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arXiv:0805.1555 · Physical Review Letters

Quantum Phase Tomography of a Strongly Driven Qubit

Mark Spencer Rudner, A. V. Shytov, Leonid S Levitov, David M. Berns, William D. Oliver, Sergio O. Valenzuela, Terry P. Orlando

Published Nov 7, 2008• 69 citations
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Abstract

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.

DOI
10.1103/physrevlett.101.190502

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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.
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.
A new regime of coherent quantum dynamics of a qubit is realized at low driving frequencies in the strong driving limit. Coherent transitions between qubit states occur via the Landau-Zener process when the system is swept through an energy-level avoided crossing. The quantum interference mediated by repeated transitions gives rise to an oscillatory dependence of the qubit population on the driving-field amplitude and flux detuning. These interference fringes, which at high frequencies consist of individual multiphoton resonances, persist even for driving frequencies smaller than the decoherence rate, where individual resonances are no longer distinguishable. A theoretical model that incorporates dephasing agrees well with the observations.
The device for the Josephson flux qubit can be considered as a solid-state artificial atom with multiple energy levels. When a large-amplitude harmonic excitation is applied to the system, transitions at the energy levels avoided crossings produce visible changes in the qubit population over many driven periods that are accompanied by a rich pattern of interference phenomena. We present a Floquet treatment of the periodically time-dependent Schr\"odinger equation of the strongly driven qubit beyond the standard two-level approach. For low amplitudes, the average probability of a given sign of the persistent current qubit exhibits, as a function of the static flux detuning and the driving amplitude, Landau-Zener-St\"uckelberg (LZS) interference patterns that evolve into complex diamondlike patterns for large amplitudes. In the case of highly coherent flux qubits we show that the higher-order diamonds can not be simply described relying on LZS transitions in each avoided crossing considered separately. In addition we propose a spectroscopic method based on starting the system in the first excited state instead of in the ground state, which can give further information on the energy-level spectrum and dynamics in the case of highly coherent flux qubits. We compare our numerical results with recent experiments that perform amplitude spectroscopy to probe the energy spectrum of the artificial atom.
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.
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.
Controlling coherent interaction at avoided crossings and the dynamics there is at the heart of quantum information processing. A particularly intriguing dynamics is observed in the Landau-Zener regime, where periodic passages through the avoided crossing result in an interference pattern carrying information about qubit properties. In this Letter, we demonstrate a straightforward method, based on steady-state experiments, to obtain all relevant information about a qubit, including complex environmental influences. We use a two-electron charge qubit defined in a lateral double quantum dot as test system and demonstrate a long coherence time of T2 ≃ 200 ns, which is limited by electron-phonon interaction.