## Abstract

A major challenge in the development of atom–photon and photon–photon quantum gates is to provide output photons in a pure quantum state, as opposed to incoherent superpositions. Here we introduce a tomography approach to describe the optical response of a cavity quantum electrodynamics device, by analyzing the polarization density matrix of the reflected photons in the Poincaré sphere. Applying this approach to an electrically controlled quantum dot (QD)-cavity device, we show that the superposition of emitted single photons with directly reflected photons leads to a large rotation of the output polarization, by 20° both in latitude and longitude in the Poincaré sphere, with a polarization purity remaining above 84%. The QD resonance fluorescence is shown to contribute to the polarization rotation via its coherent part, while its incoherent part contributes to degrading the polarization purity. This polarization tomography technique allows discriminating between various decoherence processes, a powerful tool for solid-state quantum technologies.

© 2017 Optical Society of America

## 1. INTRODUCTION

In the development of quantum photonic networks, a crucial challenge is to demonstrate highly efficient interfaces allowing the coherent transfer of quantum information between a stationary qubit and a flying one [1]. In this context, it has been shown that a single natural or artificial atom in a cavity quantum electrodynamics (CQED) system can induce giant phase shifts on incoming photons [2–5]: this allowed the recent implementation of atom–photon gates [6–8] and, subsequently, photon–photon gates [9–12]. Most of these achievements are based on polarization-encoding protocols [13], in which the optical phase shift is used to rotate the polarization state of the outgoing photons. Polarization rotation is also at the heart of numerous quantum computation proposals based on spin–photon logic gates, where a stationary spin qubit is used as a quantum memory with a long coherence time [14–18].

By essence, these concepts of optical phase shift and polarization rotation rely on a semiclassical image, considering that the optical response of the atom-based device is entirely coherent. Yet, the output photonic field includes a contribution from the resonance fluorescence emitted by the natural or artificial atom, which can be partially incoherent with respect to the incoming laser [19]. In quantum optics this is described by an output field operator $\widehat{b}$ that cannot be reduced to its expectation value. The coherent contribution is governed by the average field $\u27e8\widehat{b}\u27e9$, which keeps a defined phase with respect to the incoming field; the incoherent component, however, arises from quantum fluctuations around the average field, and has no specific phase [19]. A direct consequence of this is that the optical response of a device cannot, in general, be solely described by a reflection coefficient with a well-defined amplitude and phase. Similarly, a general output polarization is not necessarily a pure polarization state.

The distinction between coherent and incoherent responses is especially important in solid-state quantum technologies, where artificial atoms, such as semiconductor quantum dots (QDs), interact with a fluctuating environment. As was experimentally demonstrated using spectrally resolved [20,21] or interferometry-based [21–23] measurements, the fluorescence emitted by a resonantly excited QD has its coherent fraction at best equal to $\frac{{T}_{2}}{2{T}_{1}}$, with ${T}_{1}$ the lifetime of the photon wavepacket and ${T}_{2}$ its coherence time [19]. QD-based CQED structures have also been used to demonstrate QD-induced optical phase shifts [4,24–26], and a spin-dependent polarization rotation of up to $\pm 6\xb0$ [27]. None of the reported works on optical phase shifts, however, have gone beyond the semiclassical image to describe the global optical response of a CQED device, where the response of the cavity itself is superposed to the extracted resonance fluorescence.

Here we introduce a polarization tomography approach to investigate the polarization rotation of coherent light interacting with an electrically controlled QD-cavity device [28–32]. We analyze the polarization density matrix in the Poincaré sphere [33], which allows distinguishing between a general mixed polarization and a pure polarization state. We show that the superposition of emitted single photons ($H$-polarized) with reflected photons ($V$-polarized) leads to a large rotation of the output polarization, by 20° both in latitude and longitude in the Poincaré sphere, with a polarization purity above 84%. We implement a CQED model with which we analyze the complete information set provided by the polarization tomography. We demonstrate that the coherent part of the QD emission contributes to the polarization rotation, while its incoherent part contributes to degrading the polarization purity. Compared to previous works on phase-shift measurements [2–6,10,11,24–26], polarization tomography allows extracting qualitatively richer information, thus bringing an insight into the physics of the probed CQED devices, and suggesting paths for potential improvements. In particular, we show that the tomography technique is sensitive to the nature of the decoherence process.

