Spin-only dynamics of the multi-species nonreciprocal Dicke model

Abstract

The Hepp-Lieb-Dicke model is ubiquitous in cavity quantum electrodynamics, describing spin-cavity coupling which does not conserve excitation number. Coupling the closed spin-cavity system to an environment realizes the open Dicke model, and by tuning the structure of the environment or the system-environment coupling, interesting spin-only models can be engineered. In this work, we focus on a variation of the multi-species open Dicke model which realizes mediated nonreciprocal interactions between the spin species and, consequently, a dynamical limit-cycle phase. In particular, we improve upon adiabatic elimination and, instead, employ a Redfield master equation in order to describe the effective dynamics of the spin-only system. We assess this approach at the mean-field level, comparing it both to adiabatic elimination and the full spin-cavity model, and find that the predictions are sensitive to the presence of single-particle incoherent decay. Additionally, we clarify the symmetries of the model and explore the dynamical limit-cycle phase in the case of explicit PT-symmetry breaking, finding a region of phase coexistence terminating at an codimension-two exceptional point. Lastly, we go beyond mean-field theory by exact numerical diagonalization of the master equation, appealing to permutation symmetry in order to increase the size of accessible systems. We find signatures of phase transitions even for small system sizes.

Figures (6)

• Figure 1: Schematic depiction of the system studied in this paper. A single cavity mode (red beam) coherently interacts with various species $m$ and $m'$ of collective spins (blue spheres) through phase shifted photons $e^{- i \phi_m} a$ and $e^{- i \phi_{m'}} a$ (red ellipsoids) with coupling strengths $g_m$ and $g_{m'}$. The cavity mode is coupled (dark red squiggles) to a bath of extra-cavity modes (pale red beams) leading to decay at the rate $\kappa$. This setup generically realizes effective nonreciprocal interactions between spins of different species $m$ and $m'$ (dashed blue lines). Similarly, the spins experience single-particle incoherent decay at the rate $\Gamma$.
• Figure 2: Effective coupling parameters in the spin-only system. (a) Coupling parameters $J_1 / \epsilon$, $K_1 / \epsilon$, $J_2 / \epsilon$, and $K_2 / \epsilon$ from Eq. \ref{['eq: spin only adjoint Redfield']} as a function of the cavity decay rate $\kappa / \epsilon$ with the cavity energy $\omega_c / \epsilon = 10$ fixed (dashed gray line). $J_2$ is negative (dashed line plotting $- J_2$) for $\kappa^2 < \omega_c^2 - \epsilon^2$. (b) Coupling parameters $J_0 / \epsilon$ and $K_0 / \epsilon$ from Eq. \ref{['eq: spin only Redfield fast-cavity']} as a function of the cavity decay rate $\kappa / \epsilon$ with the cavity energy $\omega_c = 2 \epsilon$ fixed (dashed gray line) compared to $J_1 / \epsilon$ and $K_1 / \epsilon$. There is no analogy to $J_2 / \epsilon$ and $K_2 / \epsilon$ in Eq. \ref{['eq: spin only Redfield fast-cavity']}.
• Figure 3: Comparison of the linear stability analyses of the normal state fixed-point $\bm{\tau}_\text{ns}$ for a two-species model. (a) Boundary of the superradiant phase (lower branch, see text) for $\kappa / \epsilon = 20$, comparing the predictions from Eq. \ref{['eq: Redfield nonlinear mean-field EOM']} (blue), Eq. \ref{['eq: AE mean-field EOM']} (red), and Eq. \ref{['eq: spin and cavity adjoint GKSL']} (black) which retains the cavity mode. In the limit $\kappa / \epsilon \to \infty$, the region formally vanishes in the parameter $\phi$. (b) Linear stability analysis Eq. \ref{['eq: linearized Redfield EOM']} and (c) linear stability analysis Eq. \ref{['eq: linearized AE EOM']} for $\kappa = \omega_c = 20 \epsilon$. The main figures vary the phase $\phi$ and the coupling strength $g / \epsilon$ (fixing $\Gamma / \epsilon = 5 \times 10^{-3}$), while the insets vary the single-particle incoherent decay rate $\Gamma / \epsilon$ and the coupling strength $g / \epsilon$ (fixing $\phi = 0.3$). Color indicates the (in)stability of the normal state fixed-point $\bm{\tau}_\text{ns}$: Stable ("NS," tan), unstable with a pair of complex-conjugate eigenvalues ("DS," blue), unstable with one real eigenvalue ("SR1," green), unstable with two real eigenvalues ("SR2," light-blue), unstable with a real eigenvalue and a pair of complex-conjugate eigenvalues (unlabeled, dark-purple).
• Figure 4: Analysis of the dynamical limit-cycle phase for a two-species model. (a) Behavior of the limit-cycle state for increasing coupling strength $g / \epsilon$ as characterized by the time-averaged magnetization, time-averaged oscillation radius, and the oscillation frequency for Eq. \ref{['eq: Redfield nonlinear mean-field EOM']} (solid) and Eq. \ref{['eq: AE mean-field EOM']} (dashed) and each species $m = + 1$ (blue) and $m = - 1$ (red). The phase $\phi = \pi / 4 - 0.1 \approx 0.68$ is slightly tuned away from $\phi = \pi / 4$ to differentiate the species (see b-d). (Inset) Example limit-cycle state in the Bloch sphere for $g / \epsilon = 2.8$. (b) Time-averaged magnetization and time-averaged oscillation radius for each species $m = + 1$ (blue) and $m = - 1$ (red) and symmetric (solid) and anti-symmetric (dashed) initial perturbations as a function of the coupling strength $g / \epsilon$ (using Eq. \ref{['eq: Redfield nonlinear mean-field EOM']}). At $g / \epsilon \approx 1.51$, the system reaches a codimension-two exceptional point for $\Gamma \to 0$ and the limit-cycles for each species differentiate. (Inset) Floquet multipliers inside the complex unit circle (dashed, center at red point) calculated as the eigenvalues of the monodromy matrix just below the transition. (c) Tuning the "chirality breaking" parameter $\phi$ uncovers a narrow region about $\phi = \pi / 4$ exhibiting hysteresis, growing wider for increasing coupling strength $g / \epsilon$. Symmetric (solid) and anti-symmetric (dashed) initial perturbations can converge to different stable limit-cycle states. (d) Example of the limit-cycle state in the Bloch sphere below the transition (upper) and above (lower). All plots use the parameters $\kappa / \epsilon = 20$ and $\omega_c / \epsilon = 20$ with (a-b) using $\Gamma / \epsilon = 1 \times 10^{-2}$ and (c-e) using $\Gamma / \epsilon = 5 \times 10^{-3} \epsilon$.
• Figure 5: Superradiant steady-states in the fast-cavity limit $\kappa, \omega_c \gg \epsilon$ for phases $\phi = 0.01$, $\pi / 8 \approx 0.39$, and $\pi / 4 - 0.1 \approx 0.69$ and coupling strength $g / \epsilon = 8$. (Top row) Fixed-point solutions to the mean-field equations Eq. \ref{['eq: AE mean-field EOM']} showing species $m = 1$ (filled circles) and $m = - 1$ (stars), connected by lines. There are three solutions (two stable) for $\phi = 0.01$, five solutions (two stable) for $\pi / 8$, and nine solutions (four stable) for $\pi / 4 -0.1$, distinguished by color, which all lie in a single plane (see text). (Middle and bottom row) For a finite-sized system (six particles each), we visualize the density matrix via the spin-averaged Wigner distribution $W^{(m)} \left( \vartheta, \varphi \right)$, species $m = + 1$ (middle row) and $m = - 1$ (bottom row), where $\vartheta$ is the polar angle and $\varphi$ the azimuthal angle, which is consistent with the mean-field predictions. The value is indicated by the color, with the completely mixed state $1/4 \pi \approx 0.08$ (green) for reference. We find a unique steady-state despite the change in the number of mean-field fixed-points. All plots use the parameters $\Gamma / \epsilon = 5 \times 10^{-3}$, $\kappa / \epsilon = 20$, and $\omega_c / \epsilon = 20$, as in Fig. \ref{['fig: liouvillian gap']}.