Phonon coupling versus pure dephasing in the photon statistics of cooperative emitters

Realizing scalable quantum networks requires a meticulous level of understanding and mitigating the deleterious effects of decoherence. Many quantum device platforms feature multiple decoherence mechanisms, often with a dominant mechanism seemingly fully masking others. In this paper, we show how access to weaker dephasing mechanisms can nevertheless be obtained for optically active qubits by performing two-photon coincidence measurements. To this end we theoretically investigate the impact of different decoherence mechanisms on cooperatively emitting quantum dots. Focusing on the typically dominant deformation-potential coupling to longitudinal acoustic phonons and typically much less severe additional sources of pure dephasing, we employ a numerically exact method to show that these mechanisms lead to very different two-photon coincidence signals. Moreover, surprisingly, the impact of the strongly coupled phonon environment is weak and leads to long-lived coherences. We trace this back to the superohmic nature of the deformation-potential coupling causing interemitter coherences to converge to a nonzero value on a short timescale, whereas pure dephasing contributions cause a complete decay of coherence over longer times. Our approach provides a practical means of investigating decoherence processes on different timescales in solid-state emitters, and thus contributes to understanding and possibly eliminating their detrimental influences.

Figure 1

Two-photon coincidence measurement setups for different configurations of two QDs coupled to two environments E1 and E2. (a) Energetically detuned emitters produce photons of different wavelengths (differently colored arrows). The detection of a photon at detector D1 or D2 determines which QD has decayed (one possible pathway is shown). (b) Spectrally indistinguishable emitters at distance $\left|\mathbf{r}\right|\gg \lambda$. Collecting the emitted light erases information about the source of a measured photon. (c) Superradiance: Two resonant emitters at a distance $\left|\mathbf{r}\right|\ll \lambda$ build a combined system (symbolized by the ellipse encircling both emitters) and photons cannot be traced back to a single QD.

Figure 2

Numerical method to calculate two-time correlation functions. (a) For a single environment, unitary and Markovian contributions to the time evolution of the reduced system are captured by the tensors $\mathcal{M}$, while the tensors $\mathcal{Q}$ capture the influence of the phonon environment. Information about the environment states is propagated through time by matrix products over the inner indices of the PT. A correlation function $\langle \mathcal{B}\left(t_n\right)\mathcal{A}\left(t_m\right)\rangle$ can be calculated by inserting the tensor representation of the operators at the corresponding time step. (b) Two-time correlation function for a combined two-emitter system with separate environments. The PTs for the two environments act only on a specific subset of the indices of the whole two-emitter system, while the tensors $\mathcal{M}$ can introduce an effective interemitter coupling.

Figure 3

(a) Two-photon coincidences from spatially separated emitters with radiative decay rate ${\gamma }^{-1}=1.76$ ns for SPC (blue) and fits of Eq. (23) (cyan) and the PPD model, Eq. (22) (red). The values obtained from fits are $a=0.0854$ for $g_{\text{id}}^{\left(2\right)}\left(\tau \right)$ and ${\gamma }_d^{-1}=3.9$ ns for $g_{\text{pd}}^{\left(2\right)}\left(\tau \right)$. The inset provides a closer look into the region around $\tau \approx 0$. Additionally, the two-photon coincidences for an ohmic spectral density are shown. (b) $g^{\left(2\right)}\left(\tau \right)$ for SPC (blue), SPC with additional PPD with rate ${\gamma }_d^{-1}=221$ ps (green line), and PPD with a rate of ${\gamma }_d^{-1}=0.199$ ps (pink) consistent with experiments [28]. The region $\tau \approx 0$ is shown in the inset. (c) Interemitter coherences of a two-emitter system initially prepared in the Dicke state $|{\mathrm{\Psi }}_S\rangle$ for the types of dephasing considered in (a): SPC (blue line), PPD with rate ${\gamma }_d^{-1}=3.9$ ns (red), and ohmic (yellow). Note the logarithmic scale on the time axis. (d) Interemitter coherences of a two-emitter system initially prepared in the Dicke state $|{\mathrm{\Psi }}_S\rangle$ for the types of dephasing considered in (c): SPC (blue line), combined SPC and PPD (green line), and PPD (pink line).

Figure 4

Two-photon coincidences for superradiant emitters for different pumping rates. (a) In the case of no dephasing, (b) for SPC, and (c) for a PPD with rate ${\gamma }_d^{-1}=199$ ps. The decay rate of the individual QDs is ${\gamma }^{-1}=1.76$ ns across all panels.

Figure 5

PPD approximation to the SPC in the case of superradiant emitters. The pure dephasing rate has been determined to be ${\gamma }_d^{-1}=3.9$ ns in the previous section. The individual emitter decay rate is ${\gamma }^{-1}=\left(2{\gamma }_p^{-1}\right)=1.76$ ns.

Figure 6

Two-photon coincidence results for a finite instrument resolution of $\approx 240$ ps. (a) Two-photon coincidences for SPC and PPD approximation [for the parameters we refer to Fig. 3]. (b) Two-photon coincidences for SPC, PPD with an experimentally obtained rate, and a combined SPC-plus-PPD model that reproduces the experiment [cf. Fig. 3 for parameters]. (c) Analog to Fig. 4. (d) Analog to Fig. 5.