Polariton states in circuit QED for electromagnetically induced transparency

Electromagnetically induced transparency (EIT) has been extensively studied in various systems. However, it is not easy to observe in superconducting quantum circuits (SQCs) because the Rabi frequency of the strong-controlling field corresponding to EIT is limited by the decay rates of the SQCs. Here, we show that EIT can be achieved by engineering decay rates in a superconducting circuit QED system through a classical driving field on the qubit. Without such a driving field, the dressed states of the system, describing a superconducting qubit coupled to a cavity field, are approximately product states of the cavity and qubit states in the large-detuning regime. However, the driving field can strongly mix these dressed states. These doubly dressed states, here called polariton states, are formed by the driving field and dressed states, and are a mixture of light and matter. The weights of the qubit and cavity field in the polariton states can now be tuned by the driving field, and thus the decay rates of the polariton states can be changed. We choose the three lowest-energy polariton states with a $\mathrm{\Lambda }$-type transition in such a driven circuit QED system, and demonstrate how EIT and Autler-Townes splitting can be realized in this compound system. We believe that this study will be helpful for EIT experiments using SQCs.

Figure 1

A driven qubit coupled to a resonator mode. Here, $\left\{\right|g\rangle ,|e\rangle \}$ are the ground and excited states of the qubit, $g$ is the coupling strength between the qubit and the cavity field, and ${\gamma }_q$ (${\gamma }_c$) is the decay rate of the qubit (cavity field).

Figure 2

A schematic energy diagram for the Jaynes-Cummings model vs the detuning between the qubit and cavity field in the rotating frame. The eigenstates are denoted by $|\pm ,n\rangle$. In the large-detuning regime, the coupled states $|\pm ,n\rangle$ are approaching the bare qubit and photon states. Here, $\mathrm{\Delta }={\tilde{\omega }}_r-{\tilde{\omega }}_q$. The nesting regime, defined by ${\omega }_{|g,0\rangle }\le {\omega }_{|e,0\rangle }<{\omega }_{|e,1\rangle }\le {\omega }_{|g,1\rangle }$, as in Refs. [43, 44, 45], is between the two points linked by a line with two arrows. The frequency of the driving field ${\omega }_d$ sets the boundary of the nesting regime. The lower limit is at ${\omega }_{|e,1\rangle }={\omega }_{|g,1\rangle }$, where ${\omega }_d={\omega }_q-3\chi$. The upper limit is at ${\omega }_{|g,0\rangle }={\omega }_{|e,0\rangle }$, where ${\omega }_d={\omega }_q-\chi$. Here, $\chi =g^2/\mathrm{\Delta }$, with $g$ the coupling strength between the qubit and cavity field. We note that when the zero-point fluctuation of the cavity field is taken into account, the energy of the ground state $|g\rangle |0\rangle$ is $\mathrm{\Delta }/2$.

Figure 3

(a) The dressed states of the Jaynes-Cummings mode for large detuning are mixed further by a driving field applied to the qubit, which only connects $|g,n\rangle$ and $|e,n\rangle$. In the unnesting regime, without the driving field, the lowest four levels are approximately $|g,0\rangle ,|e,0\rangle ,|g,1\rangle ,|e,1\rangle$, which are mixed when the driving field is applied to the qubit. (b) Four polariton states, i.e., energy levels $|i\rangle$ with ($i=1,2,3,4$) expressed in Eqs. (11), (12) and Eqs. (14), (15), in which we choose the three lowest-energy levels to study EIT and ATS in Sec. 4.

Figure 4

Transition frequencies ${\omega }_{ij}$ vs both the frequency ${\omega }_d$ and the strength $\mathrm{\Omega }$ of the driving field in the rotating reference frame. Here, ${\omega }_{ij}={\omega }_i-{\omega }_j$. The parameters used here are ${\omega }_q/2\pi =5\text{GHz}, {\omega }_r/2\pi =10\text{GHz}, g/2\pi =500\text{MHz}$, and $\chi /2\pi =50\text{MHz}$. (a) Two $\mathsf{V}$-shaped surfaces with the minimum values at ${\omega }_d/2\pi =4.95$ and ${\omega }_d/2\pi =4.85\text{GHz}$ represent ${\omega }_{21}$ and ${\omega }_{43}$, respectively. (b) The upper surface represents ${\omega }_{31}$ and the lower one represents ${\omega }_{32}$.