## 2. SAMPLE AND EXPERIMENTAL SETUP

We study a deterministically coupled, electrically controlled QD-cavity system. The sample, grown by molecular beam epitaxy, consists of a $\lambda $-GaAs cavity embedding InGaAs QDs, surrounded by two distributed Bragg reflectors (GaAs and ${\mathrm{Al}}_{0.9}{\mathrm{Ga}}_{0.1}$ As, with 30 and 20 pairs for the bottom and top mirrors). *In-situ* lithography was used to deterministically fabricate a micropillar cavity around a single InGaAs QD [34]. The 3 μm diameter micropillar is connected by four ridges to a circular frame where metallic ohmic contacts are defined [Fig. 1(a)]. This cavity presents a small anisotropy leading to linearly polarized modes considered $H$ (horizontal) and $V$ (vertical) polarizations, with a 70 μeV splitting. The metallic contacts allow the electrical tuning of the QD transition with respect to these cavity modes [35].

The experimental setup is sketched in Fig. 1(b): the device is kept inside a helium exchange gas cryostat at $\sim 10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{K}$, and a tunable continuous-wave laser with 1 MHz linewidth excites the cavity from the top mirror. The reflected beam can be separated in two orthogonally polarized components in various polarization bases, using calibrated waveplates and a Wollaston polarizing prism. The input and output field intensities can then be measured with avalanche photodiodes (APDs), as displayed in Fig. 1(b), or analyzed via spectral or autocorrelation measurements (not shown).

As sketched in Fig. 1(c), the experiment we perform consists of exciting the device with a $V$-polarized input field (denoted ${\widehat{b}}_{V}^{(\text{in})}$), corresponding to one of the cavity eigenaxes. In the case of a fully detuned QD, this would imply a $V$-polarized reflected beam; yet the interaction with a QD optical transition leads to a more complex output. Indeed, the neutral QD can be described as a three-level system with a ground state $|g\u27e9$ and two excitonic states $|{e}_{V}\u27e9$ and $|{e}_{H}\u27e9$, that can respectively be excited with $V$- and $H$-polarized light. However, $|{e}_{H}\u27e9$ and $|{e}_{V}\u27e9$ are not the system eigenstates, as the QD displays an anisotropy along two axes $X$ and $Y$, differing from $H$ and $V$ by an angle $\theta $ (see Supplement 1). As a consequence, an initially excited state $|{e}_{V}\u27e9$ coherently oscillates between $|{e}_{H}\u27e9$ and $|{e}_{V}\u27e9$ [Fig. 1(c)]. By exciting the system with $V$ polarized light one thus populates the $|{e}_{V}\u27e9$ state, which rotates into $|{e}_{H}\u27e9$: this leads to resonance fluorescence emitted in both $V$ and $H$ polarizations. As sketched in Fig. 1(c), polarization tomography provides a global analysis of the $V$- and $H$-polarized output fields, denoted ${\widehat{b}}_{V}$ and ${\widehat{b}}_{H}$. In particular, if both the $H$- and $V$-polarized output fields are coherent with the incoming laser, they superpose with a well-defined phase, resulting in a pure polarization state.

## 3. EXPERIMENTAL RESULTS

We first analyze the device in the cavity eigenbasis $H/V$. Setting the polarization analyzer to separate the $H$ and $V$ polarizations, we measure the spectrum of the $H$-polarized output field with a spectrometer coupled to a CCD camera, filtering out the $V$-polarized light. Figure 2(a) displays the $H$-polarized signal as a function of the bias voltage applied to the device and of the emission wavelength, for a fixed laser wavelength ${\lambda}_{\text{laser}}=927.29\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ (in resonance with the bias-independent $V$-polarized cavity mode) and a fixed incoming power ${P}_{\text{in}}=200\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{pW}$ (low incoming power ensuring that the QD excitonic state is negligibly populated). When the bias is tuned to $-2.33\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{V}$, which tunes the QD transition wavelength ${\lambda}_{\mathrm{QD}}$ in resonance with ${\lambda}_{\text{laser}}$, the emitted intensity is strongly increased, as expected for a resonance fluorescence process. The logarithmic scale used in Fig. 2(a) also allows observing two residual features at biases above $-2\mathrm{V}$: one corresponding to residual light at the laser wavelength, and one corresponding to Raman-assisted QD emission.

