Transmon in a semi-infinite high-impedance transmission line: Appearance of cavity modes and Rabi oscillations

In this paper, we investigate the dynamics of a single superconducting artificial atom capacitively coupled to a transmission line with a characteristic impedance comparable to or larger than the quantum resistance. In this regime, microwaves are reflected from the atom also at frequencies far from the atom's transition frequency. Adding a single mirror in the transmission line then creates cavity modes between the atom and the mirror. Investigating the spontaneous emission from the atom, we then find Rabi oscillations, where the energy oscillates between the atom and one of the cavity modes.

Figure 1

(a) Circuit model of a transmon coupled through the coupling capacitance $C_c$ to a semi-infinite 1D TL with impedance $Z_0$. The Josephson energy, flux, and capacitance of the transmon are denoted by $E_J$, ${\varphi }_J$, and $C_J$. The flux on the coupling capacitance $C_c$ is denoted by ${\varphi }_0$, with the corresponding voltage $V_0={\dot{\varphi }}_0$. (b) Sketch of the system depicting an atom in front of a mirror coupled to incoming and outgoing microwave fields to the left and right characterized by their respective voltages $V_{L/R}^{\text{in/out}}$, respectively, at the transmon. The mirror couples the fields to the right $V_R^{\text{in}}\left(t\right)=-V_R^{\text{out}}(t-T)$, introducing the time of propagation to the mirror and back $T=2L/v$.

Figure 2

Reflection of a transmon in an open TL for different ratios of the TL and qubit impedance $Z_0/Z_J$. The curves show the reflection for $\frac{C_c}{C_c+C_J}=0.1$ and $Z_0/Z_J=0.1$ (purple), $Z_0/Z_J=1$ (blue), $Z_0/Z_J=10$ (green), $Z_0/Z_J=100$ (yellow), and $Z_0/Z_J=1000$ (red) from low to high TL impedance. For low impedance, the qubit reflects only at its low-impedance frequency ${\omega }_0$, but for high impedance, it reflects everywhere but its bare frequency ${\omega }_J$.

Figure 3

Amplitude of the electromagnetic field between the qubit and the mirror $\left|f\left(\omega \right)\right|=\left|V_R^{\text{out}}\left(\omega \right)/V_L^{\text{in}}\left(\omega \right)\right|$ for $\frac{C_c}{C_J}=0.1$ and $Z_0/Z_J=1000$. The delay time is given by $T=2\pi n/{\omega }_c,n=1$, and ${\omega }_c$ varies with respect to ${\omega }_J$. The different colors show ${\omega }_c/{\omega }_J=0.98$ (red), ${\omega }_c/{\omega }_J=0.99$ (yellow), ${\omega }_c/{\omega }_J=1$ (green), ${\omega }_c/{\omega }_J=1.01$ (blue), and ${\omega }_c/{\omega }_J=1.02$ (purple). All curves show a splitting which can be understood as a vacuum Rabi splitting. Without detuning between ${\omega }_J$ and ${\omega }_c$, the splitting is centered exactly around $\omega /{\omega }_J=1$. The inset shows the appearance of higher cavity modes appearing at $\omega \approx n{\omega }_c$, where ${\omega }_{c}T=2\pi$ for ${\omega }_c/{\omega }_J=1$, corresponding to the green curve in the main plot.

Figure 4

Energy of the transmon (blue), field between the mirror and transmon (pink), outgoing field to the left (orange), sum of all energies (red), approximated decay ${\gamma }_J^m$ (dashed cyan), and approximated Rabi frequency ${\mathrm{\Omega }}_T$ and decay $e^{-{\gamma }_J^{m}t/2}{cos}^2({\mathrm{\Omega }}_{T}t/2)$ (dashed yellow) for $C_c/C_J=0.5,Z_0/Z_J=100,T=2\pi /{\omega }_c,n{\omega }_c={\omega }_J,n=1$ as a function of periods of the Rabi oscillations. The inset shows the energy of the transmon (blue) and the flux ${\varphi }_J^2$ (green) for the first oscillation of the frequency ${\mathrm{\Omega }}_T$. This clearly shows the different timescales of the energy compared to the variables of the transmon.

