Fig. 1
Device and scheme for measuring geometric phase. a Device schematic and image illustrating the five frequency-tunable qubits, labeled from Q1 to Q5, and the bus resonator R, which has a fixed bare frequency (resonator frequency in absence of qubits) ω rb/2π ≈ 5.585 GHz. The color-coded icons identify the pads where pulses are injected onto the circuit chip. The transmission line (TL) carries the multi-tone microwave pulse through the circuit chip, which is amplified by a Josephson parametric amplifier (JPA) at low temperature and then demodulated at room temperature to yield the state of all qubits. b Energy level configuration of the qubit-resonator system. The strong coupling between \(\left| {2,0} \right\rangle\) and \(\left| {1,1} \right\rangle\) produces the dressed states \(\left| {{\phi _ \pm }} \right\rangle\) whose energy levels are well separated. A microwave drive with a tone of ω d that is slightly detuned from ω r by δ can or cannot excite the resonator depending on whether the qubit is in the state \(\left| 0 \right\rangle\) or \(\left| 1 \right\rangle\). c Resonator’s phase-space displacement conditional on the qubit state \(\left| 0 \right\rangle\). In the drive frame, the resonator, initially in its ground state, is displaced by the microwave drive of an amplitude Ω along a circle in phase space with the radium Ω /δ and the angular velocity δ conditional on the qubit state \(\left| 0 \right\rangle\). At time T = 2π/δ, the resonator makes a cyclic evolution, returning to the ground state, but acquires a conditional geometric phase proportional to the enclosed phase-space area. d Ramsey interference sequence plotted in the frequency vs. time plane. The geometric operation, resulting from the combination of the microwave drive (green sinusoid) and the qubit-resonator coupling, is sandwiched in between the two π/2 rotations (blue sinusoids with Gaussian envelopes), X π/2 and θ π/2, whose rotation axes are in the xy plane of the Bloch sphere and differ by an angle of θ. The corresponding geometric phase β is revealed by measuring the qubit \(\left| 1 \right\rangle\)-state probability as a function of θ, using the microwave pulse through the TL readout line (light brown sinusoid with a ring-down shape at the beginning)
