The complex-time picture of Shortcuts to Adiabaticity: interference in the Landau-Zener-Stückelberg-Majorana model

When a Hamiltonian is time-dependent, the instantaneous eigenstates are not exact solutions of the time-dependent Schrödinger equation, except for the limit when the time-dependence is very slow compared to the time-scale set by the gap (the adiabatic theorem). Otherwise, a particle prepared in a given eigenstate will undergo transitions to other eigenstates. In the method of shortcuts to adiabaticity, one completely suppresses such transitions by adding a properly designed correction to the dynamics, the counterdiabatic Hamiltonian. We investigate the physical mechanism by which the counterdiabatic Hamiltonian suppresses transitions in the Landau-Zener-Stückelberg-Majorana model. In the complex-time picture of this model, the probability of transitions is determined by the position of the branch point of the eigenvalues which lies closest to the real-time axis, a fact known as the Dykhne formula. We consider a one-parameter deformation of the Hamiltonian corresponding to ``turning on" the counterdiabatic Hamiltonian and show that this introduces extra relative phases between the different paths in the complex plane. We then derive a new non-perturbative formula for the probability of non-adiabatic transitions in this extended model, and show that in the case of counterdiabatic driving the probability of transitions vanishes due to the exactly destructive interference in the complex plane. This intuition further extends to a whole class of models known as integrable time-dependent quantum Hamiltonians, a fact which we demonstrate by proving that the addition of counterdiabatic Hamiltonians preserves their characteristic integrability condition.