Emission-frequency separated high quality single-photon sources enabled by phonons

TL;DR

This paper theoretically shows that phonon-assisted off-resonant excitation of quantum dots can produce high-purity, high-brightness single-photon sources with spectral separation from the laser, outperforming resonant methods.

Contribution

It introduces a phonon-assisted off-resonant excitation scheme that enhances single-photon purity while maintaining brightness and spectral separation, surpassing resonant excitation techniques.

Findings

Single-photon purity is higher with phonon-assisted off-resonant excitation.

Brightness remains high despite off-resonant excitation.

Spectral separation enables high-quality photon emission free from laser background.

Abstract

We demonstrate theoretically that the single-photon purity of photons emitted from a quantum dot exciton prepared by phonon-assisted off-resonant excitation can be significantly higher in a wide range of parameters than that obtained by resonant preparation for otherwise identical conditions. Despite the off-resonant excitation the brightness stays on a high level. These surprising findings exploit that the phonon-assisted preparation is a two-step process where phonons first lead to a relaxation between laser-dressed states while high exciton occupations are reached only with a delay to the laser pulse maximum by adiabatically undressing the dot states. Due to this delay, possible subsequent processes, in particular multi-photon excitations, appear at a time when the laser pulse is almost gone. The resulting suppression of reexcitation processes increases the single-photon purity. Due to the spectral separation of the signal photons from the laser frequencies this enables the emission of high quality single photons not disturbed by a laser background while taking advantage of the robustness of the phonon assisted scheme.

Equations

$$p=\frac{{\int }_{-T_{\text{Pulse}}/2}^{T_{\text{Pulse}}/2}d\tau G^{(2)}(\tau )}{{\int }_{T_{\text{Pulse}}/2}^{3T_{\text{Pulse}}/2}d\tau G^{(2)}(\tau )},$$

Full text

On-demand single-photon sources continue to gain attention as key building blocks in quantum technological applications, ranging from novel metrology over quantum communication to quantum computing. Semiconductor quantum dots (QDs) have proven to be suitable single-photon emitters Michler et al. (2000); Santori et al. (2001, 2002); He et al. (2013); Somaschi et al. (2016); Wei et al. (2014); Ding et al. (2016); Schweickert et al. (2018) that due to their high compatibility with existing semiconductor technology are promising candidates for device applications. In contrast to atomic systems, these nanoscale structures are prone to the influence of the surrounding solid state crystal matrix. Longitudinal acoustic (LA) phonons are the main source of decoherence of excitons in semiconductor QDs even at cryogenic temperatures of a few Kelvin Förstner et al. (2003); Vagov et al. (2007); Ramsay et al. (2010); Kaer et al. (2010); Reiter (2017). Nevertheless, phonon-assisted off-resonant QD excitations have been shown to provide a robust alternative to resonant exciton preparation schemes Glässl et al. (2013); Ardelt et al. (2014); Reiter et al. (2014); Bounouar et al. (2015); Quilter et al. (2015). In this letter, we demonstrate theoretically that, quite unexpectedly, the coupling to LA phonons combined with off-resonant driving can be extremely beneficial for a single-photon source based on a QD-cavity system, allowing for the generation of high-quality single-photons that are easily detectable due to their spectral separation from the laser pulses used for the excitation of the QD.

Placing a QD in a cavity strongly enhances the photon emission by enlarging the effective dot-cavity coupling and by setting a preferable emission axis. When exciting the QD exciton resonantly, the frequencies of the excitation and the signal are identical - separating the two is a formidable experimental challenge. In fact, spectral separability is achievable, e.g., by wetting layer excitation or by exciting the biexciton via the two-photon resonance and subsequently exploiting the biexciton-exciton cascade Schweickert et al. (2018); Hanschke et al. (2018). But while the former introduces a time jitter that reduces the on-demand character of the photon source, the latter is sensitive to small fluctuations of excitation parameters such as the laser energy and the pulse area. Both problems are overcome by an off-resonant excitation of the quantum dot, which is thus extremely advantageous. Indeed, it has recently been shown that the robustness of off-resonant excitation schemes paves the way to excite two spatially separated QDs with different transition energies simultaneously with the same laser pulse, which is a milestone towards the scalability of complex quantum networks Reindl et al. (2017).

