Unraveling Dicke Superradiant Decay with Separable Coherent Spin States

Abstract

We show that idealized Dicke superradiant decay from a fully inverted state can at all times be described by a positive statistical mixture of coherent spin states (CSS). Since CSS are separable, this implies that no entanglement is involved in Dicke decay. Based on this result, we introduce a new numerical quantum trajectory approach leading to low-entanglement unravelings. This opens up new possibilities for employing matrix product state (MPS) techniques for large-scale numerical simulations with collective decay processes.

Figures (5)

• Figure 1: (a) Superradiant decay represented as a fall through the collective Bloch sphere. This leads to mixed states (grey points) on the $S_z$-axis. Our ansatz consists in representing those states as mixtures of coherent spin states (CSS), Eq. \ref{['eq:CSS_ansatz']}, here sketched for $N=2$ (12 arrows pointing to the surface of the sphere). (b) Evolution of the coefficients $\rho_m$ in the entangled Dicke state expansion \ref{['eq:DS_ansatz']} ($N=5$). (c) Equivalent positive evolution of expansion coefficients $P_a$ in the CSS ansatz \ref{['eq:CSS_ansatz']}, which demonstrates the absence of entanglement.
• Figure 2: (a) Logarithm of negativity $\log_{10}(\mathcal{N})$ in the range $[-10,0]$ as function of time and $\eta$, for $N=30$. Before the burst time $t_B = \ln (N)/\Gamma$ (dashed lines) many positive solutions exist. Insets: Zooms in the ranges $0 \leq t\Gamma \leq 0.5$, $10^{-3} \leq \eta \leq 0.7$ and $9 \leq t\Gamma \leq 9.02$, $0.97332 \leq \eta \leq 0.97344$. After $t_B$, two passages to late times with positive solutions are highlighted: star (lower, at all time remaining $\eta \leq 1$), triangle (upper). (b) $\log_{10}(\mathcal{N})$ as function of $\eta$ at $t\Gamma=8$. For different $N$, $\mathcal{N} \to 0$ in the areas of the two passages. (c) Evolution with $P_a >0$ for a numerically extracted $\eta(t)$ along the lower passage ($N=30, \mathcal{N} < 10^{-6}$). (d) $\{P_{a}\}_{a=0}^{N}$ distribution as function of $\theta$ at different times ($N=30$). At $t_B$, the peak centers around $\theta=\pi/2$.
• Figure 3: Trajectory averaged evolution ($N=50$) of (a) normalized Bloch vector length $\overline{\xi}$ and (b) TE. We compare naive [Kraus operators \ref{['eq:kraus_E']}], $\phi$-randomized, and $\phi$-optimized unravelings [using randomized/optimized choices of Kraus operators \ref{['eq:kraus_F']}, see text]. The scaling with $N$ is analyzed for averaged: (c) minimum Bloch vector length, $\xi_\chi$; and (d) maximum TE, $\overline{S}_{\rm max}$. Dashed lines highlight the burst time.
• Figure S1: (a) Numerically extracted positive-$\bm{P}$ solutions close to the lower passage for different values of $N$. All solutions remain close to the analytical result for $N=2$ (red line). The smooth behavior may suggest the existence of a a general smooth solution for $N>2$. (b) $P_a$ evolution for $N=20$ and (c) $N=40$ ($\mathcal{N} < 10^{-6}$). The vertical dashed lines indicate the burst time.
• Figure S2: QT simulations for $N=50$ with randomized $\phi_F$ taking the average over 100 trajectories: (a) TE evolution using various choices of $\theta_F$. No major variation in TE is observed when comparing $\theta_F=\pi/8, \pi/4, 3\pi/4$; (b) TE evolution for different timesteps, showing invariance with respect to $\Delta t$