Engineering photonic dispersion relation and atomic dynamics in waveguide QED setup via long-range hoppings

TL;DR

The paper demonstrates that long-range ${\rm JNN}$ hoppings in a 1D coupled-resonator waveguide can engineer photonic dispersion with a chiral, linear region, enabling high-fidelity, dispersion-controlled propagation of Gaussian wave packets. By tuning the phases $\theta_j$, the authors realize chiral dispersions where $\omega(k) \neq \omega(-k)$ and derive explicit hopping parameters that suppress higher-order terms, yielding an effectively linear spectrum with constant group velocity $v_g$. They show practical atomic control: directional radiation and efficient absorption between a radiating and a target atom coupled to neighboring resonators, with weak, moderate, and wave-packet-shaping protocols achieving exponential decay, high absorption probability (up to $\sim0.88$), and complete excitation transfer, respectively. Furthermore, the framework generalizes to symmetric and arbitrary dispersion forms, including quadratic and cubic dispersions, by appropriate choice of ${\rm JNN}$ hoppings, offering a unified approach to simulate atom–environment couplings with tailored dispersion in platforms such as superconducting circuits.

Abstract

Non-trivial dispersion relations engineered in photonic waveguide for the precise control of atomic dynamics has recently attracted considerable attention. Here, we study a system in which atoms are coupled to one-dimensional coupled-resonator waveguides with long-range hoppings. By carefully engineering the jth-order nearest neighbor (JNN) hoppings between resonators, we construct linear dispersion relations with the chiral characteristic. To quantify the degree of linearity, we analyze the propagation fidelities of Gaussian wave packets in these waveguides. Furthermore, we demonstrate that such coupled-resonator waveguides can serve as versatile platforms for enabling directional atomic radiation and absorption. Beyond linear dispersion relations, more general forms, including quadratic and cubic relations, can also be achieved through tailored JNN-hoppings. Our study thus provides a unified framework for simulating atom-environment couplings with arbitrary dispersion relations.

Figures (7)

• Figure 1: Schematic diagram for the 1D coupled resonator waveguide with the long-range hoppings coupled to the two atoms.
• Figure 2: (a) and (b) The linear dispersion relation $\omega(k)$ and its group velocity $v=\partial \omega(k)/\partial k$ versus $k$ for the waveguide with ${\rm JNN}$-hoppings and ${\rm NN}$-hoppings, respectively. (c) The real part ${\rm Re}(\langle\psi_i(0)|a^{\dag}_{k}|0\rangle)$ and imaginary part ${\rm Im}(\langle\psi_i(0)|a^{\dag}_{k}|0\rangle)$ in the momentum space for the wave packet in Eq. (\ref{['wavepacket0']}). (d) and (e) The photon population distribution of $|\langle \psi_f(t)|a^{\dag}_{l}|0\rangle|^2$ for different time $t$ in the waveguide with ${\rm JNN}$-hoppings and ${\rm NN}$-hoppings, respectively. (f) The propagating fidelity ${\rm PF}$ of the initial wave packet $|\psi_i(0)\rangle$ versus time $t$. The parameters are set as: $L=300$, $\omega_0=0$, $\sigma=3$, $l_0=10$, $k_0=0$, $J=5$. $h_1=5v_g/6$, $h_2=-5v_g/21$, $h_3=5v_g/84$, $h_4=-5v_g/504$, $h_5=v_g/1260$ for (a), (d), (f). $h_1=v_g/2$, $h_2=0$, $h_3=0$, $h_4=0$, $h_5=0$ for (b), (e), (f).
• Figure 3: (a) The photon population distribution of $|\langle \psi_{w}(t)|a^{\dag}_{l}|0\rangle|^2$ versus time $t$ in the waveguide. (b) The population distribution of $|\langle \psi_{w}(t)|\sigma_{1,2}^{+}|0\rangle|^2$ versus time $t$. (c) The photon population distribution of $|\langle \psi_{m}(t)|a^{\dag}_{l}|0\rangle|^2$ versus time $t$ in the waveguide. (d) The population distribution of $|\langle \psi_{m}(t)|\sigma_{1,2}^{+}|0\rangle|^2$ versus time $t$. The parameters are set as: $L=400$, $l_1=50$, $l_1=150$, $g_1=g_2=0.5v_g$, $\omega_0=\omega_1=\omega_2$, $J=5$, $h_1=5v_g/6$, $h_2=-5v_g/21$, $h_3=5v_g/84$, $h_4=-5v_g/504$ and $h_5=v_g/1260$. $g_1=g_2=0.1v_g$ for (a) and (b). $g_1=g_2=0.5v_g$ for (c) and (d).
• Figure 4: (a) $g_1(t)$ and $g_2(t)$ versus time $t$. (b) The photon population distribution of $|\langle \psi_{u}(t)|a^{\dag}_{l}|0\rangle|^2$ versus time $t$ in the waveguide. (c) The population distribution of $|\langle \psi_{u}(t)|\sigma_{1,2}^{+}|0\rangle|^2$ versus time $t$. (d) The photon population distribution of $|\langle \psi_u(t)|a^{\dag}_{l}|0\rangle|^2$ for different time $t$ in the waveguide with ${\rm JNN}$-hoppings. The parameters are set as: $L=200$, $l_1=50$, $l_1=150$, $\omega_0=\omega_1=\omega_2$, $g_{\rm max}=0.2$, $J=5$, $h_1=5v_g/6$, $h_2=-5v_g/21$, $h_3=5v_g/84$, $h_4=-5v_g/504$ and $h_5=v_g/1260$.
• Figure 5: (a) The quadratic dispersion relation $\omega(k)$ and the function $f=k^2$ versus $k$. (b) The cubic dispersion relation $\omega(k)$ and the function $f=k^3$ versus $k$. The parameters are set as: $\omega_0=0$, $J=5$. $h_1=-826v_g/1069$, $h_2=586v_g/507$, $h_3=-242v_g/501$, $h_4=140v_g/1259$ and $h_5=-2v_g/177$ for (a). $h_1=843v_g/1091$, $h_2=-293v_g/507$, $h_3=-280v_g/1739$, $h_4=-35v_g/1259$ and $h_5=2v_g/885$ for (b).