Spectrum of the Dicke model in a superconducting qubit-oscillator system

We calculate the transmission spectrum of a superconducting circuit realization of the Dicke model and identify spectroscopic features that can serve as signatures of the superradiant phase. In particular, we calculate the resonance frequencies of the system as functions of the bias term, which is usually absent in studies on the Dicke model but is commonly present in superconducting qubit circuits. To avoid over-complicating the proposed circuit, we assume a fixed coupling strength. This situation precludes the possibility of observing signatures of the phase transition by varying the coupling strength across the critical point. We show that the spectrum obtained by varying the bias point under fixed coupling strength can contain signatures of the normal and superradiant phases: in the normal phase one expects to observe two spectral lines, while in the superradiant phase four spectral lines are expected to exist close to the qubits' symmetry point. Provided that parameter fluctuations and decoherence rates are sufficiently small, the four spectral lines should be observable and can serve as a signature of the superradiant phase.

Figure 1

Schematic diagram showing the stable points in the potential $V\left(x\right)=\varepsilon x+\eta x^2+x^4$. The local minima of the potential are marked by triangles. The circles mark local maxima in the potential. Panel (a) shows the case of positive $\eta$, which gives a single local minimum. Here we have taken $\varepsilon =0$. Panels (b)–(d) show the potential for negative values of $\eta$ (specifically $\eta =-1$), which generally gives a double-well potential and hence two local minima of the potential. Panel (b) shows the symmetric case $\varepsilon =0$, where two equivalent local minima exist. Panel (c) shows the weakly asymmetric case $\varepsilon =-0.3$. In this case, two local minima exist. However, one of them corresponds to a metastable excited state. The curvature, and hence the excitation frequency, around the metastable excited state is smaller than that around the ground state. Panel (d) shows the strongly asymmetric case $\varepsilon =-0.6$, where the potential has only one minimum and there is no metastable excited state.

Figure 2

Frequencies of the excitation modes ${\nu }_{\pm }$ as functions of the bias parameter $\epsilon /\left(\hslash \omega \right)$. The red solid lines are the spectral lines that correspond to excitation from the ground state, while the green dashed lines correspond to excitation from the metastable excited state. Here we set $\mathrm{\Delta }=\hslash \omega$. The coupling strength is given by (a) $\sqrt{N}g/\left(\hslash \omega \right)=0.2$, (b) 0.4, (c) 0.6, and (d) 0.8. In panels (a) and (b), $\sqrt{N}g/\left(\hslash \omega \right)<0.5$, which corresponds to the normal phase, while in panels (c) and (d) $\sqrt{N}g/\left(\hslash \omega \right)>0.5$, which corresponds to the superradiant phase. In the normal phase only two spectral lines are obtained, whereas in the superradiant phase the existence of a metastable excited state results in two additional spectral lines. As we move away from the symmetry point, there is a critical value of $\epsilon$ at which the metastable excited state does not exist any more, and its spectral lines disappear. In particular, we note that the high-frequency dashed line disappears at the same value of $\epsilon$ where the frequency of the lowest spectral line reaches zero, even though this fact might not be entirely clear from the appearance of the lines in the figure.

Figure 3

Slopes of the frequencies ${\nu }_{-}$ (red solid line) and ${\nu }_{+}$ (green dashed line) with respect to $\epsilon$ (i.e., $\hslash d{\nu }_{\pm }/d\epsilon$) at the symmetry point as functions of coupling strength $g$ (measured relative to the critical value $g_c$). We take the slopes of the spectral lines that correspond to the ground state for positive $\epsilon$, i.e., those that extend to $\epsilon \to \infty$. As in Fig. 2 we set $\mathrm{\Delta }=\hslash \omega$. In the normal phase (i.e., when $g/g_c<1$), the slopes vanish, because the spectral lines have their minima at the symmetry point. In the superradiant phase, two pairs of spectral lines cross with slopes that are equal in magnitude but with opposite signs. As we approach the transition point $g=g_c=\sqrt{\mathrm{\Delta }\hslash \omega /\left(4N\right)}$ from above, the slope of ${\nu }_{-}$ diverges as ${(g-g_c)}^{-1}$.

