All-Optical Manipulation of Electron Spins in Carbon-Nanotube Quantum Dots

We demonstrate theoretically that it is possible to manipulate electron or hole spins all optically in semiconducting carbon nanotubes. The scheme that we propose is based on the spin-orbit interaction that was recently measured experimentally; we show that this interaction, together with an external magnetic field, can be used to achieve optical electron-spin state preparation with a fidelity exceeding 99%. Our results also imply that it is possible to implement coherent spin rotation and measurement using laser fields linearly polarized along the nanotube axis, as well as to convert spin qubits into time-bin photonic qubits. We expect that our findings will open up new avenues for exploring spin physics in one-dimensional systems.

Figure 1

(a) Energy diagram of the lowest electron (subscript $e$) and hole (subscript $h$) states in a nanotube quantum dot as a function of the applied axial magnetic field $B_{\Vert }$ [10, 18]. (b) Energy diagram of a singly charged nanotube quantum dot for large $B_{\Vert }$ showing the relevant optical transitions coupled by a laser field polarized along the CNT axis with Rabi frequency ${\Omega }_L$. The orthogonal magnetic field causes a coherent $\uparrow -\downarrow$ coupling with strength $\hslash {\Omega }_T=g{\mu }_B{B}_{\perp }$. The decay of the excited states is assumed to be spin-conserving and monoexponential with rate $\Gamma$. Spin relaxation rates for electrons and holes are denoted by ${\xi }_e$ and ${\xi }_h$, respectively. $K\mathrm{\text{−}}{K}^{\prime }$ mixing can cause optically assisted valley flip to the state $D{\downarrow }_e$ at an effective rate ${\gamma }_{K{K}^{\prime }}$.

Figure 2

The contour plot of the population difference between states $U{\uparrow }_e$ and $U{\downarrow }_e$ as a function of the magnetic field components when the laser is kept resonant with the $U{\downarrow }_e\to U{\downarrow }_{e}U{\uparrow }_{e}U{\downarrow }_h$ transition (zero of the detuning scale in the inset). We take ${\xi }_e={\xi }_h=0.1\mu {\mathrm{s}}^{-1}$ and ${\Gamma }^{-1}=40\mathrm{ps}$. In (a), we assume that the optical transitions are broadened by Markovian dephasing with rate $\hslash {\gamma }_{\mathrm{deph}}=0.25\mathrm{meV}$. In (b), we consider a scenario where the coherence between states $U{\downarrow }_{e}U{\uparrow }_{e}U{\downarrow }_h$ and $U{\downarrow }_{e}U{\uparrow }_{e}U{\uparrow }_h$ also undergoes fast dephasing at the same rate (0.25 meV). In the inset to part (b), we show the laser absorption (black line) and population difference (red line) between $U{\uparrow }_e$ and $U{\downarrow }_e$ states for external magnetic fields $B_{\Vert }=B_{\mathrm{SO}}^h$ and $B_{\perp }=0.2\mathrm{T}$. The absorption (right scale) is normalized to its maximum value in the absence of spin pumping (when $B_{\perp }=0\mathrm{T}$).

Figure 3

(a) Schematics of the cavity QED setup discussed in the text. Both laser pulses are polarized along the nanotube axis. Provided that the pulse separation $t_2-t_1$ is larger than all of the other time scales (inverse Rabi frequency and cavity-enhanced exciton decay rate) quantum information can be efficiently encoded in a photon time-bin qubit. (b) Energy diagram of the nanotube quantum dot with the relevant transitions and rates used in the scheme.