All-cavity electromagnetically induced transparency and optical switching: Semiclassical theory

The transmission of a probe field experiencing electromagnetically induced transparency and optical switching in an atomic medium enclosed in an optical cavity is investigated. Using a semiclassical input-output theory for the interaction between an ensemble of four-level atoms and three optical cavity fields coupled to the same spatial cavity mode, we derive the steady-state transmission spectra of the probe field and discuss the dynamics of the intracavity field buildup. The analytical and numerical results are in good agreement with recent experiments with ion Coulomb crystals [M. Albert et al., Nature Photon. 5, 633 (2011)].

Figure 1

(a) Atomic four-level structure considered. (b) System considered: an ensemble of four-level atoms, positioned in a linear optical cavity, interact with three optical fields. In the all-cavity geometry all three fields are injected into the cavity and coupled to the same cavity mode. In an alternative standard geometry, only the probe field $a_p$ is injected into the cavity, while the control and switching fields, $a_c$ and $a_s$, are free propagating with waists much larger than that of the ensemble (see text for details).

Figure 2

Real (blue) and imaginary (red) parts of the probe first-order susceptibility as a function of the probe detuning $\Delta$ (solid line: all-cavity EIT [Eq. (23)], dashed line: standard EIT [Eq. (25)]). The parameters $(g_p\sqrt{N},\gamma ,{\gamma }_0,{\Omega }_c)=\left(2\pi \right)\times (16,11.2,6\times {10}^{-4},6)$ MHz are similar to those used in the experiments of Ref. [61].

Figure 3

Upper panel: Probe field normalized transmission spectra for an empty cavity (dotted line), a cavity containing a uniform density ensemble interacting with the probe field only (dashed black line) and a cavity containing a uniform density ensemble interacting with a probe and a control field in an all-cavity (solid black line) and standard EIT situation (gray solid/dashed lines). Parameters: $(g_p\sqrt{N},\gamma ,{\gamma }_0,{\Omega }_c,\kappa )=\left(2\pi \right)\times (16,11.2,6\times {10}^{-4},6,2.2)$ MHz. The solid and dashed grey curves show the standard EIT situation with control field Rabi frequencies ${\Omega }_c=\left(2\pi \right)6$ MHz and ${\Omega }_c={\Omega }_{c,\text{eff}}=\left(2\pi \right)6/2.2$ MHz, respectively. Lower panel: Spectra enlarged around $\Delta =0$. As discussed in the text, the latter value of the effective Rabi frequency was chosen to illustrate the lineshape difference in a situation when the all-cavity and standard EIT resonance curves have comparable halfwidths at half-maximum.

Figure 4

Reflection spectra for the same configurations and parameters as in Fig. 3, for a cavity with non negligible roundtrip absorption losses [$({\kappa }_H,{\kappa }_L,{\kappa }_A)=\left(2\pi \right)\times (1.5,0,0.7)$ MHz].

Figure 5

Probe field normalized transmission as a function of time (solid black line: all-cavity case, solid grey line: standard case, dotted grey line: empty cavity) for a resonant probe field ($\Delta =0$). Parameters as in Fig. 3. The control field Rabi frequency is scaled by a factor 2.2 in the standard situation.

Figure 6

Optical switching susceptibility: real (blue) and imaginary (red) parts around two-photon resonance. The solid line shows the all-cavity switching for $(g_p\sqrt{N},\gamma ,\kappa ,{\gamma }_0,{\Omega }_c,{\gamma }_s,{\Delta }_s,{\Omega }_s)=\left(2\pi \right)\times (16,11.2,2.2,0.0006,4,11,4300,40)$ MHz. The dashed line shows the standard switching situation for the same parameters, but with the control and switching field Rabi frequencies scaled by a factor 2.2 for comparison.

Figure 7

Probe field normalized transmission spectrum for different switching field Rabi frequencies ${\Omega }_s$: (a) All-cavity switching. ${\Omega }_s=\left(2\pi \right)\times (0,15,25,40)$ MHz from left to right. Other parameters as in Fig. 6. (b) Standard switching for the same parameters but with Rabi frequencies ${\Omega }_c$ and ${\Omega }_s$ scaled by a factor 2.2.

Figure 8

Probe field normalized transmission on resonance ($\Delta =0$) as a function of intracavity switching photon number $n_s$ and for different switching field detunings ${\Delta }_s$ (blue (left) curve: ${\Delta }_s=\left(2\pi \right)0$ MHz, red (middle) curve: ${\Delta }_s=\left(2\pi \right)110$ MHz, green (right) curve: ${\Delta }_s=\left(2\pi \right)4300$ MHz). Parameters: $(g_p\sqrt{N},\gamma ,{\gamma }_s,{\gamma }_0,\kappa ,{\Omega }_c,\Delta )=\left(2\pi \right)\times (16,11.2,11,6\times {10}^{-4},1.5,2,0)$ MHz.

Figure 9

Probe field transmission spectrum around two-photon resonance for the three situations: standard EIT (dotted line) and all-cavity EIT with localized (dashed line) and delocalized (solid line) atoms. Parameters as in Fig. 3 with the control field Rabi frequency scaled by a factor 2.2 in the standard situation.