Holonomic Quantum Control by Coherent Optical Excitation in Diamond (2024)

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  Figure 1

  Experimental system and holonomic concept. (a)Optical NV $\mathrm{\Lambda }$ system. The $|\pm 1⟩$ spin states of the NV triplet ground state (GS) are linked to $|A_2⟩$ 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 $|d⟩$ undergoes trivial dynamics, while the bright state $|b⟩$ undergoes precession around a tilted axis with angle $\alpha$ on the $|b⟩/|A_2⟩$ Bloch sphere. After one nonadiabatic cycle, $|b⟩$ acquires a geometric phase $\gamma$ proportional to the enclosed solid angle.

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  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 $|A_2⟩$ population, as a function of detuning $\mathrm{\Delta }$, showing oscillations between $|b⟩=|+1⟩$ and $|A_2⟩$ ($\theta =0$). (c)Measured phase shifts $\gamma$, averaged over $|x⟩$ and $|y⟩$ 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 ${\sigma }_{\mathrm{\Delta }}$.

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  Figure 3

  Resonant holonomic gates. (a)Gates with variable $\theta$ after initializing $|z⟩$. The probabilities of the final states to be measured in $|z⟩$ and $|-z⟩$ are plotted. (b)Gates with variable $\varphi$ after initializing $|x⟩$. 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 $|d⟩$. (c)–(e)Process matrices for $X$, $Y$, and $H$ at $\mathrm{\Omega }/2\pi =168\mathrm{MHz}$. (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.

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  Figure 4

  Variable rotations around the $x$ and $y$ axes. (a)After initializing $|z⟩$, a holonomic gate with variable detuning $\mathrm{\Delta }$ and a fixed rotation axis along $|x⟩$ is applied. The data show Rabi oscillations between $|z⟩$ and $|-z⟩$ 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 $\mathrm{\Omega }/2\pi =152\mathrm{MHz}$ and $\mathrm{\Delta }/2\pi =\pm 88\mathrm{MHz}$.

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