Robust electric dipole transition at microwave frequencies for nuclear spin qubits in silicon

Guilherme Tosi, Fahd A. Mohiyaddin, Stefanie Tenberg, Arne Laucht, and Andrea Morello

Phys. Rev. B 98, 075313 – Published 30 August 2018

Abstract

The nuclear spin state of a phosphorus donor ( ${}^{31}P$ ) in isotopically enriched silicon-28 is an excellent host to store quantum information in the solid state. The spin's insensitivity to electric fields yields a solid-state qubit with record coherence times but also renders coupling to other quantum systems very challenging. Here, we describe how to generate a strong electric dipole ( $> 100$ D) at microwave frequencies for the nuclear spin. This is achieved by applying a magnetic drive to the spin of the donor-bound electron while simultaneously controlling its charge state with electric fields. Under certain conditions, the microwave magnetic drive also renders the nuclear spin resonance frequency and electric dipole strongly insensitive to electrical noise, yielding long ( $> 1$ ms) dephasing times and robust gate operations. The nuclear spin could then be strongly coupled to microwave resonators, with a vacuum Rabi splitting of the order of 1 MHz, or to other nuclear spins, nearly half a micrometer apart, via strong electric dipole-dipole interaction. This work brings the ${}^{31}P$ nuclear qubit into the realm of hybrid quantum systems and opens up new avenues in quantum information processing.

Received 26 June 2017

DOI:https://doi.org/10.1103/PhysRevB.98.075313

Physics Subject Headings (PhySH)

Defects Hybrid quantum systems Quantum information with solid state qubits

Nuclear magnetic resonance

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Guilherme Tosi, Fahd A. Mohiyaddin^*, Stefanie Tenberg, Arne Laucht, and Andrea Morello^†

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

^*Present address: Department of Electrical Engineering (ESAT), Katholieke Universiteit Leuven, Leuven 3000, Belgium, and IMEC, Leuven 3001, Belgium.
^†a.morello@unsw.edu.au

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Vol. 98, Iss. 7 — 15 August 2018

