgang Löffler², Nadia Belabas¹, Dario A Fioretto¹,⁴ and Pascale Senellart¹

¹ C2N, Photonics Department, Université Paris-Saclay, 10 Bd Thomas Gobert, 91120 Palaiseau, France

² Leiden Institute of Physics, Leiden University, 2333 CA Leiden, The Netherlands

³ Imperial College London, Quantum Optics and Laser Science, Exhibition Rd, London SW7 2BX, United Kingdom

⁴ Quandela, 7 Rue Léonard de Vinci, 91300 Massy, France

⁵ Currently at LCTI, Telecom Paris, IP Paris, 19 Place Marguerite Perey, 91120 Palaiseau, France.

* Author to whom any correspondence should be addressed.

E-mail: hubert.lam@universite-paris-saclay.fr

**Keywords:** quantum interference, homodyne photon correlation measurement, quantum dot single-photon source

Supplementary material for this article is available online

## Abstract

The quantum interference between a coherent state and a single photon is an important tool in continuous variable optical quantum technologies to characterize and engineer non-Gaussian quantum states. Semiconductor quantum dots (QDs), which have recently emerged as a key platform for efficient single-photon generation, could become interesting assets in this context. An essential parameter for interfering single photons and classical fields is the mean wavepacket overlap between both fields. Here, we report on two homodyne photon-correlation techniques enabling the precise measurement of the overlap between a single photon generated by a QD-cavity device and pulsed laser light. The different statistics of interfering fields lead to specific signatures of the quantum interference on the photon correlations at the output of the interfering beam splitter. We compare the behavior of maximized overlap, measuring either the Hong-Ou-Mandel visibility between both outputs or the photon bunching at a single output. Through careful tailoring of the laser light in various degrees of freedom, we achieve a record overlap of 76% with integrated solid-state sources, which evidences the very low level of noise in our integrated single-photon sources.

## 1. Introduction

Leveraging light to process quantum information is a continuously growing field, with foreseen applications in quantum computing [1, 2], quantum communications [3–5], and quantum sensing [6]. Single-photon states are central to these technological developments, both when encoding the information on single photons (discrete variable (DV)) framework) or in the quadrature of the quantum light field (continuous variable (CV) framework). Single photons have been extensively exploited to implement DV quantum information processing protocols [1]. Having also intrinsically non-Gaussian character, single photons can be used to implement non-Gaussian operation on a Gaussian state—a necessary gate for universal CV quantum computing [7]. Such a gate can be realized by a heralded interference of the single photon and the Gaussian state, effectively leading to single-photon addition [8] and generation of a non-Gaussian state [9–11].

Concretely, interfering a single photon and a coherent state on a beam splitter is a powerful tool in this framework. For instance, such interference combined with a proper heralding scheme allows the generation of a variety of non-Gaussian states, including displaced Fock states and photon-number superpositions [12–14]. Finally, note that the non-Gaussianity of a single-photon state in the CV framework
is assessed using homodyne detection through the interference with a laser light acting as a local
oscillator [15, 16].

So far, the vast majority of experiments interfering single photons with coherent states have been
implemented with heralded single-photon sources based on frequency conversion [17–20]. These sources
are based on a probabilistic generation process and suffer from intrinsic limitation on terms of efficiency. In
the last decade, a new technology to generate single photons in a deterministic way has emerged based on
semiconductor quantum dots (QDs) [21–25]. Exploiting these sources would bring the advantage of
on-demand deterministic generation with integrated devices to the CV framework. However, achieving a
large overlap between a coherent state and single photons emitted from a solid-state emitter represents an
important challenge: the solid-state environment of the emitter leads to slow electrical and magnetic
fluctuations [26] that reduce the quantum purity of the emitted photons over long (second-scale) timescales.

