Distilling one, two and entangled pairs of photons from a quantum dot with cavity QED effects and spectral filtering

The system under analysis consists of a quantum dot that can host up to two excitons with opposite spins. The corresponding orthonormal basis of linear polarizations, horizontal (H) and vertical (V), reads $\{\left | \mathrm {G}\right \rangle ,\left | \mathrm {H}\right \rangle ,\left | \mathrm {V}\right \rangle ,\left | \mathrm {B}\right \rangle \}$, where G stands for the ground state, H and V for the single exciton states and B for the biexciton or doubly occupied state. The four-level scheme that they form is depicted in figure 2(a). The Hamiltonian of the system reads (ℏ = 1)

$$\begin{equation} H_{\mathrm{dot}}=\Big(\omega_{\mathrm{X}}+\frac{\delta}{2}\Big)\left| \mathrm{H}\right\rangle\left\langle \mathrm{H}\right|+\Big(\omega_{\mathrm{X}}-\frac{\delta}{2}\Big)\left| \mathrm{V}\right\rangle\left\langle \mathrm{V}\right|+\omega_{\mathrm{B}}\left| \mathrm{B}\right\rangle\left\langle \mathrm{B}\right|, \end{equation} \noindent \tag{ 1 }$$

where we allow the excitonic states to be split by a small energy δ, as is typically the case experimentally, through the so-called fine structure splitting [24]. The biexciton at ω_B = 2ω_X − χ is far detuned from twice the exciton thanks to the binding energy, χ, typically the largest parameter in the system. We include the dot losses, at a rate γ, and an incoherent continuous excitation (off-resonant driving of the wetting layer), at a rate P, in both polarizations x = H, V, in a master equation

$$\begin{equation} \partial_t \rho= \mathrm{i}[\rho, H_{\mathrm{dot}}]+\sum_{x=\mathrm{H,V}}\Big[\frac{\gamma}{2}\Big( \mathcal{L}_{\left| \mathrm{G}\right\rangle\left\langle x\right|}+\mathcal{L}_{\left| x\right\rangle\left\langle \mathrm{B}\right|}\Big)+\frac{P}{2}\Big(\mathcal{L}_{\left| x\right\rangle\left\langle \mathrm{G}\right|}+\mathcal{L}_{\left| \mathrm{B}\right\rangle\left\langle x\right|}\Big)\Big](\rho), \end{equation} \tag{ 2 }$$

where  $\mathcal {L}_c(\rho ){=}2c\rho c ^{\dagger }-c^{\dagger }c\rho -\rho c^{\dagger }c$ is in the Lindblad form. In this work, we are interested in the steady state of the system resulting from the interplay between the incoherent continuous pump and decay of the quantum dot levels.

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We assume in what follows an experimentally relevant situation, χ = 100γ, δ = 20γ, and study the steady state under P = γ, in which case all levels are equally populated (the populations read ρ_G = γ²/(P + γ)², ρ_H = ρ_V = Pγ/(P + γ)² and ρ_B = P²/(P + γ)²). The photoluminescence spectra of the system, S(ω), are shown in figure 2(b) for the H- and V-polarized emission with orange and green lines, respectively. The four peaks are well separated thanks to the binding energy and fine structure splitting, corresponding to the four transitions depicted in panel (a) with the same colour code

$$\begin{equation} \omega_{\mathrm{BH}}=\omega_{\mathrm{B}}-\omega_{\mathrm{H}}=\omega_{\mathrm{X}}-\chi-\delta/2=-110\gamma, \end{equation} \tag{ 3 }$$

$$\begin{equation} \omega_{\mathrm{H}}=\omega_{\mathrm{X}}+\delta/2=10\gamma, \end{equation} \tag{ 4 }$$

$$\begin{equation} \omega_{\mathrm{BV}}=\omega_{\mathrm{B}}-\omega_{\mathrm{V}}=\omega_{\mathrm{X}}-\chi+\delta/2=-90\gamma, \end{equation} \tag{ 5 }$$

$$\begin{equation} \omega_{\mathrm{V}}=\omega_{\mathrm{X}}-\delta/2=-10\gamma \end{equation} \tag{ 6 }$$

with ω_X → 0 as the reference and full-width at half-maximum γ_BH = γ_BV = 3γ + P, γ_H = γ_V = 3P + γ. We concentrate on the H-mode emission because it has the largest peak separation and allows for the best filtering but all results apply similarly to the V polarization. The emission structure at the single-photon level is very simple: two Lorentzian peaks are observed corresponding to the upper and lower transitions

$$\begin{equation} S [\sigma_{\mathrm{H}}] (\omega)=\frac{1}{\pi}\Big[ \rho_{\mathrm{B}}\frac{\gamma_{\mathrm{BH}}/2}{(\gamma_{\mathrm{BH}}/2)^2+(\omega-\omega_{\mathrm{BH}})^2}+\rho_{\mathrm{H}}\frac{\gamma_{\mathrm{H}}/2}{(\gamma_{\mathrm{H}}/2)^2+(\omega-\omega_{\mathrm{H}})^2}\Big]\,. \end{equation} \noindent \tag{ 7 }$$

