Route from spontaneous decay to complex multimode dynamics in cavity QED

We study the non-Markovian quantum dynamics of an emitter inside an open multimode cavity, focusing on the case where the emitter is resonant with high-frequency cavity modes. Based on a Green's-function technique suited for open photonic structures, we study the crossovers between three distinct regimes as the coupling strength is gradually increased: (i) overdamped decay with a time scale given by the Purcell modified decay rate, (ii) underdamped oscillations with a time scale given by the effective vacuum Rabi frequency, and (iii) pulsed revivals. The final multimode strong-coupling regime (iii) gives rise to quantum revivals of the atomic inversion on a time scale associated with the cavity round-trip time. We show that the crucial parameter to capture the crossovers between these regimes is the nonlinear Lamb shift, accounted for exactly in our formalism.

Figure 1

Two-level system with transition frequency ${\omega }_a$ inside an open cavity.

Figure 2

Route from the single-mode to the multimode coupling regime for different coupling strengths $\gamma$. The top row shows the dimensionless kernel function $U\left(\omega \right)$ [Eq. (4)]. The bottom row shows the dimensionless nonlinear Lamb shift $\delta \left(\omega \right)$ [Eq. (5)] for the same $\omega$ interval as above (note the different zooms for the three columns). The left column shows the weak-coupling regime for $\gamma =4\times {10}^{-6}$ with a single peak in $U\left(\omega \right)$ (Purcell modified spontaneous decay). The middle column shows the strong-coupling regime for $\gamma =2.5\times {10}^{-3}$ with a well-resolved Rabi splitting in $U\left(\omega \right)$ (regime of damped Rabi oscillations). The right column shows the multimode strong-coupling regime for $\gamma =1.44$ with a multipeak structure in $U\left(\omega \right)$ consisting of almost equidistant peaks (regime of revivals). Closed circles label resonance values ${\omega }_r$ of the kernel $U\left(\omega \right)$ occurring at the intersections between the Lamb shift $\delta \left(\omega \right)$ and the dashed line $(\omega -{\omega }_a)/\gamma$. At open circles (not shown in right column) such intersections are nonresonant and do not lead to a corresponding peak in $U\left(\omega \right)$ (see the text). The transition frequency ${\omega }_a\approx 19\pi$ of the TLS coincides with the tenth resonance of the spectral function $F\left(\omega \right)$ [Eq. (2)]. The reflectivity parameter $\eta =0.1$ is such that the mirror reflectivity $|r\left({\omega }_a\right){|}^2=0.9$. Frequency $\omega$ is measured here in units of the inverse half the cavity round-trip time.

Figure 3

Temporal evolution of the excited-state probability ${\left|c\left(t\right)\right|}^2$ of the TLS for the three cases shown in Fig. 2. Time $t$ is measured here in units of half the cavity round-trip time. The left panel shows the weak-coupling regime ($\gamma =4\times {10}^{-6}$) featuring spontaneous decay (also shown on a log-lin scale in the inset). The middle panel shows the strong-coupling regime ($\gamma =2.5\times {10}^{-3}$) with damped Rabi oscillations. The right panel shows the multimode strong-coupling regime ($\gamma =1.44$) featuring pulsed revivals at multiple integers of half the cavity round-trip time.

Figure 4

Destruction of the multimode strong-coupling regime by broadening of the peaks in the spectral function (2). The left column shows the mirror reflectivity parameter $\eta =0.9$ (as in right panel of Fig. 2). The middle column shows $\eta =0.3$. The right column shows $\eta =0.015$. The top row shows the dimensionless kernel function $U\left(\omega \right)$. The bottom row shows the corresponding excited-state probability ${\left|c\left(t\right)\right|}^2$ of the TLS versus normalized time $t$. The transition frequency ${\omega }_a\approx 19\pi$ and the coupling strength $\gamma =1.44$ are the same as in the right panels of Figs. 2 and 3.

Figure 5

Comparison between the results obtained from a single Volterra equation [dark gray (red) curves] and from the system-and-bath formalism [gray (orange) curves]. The calculations are performed for the 1D geometry presented in Fig. 1 with the mirror reflectivity parameter set to $\eta =0.18$. The left panel shows $\gamma =2.5\times {10}^{-3}$ (regime of Rabi oscillations) and the right panel $\gamma =1.44$ (multimode strong-coupling regime). Time $t$ is measured in units of half the cavity round-trip time.

Figure 6

Contour completion in the complex plane $s=\sigma +i\omega$ for the calculation of the inverse Laplace transform (A2). Those contours that give nonzero contribution are designated by numbers.