In the present work, we use state-of-the-art QD sources and take a first step toward bringing QD sources to the continuous-variable paradigm. We quantify the interference between a laser pulse and the single photons generated by a QD in a cavity [21]. We report on two complementary experimental methods to assess and optimize their mode overlap *M* (called mean wavepacket overlap) [27]. The first is reminiscent of the Hong–Ou–Mandel (HOM) interference performed with two single photons, where *M* is directly related to the interference visibility determined by cross-correlation measurements between the two outputs of the beam splitter [28, 29]. The second method monitors the photon bunching through two-photon correlation at a single output of the beam splitter. Here again, the difference in photon-number statistics between both interfering fields leads to a distinctive signature for maximal overlap, which is obtained by adjusting the power of the laser light. Doing so, we report a record value approaching 80% for overlap between QD emitted photons and laser light, evidencing the maturity of our source technology for further applications.

## 2. Experimental procedure

Our single-photon sources are based on InGaAs QDs grown by molecular beam epitaxy embedded in the center of a λ GaAs optical cavity between two AlGaAs/GaAs distributed Bragg reflectors with 18 top and 36 bottom pairs. Furthermore, a single QD is deterministically positioned at the electric field antinode of the subsequently etched micropillar cavity with approximately 50 nm accuracy using the in-situ lithography technique [30]. The QD is embedded in a p–i–n junction, allowing stabilization of the solid-state environment by sweeping away local charges and tuning of the optical transition of the QD to cavity resonance around ~925 nm by the quantum-confined Stark effect, leading to a suppression of the phonon sideband via Purcell acceleration of the spontaneous emission [31] and Purcell-enhanced radiative lifetime τ<sub>LT</sub> of (170 ± 5) ps. The QD-cavity system operates at 5 K in a closed-cycle cryostat. We create single photons by coherently exciting the QD micropillar system on-resonance using a mode-locked Ti:Sapphire laser with ~10 ps pulse duration and 1/τ<sub>rep</sub> ≈ 82 MHz repetition rate. The maximum population inversion of the QD excitonic states is achieved with a pulse area corresponding to a π. This results in a normalized second-order intensity correlation of g<sub>ψ</sub><sup>(2)</sup>(0) = (4.12 ± 0.05)% where ψ indicates the averaged wavepacket emitted by the QD device. The indistinguishability of successively emitted photons is measured to reach M<sub>ψ</sub> = (90.5 ± 0.1)% [29].

The emitted photonic state is separated from the excitation laser by polarization filtering in a cross-polarization setup [21, 32]. The collected photons are directed to a single-mode fiber-based beam splitter (FBS) of near-balanced splitting ratio, ensuring near-ideal spatial overlap, where the photons are mixed with a power-controlled coherent state |α⟩ supplied from the same excitation laser. The coherent state is spectrally shaped to ensure mode overlap in the spectral-temporal domain with the single photons, as illustrated in figure 1(a). The spectral shaping of the coherent state is performed by a pair of Fabry–Pérot etalons (FWHM =14.5 pm) to approach the spectral-temporal properties of ψ —details in the supplementary. The power of the local oscillator is controlled using two calibrated variable optical attenuators. We control the polarization of each field with a separate polarizer followed by a set of free-space zero-order waveplates. The timing of laser and single photons arrival is adjusted finely using free-space delay lines.

We detect light at both FBS outputs using superconducting nanowire single-photon detectors (SNSPDs) and simultaneously measure cross- and auto-correlations [27] between the FBS outputs using three detectors, as illustrated in figure 1(a). Examples of the homodyne correlations $G^{(2)}(\tau)$ measured in cross-correlation and auto-correlation configurations with a local oscillator power of $\mu_\alpha = 0.28 \pm 0.01$ are shown in figures 1(b) and (c). Note that, since single photons exhibit no intrinsic phase dependence in quadrature space, homodyne measurements can be performed without the need for active phase locking.
<figure><img src="image_7.png" /></figure>

### 3. HOM visibility analysis

First, we consider the cross-correlation measurement where the overlap $M$ between two optical modes can be deduced from the strength of photon bunching on a balanced beam splitter. This method is commonly used to determine the indistinguishability of photons successively emitted by a QD source, where the total mean wavepacket overlap $M_{\psi} = V_{\text{HOM}} (1 + g_{\psi}^{(2)}(0))$ [29] can be quantified by the HOM visibility $V_{\text{HOM}}$ and the purity of the single-photon stream $g_{\psi}^{(2)}(0)$. This approach corrects for multi-photon contamination in the QD emission, which reduces the strength of photon bunching [29, 33].

