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Abstract
Entangled photon pairs are essential for quantum communication technology. They can be generated on-demand by semiconductor quantum dots, but several mechanisms are known to reduce the degree of entanglement. While some obstacles like the finite fine-structure splitting of the exciton states can currently be overcome, the excitation scheme itself can impair the entanglement fidelity. Here, we demonstrate that the swing-up of quantum emitter population (SUPER) scheme, using two red-detuned laser pulses applied to a quantum dot in a cavity, yields almost perfectly entangled photons. The entanglement remains robust against phonon influences even at elevated temperatures, due to decoupling of the excitation and emission process. With this achievement, quantum dots are ready to be used as entangled photon pair sources in applications requiring high degrees of entanglement up to temperatures of approximately 80 K.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With their ability to generate entangled photons on-demand [1–3], quantum dots offer exciting possibilities for advancing the field of quantum communication [4]. To harness their usefulness for quantum applications, considerable efforts have been dedicated to achieving perfect photon entanglement. The generation process in quantum dots relies on the biexciton–exciton cascade. A major obstacle to obtain perfect entanglement is the fine-structure splitting (FSS) between the quantum dot’s single exciton states [5]. An impaired fidelity can be quantified by the concurrence, which becomes unity only in the ideal case. The issue of the FSS has been successfully addressed by applying external fields [2,6,7], advanced quantum dot growth [8,9], or via strain tuning [10]. These methods have enabled the generation of entangled photons with a remarkably high concurrence (approximately $97{\% }$ at ${5}\,\textrm{K}$) [11]. To achieve perfect entanglement, the preparation process of the biexciton is likewise of paramount importance. The two-photon excitation (TPE) assures the ultrafast, on-demand preparation of the biexciton. However, during the action of the TPE pulse, an optical Stark shift is induced on the exciton levels, which acts as an effective FSS. Accordingly, TPE sets a fundamental limit to the achievable concurrence [12,13]. This calls for a new scheme to excite the biexciton in an ultrafast way, yet without affecting the degree of entanglement.
The recently proposed swing-up of quantum emitter population (SUPER) scheme [14] is a candidate to address the excitation issue. In the SUPER scheme, off-resonant excitation with two pulses is employed to address the desired state [14–16]. The off-resonant excitation induces an optical Stark shift of the energies of the target states during the excitation. In Ref. [17], it was shown that the combination of SUPER with a photonic cavity, as available in different geometries [18–24], leads to improved photon properties, because emission into the cavity is suppressed during the pulse. The question remains open if by using SUPER, the limitation to the degree of entanglement imposed by TPE can be overcome and perfectly entangled photons can be created.
Another obstacle to overcome for solid-state quantum emitters is the interaction with lattice vibration of the crystal lattice, in particular longitudinal acoustic (LA) phonons [25–28]. For entangled photons, if the biexciton is initially prepared and there is no FSS, the concurrence is unaffected by LA phonons even at elevated temperatures [29,30]. As soon as this situation is broken, phonons degrade the entanglement, in particular at elevated temperatures [31–33].
In this paper, we show that entangled photons after excitation of a quantum dot with the SUPER scheme can reach ${99.8}{\% }$ concurrence even under the influence of phonons. More remarkably, the concurrence of over ${99}{\% }$ remains for increasing temperatures, up to the temperature of liquid nitrogen at ${77}\,\textrm{K}$. Using SUPER for entangled photon generation is therefore highly promising, even at elevated temperatures, which can for example be employed for satellite-based quantum communication [34].
2. Background and Model
Here, we give a brief summary of our model and the simulations. Details of the model and its Hamiltonian alongside the parameters used in the calculations can be found in Supplement 1. A sketch of the system is shown in Fig. 1. Our model consists of the quantum dot modeled as a four-level system placed inside a photonic cavity. The quantum dot is excited using a diagonally polarized external laser field, treated semi-classically, which addresses the quantum dot without directly populating the cavity. To maximize the concurrence, our calculations are performed for a quantum dot with zero FSS. In this scenario, any linear polarization for the laser would yield identical results for the concurrence [12]. The biexciton energy is reduced from twice the single exciton by the biexciton binding energy (BBE), for which we take $\Delta _B= {1}\,\textrm{meV}$ unless stated otherwise. Both polarization modes of the cavity are set resonant to the two-photon energy. While this makes it harder to distinguish the photons in an experimental setting, it enables direct simultaneous emission of two photons [18,35]. We assume a cavity coupling of $g={0.06}\,\textrm{meV}$ which, for the SUPER scheme, yields high concurrence values as discussed in Supplement 1 and is within reach for experimentally realized cavities [36,37]. The coupling to LA phonons via the deformation potential coupling, as identified as the main hindering mechanism for state preparation [25,26], is included via the standard Hamiltonian [38]. We further account for radiative decay that does not feed into the cavity via the rate $\gamma$ and cavity losses via the rate $\kappa$.
