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arXiv:1509.04652 · Journal of Physics A Mathematical and Theoretical

Wei–Norman–Kolokolov approach for Landau–Zener problems

M B Kenmoe, L C Fai

Published Oct 31, 2014• 5 citations
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Abstract

The tunneling between the $2S+1$ Zeeman multiplets of an arbitrary spin $S$ using the $SU(2)$ Lie group exponential ordering Wei-Norman technique is investigated. A spin subjected to a regular time-dependent magnetic field and a transverse colored noise is considered and the regimes of fast and slow noise examined.

DOI
10.1088/1751-8113/47/46/465202

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The tunneling between the 2S+12S+12S+1 Zeeman multiplets of an arbitrary spin SSS using the SU(2)SU(2)SU(2) Lie group exponential ordering Wei-Norman technique is investigated. A spin subjected to a regular time-dependent magnetic field and a transverse colored noise is considered and the regimes of fast and slow noise examined.
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We investigate the Landau-Zener transition in two- and three-level systems subject to a classical Gaussian noise. Two complementary limits of the noise being fast and slow compared to characteristic Landau-Zener tunnel times are discussed. The analytical solution of a density matrix (Bloch) equation is given for a long-time asymptotic of transition probability. It is demonstrated that the transition probability induced or assisted by the fast noise can be obtained through a procedure of Bloch's equation averaging with further reducing it to a master equation. In contrast to the case of fast noise, the transition probability for LZ transition induced, or assisted, by the slow classical noise can be obtained by averaging the solution of Bloch's equation over the noise realization. As a result, the transition probability is described by the activation Arrhenius law. The approximate solution of the Bloch's equation at finite times is written in terms of Fresnel's integrals and interpreted in terms of interference pattern. We discuss consequences of a local isomorphism between SU(2) and SO(3) groups and connections between Schr\"odinger and Bloch descriptions of spin dynamics. Based on this isomorphism, we establish the relations between S=1/2S=1/2S=1/2 and 1 transition probabilities influenced by the noise. A possibility to use the slow noise as a probe for tunnel time is discussed.
Stimulated by the experimental realization of spin-dependent tunneling via a gradient magnetic field [C. J. Kennedy et al., Phys. Rev. Lett. 111, 225301 (2013); M. Aidelsburger et al., Phys. Rev. Lett. 111, 185301 (2013)], we investigate the dynamics of Bloch oscillations and Landau-Zener tunneling of single spin-half particles in a periodic potential under the influence of a spin-dependent constant force. In analogy to the Wannier-Stark system, we call our system the Wannier-Zeeman system. If there is no coupling between the two spin states, the system can be described by two crossing Wannier-Stark ladders with opposite tilts. The spatial crossing between two Wannier-Stark ladders becomes a spatial anticrossing if the two spin states are coupled by external fields. For a wave packet away from the spatial anticrossing, due to the spin-dependent constant force, it will undergo spatial Landau-Zener transitions assisted by the intrinsic intraband Bloch oscillations, which we call the Bloch-Landau-Zener dynamics. If the interspin coupling is sufficiently strong, the system undergoes adiabatic Bloch-Landau-Zener dynamics, in which the spin dynamics follows the local dressed states. Otherwise, for nonstrong interspin couplings, the system undergoes nonadiabatic Bloch-Landau-Zener dynamics.