Figure 5

Response function (log color scale) versus normalized frequency $\omega /{\omega }_J$ and linearly varied time delay ${\omega }_{J}T$. The eigenvalues ${\mathrm{\Omega }}_{\alpha }$ are computed for $N=8$ cavity modes and are shown as superimposed white dashed lines. The parameters are chosen to be $Z_0/Z_J=100$ and $C_c/C_J=0.3$.

Figure 6

Energy of the transmon (blue), energy of the field trapped between the transmon and the mirror (magenta), outgoing field to the left side of the transmon (orange), total energy of the system (red), and semi-numerically calculated energy of the transmon using Eq. (A2) (yellow dashed) as a function of the period of the Rabi oscillations. The parameters for the panels are the following: (a) $Z_0/Z_J=1000$, $C_c/C_J=0.02$, $T=2\pi /{\omega }_c$, and ${\omega }_J=n{\omega }_c$, $n=1$. Since we chose $C_c/C_J=0.02$, the decay is very slow. (b) $Z_0/Z_J=1000$, $C_c/(C_J+C_c)=0.02$, $T=2\pi /{\omega }_c$, and ${\omega }_0=n{\omega }_c$, $n=1$. Here, the parameters are almost the same as in (a), but now the dark state condition ${\omega }_c={\omega }_0$ is fulfilled. The system converges into a dark state with a finite-excitation probability of both the transmon and the field between the transmon and the mirror. The green line indicates the analytical value of the dark state energy $\frac{E_{DS}}{E_0}=\frac{1}{{(1+\frac{T}{2}{\gamma }_0)}^2}$, with ${\gamma }_0=\frac{Z_0{\omega }_0^2}{2}\frac{C_c^2}{C_c+C_J}$. (c) $Z_0/Z_J=1000$, $C_c/(C_J+C_c)=0.3$, $T=2\pi /{\omega }_c$ and ${\omega }_0=n{\omega }_c$, $n=1$. Here, although ${\omega }_0={\omega }_C$ as in (b), the system does not seem to converge into a dark state, and the decay rate is given by ${\gamma }_J=1/C_J{Z}_0$. The difference from (b) is that the ratio between $\frac{C_c}{C_J}\frac{Z_0}{Z_J}$ here is much bigger than in (b), which also means that ${\omega }_J$ is not close to ${\omega }_C$ and the Rabi oscillations are barely visible. (d) $Z_0/Z_J=1000$, $C_c/C_J=0.3$, $T=2\pi {\omega }_c$, and ${\omega }_J=n{\omega }_c$, $n=1$. Here, as in (c) $\frac{C_c}{C_J}\frac{Z_0}{Z_J}\gg 1$, but the “Rabi condition” ${\omega }_J=n{\omega }_C$ is still fulfilled. We see clear Rabi oscillations, the decay is slower compared to (c), and the decay rate is given by ${\gamma }_J^m/2=1/2C_J{Z}_0$.

Figure 7

Rabi frequency $\mathrm{\Omega }$ as a function of $Z_0/Z_J$ calculated analytically ${\mathrm{\Omega }}_T$ (solid yellow line) and numerically ${\mathrm{\Omega }}^N$ (dots) for different values of $C_c/C_J$, $C_c/C_J=0.01$ (purple), $C_c/C_J=0.05$ (red), $C_c/C_J=0.3$ (orange), for $T=2\pi /{\omega }_c$, ${\omega }_J=n{\omega }_c$, $n=1$. We see that the higher $\frac{C_c}{C_J}\frac{Z_0}{Z_J}$ becomes, the closer the approximated analytical frequency ${\mathrm{\Omega }}_T$ and the numerically calculated frequency ${\mathrm{\Omega }}^N$ become.