The $H$-polarized signal arises from cross-polarized resonance fluorescence, i.e., from single photons emitted by the resonantly driven QD. This is evidenced by the measurement of the second-order autocorrelation of the $H$-polarized field, using a standard Hanbury-Brown–Twiss experiment with two single-photon avalanche diodes [36], with ${\lambda}_{\text{laser}}={\lambda}_{\mathrm{QD}}$ both fixed. Figure 2(b) displays the corresponding histogram of ${g}_{H}^{(2)}(\tau )$, with $\tau $ the delay between photon detection events. The raw value of ${g}_{H}^{(2)}(0)$ decreases to $7\pm 5\%$, which is compatible with a single-photon emission by the $|{e}_{H}\u27e9$ state if one takes into account the finite time response of the single-photon diodes. As is also displayed in Fig. 2(b), a good agreement is found between the experimental data and our numerical prediction, obtained with CQED parameters that will be discussed later on.

We then measure the intensities of the $V$- and $H$-polarized components of the output field, denoted ${I}_{V}$ and ${I}_{H}$. In the following we work at a fixed bias of $-2.33\mathrm{V}$, fixing ${\lambda}_{\mathrm{QD}}=927.29\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ to be in resonance with the $V$-polarized cavity mode, while scanning the laser wavelength. Figure 2(c) displays the evolution of the normalized intensities ${I}_{V}/{I}_{\text{in}}$ and ${I}_{H}/{I}_{\text{in}}$, with ${I}_{\text{in}}$ the input field intensity, as a function of ${\lambda}_{\text{laser}}$. A peak in ${I}_{H}/{I}_{\text{in}}$ is observed at ${\lambda}_{\text{laser}}={\lambda}_{\mathrm{QD}}$, indicating that up to 7% of the incident light has been reconverted by the QD into $H$-polarized resonance fluorescence. The $V$-polarized output intensity also presents a peak at ${\lambda}_{\mathrm{QD}}$, though superposed to the broader reflectivity dip of the $V$-polarized cavity mode. This behavior is understood by considering that the $V$-polarized light arises from the superposition of two fields: light directly reflected upon the top-mirror surface, and light extracted from the cavity via the top mirror [37]. The high QD-induced peak is thus related to the $V$-polarized resonance fluorescence, emitted by the $|{e}_{V}\u27e9$ exciton, which is large enough to strongly change the amount of intracavity light, and thus the resulting output field.

To perform a complete polarization tomography we now analyze our device in the other two bases. By adjusting the waveplates of the polarization analyzer introduced in Fig. 1(b), we measure the intensities ${I}_{D}$ and ${I}_{A}$ in the diagonal/anti-diagonal polarization basis, and ${I}_{R}$ and ${I}_{L}$ in the right-handed/left-handed circular polarization basis. For a given set of intensities ${I}_{\Vert /\perp}$, we define the corresponding Stokes component as ${s}_{\Vert \perp}=({I}_{\Vert}-{I}_{\perp})/({I}_{\Vert}+{I}_{\perp})$. This allows measuring the density matrix of the polarization state, and representing it in the Poincaré sphere as a vector with coordinates of ${s}_{HV}$, ${s}_{DA}$, and ${s}_{RL}$, ranging between $-1$ and $+1$. The purity of the polarization density matrix is given by the norm of the Poincaré vector, $\sqrt{{s}_{HV}^{2}+{s}_{DA}^{2}+{s}_{RL}^{2}}$. This norm is equal to 1 only for a pure polarization state, as would be given by a coherent superposition of $H$- and $V$-polarized electromagnetic fields.

