Disorder-Induced Strongly Correlated Photons in Waveguide QED

TL;DR

This work shows that disorder in qubit transition frequencies can enable strongly correlated photons in a 1D waveguide QED setting, producing antibunching and nearly perfect photon blockade (NPPB) events that are absent in ordered chains. By combining a Lindblad master equation with input-output theory, the authors derive exact expressions for the zero-delay second-order correlations and study their disorder-averaged statistics $P(s)$ across few- and many-qubit chains, in transmission and reflection. They uncover a disorder-enabled mechanism where destructive interference among scattering paths creates NPPB in transmission and enhanced PA/NPPB in reflection, with key insights into weak vs strong disorder limits, losses, chirality, and finite input bandwidth. The results suggest disorder engineering as a viable route to robust, scalable sources of strongly correlated photons for quantum technologies. The findings are supported by analytical results for small $N$, and extensive Monte Carlo simulations for many-qubit arrays, revealing scaling laws such as $\,\mathbb{P}(s<1)\sim\,1-0.97N^{-1/40}$ and $P(10^{-3})\sim 0.04\,N^{1/2}$ in the transmission, as well as disorder-assisted amplification of NPPB in reflection.

Abstract

Strongly correlated photons play a crucial role in modern quantum technologies. Here, we investigate the probability of generating strongly correlated photons in a chain of N qubits coupled to a one-dimensional (1D) waveguide. We found that disorder in the transition frequencies can induce photon antibunching, and especially nearly perfect photon blockade events in the transmission and reflection outputs. As a comparison, in ordered chains, strongly correlated photons cannot be generated in the transmission output, and only weakly antibunched photons are found in the reflection output. The occurrence of nearly perfect photon blockade events stems from the disorder-induced near completely destructive interference of photon scattering paths. Our work highlights the impact of disorder on photon correlation generation and suggests that disorder can enhance the potential for achieving strongly correlated photon.

Figures (17)

• Figure 1: (a) Schematics of a chain of qubits coupled to a 1D waveguide. The qubits have different transition frequencies denoted by $\omega_k=\omega_0+\Delta_k$. The decay rate of each qubit is $\gamma$, and the qubits are uniformly-spaced with distance $d$. For a weak classical input light with frequency $\omega_0$, both the reflection and the transmission outputs can generate strongly correlated photons. (b) PA, PPB, and NPPB events for the reflection and transmission outputs. Here PPB (NPPB) corresponds to $N\le2$ ($N>2$). The corresponding event is possible (denoted by "✓") or impossible ("✗"). Blue (red) color denotes the events for systems without (with) disorder.
• Figure 2: Correlation statistics of the transmission output. Probability of PA versus $\varphi$ and $W$ in (a), and versus $N$ and $W$ in (b). Inset in (a) displays the zoom-in of (a). $N=10$ in (a). In (b), $\varphi=0.0125\pi$, corresponding to the position where $\mathbb{P}(s<1)$ in (a) reach its maximum value. In all plots, the results are obtained from Monte Carlo integration. The numerical details are discussed in the Supplemental Materials supp.
• Figure 3: Correlation statistics of the transmission output. (a,b) Probability density functions. The chosen parameters are $\qty{N=3,\varphi=0.04\pi,W=0.14}$ in (a) and $\qty{N=3,\varphi=0,W=1}$ in (b). Inset of (a) shows the solutions $\qty{\Delta_1,\Delta_2,\Delta_3}$ that satisfy $g_{\rm T}=10^{-10}$. The solutions are constrained to $|\Delta_j|\le 1$. Inset of (b) shows $g^{\rm min}_{\rm T}$, corresponding to the minimum value of $g_{\rm T}$, for $\varphi=0$ (blue line) and for $\varphi=0.04\pi$ (orange line) with respect to chain size, dashed line is the numerical fit $N^{-2}$. (c,d) $P(10^{-3})$ versus $\varphi$ and $W$ in (c), and versus $N$ and $W$ in (d). $N=10$ in (c). In (d), $\varphi=0.006\pi$, corresponding to the position where $P(10^{-3})$ in (c) reach its maximum value.
• Figure 4: Correlation statistics of the transmission output. (a) The maximum values of $\mathbb{P}(s<1)$ and $P(10^{-3})$ versus $N$. Green circles (purple squares) display the maximum values of $\mathbb{P}(s<1)$ ($P(10^{-3})$) versus $N$. Green and purple solid lines are the numerical fits of $1-0.97N^{-1/40}$ and $0.04N^{1/2}$, respectively. (b) and (c) display $W$ and $\varphi$, respectively, where $\mathbb{P}(s<1)$ and $P(10^{-3})$ reach their maximum values. Black arrows point in the direction of increasing $N$. Inset of (c) displays the zoom-in of (c).
• Figure 5: Correlation statistics of the reflection output. $\mathbb{P}(s<1)$ versus $\varphi$ and $W$ in (a), and versus $N$ and $W$ in (b). The white solid curve in (a) denotes the regime where $\mathbb{P}(s<1)=1/2$. $N=10$ in (a). $\varphi=0.4\pi$ ($\varphi=0.1\pi$) in the top (bottom) of (b). (c) $\tilde{\mathbb{P}}(s<1)$ for different chain sizes and phases, with $\tilde{\mathbb{P}}(s<1)=\mathbb{P}(s<1)-2/3$. $W=100$ in (c). (d) $P(10^{-\mu})$ versus disorder strength. Solid lines in different colors correspond to different values of $\mu$, with $\mu=-3,\ -5,\ -7$. Dashed lines are the numerical slopes. In (d), we only consider the contributions from non-interacting transition paths, whose validation is discussed in supp. Results are obtained from $10^{10}$ disorder realizations.