Abstract
Transition amplitudes between instantaneous eigenstates of a quantum two-level system are evaluated analytically on the basis of a new parametrization of its evolution operator, which has recently been proposed to construct exact solutions. In particular, the condition under which the transitions are suppressed is examined analytically. It is shown that the analytic expression of the transition amplitude enables us, not only to confirm the adiabatic theorem, but also to derive the necessary and sufficient condition for quantum two-level system to remain in one of the instantaneous eigenstates.
Highlights
Our approach allows us to exhibit an example in which the system stays in its instantaneous eigenstate, even though the Hamiltonian varies with a nonvanishing rate
The adiabatic theorem is confirmed on the basis of the explicit expressions of the transition amplitudes (see [8] and refer to35), which are derived from the new parametrization of the evolution operator for the quantum two-level system29
What is stressed here is that these expressions enable us to evaluate such transition amplitudes directly in any physical situation, from the adiabatic to diabatic cases
Summary
Where Ω and ω are time-dependent real and complex functions, respectively. The instantaneous eigenvalues E±(t) = ± Ω2(t) + ω(t) 2 and eingenstates ± t,. The Hamiltonian itself varies infinitesimally slowly, realizing a process where the instantaneous eigenstates do not make transitions. We may choose such a χ that realizes φ = ν0t with a finite parameter variation of χ is essentially ν0, a νfi0naitnedvcal=ue−irr21e(lTet v)3anact ctoorTd,inwghtiole [17] and φω = 0, which means that the other parameter c varies slowly the speed of in time with c proportional to 1/T In this case, as Eqs [19] and [20] show, the characteristic frequency of the magnetic field (i.e., Hamiltonian) is ν0, while its instantaneous eigenstates become slowly varying for large T. Our approach allows us to exhibit an example in which the system stays in its instantaneous eigenstate, even though the Hamiltonian varies with a nonvanishing rate
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