Visibility study of $S\text{\ensuremath{-}}{T}_{+}$ Landau-Zener-St\"uckelberg oscillations without applied initialization

Visibility study of $S − T_{+}$ Landau-Zener-Stückelberg oscillations without applied initialization

G. Granger, G. C. Aers, S. A. Studenikin, A. Kam, P. Zawadzki, Z. R. Wasilewski, and A. S. Sachrajda

Phys. Rev. B 91, 115309 – Published 18 March 2015

Abstract

Probabilities deduced from quantum information studies are usually based on averaging many identical experiments separated by an initialization step. Such initialization steps become experimentally more challenging to implement as the complexity of quantum circuits increases. To better understand the consequences of imperfect initialization on the deduced probabilities, we study the effect of not initializing the system between measurements. For this we utilize Landau-Zener-Stückelberg oscillations in a double quantum dot circuit. Experimental results are successfully compared to theoretical simulations.

Received 18 December 2013
Revised 24 November 2014

DOI:https://doi.org/10.1103/PhysRevB.91.115309

Published by the American Physical Society

Authors & Affiliations

G. Granger, G. C. Aers, S. A. Studenikin, A. Kam, P. Zawadzki, Z. R. Wasilewski^*, and A. S. Sachrajda^†

National Research Council Canada, Ottawa, ON K1A 0R6, Canada

^*Present address: Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada.
^†andrew.sachrajda@nrc.ca

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Vol. 91, Iss. 11 — 15 March 2015

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Figure 1
(a) Electron micrograph of a device similar to the one measured. The gates define the triple quantum dot potential and two QPCs used as charge detectors. The blue circles schematically indicate where the dots under study are located and the red shows the unused dot. Gates 1 and 2 receive short voltage pulses shown in (c) in addition to dc voltages $V_{1}$ and $V_{2}$ . The left QPC current $I_{QPC}$ is used to detect the electron number, plot the stability diagram, and perform the spin readout. (b) Stability diagram from the measured transconductance of the left QPC in the $V_{1} − V_{2}$ plane. The transconductance is the numerical derivative of the left QPC conductance with respect to $V_{2}$ . The electronic configurations are indicated, and LZS fringes are seen in the (2,0) region. The pulse size and direction are indicated by the blue arrow. $M$ is the measurement point, while $P$ is the maximum detuning point during the pulse. $(δ V_{1}, δ V_{2}) = (- 5, 10)$ mV, $τ = 16$ ns, $τ_{m}$ $= 2$ $μ s$ , and $B = 80$ mT. The rise time is 8 ns. The initial detuning line relevant to the data in Fig. 3 is shown with a white line. (c) Schematic showing the definition of the rectangular pulses (black) and the resulting Gaussian-convoluted pulses used here (blue). $τ$ is the pulse duration, while $τ_{m}$ is the pulse period. The pulse amplitudes $δ V_{1}$ and $δ V_{2}$ applied to gates 1 and 2 are also indicated prior to the convolution. It is expected that for the rise times used here, the effective amplitude will be a little smaller than what is shown on the rectangular pulse. (d) Schematic of $S − T_{+}$ energy diagram as a function of detuning showing the anticrossing between $S$ and $T_{+}$ . During the pulse, there is a Landau-Zener probability $P_{LZ}$ of reaching the upper branch of the $S − T_{+}$ anticrossing. “CT” indicates the charge transfer line. Inset: Gaussian-convoluted pulse.
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Figure 2
(a) Calculated 2D map of $P (S)$ vs initial detuning for 1000 consecutive pulses. Black is $P (S) = 0$ and white is $P (S) = 1$ . The arrow indicates where the $S − T_{+}$ anticrossing is observed on the initial detuning axis [as in the lower horizontal yellow line in Fig. 1]. (b) Calculated $P (S)$ vs initial detuning for hyperfine splitting $| g | μ_{B} Δ B_{x y} = 0.2$ $μ eV$ of the IS case (red). The blue curve is the NIS case from the average over 1000 pulse repeats. The green dashed curve is NIS derived from IS (see text).
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Figure 3
(a) LZS oscillations of $P (S)$ in the $τ_{m}$ -initial detuning plane from the left QPC conductance (plane subtracted) taken along the white line from Fig. 1. The initial detuning is with respect to the position of the charge transfer line in Fig. 1. $(δ V_{1}, δ V_{2}) = (- 5, 10)$ mV, $τ = 16$ ns, and $B = 80$ mT. The rise time is 8 ns. Yellow is low, red is medium, and black is high. (b) Calculated $P (S)$ to compare to (a).
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Figure 4
Visibility as a function of $τ_{m} / T_{1}$ from the experiments in Fig. 3, where we assume $T_{1}$ $= 60$ $μ s$ as a fitting parameter (filled circles) and NIS calculations (blue line). Calculation parameters are the same as for the calculation in Figs. 2 and 2; in particular, $| g | μ_{B} Δ B_{x y} = 0.2$ $μ eV$ .
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Figure 5
(a) Calculated IS (red squares) and NIS (blue circles) visibility for $τ_{m} / T_{1}$ $\sim 0.1$ as a function of $| g | μ_{B} Δ B_{x y}$ . (b) Calculated $P (S)$ vs initial detuning for hyperfine splitting $| g | μ_{B} Δ B_{x y} = 0.11$ $μ eV$ of the IS case (red). The blue curve is the NIS case for comparison.
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Physical Review B

covering condensed matter and materials physics

Visibility study of $S − T_{+}$ Landau-Zener-Stückelberg oscillations without applied initialization

G. Granger, G. C. Aers, S. A. Studenikin, A. Kam, P. Zawadzki, Z. R. Wasilewski, and A. S. Sachrajda

Phys. Rev. B 91, 115309 – Published 18 March 2015

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Physical Review B

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Visibility study of S−T+ Landau-Zener-Stückelberg oscillations without applied initialization

G. Granger, G. C. Aers, S. A. Studenikin, A. Kam, P. Zawadzki, Z. R. Wasilewski, and A. S. Sachrajda

Phys. Rev. B 91, 115309 – Published 18 March 2015

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Visibility study of $S − T_{+}$ Landau-Zener-Stückelberg oscillations without applied initialization