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### Holonomic Quantum Control by Coherent Optical Excitation in Diamond

##### Brian B. Zhou, Paul C. Jerger, V. O. Shkolnikov, F. Joseph Heremans, Guido Burkard, and David D. Awschalom

##### Phys. Rev. Lett. **119**, 140503 – Published 2 October 2017

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#### Abstract

Although geometric phases in quantum evolution are historically overlooked, their active control now stimulates strategies for constructing robust quantum technologies. Here, we demonstrate arbitrary single-qubit holonomic gates from a single cycle of nonadiabatic evolution, eliminating the need to concatenate two separate cycles. Our method varies the amplitude, phase, and detuning of a two-tone optical field to control the non-Abelian geometric phase acquired by a nitrogen-vacancy center in diamond over a coherent excitation cycle. We demonstrate the enhanced robustness of detuned gates to excited-state decoherence and provide insights for optimizing fast holonomic control in dissipative quantum systems.

- Received 1 May 2017

DOI:https://doi.org/10.1103/PhysRevLett.119.140503

© 2017 American Physical Society

#### Physics Subject Headings (PhySH)

- Research Areas

Color centersQuantum controlQuantum gatesQuantum information with solid state qubitsSpintronics

- Physical Systems

Nitrogen vacancy centers in diamond

- Techniques

Coherent control

General PhysicsCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

#### Authors & Affiliations

Brian B. Zhou^{1}, Paul C. Jerger^{1}, V. O. Shkolnikov^{2}, F. Joseph Heremans^{1,3}, Guido Burkard^{2}, and David D. Awschalom^{1,3,*}

^{1}Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA^{2}Department of Physics, University of Konstanz, D-78457 Konstanz, Germany^{3}Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

^{*}awsch@uchicago.edu

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##### Issue

Vol. 119, Iss. 14 — 6 October 2017

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Download Accepted Manuscript#### Images

###### Figure 1

Experimental system and holonomic concept. (a)Optical NV $\mathrm{\Lambda}$ system. The $|\pm 1\u27e9$ spin states of the NV triplet ground state (GS) are linked to $|{A}_{2}\u27e9$ within the spin-orbit excited-state (ES) manifold by a two-tone optical field with one-photon detuning $\mathrm{\Delta}$ and strength $\mathrm{\Omega}(t)$. (b)Photoluminescence excitation spectrum taken by scanning a single laser frequency across the GS to ES transitions while two microwave tones mix the population among the three GS levels. (c)Geometric interpretation of the holonomic gates. The dark state $|d\u27e9$ undergoes trivial dynamics, while the bright state $|b\u27e9$ undergoes precession around a tilted axis with angle $\alpha $ on the $|b\u27e9/|{A}_{2}\u27e9$ Bloch sphere. After one nonadiabatic cycle, $|b\u27e9$ acquires a geometric phase $\gamma $ proportional to the enclosed solid angle.

###### Figure 2

Phase shift gates $Z(\gamma )$. (a)Experimental sequence consisting of state preparation, optical excitation with control parameters ($\theta $, $\varphi $, $\mathrm{\Delta}/\mathrm{\Omega}$), and state tomography. (b)Time-resolved photoluminescence, proportional to the $|{A}_{2}\u27e9$ population, as a function of detuning $\mathrm{\Delta}$, showing oscillations between $|b\u27e9=|+1\u27e9$ and $|{A}_{2}\u27e9$ ($\theta =0$). (c)Measured phase shifts $\gamma $, averaged over $|x\u27e9$ and $|y\u27e9$ input states, as a function of $\mathrm{\Delta}$ for $Z(\gamma )$ at two different optical powers. The solid lines delineate the prediction according to Eq.(2). (d)Gate fidelities via process tomography of the same $Z(\gamma )$ gates. The solid lines are simulated fidelities incorporating an excited-state lifetime ${T}_{1}$ and dephasing ${T}_{\varphi}$. (e)Dependence of the fidelity of the resonant gate $Z(\gamma =\pi )$ on $\mathrm{\Omega}$. The purple, orange, and teal lines represent the simulated fidelities by sequentially adding the effects of ${T}_{1}$, ${T}_{\varphi}$, and the spectral hopping ${\sigma}_{\mathrm{\Delta}}$.

###### Figure 3

Resonant holonomic gates. (a)Gates with variable $\theta $ after initializing $|z\u27e9$. The probabilities of the final states to be measured in $|z\u27e9$ and $|-z\u27e9$ are plotted. (b)Gates with variable $\varphi $ after initializing $|x\u27e9$. The Bloch vector projections of the final states along the $x$ and $y$ axes are plotted. For both (a) and (b), solid lines indicate simulated behaviors with ${T}_{1}$ and ${T}_{\varphi}$ decoherence; the dashed lines in (a)indicate decoherence-free behavior. Decoherence is minimized when the initialized state aligns with the gate’s dark state $|d\u27e9$. (c)–(e)Process matrices for $X$, $Y$, and $H$ at $\mathrm{\Omega}/2\pi =168\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$. (f)Dependence of $X$, $Y$, $H$ process fidelities on $\mathrm{\Omega}$. The solid lines are simulated fidelities incorporating all excited-state decoherence effects. $X$ and $Y$ are identical in the simulation.

###### Figure 4

Variable rotations around the $x$ and $y$ axes. (a)After initializing $|z\u27e9$, a holonomic gate with variable detuning $\mathrm{\Delta}$ and a fixed rotation axis along $|x\u27e9$ is applied. The data show Rabi oscillations between $|z\u27e9$ and $|-z\u27e9$ as $\mathrm{\Delta}$ is swept across resonance, tuning the rotation angle $\gamma $. The solid lines are simulations with ${T}_{1}$ and ${T}_{\varphi}$ decoherence; the dashed lines are decoherence-free behaviors. (b)–(d) Process matrices for the $X(\pi /2)$ and $Y(\pm \pi /2)$ gates at $\mathrm{\Omega}/2\pi =152\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ and $\mathrm{\Delta}/2\pi =\pm 88\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$.