The quality of a QD-cavity system as an on-demand single-photon source is typically quantified by several key figures of merit, such as the single-photon purity $\mathcal{P}$ and the brightness $\mathcal{B}$. While the former measures whether indeed a single photon is emitted by the source, the latter characterizes its total photon yield Somaschi et al. (2016). When $\mathcal{P}=\mathcal{B}=1$, the source emits a single photon with a probability of unity at every excitation pulse via the cavity. The single-photon purity (SPP) can be extracted from a Hanbury Brown-Twiss coincidence experiment Hanbury Brown and Twiss (1956); Santori et al. (2002); Bentham et al. (2016); Prtljaga et al. (2016); Weiß et al. (2016); Ding et al. (2016); Schweickert et al. (2018), which gives a conditional probability to detect a second photon when a first one has already been detected. Suppressing this probability is possible, e.g., by parametric down-conversion, which enhances the SPP, albeit at the cost of a severely reduced brightness of the photon source Pan et al. (2012). Maximizing both SPP and brightness is of utmost importance to create efficient single-photon emitters.

Simultaneously large $\mathcal{P}$ and $\mathcal{B}$ in a QD-cavity system can be achieved by exciting the dot resonantly by ultra-short laser pulses Santori et al. (2002); He et al. (2013); Ding et al. (2016). However, shortening the pulse duration is equivalent to widening it spectrally. The detrimental influence of exciting higher-lying states, especially the biexciton state of the QD by short pulses is discussed in Ref. Gustin and Hughes (2018). In view of the various advantages of phonon-assisted off-resonant excitations listed above, the question arises how photonic characteristics such as SPP and brightness perform under off-resonant schemes. In short, we want to explore whether all of the advantages of phonon-assisted off-resonant schemes come at the cost of severely reduced photonic properties.

It is expected that driving a QD off-resonantly is much less efficient. For longer and stronger pulses the resulting quantum state of a QD-cavity system contains an admixture of multi-photon states, which reduces the SPP. Phonon-induced dephasing is expected to degrade the quantum state even further. But paradoxically quite the opposite can take place: a combination of off-resonant driving with the phonon-induced relaxation between laser-dressed QD states leads eventually to high exciton occupations in a subsequent adiabatic undressing process Barth et al. (2016a). In this letter, we demonstrate that the delay of the exciton creation caused by the undressing suppresses the probability for multi-photon generation. Therefore, comparing off-resonant and resonant excitation with otherwise same conditions may, quite unexpectedly, yield enhanced SPPs in the off-resonant case. The best values predicted in this letter are even comparable to the best values obtained so far within resonant schemes addressing the exciton.

We model the QD-cavity system as a laser-driven two-level system with a ground state $|G\rangle$ and an excited state $|X\rangle$, $H_{\textrm{DL}}=-\hbar\Delta\omega_{\textrm{LX}}|X\rangle\langle X|-\frac{\hbar}{2}f(t)\left(|X\rangle\langle G|+|G\rangle\langle X|\right)$, coupled to a single-mode microcavity (cf. Fig. 1), $H_{\textrm{C}}=\hbar\Delta\omega_{\textrm{CL}}a^{\dagger}a+\hbar g\left(a^{\dagger}|G\rangle\langle X|+a|X\rangle\langle G|\right)$, which is on resonance with the QD exciton. Here, $\Delta\omega_{\textrm{LX}}$ and $\Delta\omega_{\textrm{CL}}$ are the laser-exciton and cavity-laser detuning, respectively, and $a$ is the photon annihilation operator in the cavity, which is coupled to the dot by the coupling constant $g$. A train of Gaussian pulses is assumed represented by the laser envelope function $f(t)$. The excitation can leave the system either via radiative decay or cavity losses modeled by Lindblad rates $\gamma$ and $\kappa$, respectively. Finally, the exciton is coupled to a continuum of LA phonons in a pure-dephasing manner Krummheuer et al. (2002), $H_{\textrm{Ph}}=\hbar\sum_{\mathbf{q}}\omega_{\mathbf{q}}b_{\mathbf{q}}^{\dagger}b_{\mathbf{q}}+\hbar\sum_{\mathbf{q}}\left(\gamma_{\mathbf{q}}^{\textrm{X}}b_{\mathbf{q}}^{\dagger}+\gamma_{\mathbf{q}}^{\textrm{X}*}b_{\mathbf{q}}\right)|X\rangle\langle X|$. $b_{\mathbf{q}}$ annihilates a phonon in the mode $\mathbf{q}$ coupled to the dot by the coupling constant $\gamma_{\mathbf{q}}^{\textrm{X}}$. Full details of the model and of our numerical approach are given in the supplemental material sup . It is worthwhile to note that we use path-integral methods for our simulations that allow us to perform all simulations without approximation to the model Barth et al. (2016b); Cygorek et al. (2017); Cosacchi et al. (2018); sup .