In Fig. 3(a) the three Stokes components ${s}_{HV}$, ${s}_{DA}$, and ${s}_{RL}$, as well as the polarization purity $\sqrt{{s}_{HV}^{2}+{s}_{DA}^{2}+{s}_{RL}^{2}}$, are displayed as a function of ${\lambda}_{\text{laser}}$. The same set of data is also illustrated in Fig. 3(b), where the Stokes components are used as three-dimensional Cartesian coordinates on the Poincaré sphere, for the various values of ${\lambda}_{\text{laser}}$. Far from the QD resonance we obtain ${s}_{HV}\approx -1$ together with ${s}_{DA}\approx 0$ and ${s}_{LR}\approx 0$. This corresponds to a $V$-polarized reflected field, as also illustrated by the accumulation of experimental points at the $V$ polarization in the Poincaré sphere. The fact that ${s}_{DA}$ and ${s}_{RL}$ do not return exactly to zero on both sides of the resonance is mainly because the input polarization is not perfectly aligned with the cavity eigenaxis $V$. At resonance a maximal value of ${s}_{HV}=-0.77$ is obtained, and at the same time the Stokes components ${s}_{DA}$ and ${s}_{RL}$ become non-zero in the region of the QD resonance. As seen in the Poincaré sphere, this corresponds to a rotation between 0 and more than 20 deg, both in longitude and latitude, when ${\lambda}_{\text{laser}}$ is tuned around ${\lambda}_{\mathrm{QD}}=927.29\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$.

These results, translated in terms of angles in the polarization ellipse, correspond to a rotation of the orientation and ellipticity angles of more than 10 deg. They do not mean, however, that the output polarization is a pure polarization state: indeed a degradation of the polarization purity is observed around the QD resonance, though the purity remains above 84% [Fig. 3(a), top panel]. This is also illustrated in the Poincaré sphere of Fig. 3(b), where each experimental point is colored following a scale indicating the corresponding polarization purity, i.e., the distance between the data point and the center of the sphere.

## 4. THEORETICAL MODELING AND DISCUSSION

We now analyze our experimental results based on the model sketched in Fig. 1(c), using the input–output formalism [38,39] with a single input operator ${\widehat{b}}_{V}^{(\text{in})}$ and two output operators ${\widehat{b}}_{V}$ and ${\widehat{b}}_{H}$ (see Supplement 1). This allows computing the intensities ${I}_{\Vert}$ and ${I}_{\perp}$ in any polarization basis, which gives access to the three Stokes components and to the expected polarization purity. Fitting simultaneously the experimental curves in Figs. 2(c) and 3(a) allows determining all the parameters at play, more precisely than what was previously possible with the analysis of optical nonlinearities in QD-cavity devices [30,37]. Using $\hslash =1$ units, we first extract a QD-cavity light–matter coupling $g=18\pm 3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu eV}$, a total cavity damping rate $\kappa =106\pm 4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu eV}$ for both $H$ and $V$ modes, and a pure dephasing rate ${\gamma}^{*}=3.7\pm 0.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu eV}$ (see Section 5 for a discussion of the decoherence mechanisms). The spontaneous emission rate in the external (leaky) optical modes is also estimated at ${\gamma}_{\mathrm{sp}}=0.6\pm 0.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu eV}$, consistent with a standard emission time around 1 ns. The top-mirror output coupling ${\eta}_{\mathrm{top}}$, which is the ratio between the top-mirror damping rate ${\kappa}_{\mathrm{top}}$ and the total damping rate $\kappa $, is best fitted at ${\eta}_{\mathrm{top}}=55\pm 5\%$. The QD anisotropy is also found to be characterized by a fine structure splitting ${\mathrm{\Delta}}_{\mathrm{FSS}}=9\pm 2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu eV}$ between the excitonic eigenstates $|{e}_{X}\u27e9$ and $|{e}_{Y}\u27e9$, with the $X/Y$ axes rotated by an angle $\theta =17\pm 5\xb0$ with respect to the $H/V$ cavity axes. Both these parameters play a crucial role in the observation of polarization rotation: indeed, in the absence of QD anisotropy (${\mathrm{\Delta}}_{\mathrm{FSS}}=0$) or in the special case of aligned QD and cavity eigenaxes ($\theta =0$), no polarization rotation could be obtained (see Supplement 1). To perform a consistency check, this set of parameters has also been used to provide the numerical predictions in Fig. 2(b), without any additional fitting, leading to a good agreement with the measured autocorrelation function.

