Disorder-Induced Exponential Scaling of Subradiant Decay Rates

Abstract

Subradiance, a hallmark cooperative phenomenon in waveguide QED, is characterized by a universal power-law scaling of decay rates with system size and underpins many applications in quantum information storage. Here, we demonstrate that disorder drives a sharp transition in the typical subradiant decay rates from power-law to exponential scaling, a phenomenon we term the subradiant scaling transition (SST). Through rigorous finite-size scaling analysis, we establish the SST as a critical phenomenon, characterized by a diverging characteristic scale of the decay rates at the transition point $W_c=0$. Physically, the SST originates from Anderson localization, manifested by the physical equivalence between the characteristic scale and the localization length of the subradiant states. Our findings provide deep insights into the interplay between disorder and collective dynamics, unifying the underlying physical mechanisms of exponentially-scaled subradiant decay rates and Anderson localization in waveguide QED.

Figures (4)

• Figure 1: (a) Schematic of a waveguide-coupled qubit array with positional disorder $\delta_j d$. Ordered subradiant states (dashed line) are delocalized standing waves $\phi_k(x)\propto \sin(kx)$. (b) Scaling laws for decay rates in ordered and disordered chains. We compare strong subradiant states (with $k \to 0,\pi$) and weak subradiant states (with $k \neq 0,\pi$).
• Figure 2: Scaling of $\Gamma^{\rm typ}_{k=0.75\pi}$ [(a)] and $\Gamma^{\rm typ}_{k\to0}$ [(b)] with system size $N$ for $W=0$ (orange diamond) and $W=0.4$ (blue circle). (c) Scaling of $\Gamma^{\rm typ}_{k\to 0}$ at $W=0.06$. Inset shows the dependence of $N_c$ on $W$. In (a-c), solid (dashed) line represents exponential (power-law) fit. (d) Finite-size characteristic scale $\xi$ for $\Gamma^{\rm typ}_{k\to 0}$ versus $W$ for different $N$. (e) The dependence of $\xi$ for $\Gamma^{\rm typ}_{k\to 0}$ on $N$ for different disorder strengths. The black dashed line represents a linear divergence $\propto N$. In (a-e), all results assume $\varphi=0.5\pi$ and are ensemble-averaged over $10^3\sim 10^4$ disorder realizations.
• Figure 3: Data collapse of the finite-size characteristic scale extracted from $\Gamma^{\rm typ}_{k\to 0}$ [$\Gamma^{\rm typ}_{k=0.75\pi}$] in (a) [(b)]. The insets show the dependence of $\xi$ on $(W-W_c)$ for different $N$. The yellow solid lines represent the fit to $\xi_{\infty}\propto (W-W_c)^{-\nu}$ using the critical parameters obtained from the collapse. Critical disorder strength [(c)], critical exponent [(d)], and cost function [(e)] versus $k$. The shaded area schematically marks the vicinity of $k=\varphi$. In (a-e), $\varphi=0.5\pi$. All results are ensemble-averaged over $10^3$ disorder realizations. The details about data collapse are provided in supp.
• Figure 4: (a) Schematics of the Hamiltonian configuration in the single-excitation subspace for $\hat{H}_{\rm eff}$ (top) and $\hat{H}^{-1}_{\rm eff}$ (bottom). Disordered subradiant states (solid line) are localized wavepacket $\phi(x)\propto e^{-|x-x_0|/\xi_{\phi}}$. (b-c) Effective potential ($N=400$) obtained from the strong subradiant state in (b), and for the weak subradiant state (with $k=0.75\pi$ and $N=200$) in (c). $W=0.2$. The blue lines denote the numerical fits. (d) Data collapse of the localization length for subradiant state with $k\to 0$ (blue diamond) and $k=0.75\pi$ (blue circle). Data collapse of the characteristic scale for $\Gamma^{\rm typ}_{k\to 0}$ (orange diamond) and $\Gamma^{\rm typ}_{k=0.75\pi}$ (orange circle). The extracted critical exponents are $\nu_{\phi}=1.51$ for strong subradiant state and $\nu_{\phi}=1.93$ for weak subradiant state. The critical disorder strength are $W^{\phi}_c\approx 0$ for both cases. In (b-d) all results assume $\varphi/\pi=0.5$.