Robust state preparation in quantum simulations of Dirac dynamics

A nonrelativistic system such as an ultracold trapped ion may perform a quantum simulation of a Dirac equation dynamics under specific conditions. The resulting Hamiltonian and dynamics are highly controllable, but the coupling between momentum and internal levels poses some difficulties to manipulate the internal states accurately in wave packets. We use invariants of motion to inverse engineer robust population inversion processes with a homogeneous, time-dependent simulated electric field. This exemplifies the usefulness of inverse-engineering techniques to improve the performance of quantum simulation protocols.

Figure 1

Systematic error sensitivity $q_s$ in Eq. (48). As in all figures we use dimensionless units with $c=\hslash =t_f=1$. For specific values of $\left|\nu \right|,q_s=0$ is satisfied, in particular at the minimal value $\left|\nu \right|=0.643$.

Figure 2

(a) The Rabi frequency $\mathrm{\Omega }$ (red, solid line) and detuning $\mathrm{\Delta }$ (blue, dotted-dashed line) in our optimal protocol. (b) Time evolution of the populations $P_1\left(0\right)$ (blue, solid line) and $P_2\left(0\right)$ (red, dotted-dashed line) during the population inversion. We have used $\nu =0.643$ and $p_0=0$.

Figure 3

Time evolution of the populations $P_1$ of a Gaussian wave packet centered at zero momentum (green, solid line and blue, dot-dashed line for $\sigma =0.3$ and $\sigma =0.9$, respectively) and $P_2$ (red, dotted-dashed line and black circles for $\sigma =0.3$ and $\sigma =0.9$, respectively) by averaging over all momenta $p_0$, see Eq. (23), during the population inversion. $H_0$ as in Fig. 2. Compare to the result for a plane wave, $p_0=0$, in Fig. 2.

Figure 4

(a) The adiabatic energies of Hamiltonian $H_0\left(t\right):E_{+}\left(t\right)$ (red, solid line) and $E_{-}\left(t\right)$ (blue, dotted-dashed line). (b) The adiabatic time evolution of the populations of level $|1\rangle$ for the positive (red, solid line) and negative (blue, dotted-dashed line) energy eigenstates of Hamiltonian $H_0\left(t\right)$. $\mathrm{\Omega }$ and $\mathrm{\Delta }$ are as in Fig. 2.

Figure 5

Populations of energy eigenstates along the invariant eigenstate $|{\varphi }_{+}\left(t\right)\rangle ,|\langle E_{+}\left(t\right)|{\varphi }_{+}\left(t\right)\rangle {|}^2$ (red, solid line) and $|\langle E_{-}\left(t\right)|{\varphi }_{+}\left(t\right)\rangle {|}^2$ (blue, dotted-dashed line), for the optimal $\mathrm{\Omega }\left(t\right)$ and $\mathrm{\Delta }\left(t\right)$ in Fig. 2.

Figure 6

The Rabi frequency ${\mathrm{\Omega }}_s$ (green, dotted-star line), and detuning and ${\mathrm{\Delta }}_s$ (black, dashed line) are determined by Eqs. (30) and (31) with angles ${\theta }_s\left(t\right)={\sum }_{j=0}^3{a}_j{t}^j$ and ${\beta }_s\left(t\right)={\sum }_{j=0}^4{b}_j{t}^j$ in simple invariant-based shortcuts, together with “optimal” $\mathrm{\Omega }\left(t\right)$ (red, solid line) and $\mathrm{\Delta }\left(t\right)$ (blue, dotted-dashed line) in Fig. 2.

Figure 7

Probability $P_2\left(p_0\right)$ at the final time $t_f=1$ vs systematic momentum noise $p_0$ by solving numerically Eq. (20) with the Hamiltonian (25) based on the optimal invariant-based shortcut of Fig. 2 (zero sensitivity, red, solid line), and simple ones (nonzero sensitivity, blue, dotted-dashed line).