Geometric phases in superconducting qubits beyond the two-level approximation

Geometric phases, which accompany the evolution of a quantum system and depend only on its trajectory in state space, are commonly studied in two-level systems. Here, however, we study the adiabatic geometric phase in a weakly anharmonic and strongly driven multilevel system, realized as a superconducting transmon-type circuit. We measure the contribution of the second excited state to the two-level geometric phase and find good agreement with theory treating higher energy levels perturbatively. By changing the evolution time, we confirm the independence of the geometric phase of time and explore the validity of the adiabatic approximation at the transition to the nonadiabatic regime.

Figure 1

(a) Schematic energy level diagram of the transmon, with the computational subspace spanned by $|0\rangle$ and $|1\rangle$, and the second excited state $|2\rangle$ with anharmonicity ${\alpha }_2<0$. (b) Lumped-element circuit diagram of the sample and the measurement setup (see text for details). (c) Optical microscope image of the sample with two transmons coupled to a coplanar waveguide resonator with individual capacitively coupled microwave gate lines. (d) Closeup of the transmon used in the experiments.

Figure 2

(a) Sketch of the microwave pulse sequence used in a geometric phase measurement, consisting of a series of resonant (${\omega }_{01}$) and off-resonant (${\omega }_{01}-\Delta$) pulses applied either on the $x$ quadrature (blue solid line) or the $y$ quadrature (green dotted line). The geometric phase is generated by the adiabatic evolution of the qubit between the resonant pulses. (b) Simulated adiabatic evolution (blue line) of the ground state subjected to the off-resonant drive along the path $C^{-}$, visualized in the Hilbert space of a three-level system and on the Bloch sphere, the approximate two-level equivalent (see text for details). (c) Extracted phase $\gamma$ as a function of solid angle $A$. Shown is the experimental data for $C^{-+}$ (circles), $C^{--}$ (diamonds), and $C^{+-}$ (triangles), as well as the geometric phase obtained with second-order perturbation theory (solid lines) and the prediction for a two-level system (dashed lines). The off-resonant pulses were applied with detuning $\Delta /2\pi =-35\mathrm{MHz}$. (d) Extracted phase $\gamma$ as a function of detuning $\Delta$. The experimental data for solid angles $A\approx \pi /4,3\pi /4,$ and $5\pi /4$ [indicated by circles, triangles, and diamonds, respectively, and also indicated by arrows in (c)] is shown alongside the geometric phase calculated using second-order perturbation theory (solid lines) and the prediction for a two-level system (dashed lines).

Figure 3

(a) Measured phase $\gamma$ as function of phase sweep time $\tau$ for the solid angle $A=\pi /4$ at detuning $\Delta /2\pi =-45\mathrm{MHz}$. The dashed line is the phase obtained by numerically calculating the unitary time evolution of a four-level transmon using the Schrödinger equation. The adiabaticity parameters $a$ for a given $\tau$ are indicated on the upper horizontal axis. (b) Fidelity of the geometric phase gates shown in (a) as a function of $\tau$.