nuHOPS: A quantum trajectory method for highly excited environments in non-Markovian open quantum dynamics

TL;DR

nuHOPS presents a near-unitary reformulation of the Hierarchy of Pure States to tackle non-Markovian open quantum systems with highly excited environments. By introducing a co-moving effective environment and an adaptive shift for the auxiliary hierarchy, the method preserves a quantum-trajectory unraveling while drastically reducing the required hierarchy depth, enabling numerically exact solutions for systems with up to roughly $ \,dim \\mathcal{H}_s \\\approx 1000$ and bath correlations described by $\\alpha(\\tau)$. The approach is validated on dephasing models and, most notably, the Dicke model inside a cavity, where it yields exact results for up to $N \\approx 1000$ emitters and reveals persistent non-Gaussian correlations in both emitters and the cavity field. The framework also accommodates finite-temperature baths and partial environmental excitations, broadening applicability to realistically excited environments and complex initial conditions. Overall, nuHOPS offers a scalable, trajectory-based tool for exploring non-Markovian quantum dynamics with strong coupling and collective effects in quantum optics and beyond.

Abstract

Systems in contact with an environment provide a ubiquitous challenge in quantum dynamics. Many fascinating phenomena can arise if the coupling is strong, leading to non-Markovian dynamics of the system, or collective, where the environment can become highly excited. We introduce a significant improvement of the Hierarchy of Pure States (HOPS) approach, which is able to efficiently deal with such highly excited, non-Markovian environments in a nearly unitary way. As our method relies on quantum trajectories, we can obtain dynamics efficiently, also for large system sizes by i) avoiding the quadratic scaling of a density matrix and ii) exploiting the localization properties of the trajectories with an adaptive basis. We provide the derivation of the nuHOPS (nearly unitary Hierarchy of Pure States) method, compare it to the original HOPS and discuss numerical subtleties based on an illustrative dephasing model. Finally, we show its true power using the Dicke model as the paradigmatic example of many emitters decaying superradiantly inside a cavity. We reach numerically exact solutions for up to 1000 emitters. We apply our method to study emerging higher order correlations in the emitter system or the cavity mode environment and their scaling with the number of emitters.

Figures (3)

• Figure 1: Dynamics of the dephasing model \ref{['eq:MasterDephasing']} for $\omega_a = \omega_c = \kappa = 1$, $g=0.5$ and $N=25$. Plot a) shows a perfect agreement in the physical expectation values between the original and the nuHOPS. Plot b) shows how far fewer auxiliary states are required in the nuHOPS, as evidenced by the significantly lower occupation of the auxiliary oscillator. Insets show the Q function of the auxiliary oscillator at the respective times of maximal occupation.
• Figure 2: Example of the instabilities and how to prevent them. Plot a) shows the dynamics resulting from a naive implementation of Eq. \ref{['eq:shiftedHOPSdeph']} for the (artificial) noise shown in the inset. The instability manifests itself in a seemingly abrupt jump of $\langle S^z\rangle$. Plots b)-d) show a detailed representation of the state at different evolution times, marked by the grey dashed lines in plot a). Expanding the full state according to Eq. \ref{['eq:defpm']} in spin-$S_z$ eigenstates $\ket{s,m}$, the plots show the value of $\bra{p_m}b\ket{p_m}/\braket{p_m}$ in the complex plane, where the value of $m$ is encoded in the color. Insets show the norm of the states $\ket{p_m}$ and states with $\braket{p_m} \leq 10^{-20}$ are shown transparent in the complex plane. Plots e) -h) show the same trajectory calculated with an adaptive basis, which prevents the instabilities (see text).
• Figure 3: Superradiant decay of an ensemble of atoms for different ensemble sizes. Plots show the number of photons in the cavity as well as the total amount of three-body correlations during the dynamics. Expectation values of the ensemble "spin" are plotted in the lower right inset on a sphere. (a) For $N=25$ a comparison to the exact master equation is possible, which shows perfect agreement. The upper inset shows how the three-body correlations scale with the number of emitters $N$, indicating strongly non-Gaussian states even for large $N$. Plots (b) and (c) show cavity and emitter dynamics for $N=500$ and $N=1000$, respectively. The upper inset shows the spin-Q function of the emitters at the point where $\mathcal{C}_3$ is maximal (indicated by a dashed grey line).