For the calculations, standard GaAs parameters are used Krummheuer et al. (2005) for a QD of $6\,$nm diameter (for details on the phonon coupling consider the supplement sup ). If not stated otherwise, the excitation pulse full width at half maximum is set to $7\,$ps, the cavity mode is resonant with the QD transition, the dot-cavity coupling is $\hbar g=50\,\mu$eV, the radiative decay rate is $\hbar\gamma=20\mu$eV, and the cavity loss rate is $\hbar\kappa=50\mu$eV. This corresponds to a Purcell factor of $F_{\textrm{P}}=g^{2}/(\gamma\kappa)=2.5$. The initial phonon distribution is assumed to be thermal with a temperature of $T=4.2\,$K.

The main target quantities of interest in this paper, the SPP $\mathcal{P}$ and the brightness $\mathcal{B}$, are obtained from path-integral simulations of the two-time photonic correlation function $G^{(2)}(t,\tau)=\langle a^{\dagger}(t)a^{\dagger}(t+\tau)a(t+\tau)a(t)\rangle$ and the time dependent photon occupation $\langle a^{\dagger}a\rangle(t)$, respectively. In order to express the SPP in terms of $G^{(2)}(t,\tau)$ one first needs to take the average over the first time argument $t$, i.e., $G^{(2)}(\tau)=\int_{-\infty}^{\infty}dt\,G^{(2)}(t,\tau)$, which yields a function with the delay time $\tau$ of the coincidence measurement as its single argument. The probability $p$ of detecting a second photon during the same excitation pulse after a first one has already been emitted thus can be obtained by

[TABLE]

where $T_{\textrm{Pulse}}$ is the separation of the pulses in the pulse train. The SPP is then defined as $\mathcal{P}=1-p$. Note that $-\infty<\mathcal{P}\leq 1$, where the lack of a lower bound is due to the possibility of bunching instead of anti-bunching.

In this work, the brightness of the source is modeled as the integrated leakage of the average photon number during the duration of one pulse, i.e., $\mathcal{B}=\kappa\int_{-T_{\textrm{Pulse}}/2}^{T_{\textrm{Pulse}}/2}dt\,\langle a^{\dagger}a\rangle(t)$. Due to the definition, this quantity formally ranges in $0\leq\mathcal{B}<\infty$ without an upper bound since in principal infinitely many photons can exist in a single electromagnetic field mode.

In Fig. 2a the brightness simulated without phonons is shown as a function of the detuning $\Delta\omega_{\textrm{LX}}$ between the central laser frequency and the transition frequency connecting the ground and the exciton state of the QD as well as the pulse area $\Theta$. An oscillatory behavior as a function of the pulse area with maxima at odd multiples of $\pi$ is observed (cf. Fig. 2a). This is a consequence of the well-known Rabi rotation of the exciton occupation since the exciton feeds the cavity photons, which in turn are measured by the brightness. As a function of the detuning, the regions of high brightness are confined to a fairly small range around resonance. The inclusion of phonons drastically changes this picture (cf. Fig. 2b). Through off-resonant excitation with detunings that can be bridged by the emission of LA phonons, a nonvaninshing brightness can be obtained in a previously dark region. Note that the asymmetry with respect to the sign of the detuning is due to the low temperature of $T=4.2\,$K considered here where phonon absorption is largely suppressed.

The SPP in the phonon-free case (cf. Fig. 2c) also displays Rabi rotational behavior but decreases with rising pulse area close to resonance, which is due to a reexcitation of the QD during the same laser pulse. This leads to the emission of more than one photon per pulse, thus diminishing the SPP. Although a SPP can always be calculated, one should be aware that it constitutes a physically meaningful quantity only for finite brightness. Therefore, the area of increased SPP in the upper right corner of Fig. 2c is of no physical relevance.

It is intuitively expected that the continuum of LA phonons reduces the quantum correlations of the system and thus the SPP by inducing a manifold of transitions between its quantum states. However, contrary to these expectations Fig. 2d reveals a huge systematic increase in $\mathcal{P}$ at $\Delta\omega_{\textrm{LX}}\gtrsim 0.5\,$meV. Moreover, the maximum $\mathcal{P}_{\textrm{max}}=98.8\%$ (red circle) is even larger than $90.7\%$ obtained for the resonantly driven system (blue circle). Combined with an appreciably large $\mathcal{B}$, this indicates a possibility to have a good quality single-photon source in the off-resonant excitation regime. Note that $\mathcal{B}=0.46$ observed at the point of $\mathcal{P}_{\textrm{max}}$ (cf. red circle in Fig. 2b) is not much smaller than the maximal value of $0.67$ achieved in the resonantly driven case (cf. blue circle in Fig. 2b). It is also noteworthy that it is possible to obtain a significantly larger brightness at the cost of a slight decrease in the SSP. For example, if we choose a trade-off by maximizing the sum of the squares of the two figures of merit in the off-resonant regime, we obtain $\mathcal{B}=0.53$ and $\mathcal{P}=98.1\%$ (red square). This value for $\mathcal{P}$ is close to typical experimental values obtained for resonant excitation of the quantum dot exciton ($98.8\%$ He et al. (2013), $99.1\%$ Ding et al. (2016)) even though the pulse lengths in Refs. He et al. (2013); Ding et al. (2016) have been slightly shorter 111Note that other system parameters in these experiments are also different from the ones used here, e.g., the Purcell factor reported in Ref. Ding et al. (2016) is 6.3 compared with 2.5 in our case..

