Absence of Entanglement Growth in Dicke Superradiance

Abstract

Dicke superradiance describes an ensemble of $N$ permutationally invariant two-level systems collectively emitting radiation with a peak radiated intensity scaling as $N^2$. Although individual Dicke states are typically entangled, the density matrix during superradiant decay is a mixture of such states, raising the subtle question of whether the total state is entangled or separable. We resolve this by showing analytically that for all $N$, starting from the fully excited state, the collective decay preserves separability for all times. This answers a longstanding question on the role of entanglement in Dicke superradiance and underscores that, despite collective dissipation, separable states remain separable under these dynamics.

Figures (3)

• Figure 1: Coherent-state decomposition of Dicke superradiance dynamics for $N = 7$. (a) Time evolution of Dicke state populations $p_k(t)$, shown as solid lines with color indicating excitation number $k = 0,1, \dots, 7$. Red dotted lines show reconstructed populations from the moment decomposition Eq. \ref{['eq:separability_criterion_intro']}. (b) Time-dependent decomposition weights $w_i(t)$ for the representation $\rho(t) = \sum_i w_i(t) \rho(\varepsilon_i(t))^{\otimes N}$, plotted in black with distinct linestyles. (c) Corresponding excitation probabilities $\varepsilon_i(t)$, shown in blue with matching linestyles. Vertical dotted lines indicate the time slices used in (d). (d) Bloch sphere visualization of reconstructed spin-coherent components at selected times. Each blue dot corresponds to a pure product state $\ket{\psi(\varepsilon_i, \phi)}^{\otimes N}$, where the azimuthal angle encodes the phase $\phi$ used for phase averaging. The polar angle gives the excitation probability, which decreases over time as seen in the distribution of the blue dots at $t=0.1\Gamma$ compared to $t=0.4\Gamma$. The full reconstruction algorithm is described in App. \ref{['app:moment_reconstruction']}. Phase averaging is implemented explicitly via a discrete set of phases as discussed in the App. \ref{['app:separability']}, ensuring compatibility with the diagonal form of the Dicke density matrix.
• Figure 2: Here we show the time evolution of the Hankel matrix based entanglement measure $\mathcal{N}(\bm p)$ starting from the Dicke state $\ket{\lfloor N/2\rfloor}$ in (a) and for the fully inverted ensemble in (b). The time axis and negativity measure have been rescaled by $N$, to show universality for $N\to\infty$ in (a). The negativity for an initially excited state in (b) is manifestly zero for all times.
• Figure 3: We show here plots of the negativity in different scenarios. In (a), we start from the Dicke state with a half-inverted ensemble and show the two-spin negativity defined in Eq. \ref{['eq:twospin_neg']} for different particle numbers $N$ and rescale the time and negativity by $N$ to show universality. In (b), we show the same quantity, but starting with the Dicke state where $3/4$ of the particles are inverted. In (c), we plot the negativity for $N=100$ for the density matrix for an effective density matrix partitioned as in $\mathcal{B}=(\mathcal{B}_1,\mathcal{B}_2)$, where $\mathcal{B}_1$ and $\mathcal{B}_1$ are the particle numbers in the respective subsystems and $\mathcal{B}_1+\mathcal{B}_2$ is the total particle number of the reduced density matrix for which the negativity is calculated. In (d), we show for $N=100$ that starting from full inversion, several bipartitions remain zero for all times. Details on how to calculate the reduced density matrices are detailed in Eq. \ref{['eq:particle loss']}.

Theorems & Definitions (2)

• proof
• proof