Dynamics of spontaneous emission in a single-end photonic waveguide

We investigate the spontaneous emission of a two-level system, e.g., an atom or atomlike object, coupled to a single-end, i.e., a semi-infinite, one-dimensional photonic waveguide such that one end behaves as a perfect mirror while light can pass through the opposite end with no backreflection. Through a quantum microscopic model we show that such geometry can cause nonexponential and long-lived atomic decay. Under suitable conditions, a bound atom-photon stationary state appears in the atom-mirror interspace so as to trap a considerable amount of initial atomic excitation. Yet this can be released by applying an atomic frequency shift, causing a revival of photon emission. The resilience of such effects to typical detrimental factors is analyzed.

Figure 1

Setup. A semi-infinite waveguide, whose end lies at $x=0$, coupled to a two-level system at $x=x_0$.

Figure 2

Atomic excitation probability $P_e\left(t\right)={\left|\varepsilon \left(t\right)\right|}^2$ vs time (in units of $1/\Gamma$) in the cases $\Gamma t_d=2$ (a) and $\Gamma t_d=0.1$ (b) and for $t_d=\infty$ (i.e., no mirror; solid black line), $\varphi =2n\pi$ (blue dashed line), $\varphi =\pi /2+2n\pi$ (red dotted), and $\varphi =(2n+1)\pi$ (green dash-dotted). In (a), only the range $t\ge t_d$ is shown; at earlier times the behavior does not depend on $\varphi$ and is the continuation of the black solid line.

Figure 3

Output field intensity vs $t^{\prime }=t-d/\upsilon$ in arbitrary units for $\varphi =2n\pi$ under an applied frequency shift $\Delta \left(t\right)$ (this is plotted in the insets in units of $\Gamma$). (a) $\Gamma t_d=2$ and a steplike $\Delta \left(t\right)$. (b) $\Gamma t_d=0.1$ and a sinusoidal $\Delta \left(t\right)$. In either case, a revival of the photon emission occurs when the frequency shift is switched on. Note that, in line with Eq. (17), at $t=t_d$ the intensity exhibits a discontinuity [particularly visible in (a)].

Figure 4

Robustness of $P_e\left(t\right)={\langle \left|\varepsilon \left(t\right)\right|}^2\rangle$ vs time for $\varphi =2n\pi$. We have studied the dynamics of the excitation amplitude via Eq. (C1), using ${\Gamma }_{\mathrm{tot}}\equiv \Gamma +{\Gamma }_{\mathrm{ext}}$, and $r=R+i\sqrt{R(1-R)}$. (a) We set ${\Gamma }_{\mathrm{tot}}{t}_d=2$, $R=0.98$, $\delta \omega =0.25{\Gamma }_{\mathrm{tot}}$ and vary the ratio between ${\Gamma }_{\mathrm{ext}}$ and $\Gamma$, keeping ${\Gamma }_{\mathrm{tot}}$ fixed. (b) We fix $R=1,{\Gamma }_{\mathrm{ext}}=0$, $\Gamma t_d=0.1$, and vary the dephasing rate $\delta \omega$.