Phonon-induced decoherence for a quantum-dot spin qubit operated by Raman passage

We study single-qubit gates performed via stimulated Raman adiabatic passage on a spin qubit implemented in a quantum-dot system in the presence of phonons. We analyze the interplay of various kinds of errors resulting from the carrier-phonon interaction (including also the effects of spin-orbit coupling) as well as from quantum jumps related to nonadiabaticity and calculate the fidelity as a function of the pulse parameters. We give quantitative estimates for an $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ system and identify the parameter values for which the error is considerably minimized, even to values below ${10}^{-4}$ per operation.

Figure 1

(a) An example of pulse shapes (solid) and the resulting structure of the dressed levels (dashed). The arrows show the phonon-assisted transitions, as described in Sec. 5: (1) the pure dephasing effect; (2), (3) the transitions between the trapped states. (b) The evolution of the functions $\dot{\theta }$ (dashed) and $\sin 2\theta$ (solid) for the pulse sequence shown in (a).

Figure 2

(a) The functions ${\mid s_n\left(\omega \right)\mid }^2$ describing the phonon-induced errors (for pulses as in Fig. 1) and the total spectral density of the phonon reservoir $R\left(\omega \right)$ at $T=0$ (solid) and $5\mathrm{K}$ (dashed) for the model $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ system (Table ). The inset shows the exact shape of one of the spectral features. (b) The contributions to the spectral density at $T=0$: deformation potential coupling to LA phonons (solid) and piezoelectric coupling to TA (dashed) and LA (dash-dotted) phonons. Inset: high-frequency behavior with the two bounds defined in Appendix .

Figure 3

The dependence of ${\log }_{10}\delta$ on the pulse parameters $\Delta$ and $\Omega$ at $T=0$ and $1\mathrm{K}$ for ${\tau }_0=50\mathrm{ps}$. Numbers refer to the parameter regimes discussed in the text.

Figure 4

(a) The dependence of the error for growing detuning with ${\lambda }_{-}=\mathrm{const}$, along the (0a) minimum in Fig. 3b The total error (solid) and the individual contributions: nonadiabatic jumps (dashed), phonon-assisted real transitions (dotted), and pure dephasing (dash-dotted) for a section of the parameter space.

Figure 5

(a), (b) The minimal achievable error as a function of the process duration corresponding to the optimal pulse parameters with both ${\lambda }_{\pm }$ in the low-frequency area (a) and with ${\lambda }_{+}$ in the high-frequency area (b) (for $T=0$ the plateau value for $\Delta \to \infty$ is shown). (c), (d) The optimal pulse parameters (detuning and Rabi frequency) realizing the minimal error for these two configurations. The legend in (d) applies to both (c) and (d).

Figure 6

The error as a function of $\Omega$ for ${\lambda }_{-}=\mathrm{const}$, along the (1) (a) and (2) (b) areas in Fig. 3. In (a) the individual contributions to the error at $T=5\mathrm{K}$ are also shown: nonadiabatic jumps (dashed), phonon-assisted real transitions (dotted), and pure dephasing (dash-dotted).

Figure 7

(a), (b) The minimal achievable error as a function of the process duration corresponding to the optimal pulse parameters with both ${\lambda }_{\pm }$ in the high-frequency parameter areas (1) (a) and (2) (b) and the optimal pulse parameters (detuning and Rabi frequency) realizing the minimal error for these two configurations (c), (d). In (a) the contributions to the error at $T=1\mathrm{K}$ are shown: nonadiabatic jumps (dashed), phonon-assisted real transitions (dotted), and pure dephasing (dash-dotted).