Coherent control of non-Markovian photon-resonator dynamics

We study the unitary time evolution of photons interacting with a dielectric resonator using coherent control pulses. We show that non-Markovianity of transient photon dynamics in the resonator subsystem may be controlled to within a photon-resonator transit time. In general, appropriate use of coherent pulses and choice of spatial subregion may be used to create and control a wide range of non-Markovian transient dynamics in photon-resonator systems.

Figure 1

(a) Sketch of a subsystem with inputs and outputs to continuum states. (b) Symmetric Fabry-Pérot resonator subsystem divided into spatial regions $A$, $B$, and $C$ separated by a quarter-wavelength lossless dielectric characterized by refractive index $n_{\mathrm{r}}$. The resonant wavelength is ${\lambda }_0$.

Figure 2

(a) Space-time photon energy density plot of a rectangular pulse incident on a Fabry-Pérot resonator. The resonator is of length $L_B$ (indicated by two arrows). (b) $\left|\psi \right(x_R,t\left)\right|$ (arbitrary scale) as a function of time detected at position $x_R$ far to the right of the resonator. The field decay constant $2{\tau }_{\mathrm{Q}}=229\mathrm{fs}$ is modulated by stepwise response at the resonant cavity round-trip time ${\tau }_{\mathrm{RT}}=30{\tau }_0=150\mathrm{fs}$. Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, and $T_0{\omega }_0=900$.

Figure 3

(a) Space-time photon energy density plot of a short rectangular pulse incident on the Fabry-Perot resonator showing ring-down. (b) Space-time resonant photon ray trace illustrating ring-down in the form of multiple transmitted and reflected amplitudes. Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, and $T_0{\omega }_0=60$.

Figure 4

(a) Space-time photon energy density plot showing lead and control pulses. The control pulse eliminates ring-down by removing all photon energy density in the cavity after exactly one round-trip time ${\tau }_{\mathrm{RT}}$. There is just one transmitted photon pulse. (b) Space-time resonant photon ray trace showing lead and control amplitudes configured to eliminate ring-down. Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, and $T_0{\omega }_0=60$.

Figure 5

(a) Transmitted photon energy density as a function of time with no control (as in Fig. 3). (b) Same as (a), but with a control pulse to eliminate ring-down by removing all photon energy density in the cavity after exactly one round-trip time ${\tau }_{\mathrm{RT}}$. There is just one transmitted photon pulse. Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, and $T_0{\omega }_0=60$. The photon energy density scale is arbitrary.

Figure 6

(a) Space-time resonant photon ray trace showing incident lead and control amplitudes configured to perform a finite geometric sum. (b) Space-time resonant photon ray trace showing incident lead photon and control amplitudes configured to create a finite divergent geometric series.

Figure 7

(a) Normalized measure $D\left(t\right)$ for rectangular photon pulse ${\Psi }_1(x,t)$ and ${\Psi }_2(x,t)={\Psi }_1(x,t+{\tau }_{\mathrm{M}})$ freely propagating through spatial region $A$ of length $L_{\mathrm{A}}=160\times {\lambda }_0$. When both pulses are simultaneously in spatial region $A$, then $D\left(t\right)=1$. (b) $ID\left(t\right)$, the measure of non-Markovian dynamics for the pulse in (a). Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, $T_0{\omega }_0=60$, and ${\tau }_{\mathrm{M}}/{\tau }_0=60$.

Figure 8

(a) $D\left(t\right)$ for rectangular photon pulse ${\Psi }_1(x,t)$ and ${\Psi }_2(x,t)={\Psi }_1(x,t+{\tau }_{\mathrm{M}})$ in spatial region $A$ [blue (dark gary)] to the left of the dielectric resonator and region $C$ [red (light gray)] to the right of the resonator. $L_{\mathrm{A}}=160\times {\lambda }_0$ and $L_{\mathrm{C}}=120\times {\lambda }_0$. (c) $ID\left(t\right)$ for uncontrolled ring-down of $D\left(t\right)$ shown in (a). The calculated duration of non-Markovianity in region $A$ to the left of the dielectric resonator is limited by the pulse leaving the domain. (b) $D\left(t\right)$ and (d) $ID\left(t\right)$ illustrate the use of a single control pulse to reduce non-Markovianity in transient dynamics by removing all photon energy in the resonator after exactly one cavity round-trip time ${\tau }_{\mathrm{RT}}$. Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, $T_0{\omega }_0=60$, and ${\tau }_{\mathrm{M}}/{\tau }_0=60$.

Figure 9

$ID\left(t\right)$ for a resonator with cavity length $L_{\mathrm{B}}=120\times {\lambda }_0$, which defines region $B$, and a half space in the resonator cavity of length $L_{\mathrm{B}}/2$ adjacent to the left mirror, which defines region $B^{\prime }$. In the absence of a coherent control pulse, $ID\left(t\right)$ is purely Markovian for subspace $B$ and is non-Markovian for subspace $B^{\prime }$. Photon pulse parameters are $\hslash {\omega }_0=0.827\mathrm{eV}$, $\hslash \Delta {\omega }_{\mathrm{s}}=0.207\mathrm{eV}$, $T_0{\omega }_0=60$, and ${\tau }_{\mathrm{M}}/{\tau }_0=30$.