The second-order coherence function of the H-emission in the steady state (set as the starting time t = 0) with a positive delay τ, reads

$$\begin{equation} g^{(2)}[\sigma_{\mathrm{H}}](\tau)=\langle \sigma_{\mathrm{H}}^+(0)\sigma_{\mathrm{H}}^+ (\tau)\sigma_{\mathrm{H}}(\tau)\sigma_{\mathrm{H}}(0)\rangle/\langle \sigma_{\mathrm{H}}^+\sigma_{\mathrm{H}}\rangle^2, \end{equation} \tag{ 8 }$$

where the H-photon destruction operator is defined as

$$\begin{equation} \sigma_{\mathrm{H}}=s_1+s_2,\quad\mathrm{with}\quad s_1=\left| \mathrm{H}\right\rangle\left\langle \mathrm{B}\right| \quad\mathrm{and}\quad s_2=\left| \mathrm{G}\right\rangle\left\langle \mathrm{H}\right|\,. \end{equation} \tag{ 9 }$$

In equations (7) and (8), we have specified the channel of emission in square brackets, since this will be an important attribute in the rest of the paper. Surprisingly, the quantum dot described with the spin degree of freedom always exhibits uncorrelated statistics in the linear polarization

$$\begin{equation} g^{(2)}[\sigma_{\mathrm{H}}](\tau)=1\,. \end{equation} \tag{ 10 }$$

One recovers the expected antibunching of a two-level system [37] by turning to the intrinsic two-level systems composing the quantum dot, namely, the spin-up and spin-down excitons: g⁽²⁾[σ_↑](0) = g⁽²⁾[σ_↓](0) = 0. The Pauli exclusion principle that holds for the spin σ_↑↓ breaks in the linear polarization, i.e. while σ²_↑↓ = 0, one has $\sigma _{\mathrm {H}}^2{=}\left | \mathrm {G}\right \rangle \left \langle \mathrm {B}\right |{\neq }0$. We can find a simple explanation for this if we write the total correlations in terms of the four contributions (different from zero)

$$\begin{equation} \langle \sigma_{\mathrm{H}}^+(0)\sigma_{\mathrm{H}}^+(\tau)\sigma_{\mathrm{H}}(\tau)\sigma_{\mathrm{H}}(0)\rangle=\sum_{i,j=1,2}\langle s_i^+(0) s_j^+(\tau) s_j(\tau) s_i(0)\rangle, \end{equation} \tag{ 11 }$$

which are given by (τ ⩾ 0)

$$\begin{eqnarray} &g^{(2)}[s_1;s_1] (\tau)=(1-\mathrm{e}^{-(\gamma+P)\tau})\left(1+\frac{\gamma}{P}\mathrm{e}^{-(\gamma+P)\tau}\right),\nonumber \\ &g^{(2)}[s_2;s_2] (\tau)=(1-\mathrm{e}^{-(\gamma+P)\tau})\left(1+\frac{P}{\gamma}\mathrm{e}^{-(\gamma+P)\tau}\right),\nonumber \\ &g^{(2)}[s_1;s_2] (\tau)=(1-\mathrm{e}^{-(\gamma+P)\tau})+\mathrm{e}^{-(\gamma+P)\tau}\left(2+\frac{\gamma}{P}+\frac{P}{\gamma}\right),\nonumber \\ &g^{(2)}[s_2;s_1] (\tau)=(1-\mathrm{e}^{-(\gamma+P)\tau})^2 \end{eqnarray} \noindent \tag{ 12 }$$

in their normalized form, g⁽²⁾[s_i;s_j](τ) = 〈s⁺_i(0)s⁺_j(τ)s_j(τ)s_i(0)〉/(〈s⁺_is_i〉〈s⁺_js_j〉). As shown in figures 3(c) and (d) with pale grey lines, all these functions are antibunched except for g⁽²⁾[s₁;s₂](τ), which corresponds to the natural order in the H-cascaded emission of two photons and is, consequently, bunched. It compensates fully for the other three terms, leading to total correlations of 1 at all τ.

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Figure 2 provides a clear picture of the physical grounds on which such a system can be used as a quantum emitter, but it lacks even a qualitative picture of how quantum correlations are distributed. Figure 2(b) merely shows where the system emits light but nothing on how correlated is this emission. All these crucial features are revealed in the two-photon spectrum g⁽²⁾_Γ(ω₁;ω₂,τ) [38]. This is the extension of the Glauber second-order coherence function—which quantifies the correlations between photons in their arrival times—to frequency. By specifying both the energy (ω₁ and ω₂) and time of arrival (t = 0 and t = τ ⩾ 0) of the photons, one provides an essentially complete description of the system. Γ denotes the linewidth of the frequency window of the filter over which this joint characterization is obtained. The corresponding time resolution is given by its inverse, 1/Γ, in accordance with the Heisenberg uncertainty principle. It is a necessary variable without which nonsensical or trivial results are obtained.

