Analytically solvable Hamiltonians for quantum two-level systems and their dynamics

Abstract

A simple systematic way of obtaining analytically solvable Hamiltonians for quantum two-level systems is presented. In this method, a time-dependent Hamiltonian and the resulting unitary evolution operator are connected through an arbitrary function of time, furnishing us with new analytically solvable cases. The method is surprisingly simple, direct, and transparent and is applicable to a wide class of two-level Hamiltonians with no involved constraint on the input function. A few examples illustrate how the method leads to simple solvable Hamiltonians and dynamics.

Analytical results for a parity-time-symmetric two-level system under synchronous combined modulations

“…In that case, from Eqs. (9)- (10) and (23), the final populations in levels |1 and |2 corresponding to the initial condition (25) can be easily found to be…”

Interaction-free evolution in the presence of time-dependent Hamiltonians

“…In general Eq. (1) is untractable and, except for some lucky cases [14][15][16], it requires special assumptions, such as for example the adiabatic one [17], or suitable approximations, like in the perturbative treatment [18,19]. Therefore, even the partial resolution of a class of timedependent problems in the presence of time-dependent Hamiltonians is of interest itself.…”

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

“…3 on the basis of Ref. [42]. Illustrative examples and conclusive remarks are finally given in Secs.…”

“…Quite recently, however, novel resolution strategies for exactly solving the quantum dynamics of the SU(2) problem under scrutiny has been reported [42,43,44,45]. The approach in Ref.…”

“…The approach of Ref. [42] to deduce the class of solvable SU(2) Hamiltonian problems is briefly reviewed in Sec. 2.…”