Dynamics of Many-Emitter Ensembles: Probing Cooperative Evolution with Scalable Quantum Circuits

Abstract

Many-particle quantum systems often give rise to exotic behaviors in their nonequilibrium dynamics that are rather challenging to reveal with analytical methods or with classical computation. Here, we consider the case of a system of many quantum emitters coupled through a radiation bath. By adopting an efficient mapping of the bosonic modes onto a set of quantum bits, we implement quantum circuits, compatible with NISQ (Noisy Intermediate-Scale Quantum) era systems, that allow us to investigate the dynamics of the ensemble as a function of various parameters, including the number of emitters, the spectral inhomogeneity in the system, the emission lifetime of independent emitters, and the spatial separation between emitters. The quantum algorithms afford us the capacity to precisely track the emergence of cooperative dynamics, manifested through superradiant emission, as the system is tuned towards optimal coupling with respect to various parameters. We are particularly able to characterize superradiant emission in an inhomogeneous ensemble as a function of the linewidth of the individual emitters. These quantum algorithms avoid approximations performed in conventional studies of many-emitter systems and provide a robust and intuitive characterization. Despite being limited to a small number of qubits, the present calculations are found to provide a reliable characterization validated by comparison with analytical solutions and classical computation results in their respective regimes of validity. These findings indicate that the approach can be employed to effectively simulate a broad variety of many-emitter systems.

Figures (10)

• Figure 1: Illustration of the circuit to simulate the time-evolution of the many atom + radiation system under Hamiltonian (\ref{['eq:HamiltonianQubitized']}).
• Figure 2: Assessing the conservation of energy in the system as a function of time for various values of Trotter step size, $\delta t$. The system consists of four resonant atoms in the long wavelength limit in a bath of seven modes.
• Figure 3: (a) Emission spectrum of a system of 5 resonant atoms in a bath of 7 radiation modes in the long wavelength limit, shown as a function of time with darker shades representing later times. (b) Occupation of the resonant mode ($\omega_{M_0} = \omega_\alpha = 100$) for ensembles of $N_A$ identical atoms as a function of time. (c) Time to $80\%$ saturation as a function of the number of atoms in the ensemble. We see the time monotonically decrease with atom number, as we expect due to the increased effective emission rate arising from the collective superradiant effects. (d) Total occupation of the radiation modes normalized by the number of emitters in the ensemble, here for $N_A = 5$ atoms.
• Figure 4: Emission intensity for an ensemble of 4 atoms as a function of time. The gray lines in each panel correspond to the master equation solution for a homogeneous ensemble with the solid gray line representing the superradiant intensity and the dashed gray line representing the intensity for independent emission. The blue circles show the emission intensity $I_4(t)= \Gamma_0 \langle S^+S^-\rangle$ calculated with the quantum algorithm. The red squares show the non-coherent contribution to the total emission intensity from the quantum algorithm $I_{4, \text{NC}}(t) = \Gamma_0\sum_\alpha\langle\sigma_\alpha^+\sigma_\alpha^-\rangle$. All curves are normalized by the emission intensity for a non-coherent ensemble of 4 emitters at $t=0$. Each panel corresponds to a different value of the relaxation rate of individual atoms in the ensemble with $\Gamma_0 = 0.1, \; 2.0, \; 3.0, \; \mathrm{and} \; 5.0$.
• Figure 5: Coherence $I_{4,C} = \Gamma_0\sum_{\alpha\neq\beta} \langle\sigma_\alpha^+ \sigma_\beta^-\rangle$ as a function of time for an ensemble of 4 atoms. The dotted gray line in each panel corresponds to the coherence of an ensemble of 4 perfectly resonant atoms that is calculated by numerically integrating the master equation using QuTiP. The coherence of the diffuse ensemble simulated by the quantum algorithm is represented by the cyan triangles. As in Figure \ref{['fig:intensityInhomogeneousEnsemble']}, all curves are normalized by $N_A\Gamma_0$, the emission rate of the ensemble at $t=0$ to allow for a direct comparison.