Transmission of a microwave cavity coupled to localized Shiba states

We consider a strongly correlated quantum dot, tunnel coupled to two superconducting leads and capacitively coupled to a single mode microwave cavity. When the superconducting gap is the largest energy scale, multiple Shiba states are formed inside the gap. The competition of these states for the ground state signals a quantum phase transition. We demonstrate that photonic measurements can be used to probe such localized Shiba states. Moreover, the quantum phase transition can be pinpointed exactly from the sudden change in the transmission signal. Calculations were performed using the numerical renormalization-group approach.

Figure 1

Sketch with the setup. An interacting quantum dot with on-site energy ${\varepsilon }_d$ and Coulomb repulsion $U$ is coupled to two superconducting leads. The dot is capacitively coupled to a microwave cavity. The cavity is assumed to support a single mode with frequency ${\omega }_0$. The superconducting gap $\mathrm{\Delta }$ is the largest energy scale.

Figure 2

(a) Shiba states inside the gap in the absence of photons. The QCP is marked by the red dot. (b) Shiba states in the presence of the photons. The other parameters are fixed to ${\omega }_0/\mathrm{\Delta }=0.5, g/\mathrm{\Delta }=0.05$, and $U/\mathrm{\Delta }=0.4$. For this set of parameters the quantum critical point corresponds to $x_{\mathrm{QCP}}={\left(\mathrm{\Gamma }/\mathrm{\Delta }\right)}_{\mathrm{QCP}}\simeq 0.17$. The shaded lines emphasize the avoided level crossing. The shaded area above $E=\mathrm{\Delta }$ represents the continuum.

Figure 3

Spectral function for the on-site creation operator $d_{\sigma }^{†}$ at $T=0$. The weights and positions for the transitions between the ground state and the excited Shiba states are indicated by vertical arrows. They correspond to the transitions between $D_{\sigma }^0\leftrightarrow \{S_{-}^{\left(0\right)},S_{+}^{\left(0\right)},S_{-}^{\left(1\right)},S_{-}^{\left(2\right)}\}$ for $\mathrm{\Gamma }/\mathrm{\Delta }=0.04$ when the ground state is $D_{\sigma }$, and $S_{-}^{\left(0\right)}\leftrightarrow \{D_{\sigma }^{\left(0\right)},D_{\sigma }^{\left(1\right)}\}$ when $\mathrm{\Gamma }/\mathrm{\Delta }=\{0.2,0.4\}$, when the ground state is $S_{-}$.

Figure 4

Bosonic spectral function for the occupation number operator of the dot, $\widehat{n}$ at $T=0$. The transitions inside the gap are between the ground state and excited states with the same parity.

Figure 5

Spectral function $A_{a+a^{†}}\left(\omega \right)$ in different regimes, as calculated by the NRG. Each peak is associated with a given transition, as indicated. The other parameters correspond to those in Fig. 2. We have used $\alpha =5\times {10}^{-4}$ in Eq. (17).

Figure 6

Density plots for the phase and absolute value of the transmission through the cavity, as calculated by the NRG. The three dashed lines correspond to the cuts along which the phase and amplitude of the transmission are displayed in Figs. 7 and 8.

Figure 7

Phase of the transmission through the cavity when the system is in the singlet ground state, corresponding to $x>x_{\mathrm{QCP}}$. The three panels correspond to the vertical cuts in Fig. 6. For $x<x_{\mathrm{QCP}}$ the system simply shows a jump of $\pi$ at $\omega /{\omega }_0=1$ [displayed as a red dashed line in (a)].

Figure 8

Absolute value for the amplitude of the transmission through the cavity for the same values of $\mathrm{\Gamma }/\mathrm{\Delta }$ as in Fig. 7. The dashed red line in (a) represents the transmission amplitude when the system is in a doublet ground state, and corresponds to the otherwise free cavity.