Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits

We study the microwave absorption of a driven three-level quantum system, which is realized by a superconducting flux quantum circuit (SFQC), with a magnetic driving field applied to the two upper levels. The interaction between the three-level system and its environment is studied within the Born-Markov approximation, and we take into account the effects of the driving field on the damping rates of the three-level system. We study the linear response of the driven three-level SFQC to a weak probe field. The linear magnetic susceptibility of the SFQC can be changed by both the driving field and the bias magnetic flux. When the bias magnetic flux is at the optimal point, the transition from the ground state to the second-excited state is forbidden and the three-level SFQC has a ladder-type transition. Thus, the SFQC responds to the probe field like natural atoms with ladder-type transitions. However, when the bias magnetic flux deviates from the optimal point, the three-level SFQC has a cyclic transition, thus it responds to the probe field like a combination of natural atoms with ladder-type transitions and natural atoms with $\Lambda$-type transitions. In particular, we provide detailed discussions on the conditions for realizing electromagnetically induced transparency and Autler-Townes splitting in three-level SFQCs.

Figure 1

(a) Schematic diagram of a SFQC with three Josephson junctions. Here, ${\Phi }_e$ is a bias magnetic flux (dc), while ${\Phi }_{\mathrm{D}}\left(t\right)={\Phi }_{c}cos\left({\omega }_{0}t\right)$ is a strong driving magnetic flux (ac) provided by the left coil (in red color), ${\Phi }_{\mathrm{P}}\left(t\right)$ is a weak probe magnetic flux (ac) provided by the right coil (in green color). (b) Schematic diagram of the three-level SFQC. The driving field is used to couple the two upper energy levels $|1\rangle$ and $|2\rangle$. However, the probe field is used to couple the energy levels $|0\rangle$ and $|1\rangle$, as well as the energy levels $|0\rangle$ and $|2\rangle$.

Figure 2

(a) Eigenvalues $E_l$ of the SFQC versus the reduced magnetic flux $f$, in units of $E_{\mathrm{J}}$, for the six lowest-energy levels. (b) Moduli of the loop current transition matrix elements $|I_{ij}|$ $(i<j)$ for the three lowest-energy levels $(|0\rangle$, $|1\rangle$, and $|2\rangle )$ versus the reduced magnetic flux $f$ for $|I_{01}|$ (blue dashed curve), $|I_{02}|$ (green dashed-dotted curve), and $|I_{12}|$ (red solid curve). (c) Loop current diagonal matrix elements $I_{ii}$ for the three lowest-energy levels $(|0\rangle$, $|1\rangle$, and $|2\rangle )$ versus the reduced magnetic flux $f$ for $I_{00}$ (blue dashed curve), $I_{11}$ (green dashed-dotted curve), and $I_{22}$ (red solid curve). In (b) and (c), the loop current matrix elements are in units of $I_0$. Here, we choose $\alpha =0.7$ and $E_{\mathrm{J}}/E_{\mathrm{c}}=48$.

Figure 3

Damping rates ${\gamma }_{11}$ (the three lowest curves) and ${\gamma }_{22}$ (the three highest curves) in (a) as functions of the reduced magnetic flux $f$, are plotted for the temperatures $T=0$ (blue dashed curves), $T=25$ mK (green dashed-dotted curves), and $T=50$ mK (red solid curves). Damping rates ${\gamma }_{11}$ (the two lowest curves) and ${\gamma }_{22}$ (the two highest curves) in (b) as functions of the temperature $T$, are plotted for the reduced magnetic flux $f=0.5$ (blue dashed curves) and $f=0.525$ (red solid curves). Damping rates ${\gamma }_{11}$ in (c), ${\gamma }_{22}$ in (d), ${\gamma }_{12}$ in (e), and ${\gamma }_{21}$ in (f) as functions of the Rabi frequency $|{\Omega }_{\mathrm{D}}|$, are plotted for the reduced magnetic flux $f=0.5$ (blue dashed curve) and $f=0.525$ (red solid curve). Here, $\Delta$ is assumed to be 0. In (a) and (b), $|{\Omega }_{\mathrm{D}}|$ is assumed to be 0. In (e) and (f), $T$ is assumed to be 25 mK. The other parameters of the three-level SFQC are provided in Appendix pp3.

Figure 4

Thresholds ${\Omega }_{\mathrm{W}}$ (dashed-dotted curve) and ${\Omega }_{\mathrm{M}}$ (solid curve) are plotted as functions of ${\gamma }_{\mathrm{R}}$. The red (yellow) area represents the EIT (ATS) regime. Six representative points $A_1(0.3,0.07)$, $A_2(0.3,0.24)$, $A_3(0.3,1)$, $B_1(1.5,0.13)$, $B_2(1.5,0.6)$, and $B_3(1.5,1.4)$ are marked by round dots.