The two-photon spectrum unravels a large class of processes hidden in single-photon spectroscopy and can be expected to become a standard tool to characterize and engineer quantum sources. The computation of such a quantity¹ has remained a challenging task for theorists since the mid-eighties [39–41], until a recent workaround [36] has been found which allows an exact numerical computation. The method consists in coupling very weakly the system to two sensors or quantized modes (such that the dot dynamics remains unaltered), which model the filtering and detection processes. The intensity–intensity cross correlations of the emission from such sensors, with operators ς₁ and ς₂, provide the two-photon spectrum in a simple form,

$$\begin{equation} g_{\Gamma}^{(2)}(\omega_1;\omega_2,\tau)=\frac{\langle n_1(0) n_2(\tau)\rangle}{\langle n_1(0)\rangle\langle n_2(0)\rangle}, \end{equation} \tag{ 13 }$$

where n_i = ς⁺_iς_i for i = 1, 2. The sensors natural frequencies, ω_i, decay rates, Γ, and the delay in the detection, τ, can be varied to scan the relevant frequency and time ranges with the appropriate resolution. This method will be applied here for the first time to the case of biexciton emission and used to understand, characterize and enhance various processes useful for its quantum emission. We start by solving the coupling of two sensors with the H-mode only, obtaining g⁽²⁾_Γ[σ_H](ω₁;ω₂,τ). A detailed discussion on even more fundamental emitters is given in  [38].

Experimentally, the two-photon spectrum corresponds to the usual Hanbury Brown–Twiss setup to measure second-order correlations through photon counting, with filters or monochromators being placed in front of the detectors to select two, in general different, frequency windows. The technique has been amply used in the laboratory [15, 16, 22, 42–49], but lacking hitherto a general theoretical description, the global picture provided here has not yet been achieved experimentally. Note finally that when considering correlations between equal frequencies, ω = ω₁ = ω₂, the result is equivalent to placing a single filter before measuring the correlations of the outcoming photon stream [36] or embedding the quantum dot in a cavity and measuring the correlations from the emission of a weakly coupled cavity mode with frequency ω.

Figure 3(a) shows the rich landscape provided by the two-photon spectrum g⁽²⁾_Γ[σ_H](ω₁;ω₂), in contrast with the one-photon spectrum S[σ_H](ω) and the colour-blind second-order correlations g⁽²⁾[σ_H], equation (10). It is shown at zero delay (τ = 0), where results are symmetric under exchange of ω₁ and ω₂. An analytical expression is found in this case but it is too lengthy to be reproduced here. The sensor linewidths are chosen to filter the full peaks (Γ = 5γ) in order to observe well-defined regions of enhancement and suppression of the correlations: sub-Poissonian values (<1) are shown in blue, Poissonian (=1) in white and super-Poissonian (>1) in red. This figure is the backbone of this paper. We now discuss in turn these different regions where the quantum dot operates as a quantum source with different properties.

When the filters are tuned to the same frequency (diagonal black line in the (ω₁;ω₂) space in figure 3(a)), there is a systematic enhancement of the bunching as compared to the surrounding regions due to the two possibilities of detecting identical photons [36, 38]. Despite this feature, which is independent of the system dynamics, when both frequencies coincide with one of the dot transitions, (ω_H;ω_H) or (ω_BH;ω_BH), there is a dip in the correlations. This is more clearly shown in the cut at equal frequencies, the black line in figure 3(b). The blue butterfly shape that is observed in the two-photon spectrum locally around each of the dot transitions is characteristic of an isolated two-level system [38]. This zero delay information is complemented by the antibunched τ dynamics, shown in figure 3(c). The two dot resonances, upper and lower, coincide in this case due to the symmetric conditions P = γ but they are typically different (the upper level being less antibunched at low pump). Filtering and detection makes it impossible to have a perfect antibunching, getting closest to the ideal correlations g⁽²⁾_Γ[σ_H](ω_H;ω_H,τ) ≈ g⁽²⁾[s₁;s₁](τ) from equations (12), at around Γ ≈ χ/2. At this point, the peaks are maximally filtered with still negligible overlapping of the filters. It is possible to derive a useful expression for the filtered correlations at τ = 0, with Γ ⩽ χ/2, in the limit

$$\begin{equation} \lim_{\chi\rightarrow \infty} g_\Gamma^{(2)}[\sigma_{\mathrm{H}}] (\omega_{\mathrm{H}};\omega_{\mathrm{H}}) =\frac{2(P+\gamma)^2(3P+\gamma+\Gamma)(2\gamma+\Gamma)}{\gamma(P+\gamma+\Gamma)(2P+2\gamma+\Gamma)(3P+\gamma+3\Gamma)} \end{equation}\noindent \tag{ 14 }$$

typically relevant in experiments.

All this shows that one can recover or optimize the quantum features of a single photon source (antibunching) in a system whose total emission is uncorrelated, by frequency filtering photons from individual transitions. Consequently, the system should exhibit two-photon blockade when probed by a resonant laser at frequency ω_L in resonance with the lower transition, ω_L = ω_H. The antibunched emission of each of the four filtered peaks of the spectrum has been observed experimentally [15].

