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

Technical Summary: Critical Dephasing Rates for the Observation of Collective Behavior in a Pair of Coupled Quantum Emitters

Problem Statement
Efficient atom-photon interfaces rely on the controlled assembly of quantum emitters to harness collective effects such as superradiance and subradiance. While recent experiments with subwavelength arrays of quantum dots have demonstrated superradiance at room temperature, a fundamental tension exists between the collective enhancement of coherent emission and pure dephasing ($\gamma^*$), which destroys phase coherence. The conventional view posits that pure dephasing is detrimental to optical collective effects. However, recent global descriptions of dephasing suggest a more complex interplay where dephasing can induce transitions between collective states, potentially preserving collective behavior under certain conditions. The central problem addressed is the quantitative characterization of how pure dephasing modifies experimentally accessible signatures of cooperative dynamics. Specifically, the authors seek to identify the critical threshold values of $\gamma^*$ beyond which collective signatures vanish, noting that these thresholds may depend sensitively on the specific observable being measured.

Methodology
The study focuses on the minimal system required to observe collective effects: a pair of coupled two-level quantum emitters ($N=2$). The system is modeled using a total Hamiltonian comprising matter, photonic, and dephasing bath components. The authors rigorously distinguish between two theoretical frameworks for describing dephasing:

1. Local Master Equation: Neglects dipole-dipole interactions when tracing over the dephasing bath. This framework assumes the system eigenstates are those of the bare emitters.
2. Global Master Equation: Retains dipole-dipole interactions in the system Hamiltonian during the derivation, accounting for transitions between the collective eigenstates (symmetric $|S\rangle$ and antisymmetric $|A\rangle$).

The authors demonstrate that for experimentally relevant parameters (specifically where the dipole-dipole coupling energy $\hbar|\Omega_{12}|$ is much smaller than the thermal energy $k_B T$), the global master equation reduces to the local master equation. Consequently, the analysis focuses on the high-temperature regime using the local master equation, where the temperature dependence is encapsulated entirely in the dephasing rate $\gamma^*$.

The study systematically investigates four experimentally accessible observables, varying $\gamma^*$ from negligible to dominant values:

• Steady-state observables:
  • Excited-state population ($n_{exc}$) as a function of laser detuning.
  • Zero-delay second-order correlation function of the zero-phonon-line field, $g^{(2)}(0)$.
• Transient (free evolution) observables:
  • Time-dependent excited-state population $n_{exc}(t)$ following initialization in single-excitation ($|S\rangle$ or $|A\rangle$) or double-excitation ($|E\rangle$) manifolds.
  • Time-resolved first-order correlation functions $G^{(1)}(t, t+\tau)$ and second-order correlation functions $g^{(2)}(t, t)$.

Analytical solutions are derived using symbolic computation (SymPy) for identical emitters, complemented by numerical simulations of the Liouvillian superoperator.

Key Contributions and Results
The paper identifies that the "critical dephasing rate" ($\gamma^*_{lim}$) is not a universal constant but depends strongly on the chosen observable and the excitation regime.

1. Steady-State Population ($n_{exc}$):

   • For two-photon transitions ($|G\rangle \to |E\rangle$), collective signatures manifest as a superlinear (quadratic) scaling of excitation with intensity.
   • The critical dephasing rate above which this scaling vanishes is derived as $\gamma^*_{lim} \approx (4\Omega_{12}^2 \Gamma_0)^{1/3}$.
   • This threshold scales with the coherent coupling strength $\Omega_{12}$, suggesting that even with significant dephasing, coupled pairs can be identified if the dipole-dipole interaction is sufficiently strong (e.g., at sub-wavelength separations).

2. Steady-State Correlations ($g^{(2)}(0)$):

   • Collective behavior is observed as photon bunching ($g^{(2)}(0) \gg 1$) for two-photon transitions and antibunching ($g^{(2)}(0) \ll 1$) for single-photon transitions.
   • The transition to Poissonian statistics ($g^{(2)}(0) \approx 1$) occurs at different thresholds depending on the transition:
     • For $|G\rangle \to |E\rangle$: $\gamma^*_{lim} \approx (4\Omega_{12}^2 \Gamma_0)^{1/3}$.
     • For $|G\rangle \to |S\rangle$: $\gamma^*_{lim} \approx 4|\Omega_{12}|$.
   • The threshold for the single-photon transition is less restrictive (higher) than for the two-photon transition due to differences in power broadening scaling.

3. Free Evolution Dynamics:

   • Single Excitation: In the absence of dephasing, decay is mono-exponential (superradiant or subradiant). With dephasing, it becomes bi-exponential. The crossover to independent emitter behavior occurs when $\gamma^* \sim \gamma_{12}$ (the dissipative coupling rate), which is typically much smaller than the coherent coupling thresholds found in steady-state.
   • Double Excitation: The system exhibits a superradiant burst followed by a slow decay. The temporal features of the second-order correlation function $g^{(2)}(t, t)$ (dips and peaks) are suppressed when $\gamma^*$ approaches the order of the dissipative coupling rate $\gamma_{12}$.
   • Crucially, the thresholds for transient observables are governed by the dissipative coupling $\gamma_{12}$ (scaling with $\Gamma_0$), whereas steady-state thresholds are governed by the coherent coupling $\Omega_{12}$.

Significance and Claims
The authors claim that their work provides a quantitative framework to guide experiments in optimizing conditions for observing collective quantum phenomena. The primary significance lies in the demonstration that the robustness of collective effects against pure dephasing is not uniform; it is highly sensitive to the specific observable and the dynamical regime (steady-state vs. transient).

• Regime Validity: The paper clarifies that for most solid-state experiments (room temperature or typical cryogenic conditions where $\hbar|\Omega_{12}| \ll k_B T$), the local master equation is the relevant and sufficient framework, despite the existence of more complex global descriptions.
• Experimental Guidance: By establishing specific threshold formulas (e.g., Eq. 16 and Table I), the work offers concrete criteria for experimentalists to determine whether collective signatures will be observable given a specific emitter separation (which dictates $\Omega_{12}$) and environmental dephasing rate.
• Scalability: While restricted to $N=2$, the authors argue that their approach is not merely a toy model. Due to the rapid decay of dipole-dipole interactions with distance, nearest-neighbor interactions often dominate in larger, disordered ensembles (such as hot vapors or random quantum dot arrays), making the $N=2$ results broadly applicable to understanding the interplay between dephasing and dipole-dipole interactions across various physical platforms.

The paper concludes modestly, noting that extending these findings to larger ensembles ($N > 2$) and exploring non-Markovian environments are necessary future steps to fully address many-body phenomena and low-temperature regimes where global dephasing descriptions become essential.