We note that ${s}_{DA}$ and ${s}_{RL}$ deviate from the simulated values, which is due to an experimental imperfection that could not be entirely corrected. Indeed, the experimental setup induces optical phase shifts, and thus additional rotations or ellipticities; this is accounted for by calibrating the reference frame of our polarization analysis, i.e., the axes of the Poincaré sphere in Fig. 3(b), at a fixed wavelength ${\lambda}_{\text{laser}}={\lambda}_{\mathrm{QD}}$. However, these phase shifts vary in a non-monotonic way when scanning the laser wavelength: this is mainly caused by the waveplates and by residual Fabry–Perot interference within the cryostat windows and within the non-polarizing beam splitter. It induces small wavelength-dependent rotations and ellipticity shifts, by up to a few degrees, when the laser wavelength is scanned through the QD resonance. A first consequence of this effect is that the input polarization deviates from the cavity eigenaxis when ${\lambda}_{\text{laser}}\ne {\lambda}_{\mathrm{QD}}$, explaining the non-zero values of ${s}_{DA}$ and ${s}_{RL}$ on both sides of the QD-induced response. A second consequence on our measurements is the presence of a residual mixing between the Stokes parameters (yet mostly between ${s}_{DA}$ and ${s}_{RL}$, as can be seen from the fact that ${s}_{HV}\approx -1$ at all wavelengths, when the QD is tuned off-resonance with the incoming laser). An interesting feature of the polarization tomography technique is that such imperfections negligibly affect the polarization purity, which constitutes a basis-independent quantity. These effects are thus treated as an increased uncertainty on ${s}_{DA}$ and ${s}_{RL}$; most care is taken to fit the polarization purity and the normalized intensities ${I}_{H}$ and ${I}_{V}$ (thus ${s}_{HV}$), while a less stringent agreement is required for the Stokes parameters ${s}_{DA}$ and ${s}_{RL}$ (see Supplement 1).

## 5. DECOHERENCE PROCESSES

If the QD resonance fluorescence field were entirely coherent with respect to the incoming laser, the output light would be completely described by only two expectation values $\u27e8{\widehat{b}}_{H}\u27e9$ and $\u27e8{\widehat{b}}_{V}\u27e9$, each having a well-defined amplitude and phase: this would lead to a coherent, classical superposition of $H$- and $V$-polarized fields, i.e., a pure polarization state. Yet partial incoherence has to be considered to interpret the reduced polarization purity: in the numerical fits of Fig. 3(a), this partial incoherence arises from the residual pure dephasing experienced by the exciton. Furthermore, polarization tomography appears sensitive to the nature of the pure dephasing process. A number of noise mechanisms can be simulated (see Supplement 1) and compared to the experimental data:

- •
*Electric/temperature noise*corresponds to perfectly correlated fluctuations for both excitonic transitions. Such a noise is typically induced by electric or temperature instabilities, which, by nature, modify both transition energies in the same way [40]. - •
*Cross-dephasing noise*corresponds to uncorrelated fluctuations for the two excitonic transitions, randomizing the relative phase between the excitonic states. It could be induced, in principle, by polarization-dependent phonon interactions [41], or anisotropic fluctuating fields induced by localized charges [42]. - •
*Magnetic noise*, finally, would be observed in the presence of a strong hyperfine interaction with nuclei [43], inducing a fluctuating Zeeman shift between the exciton transitions.

