Perturbative Approach to Landau-Zener Transitions

Ann Phys (NY) 236: 205-216 (1994)

D. Waxman

School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, Sussex, UK

Abstract

The Landau-Zener level-crossing problem is analysed perturbatively. We consider the operator H(t) = capital Phi, Greek(t) small sigma, Greek3 + capital Delta, Greeksmall sigma, Greek1, where capital Phi, Greek(t) is an external (time-dependent) force, small sigma, Greeki are the Pauli matrices, and capital Delta, Greek is taken to be a small parameter. By considering the operator U(t, -t) that evolves states in time from -t to t, it is possible to develop a perturbation series in capital Delta, Greek. For three choices of capital Phi, Greek(t), U(t, -t) is determined for small capital Delta, Greek. For capital Phi, Greek(t) periodic with period straight theta, small theta, Greek a small capital Delta, Greek approximation to U(t, -t) is obtained that is valid for times of order straight theta, small theta, Greek/(straight theta, small theta, Greek capital Delta, Greek)3. A slow modulation of this time evolution operator is found and for certain values of the parameters the slow modulation may stop, leaving U(t, -t) ~ 1 for all t. The effects of fluctuations in capital Phi, Greek(t) on the transition probability are also calculated. To O(capital Delta, Greek2) it is found that for capital Phi, Greek(t) proportional to t, white Gaussian noise modifies the probability of a transition in a finite time but leaves unaltered the long time transition probability. This is in accordance with the findings of Y. Kayanuma (Phys. Rev. Lett. 58 (1987), 1934).