Pechukas-Yukawa approach to the evolution of the quantum state of a parametrically perturbed system
Mumnuna A. Qureshi, Johnny Q. Zhong, Zihad Qureshi, Peter Mason, Joseph J. Betouras, Alexandre M. Zagoskin · Physical Review A · 2018
We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter $\ensuremath{\lambda}(t)$. Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of $d\ensuremath{\lambda}/dt$ and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.