Fig. 1. Sketch of the system. A quantum dot (QD) coupled to a two-photon resonant cavity, excited by a diagonally polarized external laser field, and subject to electron–phonon interaction. Photons from the QD can either be emitted in free-space directly or to the cavity modes which are subsequently out-coupled. Ideally, the photons coupled out via the cavity are in the maximally entangled state $|{\psi}\rangle = \frac {1}{\sqrt {2}}(|{XX}\rangle + |{YY}\rangle)$ which corresponds to the $|{\Phi ^+}\rangle$ Bell state.
To calculate the quantum dot dynamics as well as the dynamics of the cavity photons, we use a process tensor matrix product operator (PT-MPO) method, with details outlined in Ref. [39]. Within PT-MPO methods [39–43] and path integral approaches [32,44], the phonon environment can be included in a numerically complete fashion. Using the PT-MPO method, we can calculate photon properties beyond the limitations inherent to the quantum regression theorem [27]. We will compare our results with those from quantum dots without a cavity, where the concurrence is calculated via the quantum dot polarizations [12]. Calculations without phonons are performed in QuTiP [45,46]. From the corresponding dynamics, we calculate the correlation functions and the concurrence as detailed in Supplement 1.
3. Concurrence Optimization
We start with the quantum dot in a cavity without phonons. All exciting laser pulses are assumed to be Gaussian with a pulse duration of $\sigma ={2.7}\,\textrm{ps}$ (FWHM of intensity, ${4.5}\,\textrm{ps}$). For TPE without a cavity, this pulse duration results in a concurrence of ${95.1}{\% }$, in agreement with previous calculations [12]. Interestingly, for TPE, the cavity does not enhance the concurrence, but only gives a value of ${69.4}{\% }$. We attribute this to the cavity-enhanced photon emission during the pulse, leading to stronger impacts of which-path information and re-excitation.
In SUPER, two pulses with different detunings $\Delta _{1,2}$ with respect to the exciton energy and pulse areas $\alpha _{1,2}$ excite the system. We fix the detuning of the lesser detuned pulse to $\Delta _1={-5}\,\textrm{meV}$, and then scan the lesser detuned pulse area and numerically search for parameters for the second pulse yielding the highest biexciton occupation. While this does not automatically optimize the concurrence, it ensures that the resulting parameters lead to a high photon yield (see Supplement 1 for a detailed discussion). We further consider several BBEs $\Delta _B$, as previous studies revealed the influence of this property [47,48]. The results are shown in Fig. 2. We find that for $\Delta _B < {2}\,\textrm{meV}$, the concurrence reaches values above ${99}{\% }$. A maximum value of ${99.9}{\% }$ is achieved for every considered $\alpha _1$, when $\Delta _B={1}\,\textrm{meV}$. Out of these, we choose the parameters that achieve the highest biexciton preparation fidelity, which is at $\alpha _1=32\pi$, and for the higher detuned pulse, $\Delta _2=12.96$ meV and $\alpha _2=12.8\,\pi$ (cf. also Table S1 in Supplement 1). Without a cavity, these parameters give a concurrence of only ${93.1}{\% }$ due to the induced which-path information during the pulse [12].
Fig. 2. Concurrence for the SUPER scheme. Color map of the concurrence as a function of biexciton binding energy $\Delta _B$ and pulse area $\alpha _1$ of the lesser detuned SUPER pulse for a quantum dot in a cavity, calculated without phonons. The detuning of the lesser detuned pulse is fixed to $\Delta _1={-5}\,\textrm{meV}$ and the other pulse parameters are optimized numerically toward a high photon yield. For small $\Delta _B$, a concurrence over ${99}{\% }$ is achieved.
The close-to-unity concurrence for SUPER can be traced back to the decoupling of the emission during the pulse due to the optical Stark shifts of the biexciton–ground state transition. When exposed to a strong laser, the optical Stark effect leads to an energetic shift in the QD states, away from their usual resonances. Due to this shift, the cavity is now decoupled during the excitation process and the emission sets in only after the preparation is completed. For the emitted photons, the situation comes close to an initial value problem, where the biexciton is assumed to be initially populated, disregarding the excitation process. This boosts the concurrence because in the initial value problem, the situation is symmetric without any which-path information. In fact, for a true initial value problem without FSS, the concurrence is known to be exactly one [29,30].
This decoupling is visualized in Fig. 3, where the dynamics of the quantum dot states and the cavity photon number are shown. For SUPER, shown in Fig. 3(a), the quantum dot population exhibits the typical swing-up behavior, initially transitioning to the exciton states $X/Y$ before progressing to the biexciton state. For TPE, shown in Fig. 3(b), a monotonic rise of the biexciton occupation is found, while there is also a transient occupation of the exciton states. Due to the diagonal polarization and the vanishing FSS, the $X$ and $Y$ excitons (and also cavity photons) are always addressed equally. The population of the biexciton decays exponentially, accompanied by an additional oscillation of the occupation resulting from the QD–cavity coupling.
