Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms

We propose an efficient method to construct shortcuts to adiabaticity through designing a substitute Hamiltonian to try to avoid the defect in which the speed-up protocols' Hamiltonian may involve terms which are difficult to realize in practice. We show that as long as the counterdiabatic coupling terms—even only some of them—have been nullified by the additional Hamiltonian, the corresponding shortcuts to the adiabatic process could be constructed and the adiabatic process would be sped up. As an application example, we apply this method to the popular Landau-Zener model for the realization of fast population inversion. The results show that in both Hermitian and non-Hermitian systems, we can design different additional Hamiltonians to replace the traditional counterdiabatic driving Hamiltonian to speed up the process. This method provides many choices for designing additional terms of the Hamiltonian such that one can choose a realizable model in practice.

Figure 1

The time-dependent $P_2$ versus ${\alpha }_0$ (units of ${\mathrm{\Omega }}_0$) when $\{\phi =0,\zeta =3{\mathrm{\Omega }}_0,t_f=1/{\mathrm{\Omega }}_0\}$: (a) $\alpha ={\alpha }_0$ is constant; (b) $\alpha ={\alpha }_0\dot{\theta }$ is time dependent; (c) $\alpha ={\alpha }_0\dot{\theta }$ is time dependent and $\beta =0$. The evolution time in the figure is in units of $1/{\mathrm{\Omega }}_0$.

Figure 2

The time-dependent $P_2$ versus $\kappa$ (units of ${\mathrm{\Omega }}_0$) when $\{\phi =\kappa t,\zeta =3{\mathrm{\Omega }}_0,t_f=1/{\mathrm{\Omega }}_0\}$: (a) based on the original transitionless tracking algorithm that $\alpha =-(\kappa /2)sin2\theta$; (b) based on the present method with parameter $\alpha =0$; (c) based on $H_0$ without the additional term. The evolution time in the figure is in units of $1/{\mathrm{\Omega }}_0$.

Figure 3

(a) The populations $P_1$ and $P_2$ versus time when $\gamma =0.5{\mathrm{\Omega }}_0$. (b) The relative population $P_1^{\prime }$ and $P_2^{\prime }$ versus time when $\gamma =0.5{\mathrm{\Omega }}_0$.

Figure 4

(a) The time-dependent relative population $P_1^{\prime }$ versus $\gamma$. (b) The time-dependent relative population $P_2^{\prime }$ versus $\gamma$. The evolution time in the figure is in units of $1/{\mathrm{\Omega }}_0$.

Figure 5

(a) The populations $P_1$ and $P_2$ versus time when $\gamma =1/(2+t^2)$. (b) The relative population $P_1^{\prime }$ and $P_2^{\prime }$ versus time when $\gamma =1/(2+t^2)$.

Figure 6

The shapes of the imaginary part of the additional Hamiltonian's pulses when $\phi =0$. Blue dotted curve: when $\gamma =0$; red solid curve: when $\gamma =0.5{\mathrm{\Omega }}_0$; green dashed curve: when $\gamma =1/(2+t^2)$.