Coherent control via weak measurements in $^{31}\mathrm{P}$ single-atom electron and nuclear spin qubits

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Coherent control via weak measurements in ${}^{31}P$ single-atom electron and nuclear spin qubits

J. T. Muhonen, J. P. Dehollain, A. Laucht, S. Simmons, R. Kalra, F. E. Hudson, A. S. Dzurak, A. Morello, D. N. Jamieson, J. C. McCallum, and K. M. Itoh

Phys. Rev. B 98, 155201 – Published 5 October 2018

Abstract

The understanding of weak measurements and interaction-free measurements has greatly expanded the conceptual and experimental toolbox to explore the quantum world. Here we demonstrate single-shot variable-strength weak measurements of the electron and nuclear spin states of a ${}^{31}P$ single-atom donor in silicon. We first show how the partial collapse of the nuclear spin due to measurement can be used to coherently rotate the spin to a desired pure state. We explicitly demonstrate that phase coherence is preserved with high fidelity throughout multiple sequential single-shot weak measurements and that the partial state collapse can be reversed. Second, we use the relation between measurement strength and perturbation of the nuclear state as a physical meter to extract the tunnel rates between the ${}^{31}P$ donor and a nearby electron reservoir from data conditioned on observing no tunneling events. Our experiments open avenues to measurement-based state preparation, steering and feedback protocols for spin systems in the solid state, and highlight the fundamental connection between information gain and state modification in quantum mechanics.

Received 6 June 2017
Revised 26 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.155201

Physics Subject Headings (PhySH)

Quantum control Quantum gates Quantum information with solid state qubits Quantum measurements

Coherent control Weak measurements

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

Authors & Affiliations

J. T. Muhonen^*, J. P. Dehollain^†, A. Laucht, S. Simmons^‡, R. Kalra^§, F. E. Hudson, A. S. Dzurak, and A. Morello^∥

Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, New South Wales 2052, Australia

D. N. Jamieson and J. C. McCallum

Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Melbourne, Victoria 3010, Australia

K. M. Itoh

School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, 223-8522, Japan

^*Present address: Department of Physics and Nanoscience Center, University of Jyväskylä, P.O. Box 35, 40014 University of Jyväskylä, Finland; juha.t.muhonen@jyu.fi
^†Present address: QuTech and Kavli Institute of Nanoscience, TU Delft, 2628 CJ Delft, Netherlands.
^‡Present address: Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6.
^§Present address: School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia.
^∥a.morello@unsw.edu.au

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Vol. 98, Iss. 15 — 15 October 2018

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Figure 1
(a) Scanning electron micrograph of a device identical to the one used in the experiment. A broadband microwave antenna is used to provide both nuclear and electron spin resonance pulses, and a single-electron transistor (SET) detects electron tunneling events in real time. (b) Energy diagram of the electron-nuclear spin system, with labels for the transition frequencies relevant to the present experiments. (c) Pulse sequence for nuclear spin initialization and variable-strength measurement. The solid line represents the combined effect of the voltage of the electrostatic gates $V_{DG}$ adjusting the chemical potential of the donor electron with respect to the SET island, schematically shown on the left. Blue and red boxes represent ESR pulses at $ν_{e 1}$ and $ν_{e 2}$ , respectively, and yellow boxes represent NMR pulses at $ν_{n}$ . The semitransparent boxes are needed only for tomography ( $σ_{x}$ and $σ_{y}$ components). They are, from left to right, two refocusing pulses (around the $y$ axis) and one phase-modulated pulse to define the tomography axis. Timing and pulse lengths are not to scale. Single weak measurements (first column in Fig. 2) require only the first $θ$ pulse and electron readout. (d) Pulse sequence used for nuclear spin readout. An electron readout step initializes the electron spin $|↓〉$ . An electron spin $π$ pulse is applied at frequency $ν_{e 2}$ , which flips the electron spin only if the nuclear spin is $|⇑〉$ . The nuclear state is assigned $|⇑〉$ if a majority of $|↑〉$ electrons are detected after 25 repetitions.
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Figure 2
Quantum control of a nuclear spin with electron spin resonance pulses, observed through quantum state tomography of $σ_{z}$ (first row), $σ_{x}$ (second row), and $σ_{y}$ (third row) as a function of the electron spin rotation angle $θ$ on the ESR frequency $ν_{e 2}$ . The columns show, from left to right, one weak measurement, two weak measurements (each with rotation angle $θ$ ), and measurement reversal (rotation by $θ$ on $ν_{e 2}$ and then by $ϕ$ on $ν_{e 1}$ , here $ϕ = θ$ ). The circles and solid line show the experimental data and theory prediction, respectively, for the expectation value (left axis). The squares and dashed line show the experimental data and theory prediction for the success probability of the protocol. All lines are without any fitting parameters, except the solid lines for $σ_{x}$ have been scaled by a constant to match the measured asymptotic values. These are not exactly unity presumably due to dephasing-caused rotation errors in the tomography pulse. Each data point corresponds to 200 unconditional repetitions.
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Figure 3
Extracting electron tunnel rates from a data set conditioned on having no tunneling events. (a) Average nuclear polarization $〈 σ_{z} 〉$ after a $t_{m} = 1.5$ ms electron readout window as a function of donor electrochemical potential, controlled by $V_{DG}$ . (b) Electron $|↑〉$ tunnel-out time $1 / Γ_{↑, out}$ extracted from the “tunnel-less” data in (a) using Eq. (8) (circles) and measured directly from tunneling events (crosses). Data are taken by stepping $V_{DG}$ from low to high values and then doing the reverse.
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Figure 4
Final-state fidelity compared to the ideal state as a function of the rotation angle for (a) one or (b) two conditional weak measurements. The average of all points is 0.969(20) in (a) and 0.972(24) in (b). (The value in parentheses indicates the standard deviation of the data points.) The crosses show points where we have forced the state normalization $\sqrt{{〈 σ_{x} 〉}^{2} + {〈 σ_{y} 〉}^{2} + {〈 σ_{z} 〉}^{2}^{}} = 1$ ; see the text.
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Figure 5
EPR steering. Theoretical expectation values for the nuclear spin component obtained by starting from the Bell state and performing an unconditional rotation of the electron spin with angle $θ$ and then a conditional measurement of the electron. As the unconditional rotation changes the electron measurement basis, one finds a perfect correlation between measuring electron $| ↓ 〉$ and nuclear spin $Z$ components at $θ = 0, π$ . At $θ = π / 2$ , there is perfect correlation with the $σ_{x}^{n}$ component.
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Physical Review B

covering condensed matter and materials physics

Coherent control via weak measurements in ${}^{31}P$ single-atom electron and nuclear spin qubits

J. T. Muhonen, J. P. Dehollain, A. Laucht, S. Simmons, R. Kalra, F. E. Hudson, A. S. Dzurak, A. Morello, D. N. Jamieson, J. C. McCallum, and K. M. Itoh

Phys. Rev. B 98, 155201 – Published 5 October 2018

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Physical Review B

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Coherent control via weak measurements in P31 single-atom electron and nuclear spin qubits

J. T. Muhonen, J. P. Dehollain, A. Laucht, S. Simmons, R. Kalra, F. E. Hudson, A. S. Dzurak, A. Morello, D. N. Jamieson, J. C. McCallum, and K. M. Itoh

Phys. Rev. B 98, 155201 – Published 5 October 2018

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Coherent control via weak measurements in ${}^{31}P$ single-atom electron and nuclear spin qubits