Fig. 3. Dynamics of the excitation. For the (a) SUPER scheme and for (b) TPE, we show the occupations $X$ and $Y$ of the exciton states and $B$ of the biexciton state. (c) Photon numbers $N_{X/Y}$ in the two cavity modes. The shaded background indicates the exciting pulses. In SUPER, the photon occupation only rises after the pulses, while in TPE, the photon occupation already rises during the pulse.
A crucial difference of SUPER and TPE lies in the number of cavity photons $N_{X/Y}$ during the preparation process, as displayed in Fig. 3(c). For SUPER, due to the decoupling, there is minimal photon emission into the cavity during the pulses up to approximately $t={15}\,\textrm{ps}$. After that, we see that the photon number rises, as the cavity and the ground-to-biexciton two-photon transition are resonant again. However, for TPE, where the cavity is resonant to the relevant transition all the time, the photon numbers $N_{X/Y}$ already rise strongly during the pulse, resulting in the reduced concurrence [12]. This decoupling effect results in the possibility to achieve nearly perfect entanglement of photons generated from a quantum dot.
4. Temperature Dependence of the Concurrence
To use quantum dots for practical applications, it is desirable that they work at elevated temperatures. However, with increasing temperature, phonon effects become more pronounced for optical excitation schemes, unless one works in the reappearance regime [28,38,49,50]. Phonon coupling can reduce the preparation fidelity of the targeted state drastically [25,26]. As shown in Fig. 4(b), the final biexciton occupation decreases as a function of temperature. It is evident that the excitation using the SUPER scheme is less prone to disturbance by phonon interaction. For SUPER, the biexciton population drops approximately linearly with rising temperature, while for TPE, the population rapidly drops below ${50}{\% }$.
Fig. 4. Temperature dependence of the concurrence. (a) Concurrence as a function of temperature for SUPER and TPE, for the quantum dot either in a cavity or without a cavity. (b) Final population of the biexciton state for the SUPER scheme and TPE, with the same parameters as in Fig. 3. To compare the final population values, all decay mechanisms (radiative decay, coupling to the cavity) are neglected.
Let us now turn to the concurrence. Interestingly, it has been shown that in the case of zero FSS and an initially prepared biexciton, due to the highly symmetric situation, LA phonons with pure dephasing-type coupling do not affect the concurrence [29,30]. LA phonons do not introduce which-path information in a fully symmetric system since they impact both decay paths equally. However, as soon as the excitation induces an asymmetry in the exciton energies via the optical Stark shift, phonons degrade the entanglement even further [32]. Considering the case without a cavity, we clearly see in Fig. 4(a) that with increasing temperature, the concurrence drops as a function of temperature. Like for the populations, the concurrence drops more rapidly for TPE in comparison to SUPER. Including a cavity, TPE exhibits a decline in the concurrence as the temperature increases, in agreement with findings from previous studies [30,31] that identified phonons as being a substantial source of decoherence, leading to a reduction of the concurrence. It was also found in Ref. [48] that phonons cause a renormalization of the dot–cavity coupling that can improve the concurrence. We attribute the slight increase of the concurrence at approximately ${50}\,\textrm{K}$ to these effects.
The case is quite different when applying the SUPER scheme on the quantum dot in a cavity. Here, the state preparation and the photon emission are decoupled, and hence the entanglement properties should be similar to the case of an initially prepared biexciton [29,30]. Remarkably, the concurrence remains at $C\,>{99.7}{\% }$ independent of the temperature. This outcome is significant, as it suggests that using this scheme, near-perfect entanglement can be achieved even at elevated temperatures. Our model focusing on the coupling to longitudinal acoustic phonons should be valid to describe the physics up to temperatures of approximately ${80}\,\textrm{K}$. For higher temperatures, multiple longitudinal optical phonon couplings of discrete dot states to the continuum of wetting layer states have been shown to limit the concurrence even for initial value problems and vanishing FSS [51]. We also consider strongly confined quantum dots, because for weakly confined quantum dots, hot exciton states can decrease the concurrence [33].
5. Conclusions
We have shown that exciting a semiconductor quantum dot in a cavity via the SUPER scheme overcomes a significant hurdle in generating perfectly entangled photons, namely, the limit of the concurrence induced by the duration of the TPE excitation scheme. Our scheme delivers unprecedentedly high values of the concurrence. Strikingly, the almost perfect entanglement can be achieved over a broad parameter range and up to elevated temperatures which paves the way to new types of applications.
Funding
Deutsche Forschungsgemeinschaft (Project AEQuDot, project number 428026575).
Acknowledgments
We thank A. Rastelli, G. Weihs, T. Heindel, F. Kappe, Y. Karli, V. Remesh, and D. Vajner for fruitful discussion.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data are available upon reasonable request.
Supplemental document
See Supplement 1 for parameters used in the numerical simulations and supporting analysis, additionally including citations of [52–58].
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