Optimal Control of Effective Hamiltonians

We present a systematic scheme for the optimization of quantum simulations. Specifically, we show how polychromatic driving can be used to significantly improve the driving of Raman transitions in the Lambda system, which opens new possibilities for controlled driving-induced effective dynamics.

Figure 1

(a) and (b) Transition probability $P_2(t)=|⟨2|U(t)|1⟩{|}^2$ for the target dynamics (dotted line), MC dynamics (black line), and third-order optimal dynamics with $N=10$ frequency components (gray line). (c) Population of the excited state $P_3=|⟨3|U(t)|1⟩{|}^2$ as a function of time. Plot parameters are ${\mathrm{\Omega }}_{\mathrm{tg}}=0.05\omega$ and $n_0=4$.

Figure 2

Magnitude of deviations around the target unitary dynamics as function of the number $N$ of Fourier components. Blue dots depict the third-order target functional quantified by Eq. (10) and red triangles depict the numerically exact target functional, Eq. (2); plot parameters are ${\mathrm{\Omega }}_{\mathrm{tg}}=0.05\omega$ and $n_0=4$. Inset: Fourier components for $N=10$.

Figure 3

Deviations from the target dynamics ${\mathcal{F}}_n$ in the $n$th driving period [time interval [$(n-1)T$, $nT$]] for the MC pulse (blue circles), the second-order optimized pulse (brown squares), and the third-order optimized pulse (red rhombi) with $N=10$. The perturbed third-order solution [see Eq. (14)] with frequency randomness $\delta ={\delta }_{\max }/4$, averaged over 100 realizations (green triangles) shows good resilience against experimental uncertainty. ${\mathrm{\Omega }}_{\mathrm{tg}}=0.05\omega$ and $n_0=4$.