Critical dephasing rates for the observation of collective behavior in a pair of coupled quantum emitters

TL;DR

The paper models a pair of coupled quantum emitters under pure dephasing to quantify when collective emission survives in both steady-state and transient regimes. Using a Lindblad master equation with dipole-dipole coupling $\Omega_{12}$ and dissipative coupling $\gamma_{12}$ plus a separate dephasing rate $\gamma^*$, it derives observable-dependent dephasing thresholds and analyzes $n_{exc}$ and $g^{(2)}(0)$ among others. A key result is that steady-state signatures scale with $\Omega_{12}$, enabling large thresholds even at room temperature, while free-evolution signatures are limited by $\gamma_{12}$; the work provides concrete guidelines for experimental design and highlights the need to tailor observation methods to retain collectivity. These insights pave the way for observing and exploiting collective quantum phenomena in solid-state emitters and motivate extending the framework to larger ensembles and non-Markovian environments.

Abstract

Efficient atom-photon interfaces require the controlled assembly of quantum emitters, where collective effects such as superradiance and subradiance can emerge. Recent experiments with subwavelength arrays of quantum dots have observed superradiance at room temperature, revealing a delicate competition between collective enhancement of coherent emission and pure dephasing $γ^*$, which destroys it. Motivated by these results, we theoretically study $N=2$ coupled quantum emitters and identify threshold values of $γ^*$, for four experimentally accessible observables, beyond which collective effects vanish. The thresholds depend sensitively on the chosen observable, highlighting the subtlety of detecting collective behavior. Our work provides a quantitative framework to guide experiments and optimize conditions for observing collective quantum phenomena.

Figures (5)

