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arXiv:1204.6428 · Physical Review Letters

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

Simon Gustavsson, Jonas Bylander, William D. Oliver

Published Jan 2, 2013• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.110.016603

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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.
We investigate the effect of broken time reversal symmetry in flux qubits driven by a biharmonic magnetic flux signal with a phase lag. In the regime of large relaxation times, we explicitly compute the transition rate between the ground and the excited state, accounting for decoherence as a classical noise. Through a direct analogy between interference effects at the avoided level crossing and scattering events in weakly disordered electronic mesoscopic systems, the transition rate plays the role of an effective transmittance while the phase lag acts as a time reversal control parameter. Clear signatures of both weak localization and conductance fluctuations-like effects are predicted. Their behavior is studied as a function of the coherence rate, and a comparison with recent experimental results is performed. Our study shows that it is decoherence, and not the driving protocol, what limits the experimental detection of weak localization effects.
In condensed matter systems, coherent backscattering and quantum interference in the presence of time-reversal symmetry lead to well-known phenomena, such as weak localization (WL) and universal conductance fluctuations (UCFs). Here we use multipass Landau-Zener transitions at the avoided crossing of a highly coherent superconducting qubit to emulate these phenomena. The average and standard deviations of the qubit transition rate exhibit a dip and peak when the driving waveform is time-reversal symmetric, analogous to WL and UCFs, respectively. The higher coherence of this qubit enabled the realization of both effects, in contrast to the earlier work by Gustavsson et al. [Phys. Rev. Lett. 110, 016603 (2013)], who successfully emulated UCFs, but did not observe WL. This demonstration illustrates the use of nonadiabatic control to implement quantum emulation with superconducting qubits.
In condensed matter systems, coherent backscattering and quantum interference in the presence of time-reversal symmetry lead to well-known phenomena, such as weak localization (WL) and universal conductance fluctuations (UCFs). Here we use multipass Landau-Zener transitions at the avoided crossing of a highly coherent superconducting qubit to emulate these phenomena. The average and standard deviations of the qubit transition rate exhibit a dip and peak when the driving waveform is time-reversal symmetric, analogous to WL and UCFs, respectively. The higher coherence of this qubit enabled the realization of both effects, in contrast to the earlier work by Gustavsson et al. [Phys. Rev. Lett. 110, 016603 (2013)], who successfully emulated UCFs, but did not observe WL. This demonstration illustrates the use of nonadiabatic control to implement quantum emulation with superconducting qubits.
We investigate flux qubits driven by a biharmonic magnetic signal, with a phase lag that acts as an effective time reversal broken parameter. The driving induced transition rate between the ground and the excited state of the flux qubit can be thought of as an effective transmittance, profiting from a direct analogy between interference effects at avoided level crossings and scattering events in disordered electronic systems. For time scales prior to full relaxation, but large compared to the decoherence time, this characteristic rate has been accessed experimentally by Gustavsson et al. [Phys. Rev. Lett. 110, 016603 (2013)] and its sensitivity with both the phase lag and the dc flux detuning explored. In this way, signatures of universal conductance fluctuationslike effects have been analyzed and compared with predictions from a phenomenological model that only accounts for decoherence, as a classical noise. Here we go beyond the classical noise model and solve the full dynamics of the driven flux qubit in contact with a quantum bath employing the Floquet-Born-Markov master equation. Within this formalism, the computed relaxation and decoherence rates turn out to be strongly dependent on both the phase lag and the dc flux detuning. Consequently, the associated pattern of fluctuations in the characteristic rates display important differences with those obtained within the mentioned phenomenological model. In particular, we demonstrate the weak localizationlike effect in the average values of the relaxation rate. Our predictions can be tested for accessible but longer time scales than the current experimental times.
We study Landau-Zener-St\"uckelberg (LZS) interferometry in multilevel systems coupled to an Ohmic quantum bath. We consider the case of superconducting flux qubits driven by a dc+ac magnetic fields, but our results can apply to other similar systems. We find a dynamic transition manifested by a symmetry change in the structure of the LZS interference pattern, plotted as a function of ac amplitude and dc detuning. The dynamic transition is from an LZS pattern with nearly symmetric multiphoton resonances to antisymmetric multiphoton resonances at long times (above the relaxation time). We also show that the presence of a resonant mode in the quantum bath can impede the dynamic transition when the resonant frequency is of the order of the qubit gap. Our results are obtained by a numerical calculation of the finite time and the asymptotic stationary population of the qubit states, using the Floquet-Markov approach to solve a realistic model of the flux qubit considering up to ten energy levels.
Landau-Zener-Stuckelberg interferometry has been extensively investigated in quantum two-level systems, with particular interests on artificial system such as superconducting flux qubits. With increasing the driving field amplitude, more energy levels will be involved into the quantum evolution, which results in population inversion and many interesting interference patterns. These interference patterns can be used to obtain the parameters characterizing the system and probe dephasing mechanisms of the qubit. Most recently, experiments have been extended to the regime with higher-frequency and larger-amplitude driving field, in which the interference pattern exhibited more complicated characteristics. In this article, we give a universal description of the characteristics observed in both low-frequency and high-frequency regimes. Besides explaining the already observed experimental results, our theoretical model predicted many interesting phenomenon, which can be demonstrated by future experiments.