Localization and scattering of a photon in quasiperiodic qubit arrays

TL;DR

This paper investigates how localization of a single excitation in a quasiperiodically spaced qubit array affects the scattering of a single photon in a waveguide QED system. By developing both transfer-matrix and Green-function formalisms, it uncovers a continuum band of localized subradiant states near the atomic resonance, with a fractional localization up to $$(3-\sqrt{5})/2$$ at the golden-mean modulation $$(1+\sqrt{5})/2$$, linked to flat versus curved inverse energy bands in the large-period limit. The authors demonstrate that localized subradiant states block photon transmission while delocalized subradiant states enable it, leading to an emergent transmission mobility edge and a monotonic rise in overall reflection as quasiperiodicity strengthens. These findings provide new insights into localization phenomena in non-Hermitian WQED systems and offer a tunable route to control single-photon transport through engineered quasiperiodic spacings.

Abstract

We study the localization and scattering of a single photon in a waveguide coupled to qubit arrays with quasiperiodic spacings. As the quasiperiodic strength increases, localized subradiant states with extremely long lifetime appear around the resonant frequency and form a continuum band. In stark contrast to the fully disordered waveguide QED where all states are localized, we analytically find that the fraction of localized states is up to $(3-\sqrt{5})/2$ when the modulation frequency is $(1+\sqrt{5})/2$. The localized and delocalized states can be related to excitation in flat and curved inverse energy bands under the approximation of large-period modulation. When the quasiperiodic strength is weak, an extended subradiant state can support the transmission of a photon. However, as the quasiperiodic strength increases, localized subradiant states can completely block the transmission of a single photon in resonance with the subradiant states, and enhance the overall reflection. At a fixed quasiperiodic strength, we also find mobility edge in transmission spectrum, below and above which the transmission is either turned on and off as system size increases. Our work give new insights into the localization in non-Hermitian systems.

Figures (6)

• Figure 1: Schematic diagram of a photon in a waveguide scattered by qubit arrays with quasiperiodic spacing. The positions are arranged as $z_j=d[j+\delta\cos (2\pi\beta j +\theta)]$ with averaged spacing $d$, modulation strength $\delta$, modulation frequency $\beta$, and modualtion phase $\theta$.
• Figure 2: Energy spectrum as a function of (a) quasiperiodic strength by fixing $\theta=0$ and (b) modulation phase by fixing $\delta=0.5$. The colors denote the inverse participation ratio ($\textrm{IPR}$) of the corresponding eigenstates. (c) Probability distribution in the qubit array with modulation strength $\delta=0.5$ and modulation phase $\theta=0$. The colors denote the probability $P_n(j)$. The other parameters are chosen as $\omega_0=100$, $\beta=(\sqrt{5}+1)/2$, $\Gamma_0=0.01$, and $\varphi=\omega_0 d/c=1$.
• Figure 3: (a) Inverse energy band in the large periodic approximation $F_{n+1}/F_n=55/34$. (b) is the enlarged view of (a) around $\Gamma_0 S_q\in [-1,1]$. The other parameters are chosen as $\omega_0=100$, $\Gamma_0=0.01$, $\theta=0$, and $\varphi=\omega_0 d/c=1$.
• Figure 4: (a) Reflection of a single photon as a function of quasiperiodic strength and frequency of photon, and (b) its enlarged view. (c) The overall reflection as a function of quasiperiodic strength. The parameters are chosen as $\omega_0=100$, $\beta=(\sqrt{5}+1)/2$, $\Gamma_0=0.01$, $\theta=0$, $\varphi=\omega_0 d/c=1$.
• Figure 5: Comparison between properties of excitation states and the reflection of photon. The left and right panels are the cases with no quasiperiodic modulation ($\delta=0$) and strong quasiperiodic modulation ($\delta=0.5$). (a) and (b) give the lifetime of eigenstates. (c) and (d) give the $\textrm{IPR}$ of eigenstates. (e) and (f) show reflection as a function of frequency of a single photon. The blue and red solid lines are calculated via Green function method in momentum space, and the black-dashed line is calculated via transfer-matrix method in real space. The other parameters are chosen as $\omega_0=100$, $\Gamma_0=0.01$, $\theta=0$, $\beta=(\sqrt{5}+1)/2$, and $\varphi=\omega_0 d/c=1$.