Figure 5

Moduli of the transition matrix elements between the state $|i\rangle$ and the state $|j\rangle$ vs both the strength $\mathrm{\Omega }$ and the frequency ${\omega }_d$ of the driving field. ${\text{Q}}_{ij}$ denotes the transition matrix elements between the states $|i\rangle$ and $|j\rangle$, induced by an external field applied to the qubit, while ${\text{C}}_{ij}$ is the one induced by an external field applied to the cavity field. The sharp change in ${\text{Q}}_{ij}$ and ${\text{C}}_{ij}$ occurs at the boundary of the nesting regime; when $\mathrm{\Omega }=0$, this occurs when ${\omega }_q-3\chi <{\omega }_d<{\omega }_q-\chi$. Note that the nonzero coupling in (b), (d), and (e) is limited by the first order of $g/\mathrm{\Delta }$, which we omit in the theoretical analysis. The parameters are the same as in Fig. 4. Here, the states $|i\rangle$ for numerical calculations of the transition matrix elements are given by the exact eigenstates of Eq. (3).

Figure 6

Decay rates ${\gamma }_{ij}$ vs both the frequency and strength of the driving field. Here, we set ${\mathrm{\Gamma }}_{31}={\gamma }_{31}+{\gamma }_{32}\approx {\gamma }_c$. We have chosen ${\gamma }_q/2\pi =1\text{MHz}, {\gamma }_c/2\pi =20\text{MHz}$.

Figure 7

(a) The imaginary part of the susceptibility, $\mathrm{Im}\left(\chi \right)$, vs the strength of the driving field $\mathrm{\Omega }$ and the two-photon detuning $\delta$, when the ATS condition is satisfied. Here, we have chosen ${\omega }_d/2\pi =4.9$ GHz, ${\gamma }_c/2\pi =20$ MHz, ${\gamma }_q/2\pi =1$ MHz, and ${\text{A}}_c/2\pi =30$ MHz, while the rest of the parameters are identical to the ones in Fig. 4. The control field frequency is ${\omega }_c/2\pi =5.037$ GHz, which is resonant with ${\omega }_{32}$ for the $\mathrm{\Omega }/2\pi =20$ MHz case. (b) The spectral decomposition of $\mathrm{Im}\left(\chi \right)$ at resonance, i.e., ${\mathrm{\Delta }}_2=0$. Here, the blue solid curve corresponds to the absorption spectrum. The red-dotted and the black-dashed curves correspond to two Lorentzian profiles. (c) The real part of $\chi$ characterizing the refractive properties at resonance.

Figure 8

(a) The imaginary part of the susceptibility, $\mathrm{Im}\left(\chi \right)$, vs the strength of the driving field $\mathrm{\Omega }$ and the two-photon detuning $\delta$, when the EIT condition is satisfied. Here, we have chosen ${\omega }_d/2\pi =4.9$ GHz, ${\gamma }_c/2\pi =20$ MHz, ${\gamma }_q/2\pi =1$ MHz, and ${\text{A}}_c/2\pi =5$ MHz, while the rest of the parameters are identical to the ones in Fig. 4. The control field frequency is ${\omega }_c/2\pi =5.037$ GHz, which is resonant with ${\omega }_{32}$ for the $\mathrm{\Omega }/2\pi =20$ MHz case. (b) The spectral decomposition of $\mathrm{Im}\left(\chi \right)$ at resonance, i.e., ${\mathrm{\Delta }}_2=0$. Here, the blue solid curve corresponds to the absorption spectrum. The red-dotted and the black-dashed curves correspond to two Lorentzian profiles. (c) The real part of $\chi$ at resonance, which characterizes the refractive properties.