Microwave-induced amplitude- and phase-tunable qubit-resonator coupling in circuit quantum electrodynamics

In the circuit quantum electrodynamics architecture, both the resonance frequency and the coupling of superconducting qubits to microwave field modes can be controlled via external electric and magnetic fields to explore qubit-photon dynamics in a wide parameter range. Here, we experimentally demonstrate and analyze a scheme for tuning the coupling between a transmon qubit and a microwave resonator using a single coherent drive tone. We treat the transmon as a three-level system, with the qubit subspace defined by the ground and the second excited states. If the drive frequency matches the difference between the resonator and the qubit frequencies, a Jaynes-Cummings-type interaction is induced, which is tunable in both amplitude and phase. We show that coupling strengths of about 10 MHz can be achieved in our setup, limited only by the anharmonicity of the transmon qubit. This scheme has been successfully used to generate microwave photons with a controlled temporal shape [M. Pechal et al., Phys. Rev. X 4, 041010 (2014)] and can be directly implemented with superconducting quantum devices featuring larger anharmonicity for higher coupling strengths.

Figure 1

(a) Resonator transmission as a function of the transmon and resonator drive frequencies. The white curve is a fit to the data at the fixed transmon drive frequency ${\omega }_d/2\pi =8.75$ GHz. (b) Normalized transmission data for the white fit curve in (a). The amplitude of the effective coupling strength $\tilde{g}/2\pi$ extracted from this trace is 3.1 MHz. (c) Measured effective coupling $\tilde{g}$ (black diamonds) as a function of the coherent drive amplitude $\Omega$: perturbation theory prediction [thick (red) line] and numerical simulations [thin (green) line].

Figure 2

Schematic energy level diagram of the system: (a) in the laboratory frame and (b) in the rotating frame of the drive frequency ${\omega }_d$, which compensates for the energy difference between $|f0\rangle$ and $|g1\rangle$. The effective tunable coupling $\tilde{g}$ between the bare states $|f0\rangle$ and $|g1\rangle$ can be understood as a second-order cavity-assisted Raman process, indicated by the dashed arrows.

Figure 3

Measured values of ${\Delta }_{f0g1}$ for increasing drive amplitudes $\Omega$. A fit to the expression in Eq. (20) obtained from the resolvent method (solid line) where the perturbative series is truncated at the the sixth order in $\Omega$. The fit gives the conversion factor between the applied drive power and the applied drive amplitude seen by the transmon. The scale at the top indicates the drive power at the signal generator, where each tick stands for the drive power at which the data point is obtained.

Figure 4

(a) Dependence of the loss of fidelity $1-\mathcal{F}$ on the length of the $\pi$ pulse and its rise time. The transmon-resonator detuning is set to $\Delta /2\pi =0.979$ GHz, the anharmonicity is $\alpha /2\pi =-0.376$ GHz, and $g/2\pi =65$ MHz. Solid lines show the results of the numerical simulations. The theoretical prediction of Eq. (24) is shown by the dashed (red) line. (b, c) Time evolution of the populations $p_{f0},p_{g1}$ in ${|f0\rangle }_D,{|g1\rangle }_D$ and leakage to other levels given by $1-p_{g1}-p_{f0}$ for a 50-ns drive pulse with rise times of 0.2 ns (b) and 5.0 ns (c).