Mechanically induced spin resonance in a carbon nanotube

The electron spin is a promising qubit candidate for quantum computation and quantum information. Here we propose and analyze a mechanically induced single electron spin resonance, which amounts to a rotation of the spin about the $x$ axis in a suspended carbon nanotube. The effect is based on the coupling between the spin and the mechanical degree of freedom due to the intrinsic curvature-induced spin-orbit coupling. A rotation about the $z$ axis is obtained by the off-resonant external electric driving field. Arbitrary-angle rotations of the single-electron spin about any axis in the $x$-$z$ plane can be obtained with a single operation by varying the frequency and the strength of the external electric driving field. With multiple steps combining the rotations about the $x$ and $z$ axes, arbitrary-angle rotations about arbitrary axes can be constructed, which implies that any single-qubit gate of the electron-spin qubit can be performed. We simulate the system numerically by using a master equation with realistic parameters.

Figure 1

(a) Proposed system for mechanical spin manipulation. A single electron is trapped in a quantum dot (QD) in a suspended carbon nanotube (CNT) of length $L$. A magnetic field $B_{\parallel }$ parallel to the $z$ axis is applied. A gate voltage applied on the back gate can adjust the resonance frequency ${\omega }_p$ of the CNT. An external antenna or the back gate can apply an external ac electric driving field to the charged nanotube. Here we assume a CNT with length $L=400\mathrm{nm}$, resonance frequency ${\omega }_p/\left(2\pi \right)=1.5\mathrm{GHz}$, and quality factor $Q\approx 95000$. (b) The Bloch sphere representing the quantum state of the electron spin. The green line denotes electron spin resonance (ESR) and the purple line shows a rotation about the $z$ axis. (c) Energy spectrum of the QD in the CNT as a function of the parallel magnetic field $B_{\parallel }$. The two degrees of freedom of the valley ($K$, $K^{\prime }$) and single-electron spin ($\uparrow ,\downarrow$) yield a four-fold degeneracy, which is lifted by $B_{\parallel }$. A pure spin qubit is found around the magnetic field $B^{*}={\Delta }_{so}/\left(g_s{\mu }_B\right)=1.7\mathrm{T}$. Parameters used here are ${\Delta }_{so}=170\mu \mathrm{eV}$, ${\Delta }_{K{K}^{\prime }}=12.5\mu \mathrm{eV}$, ${\mu }_{\mathrm{orb}}=330\mu \mathrm{eV}$ [37].

Figure 2

The time evolution of density matrix elements ${\rho }_{ij}=\langle i\left|\rho \right|j\rangle (i,j=\uparrow ,\downarrow )$ under the conditions of (a) ESR with $\delta /\left(2\pi \right)=({\omega }_q-\omega )/\left(2\pi \right)=0.003\mathrm{MHz}$ and (b) rotation about an axis in the $x$-$z$ plane with $\delta /\left(2\pi \right)=0.012\mathrm{MHz}$ at zero temperature, with which one can achieve the Hadamard gate. The initial condition is $|0\uparrow \rangle$. The blue (red) thick lines show the population of state $|0\uparrow \rangle$ ($|0\downarrow \rangle$). The dashed lines show the coherence terms between $|0\uparrow \rangle$ and $|0\downarrow \rangle$ in the ESR. The other parameters are $\lambda /\left(2\pi \right)=0.8\mathrm{MHz}$, ${\omega }_q/\left(2\pi \right)=1.4\mathrm{GHz}$, ${\omega }_p/\left(2\pi \right)=1.5\mathrm{GHz}$, and $g/\left(2\pi \right)=0.56\mathrm{MHz}$.

Figure 3

The time evolution of the population of spin states is shown at temperature $T=100\mathrm{mK}$. The initial state is ${\rho }_{\uparrow T}$. (a) The red thick (dashed) line shows the population of $|0\uparrow \rangle$ ($|0\downarrow \rangle$) and blue (dashed) line shows $|1\uparrow \rangle$ ($|1\downarrow \rangle$). Green, orange, and gray lines show $|2\uparrow \rangle$, $|3\uparrow \rangle$, and $|4\uparrow \rangle$, separately, and the related states of spin down are damped out. (b) The total spin evolution ${\rho }_{\sigma }={\sum }_n{\rho }_{n\sigma }$, $\sigma =\uparrow ,\downarrow$ in thermal equilibrium. (Here we plot a figure with the maximum phonon number $n=4$.) The other parameters are $\lambda /\left(2\pi \right)=0.8\mathrm{MHz}$, ${\omega }_p/\left(2\pi \right)=1.5\mathrm{GHz}$, ${\omega }_q/\left(2\pi \right)=1.4\mathrm{GHz},\omega /\left(2\pi \right)=1399.997\mathrm{MHz}$ and $g/\left(2\pi \right)=0.56\mathrm{MHz}$.

Figure 4

The energy-level diagram of the combined spin-phonon states. The ac electric driving strength is denoted by $\lambda$ and its frequency by $\omega$. The driving field couples the phonon mode with a large detuning of frequency $\Delta$. The dash-dotted lines are the dressed states caused by the spin-phonon coupling $g$ (blue lines). ESR between two spin states with the same phonon numbers is obtained when $\omega$ approaches ${\omega }_q$ (green lines).