Unitary deformations of counterdiabatic driving

We study a deformation of the counterdiabatic-driving Hamiltonian as a systematic strategy for an adiabatic control of quantum states. Using a unitary transformation, we design a convenient form of the driver Hamiltonian. We apply the method to a particle in a confining potential and discrete systems to find explicit forms of the Hamiltonian and discuss the general properties. The method is derived by using the quantum brachistochrone equation, which shows the existence of a nontrivial dynamical invariant in the deformed system.

Figure 1

Potential $v\left(t\right)=v_1\left(t\right)-v_2\left(t\right)$ in Eq. (53). The phase difference ${\varphi }_1-{\varphi }_2$ is determined from Eqs. (59) and (60).

Figure 2

$v\left(t\right)$ at $c=1$ and $\Gamma =2$. $h_y=\dot{\theta }$ represents the counterdiabatic field in Eq. (61).

Figure 3

Bloch vectors $\mathit{n}\left(t\right)$ before the deformation (dashed line) and $\tilde{\mathit{n}}\left(t\right)$ after the deformation (bold). $\mathit{n}\left(0\right)=\tilde{\mathit{n}}\left(0\right)=(1,0,0)$ and $\mathit{n}\left(\infty \right)=\tilde{\mathit{n}}\left(\infty \right)=(0,0,1)$. We set $c=1$ and $\Gamma =2$.