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Figure 1
(a) Components of a Raman-enabled Si:P nuclear electric dipole transition. The electron spatial wave function (transparent gray) is shared between an interface dot $| i 〉$ and a donor-bound state $| d 〉$ , which are coupled by a tunnel rate $V_{t}$ . Metallic gates (blue) on top of the ${SiO}_{2}$ dielectric (not shown) control the electron charge state via a static vertical field $E_{dc}$ and can introduce an oscillating electric field $E_{ac}$ . In a circuit-quantum-electrodynamics setup, the electrostatic gate can be replaced by the inner conductor of a microwave resonator, and $E_{ac}$ can be replaced by the vacuum field of such a resonator. A nearby broadband antenna [23] (brown) provides the magnetic drive $B_{ac}$ . (b) Simplified energy level diagram for the Raman drive of the Si:P nuclear spin qubit, with energy splitting $ε_{ns}$ . Here, $| ↑ 〉$ ( $| ↓ 〉$ ) represents electron spin up (down), while $| ⇑ 〉$ ( $| ⇓ 〉$ ) represents nuclear spin up (down). The second-order Raman drive is obtained by combining the microwave electric and magnetic drives, having frequencies $ν_{B}$ and $ν_{E}$ , respectively, and coupling rates $g_{B}$ and $g_{E}^{ff}$ , respectively. The drive is detuned by a frequency $δ$ from the $| ↑ ⇓ 〉$ state. (c) Expanded energy diagram including the charge states $| g 〉$ and $| e 〉$ .
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Figure 2
Nuclear spin levels in the rotating frame of the magnetic drive $B_{ac}$ at frequency $ν_{B}$ . From left to right, the system eigenstates are shown while adding the electron spin state, then adding the charge state, and then increasing the strength of the magnetic drive. See the text for a detailed description.
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Figure 3
(a) Nuclear spin transition frequency $ε_{ns}^{'}$ in the rotating frame of the magnetic drive $B_{ac}$ as a function of the static vertical electric field $E_{dc}$ across the donor-dot system for vanishing magnetic drive $ε_{ns}^{'} (A)$ [Eq. (18), gray dashed line] and strong magnetic drive $ε_{ns}^{'} (A, D_{drive})$ [Eq. (22), black solid line]. We have assumed $B_{0} = 0.2$ T, $B_{ac} = 0.6$ mT, $d = 15$ nm, and $V_{t} \approx ε_{ff}$ and $ν_{B} \approx γ_{e} B_{0} - A / 4$ (since $〈 A 〉 = A / 2$ at the ionization point). Green and yellow lines show transition frequencies calculated numerically from the Hamiltonian in Eq. (17). The color indicates the degree of admixture of the bare $| g ↓ ⇓ 〉$ state into the higher $H_{n s}^{'}$ eigenstate corresponding to each transition. The nuclear spin transition (predominantly $| g ↓ ⇑ 〉 \leftrightarrow | g ↓ ⇓ 〉$ ) anticrosses a flip-flop transition (predominantly $| g ↓ ⇑ 〉 \leftrightarrow | g ↑ ⇓ 〉$ ) at $E_{dc} = 350$ V/m, with a splitting $\sim 2 g_{B}$ set by the strength of the magnetic drive. The flip-flop transition is strongly shifted by $D_{so}$ due to its coupling to the charge qubit states around $E_{dc} = 0$ . At $E_{dc} = 250$ V/m, the nuclear spin excited eigenstate has $\sim 75$ % of $| g ↓ ⇓ 〉$ and is robust against electrical noise ( $\partial ε_{ns}^{'} / \partial E_{dc} = 0$ ). (b) Nuclear electric dipole strength $p_{E}^{ns} = \partial g_{E}^{ns} / \partial E_{ac}$ obtained from Eq. (24) (theory, black line) or from the numerical diagonalization of the full Hamiltonian $H^{'}$ under $E_{ac}$ drive (numerics, light blue line). For the choice of parameters used in this figure, $p_{E}^{ns}$ peaks where $E_{dc} = 250$ V/m. (c) Nuclear spin relaxation rate $1 / T_{1, ns}$ in the presence of the magnetic drive $B_{ac}$ and the effect of coupling to phonons via charge states, Eq. (25).
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Figure 4
(a) Nuclear electric dipole strength $p_{E}^{ns}$ and (b) nuclear spin relaxation rate $1 / T_{1, ns}$ as a function of the donor-dot electric field detuning $E_{dc}$ and the magnetic drive frequency $ν_{B}$ . $E_{dc} = 0$ is the ionization point, where the electron charge is equally shared between donor and interface dot. In (a), the dashed line shows the ESR frequency $ν_{e, ⇓}$ when the nuclear spin is in the $| ⇓ 〉$ state; the dot-dashed line shows the charge qubit frequency minus the nuclear spin frequency, $ε_{o} - ε_{ns}$ ; and the double-dot-dashed line shows the electron spin resonance frequency $ν_{e, ⇑}$ when the nuclear spin is in the $| ⇑ 〉$ state. The charge and flip-flop states are detuned by $δ_{so}$ , which is close to zero at $E_{dc} = 0$ . Charge and flip-flop states then hybridize, shifting the system eigenenergies by an ac Stark shift $D_{so}$ . The plots in Figs. 3 and 3 correspond to specific line cuts of the graphs shown here for $ν_{B} = ν_{e, ⇓}$ at $E_{dc} = 0$ , i.e., $ν_{B} = 5.565$ GHz.
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Figure 5
(a) Components and (b) level diagram for long-distance coupling of two ${}^{31}P$ nuclear spins via electric dipole-dipole interactions. Each displaced electron produces an electric dipole field $E_{dip}$ (shown only for one electron). The charge dipoles induced by displacing the electron wave function partly towards the interface dot interact with a strength $g_{dd}$ [Eq. (28)], and the charge qubits interact with the flip-flop states with strength $g_{so}$ [Eq. (12)]. Adding the (global) magnetic drive of strength $g_{B}$ and tuning the system to the fully hybridized regime described in Sec. 4 results in a nuclear-nuclear coupling strength $g_{2 q}^{ns} \approx 0.55$ MHz at a 400-nm distance [Eq. (29)].
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