Hamiltonian engineering via invariants and dynamical algebra

We use the dynamical algebra of a quantum system and its dynamical invariants to inverse engineer feasible Hamiltonians for implementing shortcuts to adiabaticity. These are speeded up processes that end up with the same populations as slow, adiabatic ones. As application examples, we design families of shortcut Hamiltonians that drive two- and three-level systems between initial and final configurations, imposing physically motivated constraints on the terms (generators) allowed in the Hamiltonian.

Figure 1

Hamiltonian (left) and invariant (right) coefficients vs time. The imposed functions (symbols) are $h_1\left(t\right)$ (red squares), $h_2\left(t\right)$ (brown triangles), and $f_3\left(t\right)$ (blue circles). The derived functions (lines) are $h_3\left(t\right)$ (blue solid line), $f_1\left(t\right)$ (red short-dashed line), and $f_2\left(t\right)$ (brown long-dashed line). Parameter values: ${\tilde{h}}_1\left(t_f\right)=0.4$, ${\tilde{h}}_3\left(0\right)=1$, and ${\tilde{t}}_f=200$.

Figure 2

Hamiltonian (left) and invariant (right) coefficients vs time. The imposed functions (symbols) are $h_2\left(t\right)$ (brown triangles), $h_3\left(t\right)$ (blue squares), and $f_1\left(t\right)$ (red circles). The derived functions (lines) are $h_1\left(t\right)$ (red solid line), $f_2\left(t\right)$ (brown long-dashed line), and $f_3\left(t\right)$ (blue short-dashed line). Parameter values: ${\tilde{h}}_3\left(0\right)=1$, ${\tilde{h}}_1\left(t_f\right)=2.5$, and ${\tilde{t}}_f=200$.

Figure 3

Hamiltonian (left) and invariant (right) coefficients vs time. The imposed functions (symbols) are $h_2\left(t\right)$ (brown triangles), $h_3\left(t\right)$, (brown triangles), $h_4\left(t\right)$ (red squares), and $f_1\left(t\right)$ (blue circles). The derived functions (lines) are $h_1\left(t\right)$ (blue solid line), $f_2\left(t\right)$ (brown, short-dashed line), $f_3\left(t\right)$ (red long-dashed line), and $f_4\left(t\right)$ (red long-dashed line). Parameter values: ${\tilde{h}}_1\left(t_f\right)=-4$, ${\tilde{h}}_4\left(0\right)=1$, and ${\tilde{t}}_f=80$.