Effects of charge noise on a pulse-gated singlet-triplet $S\phantom{\rule{0.16em}{0ex}}\ensuremath{-}\phantom{\rule{0.16em}{0ex}}{T}_{\ensuremath{-}}$ qubit

Effects of charge noise on a pulse-gated singlet-triplet $S - T_{-}$ qubit

Zhenyi Qi, X. Wu, D. R. Ward, J. R. Prance, Dohun Kim, John King Gamble, R. T. Mohr, Zhan Shi, D. E. Savage, M. G. Lagally, M. A. Eriksson, Mark Friesen, S. N. Coppersmith, and M. G. Vavilov

Phys. Rev. B 96, 115305 – Published 11 September 2017

Abstract

We study the dynamics of a pulse-gated semiconductor double-quantum-dot qubit. In our experiments, the qubit coherence times are relatively long, but the visibility of the quantum oscillations is low. We show that these observations are consistent with a theory that incorporates decoherence arising from charge noise that gives rise to detuning fluctuations of the double dot. Because effects from charge noise are largest near the singlet-triplet avoided level crossing, the visibility of the oscillations is low when the singlet-triplet avoided level crossing occurs in the vicinity of the charge degeneracy point crossed during the manipulation, but there is only modest dephasing at the large detuning value at which the quantum phase accumulates. This theory agrees well with experimental data and predicts that the visibility can be increased greatly by appropriate tuning of the interdot tunneling rate.

Received 31 January 2017
Revised 16 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.115305

Physics Subject Headings (PhySH)

Quantum information with solid state qubits

Double quantum dots

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Zhenyi Qi¹, X. Wu¹, D. R. Ward^1,2, J. R. Prance¹, Dohun Kim¹, John King Gamble^1,2, R. T. Mohr¹, Zhan Shi¹, D. E. Savage¹, M. G. Lagally¹, M. A. Eriksson¹, Mark Friesen¹, S. N. Coppersmith¹, and M. G. Vavilov¹

¹Wisconsin Institute for Quantum Information, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA
²Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA

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Vol. 96, Iss. 11 — 15 September 2017

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Figure 1
(a) Micrograph of a double-dot device in a Si/SiGe heterostructure that is lithographically identical to the one used in the experiment [22]. (b) Stability diagram showing the electron occupations in the dots, obtained by measuring the current through the quantum point contact $I_{QPC}$ at different voltages $Δ V_{LP / RP}$ applied to the left and right gates labeled in (a). The numbers in parentheses are the electron occupations of the two dots.
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Figure 2
(a) Schematic energy diagram of the full five-level system. A small transverse magnetic field gradient causes an anticrossing to occur between the singlet and triplet states $S (1, 1)$ and $T_{-}$ . The left inset is an expanded view of the region in the small green box in the main figure. (b) A schematic of the pulse applied to the detuning $ɛ$ as a function of time $t$ . The system is ramped from a negative detuning $ɛ_{1}$ to a large positive detuning $ɛ_{2}$ over a ramp time $τ_{r}$ , held at $ɛ_{2}$ for a manipulation time $τ_{s}$ , and then ramped back to $ɛ_{1}$ over the time $τ_{r}$ . The pulse sequence passes through the $S - T_{-}$ anticrossing twice, giving rise to Landau-Zener-Stückelberg oscillations.
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Figure 3
Visibility of LZS oscillations as a function of ramp rate $v$ , where $v$ is defined as the slope $d ɛ / d t$ of the initial ramp in the pulse sequence of Fig. 2. The red dots are experimental data, and the blue line shows the results of theoretical simulations incorporating charge noise with a $α / ω^{0.7}$ spectrum [41]. No magnetic fluctuations are included in the model. The tunnel coupling at zero detuning used in the calculation is the same as that measured in the experiment, $t_{c 0} = 3.4 μ eV$ . The adjustable parameters used to obtain the theoretical results are $α$ , which determines the noise amplitude and yields detuning fluctuations consistent with experimental estimates [22], and $h$ , which describes the transverse magnetic field gradient and determines the optimal ramp rate. The inset shows the experimental return probability $P_{s}$ measured as a function of manipulation time $τ_{s}$ .
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Figure 4
Return probability $P_{s}$ measured in the experiment (red solid line) as a function of manipulation time $τ_{s}$ (these are the same data as in the inset of Fig. 3). A Gaussian fit to the oscillation envelope yields a decoherence time of $\sim 1.7 μ s$ [22]. The maximum visibility, or oscillation amplitude, is about 0.24. The blue dashed line corresponds to our theoretical prediction for the LZS oscillations, obtained using the same parameters as in Fig. 3 at a ramp rate of $v = 4.4 μ eV / s$ , which is the same as the experimental ramp rate. The blue solid lines indicate the envelope of the theoretical LZS oscillations, whose amplitude decays as $exp [- χ (τ_{s})]$ , as discussed in the main text.
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Figure 5
(a) Visibility of LZS oscillations at the optimum ramp rate as a function of tunnel coupling $t_{c 0}$ . (b) Semilog plot of the energy difference between the ground state $| 0 〉$ and first excited state $| 1 〉$ as a function of detuning; the dips of these curves occur at the magnetic anticrossings. (c) Transition rates $Γ$ as a function of detuning. The thin black lines in (b) and (c) are obtained for the tunnel coupling $t_{c 0} = 3.4 μ eV$ , while the thick red lines are obtained for $t_{c 0} = 20 μ eV$ . Increasing the tunnel coupling can be seen to move the magnetic anticrossing farther from the charge anticrossing, causing a decrease in the transition rates.
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Figure 6
(a) The probability of the ground state $| 0 〉$ and the first excited state $| 1 〉$ of a five-level system through a one-directional ramp, starting from the ground state at detuning $ɛ_{1} = - 100 μ eV$ . The ramp rate is $v = d ɛ / d t = 4.4 μ eV$ . Parameters $t_{c 0} = 3.4 μ eV, h = 0.042 μ eV$ , and $g μ_{B} B = 0.17 μ eV$ are the same as in Fig. 3 in the main text. Charge noise is not included in this calculation. The occupations of the other three states are too small to be visible in (a). (b) The probability of the third and fourth excited states. The second excited state is not coupled to the other states given that $h_{z} = 0$ , and the probability of being in this level remains zero throughout the ramp. Because only two of the energy levels have significant occupation at any time during the evolution, the dynamics can be described using the two-state Hamiltonian [Eq. (1) in the main text].
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Physical Review B

covering condensed matter and materials physics

Effects of charge noise on a pulse-gated singlet-triplet $S - T_{-}$ qubit

Zhenyi Qi, X. Wu, D. R. Ward, J. R. Prance, Dohun Kim, John King Gamble, R. T. Mohr, Zhan Shi, D. E. Savage, M. G. Lagally, M. A. Eriksson, Mark Friesen, S. N. Coppersmith, and M. G. Vavilov

Phys. Rev. B 96, 115305 – Published 11 September 2017

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Effects of charge noise on a pulse-gated singlet-triplet S−T− qubit

Zhenyi Qi, X. Wu, D. R. Ward, J. R. Prance, Dohun Kim, John King Gamble, R. T. Mohr, Zhan Shi, D. E. Savage, M. G. Lagally, M. A. Eriksson, Mark Friesen, S. N. Coppersmith, and M. G. Vavilov

Phys. Rev. B 96, 115305 – Published 11 September 2017

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Effects of charge noise on a pulse-gated singlet-triplet $S - T_{-}$ qubit