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Physical Review B

Noise effects on Landau-Zener-Stückelberg interferometry in multilevel artificial atoms

Lingjie Du, Yang Yu

Published Oct 25, 2010• 16 citations
Publisher

Abstract

We have developed a comprehensive theory of Landau-Zener-St\"uckelberg interferometry in a strongly driven multilevel superconducting flux qubit interacting with the environment bath. It is found that not only the dephasing but also the intrawell relaxation make important contribution to the interference fringes. We propose that Fourier-transform-based phase tomography can be used to determine the intrawell relaxation rate as well as the type of the low-frequency noise, besides dephasing rate. The quantitative agreement between our theory and the experimental results indicates that Landau-Zener-St\"uckelberg interferometry provides a useful tool to analyze the noise source of solid-state qubits.

DOI
10.1103/physrevb.82.144524

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We have developed a comprehensive theory of Landau-Zener-St"uckelberg interferometry in a strongly driven multilevel superconducting flux qubit interacting with the environment bath. It is found that not only the dephasing but also the intrawell relaxation make important contribution to the interference fringes. We propose that Fourier-transform-based phase tomography can be used to determine the intrawell relaxation rate as well as the type of the low-frequency noise, besides dephasing rate. The quantitative agreement between our theory and the experimental results indicates that Landau-Zener-St"uckelberg interferometry provides a useful tool to analyze the noise source of solid-state qubits.
In this work we propose a way to unveil the type of environmental noise in strongly driven superconducting flux qubits through the analysis of the Landau-Zener-St\"uckelberg (LZS) interferometry. We study both the two-level and multilevel dynamics of the flux qubit driven by a dc+ac\mathrm{dc}\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}\mathrm{ac}dc+ac magnetic field. We find that the LZS interference patterns exhibit well-defined multiphoton resonances whose shape strongly depends on the timescale and the type of coupling to a quantum bath. For the case of transverse system-bath coupling, the nnn-photon resonances are narrow and nearly symmetric with respect to the dc magnetic field for almost all timescales, while in the case of longitudinal coupling they exhibit a change from a wide symmetric to an antisymmetric shape for times of the order of the relaxation time. We find this dynamic behavior relevant for the interpretation of several LZS interferometry experiments in which the stationary regime is not completely reached.
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.
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.
We proposed an analytical model to analyze the Landau-Zener interference in a multilevel superconducting flux qubit driven by large-amplitude external fields. Our analytical results agree remarkably well with those of Berns et al. [Nature (London) 455, 51 (2008)]. Moreover, we studied the effect of driving frequency and dephasing rate on the interference. The dephasing generally destroys the interference, while increasing frequency rebuilds the interference at large dephasing rate. At a certain driving frequency and dephasing rate, the interference shows some anomalous features, as observed in recent experiments.
The interference between repeated Landau-Zener transitions in a qubit swept through an avoided level crossing results in Stückelberg oscillations in qubit magnetization, a hallmark of the coherent strongly driven regime in two-level systems. The two-dimensional Fourier transforms of the resulting oscillatory patterns are found to exhibit a family of one-dimensional curves in Fourier space, in agreement with recent observations in a superconducting qubit. We interpret these images in terms of time evolution of the quantum phase of the qubit state and show that they can be used to probe dephasing mechanisms.