Figure 4

The maximum value of $\left|\epsilon \right|$ at which there exists a metastable excited state, i.e., the maximum value of $\left|\epsilon \right|$ at which one expects to see four lines in the spectrum, as a function of $g/g_c$. As above $\mathrm{\Delta }=\hslash \omega$. Alternatively, this figure can be seen as a phase diagram: the labels “Normal” and “Superradiant” describe the phase that is obtained in each region in the $g\text{−}\epsilon$ parameter space, where the superradiance region is defined by the existence of two stable states.

Figure 5

Frequencies of the excitation modes as functions of $\epsilon /\left(\hslash \omega \right)$. As in Fig. 2, the red solid lines are the spectral lines that correspond to excitation from the ground state, while the green dashed lines correspond to excitation from the metastable excited state. Here we take the case of small $\mathrm{\Delta }$. Specifically we set $\mathrm{\Delta }/\hslash \omega =0.2$, which gives $g_c/\left(\sqrt{N}\hslash \omega \right)=0.224$. We set (a) $\sqrt{N}g/\left(\hslash \omega \right)=0.1$, (b) 0.2, (c) 0.3, and (d) 0.5. The inset in panel (c) shows a magnified view of the region $x\in [-0.1,0.1]$, $y\in [1.01,1.03]$.

Figure 6

The maximum frequency separation between the two ${\nu }_{+}$ spectral lines (i.e., at the maximum value of $\left|\epsilon \right|$ before one of them disappears) as a function of $g/g_c$. Here we take the resonant case $\mathrm{\Delta }=\hslash \omega$.

Figure 7

The frequencies ${\nu }_{\pm }$ at the symmetry point as functions of $g/g_c$ for the case $\mathrm{\Delta }=\hslash \omega$, where $g_c=\hslash \omega /\left(2\sqrt{N}\right)$.

Figure 8

Frequencies of the excitation modes ${\nu }_{\pm }$ as functions of $\epsilon /\left(\hslash \omega \right)$. As in Fig. 2 in the main text, the red solid lines are the spectral lines that correspond to excitation from the ground state, while the green dashed lines correspond to excitation from the metastable excited state. Here we take the nonresonant case $\mathrm{\Delta }\ne \hslash \omega$. In panel (a) we set $\mathrm{\Delta }/\left(\hslash \omega \right)=1.2$, $\sqrt{N}g/\left(\hslash \omega \right)=0.2$. In panel (b) we set $\mathrm{\Delta }/\left(\hslash \omega \right)=1.2$, $\sqrt{N}g/\left(\hslash \omega \right)=0.7$. In panel (c) we set $\mathrm{\Delta }/\left(\hslash \omega \right)=0.8$, $\sqrt{N}g=0.2$. In panel (d) we set $\mathrm{\Delta }/\left(\hslash \omega \right)=0.8$, $\sqrt{N}g/\left(\hslash \omega \right)=0.7$. The spectra are qualitatively similar to those shown in Fig. 2 in the main text.

Figure 9

Slopes of the frequencies ${\nu }_{-}$ (red solid line) and ${\nu }_{+}$ (green dashed line) with respect to $\epsilon$ (i.e., $\hslash d{\nu }_{\pm }/d\epsilon$) at the symmetry point as a function of $g/g_c$. As in Fig. 5 in the main text, we set $\mathrm{\Delta }/\left(\hslash \omega \right)=0.2$.

Figure 10

The maximum value of $\left|\epsilon \right|$ at which there exists a metastable state, i.e., the maximum value of $\left|\epsilon \right|$ at which one expects to see four lines in the spectrum, as a function of $g/g_c$. As above $\mathrm{\Delta }/\left(\hslash \omega \right)=0.2$. The labels “Normal” and “Superradiant” describe the phase that is obtained in each region in the $g\text{−}\epsilon$ parameter space.

Figure 11

The frequency separation between the two ${\nu }_{+}$ spectral lines at the maximum value of $\epsilon$ before one of them disappears as a function of $g/g_c$. The fact that the difference is negative just above $g_c$ means that the spectral lines have a shape resembling the letter W near the symmetry point, as can be seen in Fig. 5 in the main text. As above $\mathrm{\Delta }/\left(\hslash \omega \right)=0.2$.

Figure 12

The frequencies ${\nu }_{\pm }$ at the symmetry point as functions of $g/g_c$ for the case $\mathrm{\Delta }/\left(\hslash \omega \right)=0.2$, where $g_c=0.224\hslash \omega /\sqrt{N}$.