Decoherence, Autler-Townes effect, and dark states in two-tone driving of a three-level superconducting system

We present a detailed theoretical analysis of a multilevel quantum system coupled to two radiation fields and subject to decoherence. We concentrate on an effect known from quantum optics as Autler-Townes splitting, which has been recently demonstrated experimentally [M. A. Sillanpää et al., Phys. Rev. Lett. 103, 193601 (2009)] in a superconducting phase qubit. In the three-level approximation, we derive analytical solutions and describe how they can be used to extract the decoherence rates and to account for the measurement data. Better agreement with the experiment can be obtained by extending this model to five levels. Finally, we investigate the stationary states created in the experiment and show that their structure is close to that of dark states.

Figure 1

Schematic of the phase qubit, consisting of a single junction with capacitance $C$ and Josephson inductance $L_J$ inserted into a superconducting loop of inductance $L$. A nearby dc-SQUID is used for single-shot readout and an on-chip external line is employed to bias the qubit with external flux ${\Phi }_{\mathrm{ext}}$ with both dc and rf components, ${\Phi }_{\mathrm{ext}}={\Phi }_{\mathrm{dc}}+{\Phi }_{\mathrm{rf}}\left(t\right)$.

Figure 2

Potential energy of the rf-SQUID for typical parameters $L_J=276$ pH, $L=690$ pH, and ${\phi }_{\mathrm{dc}}=0.766\pi$. The minimum is located at ${\phi }_m\approx 1.551\pi$. The (red) solid curve is the exact potential energy calculated with Eq. (3); the (blue) dashed curve is the cubic approximation given by Eq. (8). The inset shows the energies of the first three metastable states $|0\rangle ,|1\rangle$, and $|2\rangle$ formed in the well centered around the local minimum ${\phi }_m$.

Figure 3

The spectroscopy traces (solid curves) taken from Fig. 1c in Ref. 11. The (black) solid trace is the single-tone spectroscopy experiment on the effective two-level system showing the $|0\rangle \to |1\rangle$ spectral line centered at 8.135 GHz. The (red) solid trace is a two-tone spectroscopy, revealing the $|1\rangle \to |2\rangle$ spectral line centered at 7.975 GHz. The dotted curves are fits to simulations of the full five-level master equation. The value of the probe field Rabi frequency used was ${\Omega }_p=2\pi \times 3.5$ MHz.

Figure 4

Autler-Townes splitting in a phase qubit. The experimental spectroscopy traces (solid lines) are taken from Fig. 3a in Ref. 11. The dotted lines are simulations of the master equation corresponding to the full five-level model, with the parameters previously extracted from the spectroscopy data. The (green) dashed lines are plotted by using Eq. (35) with the same values (the offsets of base level for ${\Omega }_c=2\pi \times 36$ MHz and ${\Omega }_c=2\pi \times 66$ MHz are 0.057 and 0.105, respectively). Note that the inclusion of higher levels at large values of the coupling results in asymmetric Autler-Townes peaks and in the displacement of the spectroscopy curve toward higher frequencies. The vertical (blue) dashed line indicates the first transition ${\omega }_{10}=2\pi \times 8.135$ GHz. At this frequency the lower curve (taken at ${\Omega }_c=2\pi \times 36$ MHz) has a minimum, while for the higher curve (taken at ${\Omega }_c=2\pi \times 66$ MHz) the minimum, marked by the blue vertical arrow, is displaced by 10 MHz to a value of 8.145 GHz.

Figure 5

The Autler-Townes frequency splitting $\delta {\omega }_{\mathrm{AT}}/\left(2\pi \right)$ as given by the simplified three-level model with dissipation Eq. (36), shown as a (red) dotted line. The straight line is the prediction of the three-level model without dissipation. The experimental data (circles) are from Fig. 3c of Ref. 11.

Figure 6

Schematic of the Autler-Townes effect as a consequence of the formation of dressed states for the $|2\rangle \leftrightarrow |1\rangle$ transition due to a resonant coupling field.

Figure 7

The fidelity of the steady state as a function of the pump and probe amplitudes. The probe and field amplitudes are directly related to the respective Rabi frequencies by ${\Omega }_p=0.69g_p$ and ${\Omega }_c=g_c$. The blue line corresponds to ${\Omega }_p=2\pi \times 3.5$ MHz ($g_p=2\pi \times 5$ MHz) and to the range of experimental values for ${\Omega }_c$ presented in Fig. 5.

Figure 8

The purity of the steady state as a function of the pump and probe amplitudes. As in Fig. 7, the blue line corresponds to ${\Omega }_p=2\pi \times 3.5$ MHz ($g_p=2\pi \times 5$ MHz) and a range of experimental values for ${\Omega }_c$ as presented in Fig. 5. Also, the probe and field amplitudes are directly related to the respective Rabi frequencies by ${\Omega }_p=0.69g_p$ and ${\Omega }_c=g_c$.

Figure 9

Numerical fitting of $|1\rangle \to |2\rangle$ spectral line (the same as traces shown in Fig. 3) with different vales of $\varepsilon$. The following parameters (as in Table ) are used for this simulation: ${\Gamma }_{10}=2\pi \times 7$ MHz, ${\Gamma }_{21}=2\pi \times 11$ MHz, ${\gamma }_{10}^{\varphi }=2\pi \times 7$ MHz, and ${\gamma }_{20}^{\varphi }=2\pi \times 16$ MHz.