Non-Markovian dynamics with a driven three-level giant atom in a semi-infinite photonic waveguide

TL;DR

The paper develops an analytical treatment of non-Markovian dynamics for a driven Λ-type three-level giant atom interacting with a semi-infinite photonic waveguide via multiple coupling points. By solving the resulting delay-differential equations with Laplace transforms, it reveals two independent bound-state conditions that give rise to a static bound state and a periodic bound-state pair, with their formation tunable by the driving field and time-delay. It also analyzes the infinite-waveguide limit, where a single bound-state condition persists, and generalizes the results to networks of multiple giant atoms, highlighting the potential to engineer multi-atom non-Markovian dynamics and quantum information storage via bound-state control. The findings illuminate how time-delayed feedback from mirror-reflected photons shapes non-Markovian decay and bound-state physics in structured waveguide environments, with implications for quantum networks and coherent feedback protocols.

Abstract

The non-Markovian effects of open quantum systems subjected to external environments are deemed to be valuable resources in quantum optics and quantum information processing. In this work, we investigate the non-Markovian dynamics of a three-level giant atom coupling with a semi-infinite photonic waveguide through multiple coupling points and driven by a classical driving field. We derive the analytical expressions for the probability amplitudes of the driven three-level giant atom and obtain two independent conditions. We find two different types of bound states (including the static bound states and the periodic equal-amplitude oscillating bound states) and discuss the physical origins of the bound states formation. Moreover, we discuss the case of the driven three-level giant atom interacting with the infinite photonic waveguide, where there is only one purely imaginary solution (i.e., only one bound state condition exists) for its complex frequency (coming from the absence of mirror at one end of the waveguide) compared to that of a driven three-level giant atom coupling with a semi-infinite photonic waveguide. With this, we also find two different types of bound states, including the static bound state and the periodic equal-amplitude oscillating bound states. Finally, the above results are generalized to a more general model involving a semi-infinite photonic waveguide coupling with an arbitrary number of noninteracting three-level giant atoms driven by the driving fields. The proposed protocol could provide a pathway to precisely elucidate the non-Markovian dynamics of driven, multi-level giant atoms coupled to semi-infinite or infinite photonic waveguides.

Figures (15)

