Objectively Discerning Autler-Townes Splitting from Electromagnetically Induced Transparency

Autler-Townes splitting (ATS) and electromagnetically induced transparency (EIT) both yield transparency in an absorption profile, but only EIT yields strong transparency for a weak pump field due to Fano interference. Empirically discriminating EIT from ATS is important but so far has been subjective. We introduce an objective method, based on Akaike’s information criterion, to test ATS vs EIT from experimental data for three-level atomic systems and determine which pertains. We apply our method to a recently reported induced-transparency experiment in superconducting-circuit quantum electrodynamics.

Figure 1

(a) $\Lambda$-type TLA with a probe (pump) driving field with Rabi frequency $\alpha$ ($\Omega$), which probes (drives) the $|a⟩\leftrightarrow |b⟩$ ($|a⟩\leftrightarrow |c⟩$) transition. (b)–(d) Absorption $A$ vs two-photon detuning $\delta$ (red dotted lines) for the resonant ($\Delta =0$) pump with ${\Gamma }_{\mathrm{ab}}=1$, ${\Gamma }_{\mathrm{bc}}=0.1$, and various $\Omega$ with best fits to $A_{\mathrm{EIT}}(C_{+},C_{-},{\gamma }_{+},{\gamma }_{-})$ (blue solid lines) and $A_{\mathrm{ATS}}(C,\gamma ,{\delta }_0)$ (green dashed lines) models calculated for (b) weak $\Omega$ with a good fit to $A_{\mathrm{EIT}}(2.14,1.89,0.581,0.520)$ and a poor fit to closest $A_{\mathrm{ATS}}(0.532,0.633,0.712)$, (c) intermediate $\Omega$ with a poor fit to closest $A_{\mathrm{ATS}}(0.472,0.512,1.03)$ as well as $A_{\mathrm{EIT}}(88.3,88.3,0.75,0.752)$, and (d) strong $\Omega$ with a poor fit to closest $A_{\mathrm{EIT}}(1.3\times {10}^3,1.3\times {10}^3,2.92,2.92)$ and a good fit to $A_{\mathrm{ATS}}(0.499,0.521,3.05)$.

Figure 2

(a) Akaike weights vs the Rabi frequency for the TLA in Fig. 1 showing a sharp transition at ${\Omega }_{\mathrm{AIC}}$ from EIT model (blue solid line) to ATS model (green dashed line); (b) Transition boundary ${\Omega }_{\mathrm{AIC}}$ with corresponding transparency values vs ${\Gamma }_{\mathrm{bc}}$.

Figure 3

(a) Per-point weights ${\overline{w}}_i$ for the conditions of Fig. 1 as a function of the pump-field Rabi frequency $\Omega$ illustrate three distinct regions: $\Omega <{\Omega }_{\mathrm{EIT}}=0.45$, where the EIT model (blue line) dominates unconditionally; $0.45<\Omega <0.86$, where the ATS model (green line) shows nonzero likelihood; $\Omega >0.86$, where the ATS model dominates. The presence of Gaussian noise with standard derivation $\sigma =0.1$ (red dots) [$\sigma =0.01$ (burgundy dots)] affects the per-point weights for EIT and ATS models leading to the absence of unconditional dominance by the EIT model. (b) In the weak-pump limit and a poor signal-to-noise ratio, both models are equally likely to fit data (red dots).

Figure 4

Transmission $\mathrm{Re}(t)$ vs two-photon detuning $\delta$ for (a) a theoretical curve (red dotted line) with parameters taken from Ref. 23 and control-field amplitude $\Omega =6\mathrm{MHz}$ compared to the best-fit $A_{\mathrm{EIT}}(25.4,24.2,6.36,6.15)$ (blue solid line) and $A_{\mathrm{ATS}}(4.42,7.1,6.1)$ (green dashed line) and (b) actual experimental data from Ref. 23 (red dots) vs best-fit $A_{\mathrm{EIT}}(11.8,9.08,6.77,5.66)$ (blue solid line) and $A_{\mathrm{ATS}}(4.59,7.29,5.49)$.