Time-Reversal Symmetry and Universal Conductance Fluctuations in a Driven Two-Level System

In the presence of time-reversal symmetry, quantum interference gives strong corrections to the electric conductivity of disordered systems. The self-interference of an electron wave function traveling time-reversed paths leads to effects such as weak localization and universal conductance fluctuations. Here, we investigate the effects of broken time-reversal symmetry in a driven artificial two-level system. Using a superconducting flux qubit, we implement scattering events as multiple Landau-Zener transitions by driving the qubit periodically back and forth through an avoided crossing. Interference between different qubit trajectories gives rise to a speckle pattern in the qubit transition rate, similar to the interference patterns created when coherent light is scattered off a disordered potential. Since the scattering events are imposed by the driving protocol, we can control the time-reversal symmetry of the system by making the drive waveform symmetric or asymmetric in time. We find that the fluctuations of the transition rate exhibit a sharp peak when the drive is time symmetric, similar to universal conductance fluctuations in electronic transport through mesoscopic systems.

Figure 1

Qubit trajectories during driven evolution. (a) Energy-level diagram of a flux qubit. Driving the qubit through the avoided crossing induces Landau-Zener transitions between the two states. The states recombine when the qubit is brought back through the avoided crossing, with the final outcome depending on the phase accumulated during the flux excursion. (b) Qubit transitions visualized as two scattering events in a Mach-Zehnder interferometer setup. (c) Illustration of a mesoscopic system with two paths related by time-reversal symmetry. (d), (e) Qubit transitions for a drive waveform that brings the qubit through the avoided crossing multiple times. The blue (solid) and red (dashed) traces mark two (out of eight) possible trajectories in the system. (d) When the waveform is symmetric in time, the trajectories acquire the same phases (since ${\phi }_1={\phi }_3$) and will interfere constructively. (e) Without time-reversal symmetry, ${\phi }_1\ne {\phi }_3$.

Figure 2

Qubit transition rate as a function of the time-reversal symmetry of the drive waveform. (a) Drive waveform for different values of the drive asymmetry parameter $\alpha$. (b) Zoom-in around the center region of (a); the waveform becomes asymmetric in time when $\alpha \ne 0$. (c) Measured transition rate versus static flux detuning $f_{\mathrm{dc}}$ and $\alpha$. (d) Qubit energy diagram. (e), (f) The case when $f_{\mathrm{dc}}>0$ [positions in (c) marked by I and II]; the qubit is driven through the avoided crossing twice, resulting in Landau-Zener oscillations as a function of the accumulated phase $\phi$. (g) The case when $f_{\mathrm{dc}}<0$ (position III); here, the qubit is driven through the avoided crossing four times, giving rise to an intricate interference pattern.

Figure 3

Fluctuations in the transition rate of a driven qubit. (a) Measured transition rate versus flux detuning and drive asymmetry $\alpha$. (b) Transition rate averaged from $-4\mathrm{m}{\Phi }_0$ to $0\mathrm{m}{\Phi }_0$. The data do not show any dependence on the drive asymmetry. (c) Standard deviation of the transition rate. The fluctuations have a sharp peak at $\alpha =0$.

Figure 4

Simulation results. (a) Simulated transition rate $\Gamma$ for qubit parameters $T_2=10\mathrm{ns}$ and $\Delta /h=19\mathrm{MHz}$. (b) Standard deviation of the simulated transition rates for different values of $T_2$, extracted from simulation results similar to those shown in (a). Note that the curves are not offset from each other.