Perturbative Approach to Landau-Zener Transitions

✍ Scribed by D. Waxman

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 379 KB
• Volume: 236
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1006/aphy.1994.1110

✦ Synopsis

The Landau-Zener level-crossing problem is analysed perturbatively. We consider the operator (H(t)=\Phi(t) \sigma^{3}+\Delta \sigma^{1}), where (\Phi(t)) is an external (time-dependent) force, (\sigma^{i}) are the Pauli matrices, and (\Delta) 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 (A). For three choices of (\Phi(t), U(t,-t)) is determined for small (\Delta). For (\Phi(t)) periodic with period (\theta) a small (\Delta) approximation to (U(t,-t)) is obtained that is valid for times of order (\theta /(\theta \Delta)^{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) \approx 1) for all (t). The effects of fluctuations in (\Phi(t)) on the transition probability are also calculated. To (O\left(A^{2}\right)) it is found that for (\Phi(t) \propto 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). O 1994 Academic Press, Inc.