Proposal for optical U(1) rotations of electron spin trapped in a quantum dot

We present a proposal to optically implement rotations of the electron spin in a quantum dot about the growth direction ($z$ axis). In particular, we make use of the analytic properties of sech pulses in two-level systems to realize spin rotations about the growth direction by an arbitrary angle, for which we give an analytical expression. We propose to use this scheme to experimentally demonstrate this spin rotation. Using realistic system and pulse parameters we find the fidelity of the rotation to be more than 96% for pulses in the picosecond regime, and robust against small errors in pulse parameters. We design a feedback (adaptive) loop to correct for errors originating from unintended dynamics. The rotation is still evident—albeit with a large fidelity loss—in the ensemble case, providing the possibility of demonstration of this optical spin rotation in an ensemble of quantum dots.

Figure 1

(Color online) Energy levels of the three-level system, comprised by the two electron spin eigenstates of ${\sigma }_x$ and by the trion.

Figure 2

(Color online) Alternative depiction of the $\Lambda$ system: the two lower levels are the eigenstates of ${\sigma }_z$ but not energy eigenstates, since they are perpendicular to the magnetic field. This basis is more suitable for imposing selection rules when circularly polarized light is used because it is just the one spin state $\left(\mid z⟩\right)$ that couples to the trion. The other one $\left(\mid \overline{z}⟩\right)$ is coupled through the magnetic field.

Figure 3

(Color online) Bloch vector representation of the pseudospin; The pulse bandwidth is fixed $(\sigma =1)$ and the detuning varies: $\Delta =0$ [red (dark grey) curve], $\Delta =1$ [blue (black) curve] and $\Delta =0.5$ [green (light grey) curve]. The plot is in the rotating frame of the laser, not that of the unperturbed system.

Figure 4

(Color online) Differential transmission signal (DTS) of transitionless pulse on mixed state. Virtually no beats are generated when $\Omega =\sigma$. Here, $\sigma =0.4\mathrm{meV}$ and $\Delta =0$.

Figure 5

(Color online) Differential transmission signal (DTS) representing rotation of the spin in a GaAs dot by $\pi$ with a resonant pulse of $\sigma =0.4\mathrm{meV}$. The time where the control pulse is centered is indicated by the arrow.

Figure 6

(Color online) DTS representing rotation of the spin in a InAs dot by $\pi$. The pulse is resonant with $\sigma =0.8\mathrm{meV}$. The arrow indicates the incidence time of the control pulse.

Figure 7

(Color online) Bloch sphere depiction of the spin generation and rotation of Fig. 5. Initially there is no SV, and its generation along the $-z$ direction is shown. During the rotation the population is moved outside the $2\times 2$ spin subspace and the SV shrinks (line inside the sphere, corresponding to the arrow of Fig. 5). The $\pi$ rotation about the $z$ axis is implemented when the SV is pointing along $-y$. Note that the Bloch sphere itself has been shrunk to a radius of 0.5 for clearer depiction.

Figure 8

(Color online) Differential transmission signal (DTS) representing spin rotation in a GaAs dot by $\pi ∕2$ with pulse of $\sigma =0.4\mathrm{meV}$ and $\Delta =\sigma$. The pump is centered at $50\mathrm{ps}$, when the beats start and the central times of the control pulses are indicated by the arrows.

Figure 9

(Color online) DTS showing the rotation of the spin in a InAs dot by $\pi ∕2$. The pulse parameters are $\sigma =\Delta =0.8\mathrm{meV}$. The arrows indicate the incidence time of the two control pulses.

Figure 10

(Color online) Fidelity of the operation as a function of the pulse bandwidth for GaAs dots. Large bandwidth corresponds to fast pulses, and therefore smaller time intervals of trion excitation. Here the angle of rotation equals $\pi$. The uncertainty in the laser electric field is 15%.

Figure 11

(Color online) Fidelity of the operation as a function of the angle of rotation for GaAs dots. Large angles correspond to pulses closer to resonance, yielding loss of fidelity due to (real) trion excitation. Here the bandwidth has been taken equal to $0.3\mathrm{meV}$ and the uncertainty in the laser electric field is 15%.

Figure 12

(Color online) Fidelity of the operation as a function of the angle of rotation for InAs dots. Large angles correspond to pulses closer to resonance, yielding loss of fidelity due to (real) trion excitation. Here the bandwidth has been taken equal to $0.8\mathrm{meV}$. The uncertainty in the laser electric field is 15%.

Figure 13

(Color online) Initialization using a sech pulse with $\sigma =0.4\mathrm{meV}$ and $\Omega =\sigma ∕2$.

Figure 14

(Color online) Initialization with a pulse with $\sigma =0.4\mathrm{meV}$ and $\Omega =1.5\sigma ∕2$.

Figure 15

(Color online) Initialization with a pulse with $\sigma =0.4\mathrm{meV}$ and $\Omega =0.5\sigma ∕2$.

Figure 16

DTS representing ensemble spin rotation by $\pi$ using a resonant pulse with $\sigma =0.4\mathrm{meV}$. The fidelity of the rotation is a lot lower than the single-qubit case. The arrow indicates the time of incidence of the control pulse.

Figure 17

DTS representing ensemble spin rotation by $\pi ∕2$. To demonstrate that the operation is a rotation, a second pulse is used to restore the beats. The two arrows indicate the incidence times of the two control pulses.