We extend this formalism to the interference between two light fields with dissimilar photon-number statistics: a stream of photons $\psi$ emitted by our QD and a coherent state $|\alpha\rangle$. We start by comparing the photon correlations measured with the polarization direction of the coherent state aligned parallel $g_{\text{cross},\parallel}^{(2)}(\tau)$ and orthogonal $g_{\text{cross},\perp}^{(2)}(\tau)$ with the polarization of $\psi$, as shown in figure 1(b). Similarly to the conventional HOM experiment, the strength of photon coincidences measured at $\tau = 0$ is the litmus test of the photon-bunching effect: the single photons in the photon stream $\psi$ bunch with the single-photon component from the local oscillator $|\alpha\rangle$ leading to reduced correlations at $g_{\text{cross},\parallel}^{(2)}(0)$ with respect to $g_{\text{cross},\perp}^{(2)}(0)$. The relative weight of the single-photon component in the coherent state during interference is inherently constrained by its Poissonian statistics, dictated by the mean photon number $\mu_{\alpha} = |\alpha|^2$. Consequently, the effective strength of the photon bunching also depends on $\mu_{\alpha}$. Figure 2(a) presents the HOM visibility $V_{\text{HOM}} = (g_{\text{cross},\perp}^{(2)}(0) - g_{\text{cross},\parallel}^{(2)}(0))/g_{\text{cross},\perp}^{(2)}(0)$ where the normalized second-order correlations $g_{\text{cross},\perp}^{(2)}(0)$ and $g_{\text{cross},\parallel}^{(2)}(0)$ are obtained from the measured correlation histograms [33–36]. The experiment is conducted for various polarizations of the interfering fields, from parallel polarization (red symbols) to increasing non-parallel polarizations (blue, yellow, green symbols). For all curves, the HOM visibility approaches zero when $\mu_{\alpha}$ approaches either zero or large values. Such behavior is expected for both limits, since the strong imbalance of the mean photon numbers between both fields results in an effective single-field interference.

To access the mean wavepacket overlap between both fields, we derive the HOM visibility $V_{\text{HOM}}$ as a function of the mean wavepacket overlap $M$ and the normalized second-order intensity correlation of each field (see supplementary). The normalized second-order correlation function of the local oscillator $|\alpha\rangle$ is
<figure><img src="image_8.png" /><figcaption>Figure 2. (a) HOM visibility $V_{HOM}$ as a function of relative mean photon number of the fields $\mu_{\alpha}/\mu_{\psi}$. Each color denotes a different $M$ value, which is controlled by altering the relative polarization of the $|\alpha\rangle$-path. The measured $V_{HOM}$ is fitted with equation (1) and the obtained $M$ is given in the legend. Identical $M$ values can be also obtained from the maximum of $V_{HOM}$ at $\mu_{\alpha}/\mu_{\psi} = \sqrt{g_{\psi}^{(2)}(0)}$ (gray vertical line). (b) The overlap $M$ extracted from individual data points match within error bars the values obtained from the fit (a) (horizontal lines).</figcaption></figure>

given by $g_{\alpha}^{(2)}(0) = 1$. For QD emission, using a short resonant $\pi$-pulse compared to the emission lifetime, we approximate the emitted state by $p_1|1\rangle\langle 1| + p_2|2\rangle\langle 2|$, where $p_1$ and $p_2$ correspond to the probability of the QD emitting one and two photons, respectively. We consider $p_1 \approx 1$, $p_2 \ll p_1$ and neglect any coherence of the photon-number [37], so that $g_{\psi}^{(2)}(0) = 2p_2/(p_1 + 2p_2)^2$. The mean wavepacket overlap of both fields is then given by [33]:

