Electromagnetically induced transparency from first-order dynamical systems

We show how a strongly driven single-mode oscillator coupled to a first-order dynamical system gives rise to induced absorption or gain of a weak probe beam and associated fast or slow light depending on the detuning conditions. We derive the analytic solutions to the dynamic equations of motion, showing that the electromagnetically induced transparency- (EIT-) like response is a general phenomenology, potentially occurring in any nonlinear oscillator coupled to first-order dynamical systems. The resulting group delay (or advance) of the probe is fundamentally determined by the system damping rate. To illustrate the practical impact of this general theoretical framework, we quantitatively assess the observable consequences of either thermo-optic or free-carrier dispersion effects in conventional semiconductor microcavities in control/probe experiments, highlighting the generality of this physical mechanism and its potential for the realization of EIT-like phenomena in integrated and cost-effective photonic devices.

Figure 1

Scheme of the physical system under analysis. An optical resonator, characterized by an intracavity field amplitude $a\left(t\right)$, bare resonance frequency ${\omega }_0$, and intrinsic decay rate $\mathrm{\Gamma }$, is coherently driven by a control (red arrow) and a probe (blue arrow) input fields. The confined mode is coupled to a first-order system (yellow), uniquely characterized by a dynamical variable $x\left(t\right)$ and an incoherent dissipation rate $\gamma$ through the coupling parameters $G$ and $\beta$.

Figure 2

Normalized intracavity energy as a function of the detuning between the control and the bare cavity frequencies $\mathrm{\Delta }={\omega }_c-{\omega }_0$ in the presence of a constant driving field ${\overline{s}}_{\mathrm{in}}$. The increased coupled power (and, consequently, resonance shift) at lower detuning results in a characteristic sawtooth lineshape typical of bistable systems. The blue line represents the system response at low control power (peak cavity energy: $|{\overline{a}}_{}{|}^2={10}^{-3}{\left|{\overline{a}}_b\right|}^2$), the red (peak $|{\overline{a}}_{}{|}^2={\left|{\overline{a}}_b\right|}^2$) and yellow (peak $|{\overline{a}}_{}{|}^2=2{\left|{\overline{a}}_b\right|}^2$) curves represent the system response at increasing input power.

Figure 3

(a) Susceptibility of the induced transparency as a function of the detuning between the control field and the shifted resonator mode frequencies. (b)–(d) Oscillation amplitude/Stokes and anti-Stokes sidebands amplitude as a function of the probe-control detuning frequency, $\mathrm{\Omega }$. The calculations were performed with system parameters $\gamma ={10}^{-3}\mathrm{\Gamma },G=-{10}^{-1}\mathrm{\Gamma },\beta ={10}^{-2}\mathrm{\Gamma }$, and driving parameters $|s_p{|}^2=0.1{\left|{\overline{s}}_{\mathrm{in}}\right|}^2,\overline{\mathrm{\Delta }}=+\mathrm{\Gamma }/2$, and $|{\overline{a}}_{}{|}^2=\left(\right|{\overline{a}}_b{|}^2,3|{\overline{a}}_b{|}^2,5|{\overline{a}}_b{|}^2)$ from the blue to the yellow curve, respectively. All curves are spaced by a vertical offset of 0.15 for clarity.

Figure 4

Output signal amplitude (normalized by the peak cavity response at $\mathrm{\Omega }=\overline{\mathrm{\Delta }}$) and phase in blue- (a)–(c) and red- (d)–(f) detuning regimes, respectively. The calculations assume the following physical parameters: $\gamma ={10}^{-3}\mathrm{\Gamma },G=-{10}^{-1}\mathrm{\Gamma },\beta ={10}^{-2}\mathrm{\Gamma }$, and driving parameters $\overline{\mathrm{\Delta }}=+\mathrm{\Gamma }/2,|{\overline{a}}_{}{|}^2=\left(\right|{\overline{a}}_b{|}^2,3|{\overline{a}}_b{|}^2,5|{\overline{a}}_b{|}^2)$ (a)–(c); $\overline{\mathrm{\Delta }}=-\mathrm{\Gamma }/2,|{\overline{a}}_{}{|}^2=\left(0.2\right|{\overline{a}}_b{|}^2,0.4|{\overline{a}}_b{|}^2,0.6|{\overline{a}}_b{|}^2)$ (d)–(f).

Figure 5

(a) Calculated trend for dip visibility as a function of the $|{\overline{a}}_{}{|}^2\tilde{\chi }$ product, which value is determined by the control field intensity and detuning conditions. Negative values indicate a peak spectral feature. (b) Calculated trend for peak phase shift Eq. (19). The $+$ sign, associated with the local maximum/minimum at $\mathrm{\Omega }=+\sqrt{\gamma {\mathrm{\Gamma }}_{IT}}$, was chosen. Red (blue) shaded regions are associated with the respective detuning regimes.

Figure 6

Discretized heat diffusion model. Each shell is characterized by a heat-capacity $C_{p,i}$, and it exhibits a heat diffusion constant $K_i$ and a temperature offset $\mathrm{\Delta }{T}_i$ with respect to the next outer shell.

Figure 7

Comparison between the TOIT response for the generalized (blue solid line) and the simplified (red dashed line) models, respectively. (a) Thermal response function $\xi \left(\mathrm{\Omega }\right)$ as a function of the probe-control detuning. (b) Temperature oscillation amplitude. (c) Amplitude and (d) phase response for the output signal. The calculation assumes the following parameters: $G=-{10}^{-1}\mathrm{\Gamma },\beta ={10}^{-2}\mathrm{\Gamma },\overline{\mathrm{\Delta }}=+\mathrm{\Gamma }/2$, and $|{\overline{a}}_{}{|}^2={\left|{\overline{a}}_b\right|}^2$. For the nonrefined model ($n=1$) we chose $\gamma ={10}^{-3}\mathrm{\Gamma }$, whereas for the refined model ($n=3$) we chose ${\gamma }_{1,1}=3\times {10}^{-3}\mathrm{\Gamma },{\gamma }_{1,2}={\gamma }_{2,2}={\gamma }_{1,1}/4,{\gamma }_{2,3}={\gamma }_{3,3}={\gamma }_{1,1}/8$. Note that by this choice of parameters, the static temperature offset $\overline{\mathrm{\Delta }T}$ is the same in both scenarios.