Characterization of Qubit Dephasing by Landau-Zener-Stückelberg-Majorana Interferometry

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 $T_2\simeq 200\mathrm{ns}$, which is limited by electron-phonon interaction.

Figure 1

Experimental setup. (a) Scanning electron micrograph showing Ti/Au gates, fabricated by electron-beam lithography, on the surface of a GaAs/AlGaAs heterostructure, grown by molecular beam epitaxy (500 nm scale bar). 85 nm beneath the surface it contains a 2DES with carrier density $n_{\mathrm{e}}=1.19\times {10}^{11}{\mathrm{cm}}^2$ and mobility $\mu =0.36\times {10}^6{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. Six of the Ti/Au gates (light yellow) are biased with negative voltages to electrostatically define a DQD in the 2DES, and the other gates are grounded. A cobalt single-domain nanomagnet (thick blue bar) produces an inhomogeneous magnetic field that slightly mixes singlet and triplet states of the DQD [12]. (b) Typical situation in our two-electron DQD: vertical lines indicate tunable tunnel barriers, horizontal lines mark the chemical potentials, blue areas are the degenerate 2DES leads. The voltage $V=({\mu }_S-{\mu }_D)/e$ causes a single-electron tunneling current [green arrow in panel (a)]. (c) Energy diagram of the relevant two-electron DQD eigenstates. Singlets (the qubit states) are represented as black and red lines; triplets, which are Zeeman split, are represented as gray lines. Rf modulation of the gate voltage $V_{\sim }$ [panel (a)] results in a modulated detuning $ϵ(t)$, indicated by gray shading.

Figure 2

LZSM interference. Measured current through the DQD as a function of mean detuning $\overline{ϵ}$ and modulation amplitude $A$ at $T\simeq 20\mathrm{mK}$ for the modulation frequencies 2.5 GHz (a) and 4.5 GHz (b). [(c), (d)] Corresponding numerically calculated current for realistic conditions. [(e)—(h)] Two-dimensional numerical Fourier transformed ($A\to {\tau }_A$, $\overline{ϵ}\to {\tau }_{ϵ}$, $I\to \widehat{I}$) of measurements (upper panels) and theory (lower panels). The shape of the sinusoidal branches of enhanced $\widehat{I}$ is determined by $\mathrm{\Omega }$; see Eq. (3). Their decay with increasing ${\tau }_{ϵ}$ encodes dephasing and decoherence. The horizontal and vertical lines of enhanced amplitude at ${\tau }_A=0$ and ${\tau }_{ϵ}=0$ are artifacts caused by the finite region of data being transformed.

Figure 3

Raw data analysis. Dots are measured at $\mathrm{\Omega }/2\pi =2.75\mathrm{GHz}$, lines numerical data for ${\alpha }_Z=1.5\times {10}^{-4}$ and ${\lambda }^{\star }=3.5\mu \mathrm{eV}$, whereas the gray line in panel (a) is for ${\lambda }^{\star }=0$. (a) Horizontal slice through a LZSM pattern: $I(\overline{ϵ})$ for a constant $A=130\mu \mathrm{eV}$. (b) Vertical slices through a LZSM pattern: $I(A)$ for $\overline{ϵ}/\hslash \mathrm{\Omega }=0$, $-2$, $-4$, $-6$. (c) Measured data as in panel (a) for various temperatures.

Figure 4

Electron-phonon coupling. (a) Decaying region of the measured $\widehat{I}({\tau }_{ϵ},{\tau }_A){|}_{\text{lemon}}$ for three temperatures (dots). Lines are generated using Eq. (4) for ${\lambda }^{\star }=3.5\mu \mathrm{eV}$ and $\lambda$ as a fit parameter. The inset shows a broader region including maxima at ${\tau }_{ϵ}=0$, $2\pi /\mathrm{\Omega }$. (b) Measured decay rate $\lambda (T)$ (black dots) and corresponding numerical data (colored circles) based on ${\lambda }^{\star }=3.5\mu \mathrm{eV}$ (indicated as horizontal line) and ${\alpha }_Z=1.0$, 1.5, $2.0\times {10}^{-4}$. (c) Analog to panel (a) but based on numerical calculations. Solid lines are identical to those in (a). The numerical resolution is based on 100 data points sampling the Gaussian broadening in $\overline{ϵ}$ of width ${\lambda }^{\star }=3.5\mu \mathrm{eV}$.