When the filtering frequencies match both the upper and lower quantum-dot transitions, i.e. (ω_BH;ω_H), the correlations are close to one at zero delay, like for the total emission g⁽²⁾[σ_H] (which is exactly one). However, although the latter is uncorrelated, since it remains equal to one at all τ, the filtered cascade emission is not uncorrelated, since it is close to unity only at zero delay and precisely because of strong correlations that are, however, of an opposite nature at positive and negative delays, i.e. showing enhancement for τ > 0 when photons are detected in the natural order that they are emitted, and suppression for τ < 0 when the order is the opposite. This is depicted in figure 3(d) where the solid line corresponds to (ω_BH;ω_H) and the dotted line to exchanging the filters, (ω_H;ω_BH). As this also corresponds to detecting the photons in the opposite time order, the two curves are exact mirror image of each other.

The identification of the upper and lower transition photons with frequency-blind operators (g⁽²⁾[s₁;s₂](τ) in equations (12)) provides crossed correlations different from our exact and general frequency resolved functions, especially at τ = 0, as shown in figure 3(d), where there is a discontinuity for the approximated functions. The frequency resolved functions have the typical smooth cascade shape that has been observed experimentally [15, 45]. The dynamics at large τ converges to the approximated functions only for Γ ≈ χ/2.

For simultaneous two-photon emission, the strongest feature lies on the antidiagonal (red line) in figure 3(a), which is also shown as the solid red line in figure 3(b). The strong bunching observed here, when both frequencies are far from the system resonances ω_BH and ω_H, corresponds to a two-photon deexcitation directly from the biexciton to the ground state without passing by an intermediate real state². Such a two-photon emission, from a Hamiltonian (equation (1)) that does not have a term to describe it, is made possible via an effective virtual state, supported by the interaction with the output fields (or detectors) and revealed by the spectral filtering. As the intermediate virtual state has no fixed energy and only the total energy ω₁ + ω₂ = ω_B needs to be conserved, the simultaneous two-photon emission is observed on the entire antidiagonal (except, again, when touching a resonance, in which case the cascade through real states takes over). We call such processes 'leapfrog' as they jump over the intermediate excitonic state [38]. The largest bunching is found at the central point, ω₁ = ω₂ = ω_B/2 = −χ/2, and at the far-ends ω₁ ≪ ω_BH and ω₁ ≫ ω_H. Among them, the optimal point is that where also the intensity of the two-photon emission is strong. The frequency resolved Mandel Q parameter takes into account both correlations and the strength of the filtered signal [50]:

$$\begin{equation} Q_\Gamma(\omega_1;\omega_2)=\sqrt{S_\Gamma(\omega_1)S_\Gamma(\omega_2)}\Big[g^{(2)}_\Gamma(\omega_1;\omega_2)-1\Big], \end{equation} \tag{ 15 }$$

(where also for the single-photon spectra, the detection linewidth, Γ, is taken into account [51]). As expected, Q_Γ[σ_H](ω₁;ω₂) becomes negligible at very large frequencies, far from the resonances of the system, and reaches its maximum at the two-photon resonance, (ω_B/2;ω_B/2) (not shown). The latter configuration is therefore the best candidate for the simultaneous and, additionally, indistinguishable, emission of two photons. The bunching is shown in figure 3(e). The small and fast oscillations are due to the effect of one-photon dynamics with the real states but are unimportant for our discussion and would be difficult to resolve experimentally. While the bunching in such a configuration has not yet been observed experimentally, recently, Ota et al [24] successfully filtered the two-photon emission from the biexciton with a cavity mode, which corroborates the above discussion.

In view of the preceding results, we now consider the possibility to further enhance the two-photon emission by filtering the cavity photons from the central peak in figure 2(c). That is, we study g⁽²⁾_Γ[a](ω₁;ω₂) for an H-polarized cavity with κ = 5γ and ω_a = ω_B/2. In this way, firstly, the cavity acts as a filter, extracting the leapfrog emission where it is most correlated, but also enhances specifically the two-photon emission and, secondly, the filtering of the cavity emission selects only those photons that truly come in pairs. Such a chain is similar to a 'distillation' process where the quantum emission is successively refined.

The results are plotted with red lines in figure 4, for the case κ = 5g pinpointed by circles on the blue lines. The filtered cavity emission is indeed generally more strongly correlated at the two-photon resonance than the unfiltered total cavity emission, plotted in blue: g⁽²⁾_Γ[a](ω_B/2;ω_B/2) ⩾ g⁽²⁾[a]. This is so for all Γ > κ for the cavity in weak coupling, where the distillation always enhances the correlations. In strong coupling, the filter must strongly overlap with the peak (Γ ≫ κ). This is because the side peaks are prominent in weak coupling (κ_2P < κ_1P), and the filtering efficiently suppresses their detrimental effect, whereas in strong coupling, two-photon correlations are already close to maximum, thanks to the dominant central peak as seen in figure 2(c), and it is therefore important for the filter to strongly overlap with it. In all cases, at large enough Γ, the full cavity correlations are recovered as expected: lim_Γ→∞g⁽²⁾_Γ[a](ω₁;ω₂) = g⁽²⁾[a]. In figure 4(a), this means that the red lines converge to the value projected by the circle on the blue lines at κ = 5γ. Here, again, filtering enhances correlations but reduces the number of counts, as shown in figure 4(b).