To better compare these decoherence processes, Fig. 4 displays zooms on the normalized intensities ${I}_{V}/{I}_{\text{in}}$ and ${I}_{H}/{I}_{\text{in}}$ from Fig. 2(c), and on the polarization purity $\sqrt{{s}_{HV}^{2}+{s}_{DA}^{2}+{s}_{RL}^{2}}$ from Fig. 3(a), with simulations for the three mechanisms. While the normalized intensities can be well fitted by considering any noise mechanism, the polarization purity cannot be accounted for in the case of cross dephasing or magnetic noise. Indeed, an important feature of these two processes is that they modify the phase between the excitonic eigenstates, i.e., they allow the incoherent conversion of the state $|{e}_{V}\u27e9$ into $|{e}_{H}\u27e9$: this contributes to the intensity ${I}_{H}$ but at the expense of a strong degradation in polarization purity. The electric/temperature noise mechanism, on the contrary, preserves the relative phase between the excitonic states and does not contribute to the intensity ${I}_{H}$; it can simply be interpreted as a spectral jitter of the QD energy. This is the only mechanism providing a satisfying fit for all the experimental curves.

The dephasing process considered in Figs. 2(c) and 3(a) is thus the electric/temperature noise mechanism, with a pure dephasing rate estimated at ${\gamma}^{*}=3.7\pm 0.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu eV}$. This mechanism is usually dominant in InAs/GaAs-based QDs [40], in which the energy of the exciton transitions is sensitive to small fluctuations at the single-charge level [44]. We note that in similar electrically controlled devices, but with different experimental configurations, much lower values of the pure dephasing rate could be obtained, down to less than 0.1 μeV, allowing the emission of single photons with near-unity indistinguishability [28]. In the present work we observe an additional dephasing due to fluctuations of the QD transition energy, which could be induced by the presence of a residual current at the operation bias of $-2.33\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{V}$, and/or by small temperature fluctuations in the helium-flow cryostat.

## 6. SEMICLASSICAL APPROXIMATION VERSUS EXACT CALCULATIONS

As we now show, pure dephasing is not the only source of fluctuations leading to a partially incoherent output: quantum fluctuations around the expectation values $\u27e8{\widehat{b}}_{V}\u27e9$ and $\u27e8{\widehat{b}}_{H}\u27e9$ are also obtained in the intermediate-power regime, i.e., when the excitation results in a population of the excitonic transition [25]. We theoretically study, in Fig. 5, the polarization tomography of an optimized device, with a pure dephasing rate ${\gamma}^{*}=0$, a top-mirror output coupling ${\eta}_{\mathrm{top}}=1$, and a QD-cavity eigenaxis angle $\theta =45\xb0$ (the other parameters being unchanged). The calculation is performed for various incoming powers ${P}_{\text{in}}=1$, 100, 500 pW, and 2 nW, corresponding to different values of the intracavity photon number ${n}_{0}$ indicated in Figs. 5(a)–5(d). In each panel, two numerical calculations are compared: the exact one considering that the intensities ${I}_{\Vert}$ and ${I}_{\perp}$, in an arbitrary basis, are given by $\u27e8{\widehat{b}}_{\Vert}^{\u2020}{\widehat{b}}_{\Vert}\u27e9$ and $\u27e8{\widehat{b}}_{\perp}^{\u2020}{\widehat{b}}_{\perp}\u27e9$, and the semiclassical one neglecting the incoherent part of the output fields, i.e., considering that ${I}_{\Vert}=\u27e8{\widehat{b}}_{\Vert}^{\u2020}\u27e9\u27e8{\widehat{b}}_{\Vert}\u27e9$ and ${I}_{\perp}=\u27e8{\widehat{b}}_{\perp}^{\u2020}\u27e9\u27e8{\widehat{b}}_{\perp}\u27e9$. At low incoming power, in Fig. 5(a), the semiclassical approximation gives the same result as the exact calculation. Maximal polarization rotations are obtained, and the QD transition even allows converting the $V$-polarized incoming light into a fully $H$-polarized output, as required for a photon–photon gate [9,12]. In this situation, the output polarization vectors stay at the surface of the sphere, as expected for fully coherent fields. At higher incoming powers, however, the incoherent resonance fluorescence emitted by the QD modifies this situation. As displayed in Figs. 5(b)–5(d), the exact calculation predicts a significant degradation of the polarization purity at increasing powers. At higher powers, the QD transition becomes saturated and the polarization purity is restored, as would be the case with an empty cavity. In Figs. 5(b)–5(d) we note the strong discrepancy between the exact calculation and the semiclassical approximation, which predicts only coherent outputs and, thus, pure polarization states.