• Figure 1: (Color online) Schematic of the setup. A $\Lambda$-type three-level giant atom (driven by a classical driving field) couples with a one-dimensional semi-infinite photonic waveguide (terminated by a perfect mirror) through $N$ coupling points. The transition between the ground state $|g\rangle$ (the eigenfrequency $\omega_g$ is set to be zero) and the excited state $|x\rangle$ (the eigenfrequency $\omega_x$) is coupled to a photonic waveguide positioned at $x_n$ with the coupling coefficient $g_{kn}$ ($n = 1, 2,\cdot \cdot \cdot,N$). The distance between the adjacent coupling points is $x_0$, which leads to $x_n=nx_0$. The transition between the excited state $|x\rangle$ and the metastable state $|e\rangle$ (the eigenfrequency $\omega_e$) is driven by a laser field with the frequency $\omega_l$ and the driving strength $G$.
• Figure 2: The blue-solid line in Fig. \ref{['y1y2']}(a) and (c) corresponds to $y_1=\cot(j\pi/N)$, while the red circle represents $y_2=4j\pi/\Gamma N^2\tau_0-2\omega_x/\Gamma N+G^2/\Gamma(N\omega_e+N\omega_l-2j\pi/\tau_0)$, where $j$ takes the integer number. The intersections between $y_1$ and $y_2$ are the solutions of the transcendental equation in Eq. (\ref{['omegak1']}). The values of $j$ for the points in Fig. \ref{['y1y2']}(a)-(d) are $j_{p_1}=4$, $j_{p_2}=3$, $j_{p_3}=23$, $j_{p_4}=26$, $j_{p_5}=27$, and $j_{p_6}=30$, respectively. The other parameters chosen are (a) $N=3$, $G \tau_0= 0.6\pi$, $\omega_x \tau_0=2\pi$, $\omega_e \tau_0=0.5818\pi$, $\omega_l \tau_0=1.2 \pi$, and $\Gamma \tau_0=0.3 \pi$; (c) $N=6$, $G \tau_0= 2.8\pi$, $\omega_x \tau_0=7.0366\pi$, $\omega_e \tau_0=0.2\pi$, $\omega_l \tau_0=2.5 \pi$, and $\Gamma \tau_0=0.1825\pi$. The blue-solid line in Fig. \ref{['y1y2']}(b) and (d) corresponds to $z_1=\cot[j\pi/(N+1)]$, while the red circle represents $z_2=4j\pi/\Gamma(N+1)^2\tau_0-2\omega_x/\Gamma(N+1)+G^2/\Gamma[(N+1)\omega_e+(N+1)\omega_l-2j\pi/\tau_0]$. The solutions of the transcendental equation given by Eq. (\ref{['omegak2']}) occur at the intersections between $z_1$ and $z_2$. The other parameters chosen are (b) $N=3$, $G \tau_0= 0.5\pi$, $\omega_x \tau_0=1.2\pi$, $\omega_e \tau_0=0.5222\pi$, $\omega_l \tau_0=0.7\pi$, and $\Gamma \tau_0=0.3 \pi$; (d) $N=6$, $G \tau_0= 2.8\pi$, $\omega_x \tau_0=6.9350\pi$, $\omega_e \tau_0=02\pi$, $\omega_l \tau_0=2.5 \pi$, and $\Gamma \tau_0=0.1079\pi$.
• Figure 3: (Color online) Static bound states for the three-level giant atom with the number of coupling points $N=3$. $j=4$ is chosen from Fig. \ref{['y1y2']}(a). The red-dashed line corresponds to the analytical solutions in Eqs. (\ref{['CxAnaN']}) and (\ref{['CeAnaN']}), while the blue-solid line represents the numerical simulation with Eqs. (\ref{['dotcx']}) and (\ref{['dotce']}), respectively. In Fig. \ref{['StaticBS']}(a) and (c), we take $G \tau_0= 0.6\pi ¦£$. In Fig. \ref{['StaticBS']}(b) and (d), we take $G \tau_0= -0.6\pi ¦£$. The other parameters chosen are $\omega_x \tau_0=2\pi$, $\omega_e \tau_0=0.5818\pi$, $\omega_l \tau_0=1.2 \pi$, $\Gamma \tau_0=0.3 \pi$, ${C_x}(0) = \sqrt {0.8}$, and ${C_e}(0) = \sqrt {0.2}$.
• Figure 4: (Color online) Static bound states for the three-level giant atom with the number of coupling points $N=3$. $j=3$ is chosen from Fig. \ref{['y1y2']}(b). The red-dashed line corresponds to the analytical solutions in Eqs. (\ref{['CxAnaN1']}) and (\ref{['CeAnaN1']}), while the blue-solid line denotes the numerical simulation with Eqs. (\ref{['dotcx']}) and (\ref{['dotce']}), respectively. In Fig. \ref{['OneSolutionN1']}(a) and (c), we take $G \tau_0= 0.5\pi ¦£$. In Fig. \ref{['OneSolutionN1']}(b) and (d), we take $G\tau_0= -0.5\pi ¦£$. The other parameters chosen are $\omega_x \tau_0=1.2\pi$, $\omega_e \tau_0=0.5222\pi$, $\omega_l \tau_0=0.7 \pi$, $\Gamma \tau_0=0.3 \pi$, ${C_x}(0) = \sqrt {0.8}$, and ${C_e}(0) = \sqrt {0.2}$.
• Figure 5: (Color online) The field intensity $P(x,t) = |\phi(x,t)|^2$ in Eq. (\ref{['phi']}) as a function of time $t$ and position $x$. In Fig. \ref{['3D']}(a) and (b), we take $G\tau_0= 0.6\pi, -0.6\pi$, respectively, while $j=4$ is chosen from Fig. \ref{['y1y2']}(a). In Fig. \ref{['3D']}(c) and (d), we take $G\tau_0= 0.5\pi, -0.5\pi$, respectively, while $j=3$ is obtained by Fig. \ref{['y1y2']}(b). The parameters chosen are (a) and (b) $\omega_x \tau_0=2\pi$, $\omega_e \tau_0=0.5818\pi$, and $\omega_l \tau_0=1.2 \pi$; (c) and (d) $\omega_x \tau_0=1.2\pi$, $\omega_e \tau_0=0.5222\pi$, and $\omega_l \tau_0=0.7 \pi$. The other parameters chosen are $\Gamma \tau_0=0.3 \pi$, ${C_x}(0) = \sqrt {0.8}$, and ${C_e}(0) = \sqrt {0.2}$. The top row corresponds to bound state condition in Eq. (\ref{['bs1']}), while the bottom row represents bound state condition in Eq. (\ref{['bs2']}).