In figure 4(c), we do the same analysis as in figure 3(b) where there was no distillation. We address the same cases but now as a function of frequency, fixing the cavity decay rate κ = 5γ and the filtering linewidth at Γ = 10γ. In blue, we consider the cavity QED case in weak and strong coupling, without filtering. Since the weak-coupling limit is identical to the single filter case, note that the blue dotted line in figure 4(c) is identical to the black solid line in figure 3(b). Off-resonance, the cavity acts as a simple filter due to the reduction of the effective coupling. The stronger coupling to the cavity has an effect only when involving the real states, where it spoils the correlations, less bunched at the two-photon resonance ω_a = ω_B/2 and less antibunched at the one-photon resonances, ω_a = ω_BH or ω_a = ω_H. This shows again that useful quantum correlations are obtained in a system where quantum processes are Purcell enhanced and quickly transferred outside, rather than stored and Rabi-cycled over within the cavity. The same is true for the red line, further filtering the output. Finally, comparing solid lines together, we see again that there is little if anything to be gained by filtering in strong coupling, whereas in weak coupling, the enhancement is considerable. As a summary, the filtering of the weakly coupled cavity (red dotted) provides the strongest correlations (at the cost of the available intensity), corresponding to distilling the photon pairs out of the original dot spectrum without any additional enhancement.

In figure 5, we show the full two-photon spectra for a quantum dot in a cavity, in both (a) strong and (b) weak coupling. The antibunching regions on the diagonal and the bunching along the antidiagonal are qualitatively similar to the filtered dot emission, but antibunching is milder and less extended in contrast to bunching that greatly dominates the two-photon spectrum as a direct consequence of the cavity. At the central point, bunching is now, respectively, weaker (stronger) than the filtered dot emission, due to the saturated (efficient) distillation in strong (weak) coupling. Another striking feature added by the cavity is the appearance of an additional pattern of horizontal and vertical lines at ω_a. While diagonal and antidiagonal features correspond to virtual processes horizontal and vertical stem from real processes that pin the correlations at their own frequency. Therefore, the new features are a further illustration that the two-photon emission becomes a real resonance of the cavity–dot system, in contrast with figure 3(a) where it was virtual. The effect is more pronounced in the strong-coupling regime since this is the case where the new state is better defined. Another qualitative difference between figures 5 and 3(a) is in the regions surrounding the cascade configuration, (ω_BH;ω_H), which has changed shape to become two antibunching blue spots. This is due to the fact that the single-photon cascade is much less likely to happen through the cavity mode than the direct deexcitation of the dot, as κ_1P = 0.05γ ≪ γ. Even if the first photon from the biexciton emission decays through the cavity, the second will most likely not.

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The properties of the output photons can be obtained from the two-photon state density matrix, θ(τ), reconstructed in the basis $\{\left | \mathrm {H1,H2}\right \rangle ,\left | \mathrm {H1,V4}\right \rangle ,\left | \mathrm {V3,H2}\right \rangle ,\left | \mathrm {V3,V4}\right \rangle \}$, denoting by $\left | xi\right \rangle$ with x = H or V and i = 1,2,3,4, the state $\left | x(\omega _i)\right \rangle$. The frequencies ω_i are, in general, different. The second photon is detected with a delay τ with respect to the first one (detected in the steady state). Let us express this matrix in terms of frequency resolved correlators, as is typically done in the literature [53, 55]. However, in contrast to previous approaches, we do not identify the photons with the transition from which they may come (using the dot operators $\left | \mathrm {G}\right \rangle \left \langle x\right |$, $\left | x\right \rangle \left \langle \mathrm {B}\right |$ with, again, x standing for either H or V) but with their measurable properties, that is, polarization, frequency and time of detection (for a given filter window). This is a more accurate description of the experimental situation where a given photon can come from any dot transition and any transition can produce photons at any frequency and time with some probability. We describe the experiments by considering four different filters, that is, including all degrees of freedom of the emitted photons in the description. Each detected filtered photon corresponds to the application of the filter operator ς_j with j = H1, H2, V3, V4 corresponding to its coupling to the H or V dot transitions with ω_i frequencies. Then, the two-photon matrix θ'_Γ(τ) (the prime refers to the lack of normalization) corresponding to a tomographic measurement is theoretically modelled as

