Supercorrelated decay in a quasiperiodic nonlinear waveguide: From Markovian to non-Markovian transitions

TL;DR

This work studies mobility edges in the two-photon subspace of a 1D Bose-Hubbard chain with a quasiperiodic mosaic potential, focusing on doublon bound states and their spectral boundaries. By deriving an effective doublon Hamiltonian and using mosaic-mechanism analyses, it reveals mobility-edge boundaries that separate localized from extended bound states and shows how emitter dynamics transitions from Markovian to non-Markovian as the bath crosses these edges. Numerical simulations corroborate the analytical predictions, and a circuit-QED implementation with transmon chains is proposed to observe the interplay between interactions and disorder in quantum baths. The results extend single-photon ME concepts to interacting multi-photon regimes and highlight tunable light-matter dynamics in structured quantum environments.

Abstract

Mobility edges (MEs) are critical boundaries in disordered quantum systems that separate localized from extended states, significantly affecting transport properties and phase transitions. Although MEs are well-understood in single-photon systems, their manifestation in many-body contexts remains an active area of research. In this work, we investigate a one-dimensional Bose-Hubbard chain with a quasiperiodic potential modulating photon-photon interactions, effectively creating a mosaic lattice. We identify MEs for doublon states (i.e, bound photon pairs resulting from strong interactions) within the two-photon subspace. Our analytical solutions and numerical simulations confirm the existence of these MEs, extending single-photon MEs theories to the two-photon regime. Additionally, we analyze the dynamics of two emitters coupled to the waveguide, enabling the emission of supercorrelated photon pairs into the waveguide. Our findings reveal that coupling to extended states results in Markovian dynamics, characterized by exponentially supercorrelated decay, while coupling to localized states gives rise to non-Markovian dynamics, marked by suppressed decay and persistent oscillations. Here, a transition from Markovian to non-Markovian behavior occurs around the MEs of the doublons. Finally, we propose a feasible experimental implementation using superconducting circuits, providing a platform to observe the interplay between interactions and disorder in quantum systems.

Figures (12)

• Figure 1: Schematic of the system Hamiltonian. The red (blue) spheres represent lattice sites modulated (unmodulated) by a quasiperiodic potential, with $\kappa = 2$ in the diagram. The red dashed lines illustrate the strength of the quasiperiodic potential, while the varying sizes of the red spheres reflect the modulation intensity. The nonlinear local potential $U$ characterizes the effective photon-photon interaction; the $U$ term of the Eq. \ref{['BH']}, only plays a role for $N \geq 2$. The solid dark blue lines represent the hopping constant between waveguide sites. The two two-level emitters, each with the same frequency $\omega_e$, have energy levels $|g\rangle_i$ and $|e\rangle_i$ ($i = 1$ or $2$). These emitters are coupled to cavities $n_1$ and $n_2$ with a coupling strength $g$, where $n_1$ and $n_2$ are fixed, and $r_e = |n_1 - n_2|$. In this work, we focus on the specific case of $r_e = 0$, meaning that the two emitters are coupled at the same position.
• Figure 2: (a) Schematic of the energy bands in the two-photon subspace, where the red and blue solid lines represent the bound states Eq. \ref{['eqEB']} and scattering states Eq. \ref{['eqES']}, respectively. The purple dashed line indicates the threshold where the bound and scattering states no longer overlap. The parameters used here are $U = \pm 4J$ and $\omega_c = 0$. (b) Variation of the bandwidths of the bound and scattering states with the nonlinear potential $U$ in two-photon subspace. The bandwidth of the scattering states remains unchanged as $U$ varies, while the bandwidth of the bound states decreases with increasing $U$, eventually approaching a flat band.
• Figure 3: (a) The second-order hopping process of the doublon pair, where $\epsilon_1$, $\epsilon_2$, and $\epsilon_3$ represent the initial, intermediate, and final states of the hopping process, respectively. (b) The 1D quasiperiodic mosaic model for $\kappa = 2$ and $\kappa = 3$ is depicted. Red spheres represent waveguide sites modulated by the quasiperiodic potential, while blue spheres indicate unmodulated sites. The black solid lines denote the effective hopping amplitude $J_\text{eff}$, and $\lambda_j$ represents the quasiperiodic mosaic potential.
• Figure 4: (a) The $\mathcal{F}_d$ of different doublon and scattering states in the two-photon subspace as a function of the quasiperiodic potential strength $\lambda$. In the scattering states, all states are extended, while MEs appear in the bound states. (b) An enlarged view of the doublon states. Since our study focuses on doublon states, subsequent figures only display the energy bands of the doublon states. The model parameters are $\omega_c = 0$, $L = F_{18} = 2584$, $\kappa = 2$, and $U = -5J$. The black dashed lines represent the analytical solutions for MEs as described in Eq. \ref{['eq14']}.
• Figure 5: (a) and (b) illustrate the doublon states and MEs for different values of $U$. In panel (a), $U = -5J$, while in panel (b), $U = -10J$. As $U$ increases, the analytical solutions for MEs align more closely with the numerical solutions of the model. Model parameters are $\omega_c = 0$, $L = F_{18} = 2584$, and $\kappa = 3$. The black dashed lines represent the analytical solutions for MEs as described in Eq. \ref{['eq15']}. In panel (a), the three distinct purple solid lines, labeled as A, B, and C, represent the different eigenfrequencies of the two emitters, as defined in Eq. \ref{['H_total']}.