$$M = V_{\text{HOM}} \left( 1 + \frac{\mu_{\alpha}}{2\mu_{\psi}} g_{\alpha}^{(2)}(0) + \frac{\mu_{\psi}}{2\mu_{\alpha}} g_{\psi}^{(2)}(0) \right), \quad (1)$$

where $\mu_{\psi} = \eta(p_1 + 2p_2)$ denotes the mean photon number of $\psi$ transmitted through the quantum channel (from QD to FBS) of total transmission $\eta$. The reconstructed $M$ is shown in figure 2(b) which evidences the maximal mode overlap $M = 0.76 \pm 0.01$ for parallel interfering fields (red). This value is found to be constant for almost four orders of magnitude of $\mu_{\alpha}$. Such a stability requires a very precise calibration of $\mu_{\alpha}$ at the beam splitter that accounts for the polarization-dependent detection efficiency of SNSPDs, as explained in the supplementary.

To the best of our knowledge, the only demonstrations of interference of QD-emitted photon and laser light so far were performed under continuous-wave excitation [34, 38, 39]. As such, the reported values correspond to an indefinitely short timescale where the QD decoherence does not play a role. The present work, reporting the first interference in the pulse regime, at maximum occupancy of the QD state, represents a milestone in this regard. The obtained mean-wavepacket overlap of 76% reflects the low level of noise reached in our devices, despite the complex nanoprocessing required for cavity-based photon extraction. The overlap benefits from the low local noise of the QD environment stabilized by the electric diode and the phonon sideband suppression via microcavity. However, it remains slightly limited due to off-resonant light leakage through the laser shaping etalons, giving that the overlap increases to $M = 0.81 \pm 0.01$ with post-processing correction (see supplementary). The remaining distinguishability arises from the mode mismatch between the spectral and temporal profiles of the two fields and the low-frequency charge noise of the single-photon source. These factors lead to a spectral and temporal overlap of $M_f = 0.85 \pm 0.02$ and $M_t = 0.910 \pm 0.005$, respectively (see supplementary).

In addition, figure 2(b) evidences a control over $M$ for several mutual polarization alignments between the two interfering fields. The relative polarization of the interfering fields is calibrated by monitoring the interference visibility using a narrow-band continuous wave calibration laser (see supplementary). For each polarization, we consistently observe the maximum of $V_{HOM} = M/(\sqrt{g_{\psi}^{(2)}(0) + 1})$ at $\mu_{\alpha}/\mu_{\psi} = \sqrt{g_{\psi}^{(2)}(0)} = 0.203 \pm 0.001$, as expected [36], marked in figure 2(a) by a gray vertical line. Note that the uncertainties of $M$ in figure 2(b) for the larger $\mu_{\alpha}$ become pronounced due to the decrease of $V_{HOM}$ to zero and the divergence of the multi-photon correction factor in equation (1). The errors at lower values of $\mu_{\alpha}$ arise from the birefringence of the power control attenuators at higher voltages, slightly altering the estimation of the mean photon number $\mu_{\alpha}$ due to the polarizer defining the local oscillator polarization, as discussed in the supplementary.
<figure><img src="image_9.png" /><figcaption>Figure 3. Photon correlations in one beam splitter output: $g_{\text{auto}}^{(2)}(0)$ for various mean photon numbers $\mu_{\alpha}$. The mean wavepacket overlap $M$ (different color) is controlled by adjusting the polarization of the mixing coherent state. In agreement with our theory (fit with equation (2), lines), with increasing power $\mu_{\alpha}$, the state in one FBS output shows a gradual transition from antibunching, through bunching, to random photon statistics, reflecting photon-number statistics changes induced by quantum interference.</figcaption></figure>