• Figure 1: (a) Representation of the system under study: a pair of two level systems coupled to each other through a photonic bath and each interacting with a separated dephasing bath. The levels just above the ground state represent vibrational energy levels typical in solid-state emitters. Some proportion $1-\alpha$ of the spontaneous decay passes through these states, leading to a red-shifted emission. On the right, we represent the hamiltonian eigenstates of the coupled system. (b) Schematic view of geometry of the system: the emitters are located on the $x$ axis with their dipole oriented along $\mathbf{\hat{e}_z}$ in a 'H' configuration. (c) Overview of the principal results of this paper: a scale of the thresholds of the maximum dephasing rate $\gamma^*$ acceptable depending on the observable considered.
• Figure 2: Dependence of the excitation spectra on the excitation intensity $\mathcal{I}$ and dephasing rate $\gamma^*$. Excitation spectra as a function of the dephasing rate for Rabi frequencies (a) $\Omega_R = 4.0\Gamma_0$ and (b) $\Omega_R = 10.0\Gamma_0$. Saturation curves at $\omega = \omega_0$ (c) for $\gamma^* = 0$ (red), $\gamma^* = 3.0\Gamma_0$ (green), $\gamma^* = 30.0\Gamma_0$ (blue) and $\gamma^*_{lim} \approx 11.8\Gamma_0$ (black, see eq. \ref{['eq:gamma^*_lim_saturation']}). Dashed lines show the analytical expression's first order series expansion in $(\Omega_R/\Gamma_0)^2$ (see Appendix \ref{['app:steady_state_saturation']}). Plain lines (d) and (e): cuts of (a) and (b) respectively. Dashed lines show the exact analytical expression with $\Delta = 0$ and all other parameters kept identical. (f) Normalized excitation spectra for $\gamma^* = 3.0\Gamma_0$, $\mathcal{I}/\mathcal{I}_{sat} = 0.5$ (light green) and $7.0$ (dark green). The dots correspond to the dots in (c). In all panels, simulations are performed with a detuning between emitters of $\Delta = 5 \Gamma_0$, while analytical results are obtained for $\Delta=0$. The emitters are placed $r \approx 0.035\lambda$ apart, in an 'H' configuration, with a Debye-Waller/Franck-Condon factor $\alpha = 0.3$, yielding a coupling strength $\Omega_{12} = 20\Gamma_0$ and a dissipative coupling rate $\gamma_{12} = 0.3\Gamma_0$.
• Figure 3: (a) Second-order correlation at zero delay $g^{(2)}(0)$ as a function of the Rabi frequency $\Omega_R$ and dephasing rate $\gamma^*$, with the excitation tuned to the two-photon transition $\omega=\omega_0$. For large $\Omega_R$ or $\gamma^*$, $g^{(2)}(0) \rightarrow 1$, indicating poissonian statistics. (b) Horizontal cuts of (a) for different Rabi frequencies $\Omega_R = 0.1$ (green), $2.0$ (light blue) and $5.0 \Gamma_0$ (dark blue). (c) Vertical cuts of (a) for different dephasing rates $\gamma^* = 0.1$ (red), $5.0$ (orange), $20.0 \Gamma_0$ (yellow). (d), (e) and (f): same as (a), (b) and (c) with the excitation tuned to the $|G\rangle \rightarrow |S\rangle$ transition. The emitters are placed $r \approx 0.035\lambda$ apart, in an 'H' configuration, with a Debye-Waller/Franck-Condon factor $\alpha = 0.3$, yielding a coupling strength $\Omega_{12} = 20\Gamma_0$ and a dissipative coupling rate $\gamma_{12} = 0.3\Gamma_0$.
• Figure 4: Study of the dynamics of the system initialized in the first manifold $n_{exc} = 1$. In (a) [respectively (d)] , we plot $n_{exc}(t)$ starting from the antisymmetric state $|A\rangle$ [respectively $|S\rangle$] for different dephasing rates. The decay is mono-exponential with $\gamma^* = 0$ and bi-exponential otherwise. In (b,c,e,d) we plot the Fourier transform of $G^{(1)}(t, t+\tau)$ with respect to the delay $\tau$ as a function of the frequency and the time $t$. We start from $|A\rangle$, with $\gamma^* = 0$ in (b) and with $\gamma^*=2.0\Gamma_0$ in (c). We start from $|S\rangle$, with $\gamma^* = 0$ in (e) and with $\gamma^*=2.0\Gamma_0$ in (f). The insets show the real & imaginary parts of $G^{(1)}(t, t+\tau)$ for $\gamma^* = 0$. The emitters are placed $r \approx 0.035\lambda$ apart, in an 'H' configuration, with a Debye-Waller/Franck-Condon factor $\alpha = 0.3$, yielding a coupling strength $\Omega_{12} = 20\Gamma_0$ and a dissipative coupling rate $\gamma_{12} = 0.3\Gamma_0$.
• Figure 5: Dynamics of the system, starting from the doubly-exited state $|E\rangle$. In (a) we plot the excited state population $n_{exc}$ as a function of time, for different dephasing rates $\gamma^*$. In (b) we plot the un-normalized first-order correlation function at zero delay $G^{(1)}(t,t)$. The inset shows a zoom on the first instants of the evolution, from $\Gamma_0t=0$ to $0.5$. In (c) we plot the normalized second-order correlation function at zero-delay $g^{(2)}(t, t)$. The inset shows a zoom on the first instants of the evolution. In all cases, the different colors (but grey) corresponds to different values of $\gamma^*$. In (b) and (c) we consider two different detection schemes: along the inter-emitter axis ($//$, plain lines) and along a perpendicular bisector of the inter-emitter axis ($\perp$, dashed lines). For comparison, in all panels we represent the dynamics of independent emitters in dashed grey line. The emitters are placed $r \approx 0.035\lambda$ apart, in an 'H' configuration, with a Debye-Waller/Franck-Condon factor $\alpha = 0.3$, yielding a coupling strength $\Omega_{12} = 20\Gamma_0$ and a dissipative coupling rate $\gamma_{12} = 0.3\Gamma_0$.