$$\begin{eqnarray}{ \fl \theta_\Gamma'(\tau)= \left( \begin{array}{@{}cccc@{}} \langle n_{\mathrm{H1}}(0)n_{\mathrm{H2}}(\tau)\rangle & \langle n_{\mathrm{H1}}(0) [\varsigma_{\mathrm{H2}}^+\varsigma_{\mathrm{V4}}](\tau)\rangle & \langle [\varsigma_{\mathrm{H1}}^+\varsigma_{\mathrm{V3}}](0)n_{\mathrm{H2}}(\tau)\rangle & \langle [\varsigma_{\mathrm{H1}}^+\varsigma_{\mathrm{V3}}](0) [\varsigma_{\mathrm{H2}}^+\varsigma_{\mathrm{V4}}] (\tau)\rangle \\ \mathrm{h.c.} & \langle n_{\mathrm{H1}}(0)n_{\mathrm{V4}}(\tau)\rangle & \langle [\varsigma_{\mathrm{H1}}^+\varsigma_{\mathrm{V3}}](0) [\varsigma_{\mathrm{V4}}^+\varsigma_{\mathrm{H2}}] (\tau)\rangle &\langle [\varsigma_{\mathrm{H1}}^+\varsigma_{\mathrm{V3}}](0) n_{\mathrm{V4}} (\tau)\rangle \\ \mathrm{h.c.} & \mathrm{h.c.} & \langle n_{\mathrm{V3}}(0)n_{\mathrm{H2}}(\tau)\rangle & \langle n_{\mathrm{V3}}(0) [\varsigma_{\mathrm{H2}}^+\varsigma_{\mathrm{V4}}] (\tau)\rangle \\ \mathrm{h.c.} & \mathrm{h.c.} & \mathrm{h.c.} &\langle n_{\mathrm{V3}}(0)n_{\mathrm{V4}}(\tau)\rangle \end{array} \right)}\nonumber\\ \end{eqnarray} \tag{ 20 }$$

where n_i = ς⁺_iς_i (we have dropped the frequency dependence in the notation, writing θ_Γ'(τ) instead of θ_Γ'(ω₁,ω₂;ω₃,ω₄,τ)). Since a weakly coupled cavity mode behaves as a filter, this tomographic procedure is equivalent to considering the four dot transitions coupled to four different cavity modes with the corresponding polarizations, central frequencies and decay rates [35, 61]. Unlike in other works where for various reasons and particular cases, some of the elements in θ_Γ'(τ) are set to zero or considered equal, here we keep the full matrix with no a priori assumptions since, in general, it may not reduce to a simpler form due to the incoherent pumping, pure dephasing, frequency filtering and fine-structure splitting.

There are essentially two ways to quantify the degree of entanglement from the density matrix θ_Γ'(τ). The most straightforward is to consider the τ-dependent matrix directly, which merely requires normalization at each time τ, yielding θ_Γ(τ) = θ_Γ'(τ)/Tr[θ_Γ'(τ)]. The physical interpretation is that of photon pairs emitted with a delay τ, that is, within the time resolution 1/Γ of the filter or cavity [34, 61]. In particular, the zero-delay matrix, θ_Γ(0), represents the emission of two simultaneous photons [59, 60]. The second approach is closer to the experimental measurement which averages over time. In this case, one considers the integrated quantity $\Theta _\Gamma (\tau _{\mathrm {max}}){=}\big (\int _0^{\tau _{\mathrm {max}}} \theta _\Gamma '(\tau )\,\mathrm {d}\tau \big )/\mathcal {N}$ that averages over all possible emitted pairs from the system [15, 53]. It is also normalized (by $\mathcal {N}$), but after integration, so that the two approaches are not directly related to each other and present alternative aspects of the problem, discussed in detail in the following. Without the cutoff delay τ_max, the integral diverges due to the continuous pumping.

The degree of entanglement of any bipartite system represented by a 4 × 4 density matrix θ can be quantified by the concurrence, C, which ranges from 0 (separable states) to 1 (maximally entangled states)³  [66]. High values of the concurrence require high degrees of purity in the system [67], it being, for instance, impossible to extract any entanglement from a maximally mixed state (in which case all the four basis states occur with the same probability). The filtered density matrices, Θ_Γ(τ_max) and θ_Γ(τ), provide each their own concurrence that we will denote by C^int_Γ(τ_max) and C_Γ(τ), respectively.