## 4. Bunching analysis

We now evaluate the mean wavepacket overlap between both fields by analyzing the bunching signature at one FBS output [20], reflecting the photon-number statistics changes induced by the mixing of the states on the FBS. In figure 3, we show different values of $g_{\text{auto}}^{(2)}(0)$ as a function of the relative optical power of the fields $\mu_{\alpha}/\mu_{\psi}$. The values are derived from second-order correlation histograms $G_{\text{auto}}^{(2)}(\tau)$ by normalizing with the uncorrelated peaks at $\tau \neq 0$, shown in figure 1(c).

By controlling $\mu_{\alpha}$, we observe a gradual transition from the quantum regime of antibunching with $g_{\text{auto}}^{(2)}(0) \approx 0$ in the single-photon stream at low $\mu_{\alpha}$ to classical coherent state statistics at high $\mu_{\alpha}$ ($g_{\text{auto}}^{(2)}(0) \approx 1$). Interestingly, in the regime $\mu_{\alpha}/\mu_{\psi} \approx 2$, where photon bunching is the strongest, $g_{\text{auto}}^{(2)}(0)$ becomes highly sensitive to $M$. For a pure single-photon stream ($g_{\psi}^{(2)}(0) = 0$) and ideal overlap ($M = 1$), this value approaches the maximum of $4/3$ for an ideal displaced single-photon Fock state [19]. Note, that similar transition is present also in three-photon correlations $g_{\text{auto}}^{(3)}(0)$ (shown in supplementary) with bunching regime between $1 < \mu_{\alpha}/\mu_{\psi} < 100$ corresponding to a significant increase of the three-photon component, induced by the quantum interference [19].

Our model (see supplementary) allows to derive the expected second-order auto-correlation:

$$g_{\text{auto}}^{(2)}(0) = \frac{\mu_{\alpha}^2 + \mu_{\psi}^2 g_{\psi}^{(2)}(0) + 2\mu_{\alpha}\mu_{\psi}(1+M)}{(\mu_{\alpha} + \mu_{\psi})^2} \quad (2)$$

Intuitively, we identify three contributions: (i) the contribution $\mu_{\alpha}^2$, arising purely from the coherent state when all photons are reflected, (ii) the second term $\mu_{\psi}^2 g_{\psi}^{(2)}(0)$, representing two-photon correlations in the single-photon stream when all photons are transmitted, and (iii) the final term, containing information about the mode overlap where photons from both sources interfere, forming displaced Fock states [13].

In figure 3, we fit $g_{\text{auto}}^{(2)}(0)$ with equation (2) and extract values of the mean wavepacket overlaps $M$. As expected, the extracted overlaps are consistent with values obtained from homodyne cross-correlation measurements [27] performed simultaneously, providing extra evidence consistent with the quantum-interference origin of the $g_{\text{auto}}^{(2)}(0)$ behavior in figure 3.

A nice feature of the auto-correlation measurement lies in the monotonic dependence of $g_{\text{auto}}^{(2)}(0)$ on $M$, shown in equation (2). This property enables real-time optimization of the mutual overlap of the fields. For a fixed relative power of the fields $\mu_{\alpha}/\mu_{\psi}$, increasing the mode overlap results in stronger photon bunching and thus a higher value of $g_{\text{auto}}^{(2)}(0)$, as shown in a recording in the supplementary material. Concretely, we first set $\mu_{\alpha}/\mu_{\psi} \approx 2$, where the photon bunching is maximal, and fine-tune the optical and mechanical elements of the setup while systematically monitoring the value of $g_{\text{auto}}^{(2)}(0)$. With an overall efficiency of the
single-photon source and the setup of $\eta \approx 3\%$, a short integration time of 1 s suffices for real-time feedback. We reliably achieve a maximum value of $g_{\text{auto}}^{(2)} = 1.243 \pm 0.002$ at $\mu_{\alpha}/\mu_{\psi} \approx 2$, corresponding to $M = 0.76 \pm 0.01$.

