Tunable electromagnetically induced transparency and absorption with dressed superconducting qubits

Electromagnetically induced transparency and absorption (EIT and EIA) are usually demonstrated using three-level atomic systems. In contrast to the usual case, we theoretically study the EIT and EIA in an equivalent three-level system: a superconducting two-level system (qubit) dressed by a single-mode cavity field. In this equivalent system, we find that both the EIT and the EIA can be tuned by controlling the level-spacing of the superconducting qubit and hence controlling the dressed system. This tunability is due to the dressed relaxation and dephasing rates which vary parametrically with the level-spacing of the original qubit and thus affect the transition properties of the dressed qubit and the susceptibility. These dressed relaxation and dephasing rates characterize the reaction of the dressed qubit to an incident probe field. Using recent experimental data on superconducting qubits (charge, phase, and flux qubits) to demonstrate our approach, we show the possibility of experimentally realizing this proposal.

Figure 1

Schematic diagram illustrating electromagnetically induced transparency. Panel (a) schematically shows a medium that strongly absorbs a probe light beam of amplitude ${\Omega }_{\mathrm{p}}$; (b) shows the same absorption, but for a control light beam of amplitude ${\Omega }_{\mathrm{c}}$. However, when both beams are applied simultaneously, as in (c), then the previously absorbing medium becomes transparent for the probe beam ${\Omega }_{\mathrm{p}}$ (i.e., transparency is electromagnetically induced by the control field ${\Omega }_{\mathrm{c}}$). The superconducting circuit analog of this phenomenon is schematically shown in (d), (e), and (f), where a “box” schematically represents the circuit. In (d) a classical voltage or current signal (in the microwave range) of amplitude ${\Omega }_{\mathrm{p}}^{\mathrm{in}}$ is fed into the input of the circuit. The signal is “absorbed” in the circuit and the amplitude ${\Omega }_{\mathrm{p}}^{\mathrm{out}}$ of the output signal is less than that of the input. Panel (e) shows the same effect with another microwave control signal of amplitude ${\Omega }_{\mathrm{c}}$. However, when both microwave signals are applied simultaneously to the circuit, as in (f), then the previously “absorbing medium,” now a circuit, becomes transparent for the first signal ${\Omega }_{\mathrm{p}}^{\mathrm{in}}$. Panel (g) shows a canonical level diagram for a three-level system coupled to a weak probe field and a strong control field.

Figure 2

Schematic diagram of the “dressing” process. The Fock number states $|n\rangle$ of a coplanar-waveguide resonator (shown in the first column) as well as the ground state $|g\rangle$ and excited state $|e\rangle$ of a qubit provide the “undressed” tensor-product states $|n,g\rangle$ and $|n,e\rangle$ (shown on the middle column). These tensor states are modified by their mutual interaction, producing the renormalized or  “dressed” states $|{\mu }_n\rangle$ and $|{\nu }_n\rangle$ (shown on the right). Mathematically, these states correspond to the eigenvectors of the circuit Hamiltonian in a nondiagonal basis (undressed states) and the diagonal basis (dressed states), respectively.

Figure 3

Coupling amplitudes of the dressed qubit to the probe field (denoted by the dashed curve) and to the control field (denoted by the solid curve).

Figure 4

Real part ${\chi }^{\prime }$ (blue) and imaginary part ${\chi }^{″}$ (red or yellow) of the normalized susceptibility of the dressed charge qubit versus the detuning $\Delta$ and the qubit level spacing ${\omega }_q$. The detuning abscissa is normalized with respect to ${\Delta }^{\prime }/2\pi =25$ MHz; the qubit-level-spacing abscissa is normalized with respect to ${\omega }_q^{\prime }/2\pi =3.5$ GHz. This plot indicates the dispersion and absorption spectra of a probe signal over different operating ranges.

Figure 5

Normalized susceptibility spectra at typical values of the qubit level spacing ${\omega }_q$ plotted versus the scaled detuning $\Delta /{\Delta }^{\prime }$: (a) the absorption ${\chi }^{″}=\mathrm{Im}[\chi ]$ and (b) the dispersion ${\chi }^{\prime }=\mathrm{Re}[\chi ]$. The curves that correspond to the resonant frequency ${\omega }_0/2=2\pi \times 3.5$ GHz are thickened and all the numbers indicate the value of ${\omega }_q$ taken (in unit of GHz) in both (a) and (b). Note that in (a) a dip appears at the center of the curve about ${\omega }_q/2\pi \approx 3.47$ GHz, which indicates the switching of the dressed qubit from being absorptive ($>3.47$ GHz) to being transparent ($<3.47$ GHz). In (b), the curves that correspond to ${\omega }_q$ below ${\omega }_0/2$ are solid and those above ${\omega }_0/2$ are dashed. Note that the solid and dashed curves with equal distance from the resonance ${\omega }_0/2$ are symmetric counterparts of each other. Their roles for positive and negative dispersion across the range of the detuning $\Delta$ are exchanged above and below the resonance frequency.

Figure 6

Plot of the normalized absorption spectrum ${\chi }^{″}=\mathrm{Im}[\chi ]$ versus the normalized detuning $\Delta /{\Delta }^{\prime }$ and the coupling amplitude ${\Omega }_{\mathrm{c}}$ to the control field.

Figure 7

Contour plots of the normalized absorption spectra ${\chi }^{″}$ versus the normalized detuning $\Delta /{\Delta }^{\prime }$ on the vertical axis and the normalized qubit level spacing ${\omega }_q/{\omega }_q^{\prime }$ on the horizontal axis (a) for a phase qubit and (b) for a flux qubit. The red end of the color spectrum (i.e., the centers of both contour plots) indicates peak values of the absorption, where the qubit is maximally dressed with a rotation angle ${\theta }_0=\pi /2$. The deep blue end of the color spectrum for $\Delta /{\Delta }^{\prime }=0$ (i.e., the middle sections of the left and right borders of both plots) indicates the minimum values of the absorption, where the qubit is dressed with a rotation angle ${\theta }_0<\pi /2$. Namely, maximum transmission due to EIT occurs at those blue ends.