To explain the mechanism behind this observation, one needs to consider the dynamics of the QD-cavity states. In Fig. 3, the time dependent occupations in the resonant and the off-resonant case (cf. the blue and red circles in Fig. 2, respectively) are compared. The considered states are product states between the QD states and a photon state with photon number $n$. After resonant $\pi$-pulse excitation (cf. Fig. 3a), the exciton state $|X,0\rangle$ without photons is occupied (blue curve). The cavity coupling rotates the dot back to its ground state and produces one photon in the cavity (orange curve). Because the driving is still nonzero at this point, the dot is reexcited to produce an occupation of the state $|X,1\rangle$ (green curve), which is shown in the inset of Fig. 3a. Finally, the cavity coupling leads to an occupation of the ground state with two photons $|G,2\rangle$ (red curve), such that the SPP is diminished.

In contrast to the $\pi$-pulse induced rotation of the Bloch vector, the off-resonant excitation scheme exploits an effect called adiabatic undressing Barth et al. (2016a). Switching on the laser transforms the dot states to a new energy eigenbasis commonly known as laser-dressed states, the gap between which can be bridged by LA phonons with typical energies of a few meV. At low temperatures, the lower dressed state becomes occupied via phonon emission. However, the phonon-induced relaxation is only efficient when both dressed states have roughly equal exciton components. Thus, the exciton state exhibits typically occupations of the order of 50% after the relaxation is completed Barth et al. (2016a). When the laser is turned off adiabatically, the lower dressed state is subsequently transformed to the exciton state in the original basis provided the detuning is positive (otherwise the ground state is reached Barth et al. (2016a)). This adiabatic undressing of the dot states therefore boosts the exciton occupation only at the end of the pulse (cf. the blue curve in Fig. 3b). This in turn means that during the phase of phonon relaxation no photon can be put into the cavity efficiently (cf. the orange curve in Fig. 3b). When finally the adiabatic undressing-induced exciton boost occurs, the occupation of $|G,1\rangle$ follows (cf. Fig. 3b). Since the excitation pulse is basically gone by then, the reexcitation of the QD is strongly suppressed (green curve), such that effectively no second photon can be put into the cavity (red curve). This implies a far higher SPP than in the resonant counterpart, as is observed in Fig. 2d. In summary, the delay of the exciton occupation caused by the two-step procedure of first relaxing to a dressed state via phonon emission and then reaching the exciton by adiabatic undressing is responsible for the enhancement of the SPP.

To quantify the robustness of the phonon-induced SPP enhancement against variations of other system parameters, the difference between the SPP after off-resonant excitation and after the resonant one is shown as a function of the radiative decay $\gamma$ and the cavity loss rate $\kappa$ in Fig. 4. A positive value (reddish shade) indicates a set of parameters where the SPP is enhanced for off-resonant excitation. We find such an enhancement for a wide parameter regime in $\kappa$ and $\gamma$ that is experimentally well accessible. Also, changing the pulse length from $7\,$ps in Fig. 4a to $14\,$ps in Fig. 4b does not change the phonon-induced SPP enhancement qualitatively. The reason why the SPP for off-resonant excitation falls below the resonant one in the bad cavity limit and/or in the limit of high radiative losses is that relaxation processes limit the time available for the adiabatic undressing which eventually becomes incomplete.

In conclusion, we have presented a seemingly paradoxical scheme for the phonon-assisted operation of a QD-cavity system as a single-photon source, where the excitation is spectrally separated from the generated photons. Two factors that would separately lead to a quality degradation - off-resonant driving and dot-phonon coupling - in combination result in a huge boost in critical characteristics of a single-photon source. We have demonstrated that the achievable single-photon purity can be noticeably higher than for resonant excitation while the brightness is still at an acceptable level. The physical mechanism of this enhancement - the adiabatic undressing - is realized in a wide interval of physically accessible parameters.