Coherent Optical Control of the Spin of a Single Hole in an $\mathrm{InAs}/\mathrm{GaAs}$ Quantum Dot

We demonstrate coherent optical control of a single hole spin confined to an $\mathrm{InAs}/\mathrm{GaAs}$ quantum dot. A superposition of hole-spin states is created by fast (10–100 ps) dissociation of a spin-polarized electron-hole pair. Full control of the hole spin is achieved by combining coherent rotations about two axes: Larmor precession of the hole spin about an external Voigt geometry magnetic field, and rotation about the optical axis due to the geometric phase shift induced by a picosecond laser pulse resonant with the hole-trion transition.

Figure 1

Preparation, coherent control, and detection of a single hole spin. (i) Resonant excitation of the neutral exciton transition by a laser pulse propagating along the $z$ axis creates a spin-polarized electron-hole pair. (ii) When the electron tunnels it leaves a spin-polarized hole that precesses about the magnetic field applied along the $x$ axis. (iii) Rotation of hole spin. The hole (trion) spin-$z$ states are coupled with in-plane Zeeman energies of $\hslash {\omega }_h$ ($\hslash {\omega }_e$), respectively. The ${\sigma }_{+}$-polarized control pulse couples the $\mid \Downarrow ⟩\leftrightarrow \mid \Downarrow \Uparrow \downarrow ⟩$ states only, imparting a phase shift on $\mid \Downarrow ⟩$. (iv) To detect the hole spin, a circularly polarized laser pulse resonant with the hole-trion transition is absorbed conditional on the spin-$z$ state of the hole. (v) When the additional carriers created in step (iv) tunnel from the dot a change in photocurrent proportional to the occupation of the hole-spin state selected by the helicity of the detection pulse is measured.

Figure 2

Precession of single hole spin ($B_x=4.7\mathrm{T}$, $V_g=0.8\mathrm{V}$). (a) Change in photocurrent vs detection-pulse detuning for co- (●) and cross- ($+$) circular excitation at various time-delays. The peak corresponds to the hole-trion transition. (b) Precession of hole spin, $s_z=\frac{I_{-}^{\mathrm{pc}}-I_{+}^{\mathrm{pc}}}{I_{-}^{\mathrm{pc}}+I_{+}^{\mathrm{pc}}}$ vs detection-pulse time-delay ${\tau }_d$. $I_{\pm }^{\mathrm{pc}}$ is the amplitude of the photocurrent peaks measured for ${\sigma }_{\pm }$ polarized detection pulse as in (a). (solid line) undamped cosine to guide the eye. (inset) Amplitude of Larmor precession vs ${\tau }_d$, the traces are Gaussian decays with $T_L=12.2$, 20.9 ns. The amplitude is determined from a sine fit to the data of (a) in the range ${\tau }_d\pm T/2$, where $T$ is the Larmor period.

Figure 3

Coherent control of hole spin. ($B=1.13\mathrm{T}$, $V=0.8\mathrm{V}$) (a) (Top,bottom) Orbits of Larmor precession of hole spin about magnetic field axis $x$, before and after the control pulse are shown as dashed lines. Top-sphere illustrates the experiments in (b), where an on-resonance control pulse rotates the hole spin about $z$ by angle $\pi$, changing phase of precession. The bottom sphere illustrates the experiments in (d). The control acts when the hole-spin points along $y$, rotating the hole spin by $\Delta {\varphi }_z({\Delta }_c)$ about $z$, reducing the amplitude of precession. [(a),middle] Pulse sequence. (b) Control of phase of precession. $\Delta I=I_{-}^{\mathrm{pc}}-I_{+}^{\mathrm{pc}}$ is plotted vs detection-time ${\tau }_d$ for various control times ${\tau }_c$. (c) Change in start time of the precession due to the control pulse: ${\tau }_s=1.99{\tau }_c-69\mathrm{ps}$. (d) Control of rotation angle $\Delta {\varphi }_z$ via the detuning ${\Delta }_c$ varies the amplitude of the hole-spin precession. The control time ${\tau }_c=234\mathrm{ps}$, where the hole spin points along $y$. (e) (●) Ratio $R$ of precession amplitude normalized to total hole population, with and without the control vs ${\Delta }_c$. (line) Calculation of $\cos \Delta {\varphi }_z$, the ideal dependence of $R$ [20].