We begin by considering the standard cascade configuration by detecting photons at the dot resonances, i.e. ω₁ = ω_BH, ω₂ = ω_H, ω₃ = ω_BV, ω₄ = ω_V, as sketched in the inset of figure 6(a). The upper density plot shows C^int_Γ(τ_max) and the density plot below shows the time-resolved concurrence C_Γ(τ), as a function of τ_max or τ and δ, for Γ = 2γ. The two concurrences, C^int_Γ(τ_max) and C_Γ(τ), are qualitatively different except at τ = τ_max = 0 where they are equal by definition. They also have in common that the maximum concurrence is not achieved at zero delay (it is most visible as the darker red area around (γτ_max,δ/γ) ≈ (0.4,0)). This is because this filtering scheme relies on the real-states deexcitation of the dot levels, and thus exhibits the typical delay from the cascade-type dynamics of correlations (see figure 3(d)). The major departure between the two is that the decay of C^int_Γ(τ_max) is strongly dependent on δ, while C_Γ(τ) is not. With no splitting, at δ = 0, the ideal symmetrical four-level structure efficiently produces the entangled state $\left | \psi \right \rangle$ and the concurrence is maximum both for the integral and the time-resolved forms. The decay time of C_Γ(τ) is the simplest to understand as, when filtering full peaks, it is merely related to the reloading time of the biexciton, of the order of the inverse pumping rate, ∼2/P [61]. The asymmetry due to a non-zero splitting in the four-level system causes an unbalanced dynamics of deexcitation via the H and V polarizations. The entanglement in the form $\left | \psi \right \rangle$ is downgraded to $\left | \psi '\right \rangle$. The fact that the concurrence C_Γ(τ) is not affected shows that it accounts for the degree of entanglement in both polarization and frequency, which is oblivious to the 'which path' information that the last variable provides. On the other hand, C^int_Γ(τ_max) is suppressed by the splitting as fast as τ_max > 2π/δ (this boundary is superimposed on the density plot). That is, as soon as the integration interval is large enough to resolve it. The integrated C^int_Γ(τ_max), therefore, accounts for the degree of entanglement in polarization only, and is destroyed by the 'which path' information provided by the different frequencies. The exact mechanism at work to erase the entanglement is shown in the lower row of figure 6, which displays the upper right matrix element [θ_Γ(τ)]_1,4 of the density matrix, which is most responsible for purporting entanglement. The modulus is shown on the left (with blue meaning 0 and red meaning 0.5) and the phase on the right (in black and white). The time-resolved concurrence C_Γ(τ) is very similar to the modulus of [θ_Γ(τ)]_1,4 which justifies the approximation often made in the literature C_Γ(τ) = 2(|[θ_Γ(τ)]_1,4| − [θ_Γ(τ)]_2,2) with [θ_Γ(τ)]_2,2 small [53]. Although each photon pair is entangled, it is so in the state $\left | \psi '\right \rangle$ with a phase, ϕ' = −π + δτ, that accumulates with τ at a rate δ [34], but this does not matter as far as instantaneous entanglement is concerned, and this is why the splitting does not affect C_Γ(τ). On the other hand, when integrating over time, the varying phase, which completes a 2π-cycle at intervals 2π/δ, randomizes the quantum superposition and results in a classical mixture. This is why the splitting does completely destroy C^int_Γ(τ_max) for τ_max > 2π/δ. And this is how the system restores the 'which path' information: given enough time, if the splitting is large enough, the photons lose their quantum coherence due to the averaging out of the relative phase between them because of their distinguishable frequency.

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Another possible configuration proposed in the literature [15, 34, 54] is sketched in the inset of figure 6(b). Photons are detected at the frequency that lies between the two polarizations, ω₁ = ω₃ = ω_BX, ω₂ = ω₄ = ω_X. For small splittings, the entanglement production scheme still relies on the cascade and real deexcitation of the dot levels. Therefore, the behaviour of C^int_Γ(τ_max) is similar to (a) (decaying with δ) but slightly improved by the fact that the contributions to the density matrix are more balanced by filtering in between the levels. For splittings large enough to allow the formation of leapfrog emission in both paths, δ ≫ Γ, there is a striking change of trend and C^int_Γ(τ_max) remains finite at longer delays τ_max when increasing δ. This is a clear sign that the entanglement relies on a different type of emission, namely simultaneous leapfrog photon pairs, rather than a cascade through real states. Accordingly, the intensity is reduced as compared to that obtained at smaller δ or to the non-degenerate cascade in (a), but the bunching is stronger, as evidenced by the two-photon spectrum, and results in a larger degree of entanglement than at δ = 0. A similar result is obtained qualitatively with the time-resolved concurrence, a strong resurgence of entanglement with δ, indicating again very high correlations in this configuration. Note finally that the phase becomes much more constant with increasing δ, resulting in the persistence of the correlation in the time-integrated concurrence. This is because the two-photon emission is through the leapfrog processes. Since the latter, by definition, involve intermediate virtual states that are degenerate in frequency, this is a built-in mechanism to suppress the splitting and not suffer from the 'which path' information as when passing through the real states.

Finally, a configuration proposed more recently in the literature [35] is the two-photon resonance, with four equal frequencies, ω₁ = ω₂ = ω₃ = ω₄ = ω_B/2, as sketched in the inset of figure 6(c). This provides a two-photon source in both polarizations that can be enhanced via two cavity modes with orthogonal polarizations [32]. Remarkably, in this case, the splitting has almost no effect on the degree of entanglement, which is maximum. Here, the leapfrog mechanism plays at its full extent: the virtual states, on top of being degenerate and thus immune to the splitting, remain always far, and are therefore protected, from the real states. This results in the exactly constant phase (black panel on the lower right end of figure 6). As a result, both C^int_Γ(τ_max) and C_Γ(τ) remain large. The only drawback of this mechanism is, being virtual-processes mediated, a comparatively weaker intensity.