Figure 5

Imaginary parts of the reduced magnetic susceptibility ${\chi }_{\mathrm{R}}^{\prime \prime }\left({\delta }_{\mathrm{R}}\right)$ (red solid curve), the resonances $\mathrm{Im}\left[R_{+}\left({\delta }_{\mathrm{R}}\right)\right]$ (blue dashed curve), and $\mathrm{Im}\left[R_{-}\left({\delta }_{\mathrm{R}}\right)\right]$ (green dashed-dotted curve) are plotted. Figures (a)–(c) correspond with the points $A_1$–$A_3$ in Fig. 4, respectively. While figures (d)–(f) correspond with the points $B_1$–$B_3$ in Fig. 4, respectively.

Figure 6

The imaginary parts of ${\chi }_q\left({\omega }_{\mathrm{P}}\right)$ (red solid curve), $R_{+}^{\left(01\right)}({\omega }_{\mathrm{P}}-{\omega }_1)$ (blue dashed curve), and $R_{-}^{\left(01\right)}({\omega }_{\mathrm{P}}-{\omega }_1)$ (green dashed-dotted curve) as functions of the detuning of the probe field ${\omega }_{\mathrm{P}}-{\omega }_1$ are plotted for $T=25$ mK and $\Delta =0$. The imaginary part of ${\chi }_q\left({\omega }_{\mathrm{P}}\right)$ is also plotted for $T=25$ mK, $\Delta /2\pi =20$ MHz (black dotted curve). Here, we assume that $f=0.5$, and $|{\Omega }_{\mathrm{D}}|/2\pi =40$ MHz. The other parameters of the three-level SFQC are provided in Appendix pp3.

Figure 7

The imaginary part of ${\chi }_q\left({\omega }_{\mathrm{P}}\right)$ (red solid curve) is shown as a function of the probe field frequency ${\omega }_{\mathrm{P}}/2\pi$. In the left part, the imaginary parts of $R_{+}^{\left(01\right)}({\omega }_{\mathrm{P}}-{\omega }_1)$ (blue dashed curve) and $R_{-}^{\left(01\right)}({\omega }_{\mathrm{P}}-{\omega }_1)$ (green dashed-dotted curve) are shown as functions of the probe field frequency ${\omega }_{\mathrm{P}}/2\pi$. In the right part, the imaginary parts of $R_{+}^{\left(02\right)}({\omega }_{\mathrm{P}}-{\omega }^{\prime })$ (blue dashed curve) and $R_{-}^{\left(02\right)}({\omega }_{\mathrm{P}}-{\omega }^{\prime })$ (green dashed-dotted curve) are shown as functions of the probe field frequency ${\omega }_{\mathrm{P}}/2\pi$. All curves are plotted with $T=25$ mK and $\Delta =0$. Here, we assume that $f=0.525$, and $|{\Omega }_{\mathrm{D}}|/2\pi =1.4$ MHz. The other parameters of the three-level SFQC are provided in Appendix pp3. Note that the left (right) vertical axis is for the left (right) part of the figure.

Figure 8

The imaginary part of ${\chi }_q\left({\omega }_{\mathrm{P}}\right)$ (red solid curve) is shown as a function of the probe field frequency ${\omega }_{\mathrm{P}}/2\pi$. In the left part, the imaginary parts of $R_{+}^{\left(01\right)}({\omega }_{\mathrm{P}}-{\omega }_1)$ (blue dashed curve) and $R_{-}^{\left(01\right)}({\omega }_{\mathrm{P}}-{\omega }_1)$ (green dashed-dotted curve) are shown as functions of the probe field frequency ${\omega }_{\mathrm{P}}/2\pi$. In the right part, the imaginary parts of $R_{+}^{\left(02\right)}({\omega }_{\mathrm{P}}-{\omega }^{\prime })$ (blue dashed curve) and $R_{-}^{\left(02\right)}({\omega }_{\mathrm{P}}-{\omega }^{\prime })$ (green dashed-dotted curve) are shown as functions of the probe field frequency ${\omega }_{\mathrm{P}}/2\pi$. All curves are plotted with $T=25$ mK and $\Delta =0$. Here, we assume that $f=0.525$, and $|{\Omega }_{\mathrm{D}}|/2\pi =40$ MHz. The other parameters of the three-level SFQC are provided in Appendix pp3. Note that the left (right) vertical axis is for the left (right) part of the figure.