Vacuum Rabi Oscillations in a Macroscopic Superconducting Qubit $LC$ Oscillator System

We have observed the coherent exchange of a single energy quantum between a flux qubit and a superconducting $LC$ circuit acting as a quantum harmonic oscillator. The exchange of an energy quantum is known as the vacuum Rabi oscillation: the qubit is oscillating between the excited state and the ground state and the oscillator between the vacuum state and the first excited state. We also show that we can detect the state of the oscillator with the qubit and thereby obtained evidence of level quantization of the $LC$ circuit. Our results support the idea of using oscillators as couplers of solid-state qubits.

Figure 1

(a) scanning electron micrograph of the sample. The square plates at the top of the picture are the top plates of the on-chip capacitors separated by an insulator from the large bottom plate. (b) A close up of the qubit and the SQUID. (c) Equivalent circuit of the sample. The Josephson junctions are indicated by crosses: three in the inner qubit loop and two in the SQUID. The $LC$ mode is indicated by the dashed line. The inductance and capacitances are calculated from the geometry to be $L=140\mathrm{pH}$ and $C=10\mathrm{pF}$, and the qubit $LC$ oscillator mutual inductance to be $M=5.7\mathrm{pH}$.

Figure 2

(a) Spectroscopic characterization of the qubit-oscillator system showing the $LC$ oscillator at ${\nu }_{\mathrm{r}}=4.35\mathrm{GHz}$ and the qubit dispersion around the gap of $\Delta =2.1\mathrm{GHz}$. (b) A close up of the region around 4.35 GHz [square in (a)] showing an avoided crossing. The lines are guides for the eye. (c) Schematic of the pulse sequence used to obtain the spectroscopy in (b): the qubit operating point is fixed at 10.5 GHz and via the MW line a shift pulse of variable height moves the operating point to the vicinity of 4.35 GHz. Two 50 ns MW pulses separated by 2 ns are added to the shift pulse. Here the second MW pulse is phase shifted by 180°. The phase shifted second pulse damps the oscillations in the $LC$ circuit [22] and is crucial in terms of resolving the relatively weak qubit signal in this region. After the MW pulses the qubit state is measured by applying a measurement pulse to the SQUID (dashed curve). The spectroscopy in Fig. 2a was obtained with the same scheme, but without the second phase shifted pulse.

Figure 3

(a) Vacuum Rabi oscillations (symbols) and a numerical fit (solid line). (b) The few lowest unperturbed and dressed energy levels when the system is in resonance. (c) The qubit energy level diagram and pulse sequence for the vacuum Rabi measurements. The $\pi$ pulse (4.6 ns long) on the qubit brings the system from state 1 to 2 and the shift pulse changes the flux in the qubit by ${\Phi }_{\mathrm{shift}}$, which, in turn, changes the operating point from 2 to 3 where the system undergoes free evolution between $|e,0⟩$ and $|g,1⟩$ at the vacuum Rabi frequency ${\Omega }_R$, until the shift pulse ends and the system returns to the initial operating point where the state is measured to be either in 2 or 4.

Figure 4

(a) Rabi oscillations when a 2 ns long pulse with frequency 4.35 GHz and power $P_{\mathrm{osc}}$ is inserted between the $\pi$ pulse and the shift pulse. (b) Measured oscillations at different drive powers (symbols), and a fit (solid curve) to $[\sum _{n=0}^3{a}_n\cos (\sqrt{n+1}{\Omega }_{R}t)]\exp (-\Gamma t)$ with $a_0,\dots ,a_3$ and ${\Omega }_R$ as the only fitting parameters ($\Gamma$ is fixed from a fit to the $-100\mathrm{dBm}$ curve). (c) The weights of the different frequency components $a_0,\dots ,a_3$ obtained from the fit as a function of the drive amplitude [the arrows show the position of the curves in (b)]. The error bars indicate the errors obtained from the fitting procedure.