The semiclassical image should thus, by no means, be considered a complete description of the optical response for a CQED device; it can be fruitfully used only in the absence of pure dephasing and for experiments under low excitation. It should be noted, however, that with near-optimal light–matter interfaces, this low-excitation regime is obtained only in the absence of two-photon components in the incoming beam. This requires either a pure single-photon excitation or, when using coherent-pulse excitation as is typically performed [9,12], ensuring an average number of photons per pulse below 0.3 [30].

In many experiments, the fact that the semiclassical approach is not exact has a limited influence: QD-induced phase shifts and polarization rotations are still predicted, though with a magnitude that is not well reproduced by the semiclassical theory. Polarization tomography, on the contrary, provides a qualitative test regarding the coherence of an output field: pure polarization states, at the surface of the Poincaré sphere, can be obtained only when the semiclassical approximation is valid. In future experiments, we thus plan to use polarization tomography as a benchmark experiment to characterize the presence or absence of undesired fluctuations in a cavity–QED system.

Finally, we note that this approach can be naturally extended to an artificial atom providing a spin degree of freedom, with two different polarization outputs for the two possible spin states [27]. In such a case, a degraded polarization purity will be obtained as soon as the spin fluctuates, and the observation of pure polarization will thus be a signature of perfect spin initialization. Moreover, it will become possible to measure directly the real-time evolution of an output polarization state in the Poincaré sphere and, hence, deduce the corresponding evolution of the spin density matrix in the Bloch sphere.

## 7. CONCLUSION

In summary, polarization tomography allows probing the optical response of a CQED device beyond the semiclassical approximation. We have shown that the superposition of $H$-polarized single photons with a $V$-polarized reflected field can be partially coherent, leading to a polarization rotation whose degree of purity can be decreased by dephasing processes. The amount of extracted information is increased compared to phase-shift measurements, bringing insight into the physics of the CQED device and the nature of the decoherence processes.

A major perspective of atom-induced shifts is the possibility to design 2-qubit quantum gates for single photons: a first incoming photon induces a $\pi $-phase shift for the second one [13]. However, the use of short single-photon pulses leads to phase noise that significantly decreases the overall efficiency of the photon–photon interaction [45,46]. As most realizations are based on polarization encoding, this directly translates into a degraded polarization purity, i.e., an imperfect output state for the rotated single photon [2,45]. Alternative protocols, based on the adiabatic switching between long-lived states [9,46,47] or quantum memories [11], have then been designed to allow high-fidelity quantum gates with a potentially pure polarization output. It will be particularly interesting to perform polarization tomography in such a framework, to evaluate the phase noise induced by a first incoming photon onto a second one.

We also note that, very recently, the reconstruction of two-photon density matrices has been performed to characterize the fidelity of photon–photon logic gates [10–12]. This was obtained through photon–photon correlation measurements in numerous configurations, and with long integration times, making it difficult to qualitatively analyze the features of the extracted matrix. Polarization tomography, by focusing on one-photon density matrices represented as Poincaré vectors, constitutes a complementary approach: the dependence of a density matrix is measured as a function of the experimental parameters, allowing its features to be qualitatively and quantitatively modeled.

The tomography approach thus provides a comprehensive measurement tool for polarization-encoded protocols, in the general framework of atom–photon, photon–photon. or spin–photon interactions. In the latter case, in particular, a single electron spin described in the Bloch sphere can be monitored by, or entangled with, a photon polarization qubit described in the Poincaré sphere. This would constitute the heart of future quantum networks where spin-based CQED devices interact by exchanging polarization-encoded photonic qubits [15].

## Funding

Agence Nationale de la Recherche (ANR) (ANR-12-BS10-0010, ANR-14-CE32-0012); H2020 European Research Council (ERC) Starting Grant QD-CQED (277885); H2020 Marie Skłodowska-Curie Actions (MSCA) Marie Curie Fellowship SQuaPh (702084); Labex NanoSaclay as part of the Investissements d'Avenir (ANR-10-LABX-0035).

See Supplement 1 for supporting content.

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