Finally, we note that the bunching analysis can be used to estimate the QD brightness at the input of an arbitrary beam splitter with transmission $T$ and reflection $R$ due to the relation between the strength of a power-calibrated laser needed to achieve the $g_{\text{auto}}^{(2)}(0)$ maximum. Concretely, the maximum of $g_{\text{auto}}^{(2)}(0)$ is observed at $T\mu_{\alpha} = 2R\mu_{\psi}$ [40], enabling to use the quantum interference as a local probe of the brightness $\mu_{\psi}$.

## 5. Conclusion and discussion

In conclusion, we have implemented two homodyne photon correlation techniques [27] to control and quantify the mean wavepacket overlap $M$ between laser light pulses and on-demand single photons emitted by the QD-micropillar device. The mode overlaps are determined through an unambiguous relationship between photon bunching and field overlap, which we obtain by measuring the second-order intensity correlations at the single output or between both outputs of the beam splitter. Both methods give consistent values and are fully accounted for by our photon correlation model, which includes the statistical properties of the single-photon source, including its brightness and purity. Benefiting from the stabilization of the QD local environment with electrical gating, we demonstrate a record value of $M = 0.76$ achieved with integrated solid-state sources, comparable to heralded single-photon technology [19, 20].

Generally, in addition to application in the CV framework, the interference between laser light and single photons is used in many quantum protocols and quantum state generation schemes. Especially due to the inherent robustness of the correlation measurements against losses, the homodyne photon correlation techniques are not constrained by detection efficiency. Therefore, these techniques can be leveraged to improve quantum communication protocols that rely on quantum interference with laser light. For instance, optimizing the overlap maximizes the fidelity of teleportation from a quantum to laser light carrier [41]. Furthermore, it enables precise tailoring and synchronization of quantum network nodes, facilitating hybrid quantum-classical protocols [42], as well as maximizing security in measurement-device-independent quantum key distribution [43] and quantum-assisted authentication protocols [44].

Finally, the optimized interference of laser light and single photons enables the improved generation of high-N NOON states [45], which are important resources for super-resolving phase measurements in quantum metrology [46], and hybrid entanglement [47] which serves as a building block for hybrid DV-CV quantum computing [48], or can be used for enhanced phase-sensitivity measurements that remains robust even in the presence of optical loss [49].

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

H L acknowledges Leonid Vidro for the discussion. This work was partially supported by the Paris Ile-de-France Région in the framework of DIM QUANTIP, the European Union's Horizon CL4 program under the grant agreement 101135288 for EPIQUE project, the Plan France 2030 through the projects ANR22-PETQ-0011, ANR-22-PETQ-0006, and ANR-22-PETQ-0013, and a public grant overseen by the French National Research Agency as part of the 'Investissements d'Avenir' programme (Labex NanoSaclay, reference: ANR10LABX0035). We acknowledge funding from NWO/OCW (Quantum Software Consortium, Nos. 024.003.037,024.003.037/3368). This work was carried out within the C2N micro-nanotechnologies platforms and was partially supported by the RENATECH network and the General Council of Essonne.

## Conflict of interest

P S (Pascale Senellart) is a scientific advisor and co-founder of the company Quandela. The other authors declare no competing interests.

## ORCID iDs

Hubert Lam ID 0000-0002-1712-9201

Juan R Álvarez ID 0000-0003-3411-8237
Petr Steindl ID 0000-0001-9059-9202

Ilse Maillette de Buy Wenniger ID 0000-0002-8809-4704

Aristide Lemaître ID 0000-0003-1892-9726

Wolfgang Löffler ID 0000-0001-6587-8947

Nadia Belabas ID 0000-0001-9561-3632

Dario A Fioretto ID 0000-0003-3829-4000

Pascale Senellart ID 0000-0002-8727-1086

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