We conclude with a more detailed analysis of what appears to be the most suitable scheme to create robust entanglement, the leapfrog photon-pair emission. The target state is always $\left | \psi \right \rangle$, equation (18), due to the degeneracy in the filtered paths. The concurrence is shown as a function of the first photon frequency, ω₁, in figure 7 (the second photon has the energy ω₂ = ω_B − ω₁ to conserve the total biexciton energy). The time-integrated (resp. instantaneous) concurrence C^int_Γ(1/γ) (resp. C_Γ(0)) is shown in red (resp. blue) lines, both for a large splitting δ = 20γ in strong tones and for δ = 0 in softer tones. In the first panel, (a), the frequency ω₁ of the filters is varied. This figure shows that concurrence is very high (with high state purity) when the filtering frequencies are far from the system resonances: ω_BH/BV and ω_H/V. These are shown as coloured grid lines to guide the eye. In this case, the real states are not involved and the leapfrog emission is efficient in both polarizations. The concurrence otherwise drops when ω₁ is resonant with any one-photon transition, meaning that photons are then emitted in a cascade in one of the polarizations, rather than simultaneously through a leapfrog process. Moreover, if at least one of the two deexcitation paths is dominated by the real state dynamics, this brings back the problem of 'which path' information, which spoils indistinguishability and entanglement. There is only one exception to this general rule, namely, the δ = 0 integrated case (soft red line) which has a local maximum at ω_BX, i.e. when touching its resonance in the natural cascade order. This is because the paths are anyway identical and the integration includes the possibility of emitting the second photon with some delay (up to τ_max). For the case of δ ≠ 0, if this is large enough, it is still possible to recover identical paths while filtering the leapfrog in the middle points, ω₁ = ω_BX and ω₂ = ω_X, to produce entangled pairs.

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Overall, the optimum configuration is, therefore, indeed at the middle point ω₁ = ω₂ = ω_B/2 (two-photon resonance), where the photons are emitted simultaneously, with a high purity, and entanglement degrees are also identical in frequency by construction. Entanglement is also always larger in the simultaneous concurrence (blue lines) as this is the natural choice to detect the leapfrog emission, which is a fast process. This comes at the price, expectedly, of decreasing the total number of useful counts and increasing the randomness of the source. Note finally that the blue curves are symmetric around ω_B/2, but the red ones are not, given that in the integrated case, the order of the photons with different frequencies is relevant. The left-hand side of the plot, ω₁ < ω_B/2, corresponds to the natural order of the frequencies in the cascade, ω₁ < ω₂. The opposite order, being counter-decay, is detrimental to entanglement.

Since of all leapfrog configurations, those in figure 6(b) and (c) are optimum to obtain high degrees of entanglement, we provide their dependence on the filter linewidths, Γ, in figures 7(b) and (c). First, we observe that in the limit of small linewidths for the filters, i.e. in the region Γ < 1/τ_max, the simultaneous and time-integrated concurrences should converge to each other, since the time resolution becomes larger than the integration time. Therefore, the integrated emission provides the maximum entanglement when the frequency window is small enough to provide the same results as the simultaneous emission. Decreasing Γ below P (which in this case is also P = γ) results in a time resolution in the filtering larger than the pumping timescale 1/P. Therefore, photons from different pumping cycles start to get mixed with each other. As a result C drops for Γ < γ in all cases. In the limit Γ ≪ P, the emission is completely uncorrelated and C = 0. This is an important difference from the cases of pulsed excitation or the spontaneous decay of the system from the biexciton state. In the absence of a steady state, entanglement is maximized with the smallest window, Γ → 0 [15, 54]. Two opposite types of behaviour are otherwise observed in these figures when increasing the filter linewidth and Γ > 1/τ_max. The simultaneous emission gains in its degree of entanglement, whereas the time-integrated one loses with increasing linewidth. In the limit Γ → ∞, this disparity is easy to understand. We recover the colour blind result in all filtering configurations, (b) or (c), that is, the decay of entanglement: from 1 corresponding to $\left | \psi \right \rangle$ at τ = 0, to 0 corresponding to a maximally mixed state at large τ. Therefore, C_Γ(0) → C(0) = 1 and C^int_Γ(τ_max) → C^int(τ_max) = 0 (for our particular choice of τ_max and P).

The decrease in C^int_Γ(τ_max) with increasing filtering window, has been discussed in the literature for the case in figure 7(b) [15, 54], and it has been attributed to a gain of 'which path' information due to the overlap of the filters with the real excitonic levels. In the light of our results, when Γ > δ the real-state deexcitation takes over the leapfrog and C^int_Γ(τ_max) is suppressed indeed due to such a gain of 'which path' information, as discussed previously. However, we find another reason why C^int_Γ(τ_max) decreases with Γ in all cases, based on the leapfrog emission: the region 1/P < Γ < δ for case (b) and the region 1/P < Γ < χ for case (c). The maximum delay in the emission of the second photon in a leapfrog processes is related to 1/Γ, due to its virtual nature. Therefore, the initial enhancement of entanglement starts to drop at delays τ ≈ 1/Γ, after which the emission of a second photon is uncorrelated to the first one (not belonging to the same leapfrog pair). For a fixed cutoff τ_max, this leads to a reduction of C^int_Γ(τ_max) with Γ. Broader filters have a smaller impact on the case of real-state deexcitation (see the zero splitting case in figure 7(b), plotted in soft red), since the system dynamics is slower than the filtering (γ, P < Γ), the filter merely emits the photons faster and faster after receiving them from the system. This results in a mild reduction of C^int_Γ(τ_max) with Γ until the detection becomes colour blind and it drops to reach its aforementioned limit of zero.