Electron spin tomography through counting statistics: A quantum trajectory approach

We investigate the dynamics of electron spin qubits in quantum dots. Measurement of the qubit state is realized by a charge current through the dot. The dynamics is described in the framework of the quantum trajectory approach, widely used in quantum optics. The relevant master equation dynamics is unraveled to simulate stochastic tunneling events of the current through the dot. Quantum trajectories are then used to extract the counting statistics of the current. We show how, in combination with an electron spin resonance field, counting statistics can be employed for quantum-state tomography of the qubit state. Further, it is shown how decoherence and relaxation time scales can be estimated with the help of counting statistics, in the time domain.

Figure 1

(Color online) Closed dot (top): chemical potentials are too small to allow an electron to tunnel onto the dot. Open dot (bottom): after excitation of the dot electron, the chemical potential ${\mu }_1$ is large enough for an electron of lead one to tunnel onto the dot and form the singlet state with the dot electron.

Figure 2

Counting statistics $P(q=1,t)$ for the first electron with initial spin-up state (dashed line, $r=1$, $\theta =0$), spin-down state (dotted line, $r=1$, $\theta =180°$), and the totally mixed state (solid curve, $r=0$). The respective coordinates refer to the Bloch sphere. The inset shows the same curves with the counting statistics of the fully mixed state subtracted. Parameters chosen are ${\Delta }_x=2\gamma =4({\Delta }_z-\omega )=2\times {10}^6{\mathrm{s}}^{-1}$, $T=20\mathrm{mK}$, $B_z=12\mathrm{T}$.

Figure 3

Counting statistics (top) $P(q=1,t)$ for eight coherent superpositions $\psi =(\mid \uparrow ⟩+e^{i\varphi }\mid \downarrow ⟩)∕\sqrt{2}$ along the equator of the Bloch sphere ($r=1$, $\theta =90°$, various angles $\varphi$) and the fully mixed state ($r=0$, solid line). The lower diagram shows the same curves with $P(q=1,t)$ of the full mixture subtracted. Same parameters as in Fig. 2.

Figure 4

Graphs taken from Fig. 3 for two pairs of opposite states along the equator of the Bloch sphere.

Figure 5

Same as Fig. 4 with $P(q=1,t)$ of the full mixture subtracted. We clearly see the symmetry of the curves for opposite states on the Bloch sphere.

Figure 6

Constructed curve for ${\vec{r}}_c$ according to Eq. (26) and curves for ${\vec{r}}_0$, ${\vec{r}}_1$, ${\vec{r}}_2$, ${\vec{r}}_3$. For comparison the curve for ${\vec{r}}_{QJ}$ simulated with the quantum jump method. See text for numerical values.

Figure 7

Counting statistics of exactly zero, one, and five electrons tunneling through the dot. Evidently, if less pronounced, all counting statistics $P(q,t)$ $(q=0,1,\dots ,5,\dots )$ carry information about the initial quantum state.

Figure 8

Same graph as Fig. 3. Here, however using spin-polarized leads with ${\gamma }^{\uparrow }=1\times {10}^6{\mathrm{s}}^{-1}$, ${\gamma }^{\downarrow }=0$.

Figure 9

Measurement of a variety of states (∣↑⟩, ∣↓⟩, coherent superposition) facilitates an estimate of $T_2$ and $T_1$.