The Landau-Zener-Stueckelberg (LZS) effect in a model system of interacting tunneling particles is studied numerically and analytically. Each of N tunneling particles interacts with each of the others with the same coupling J. This problem maps onto that of the LZS effect for a large spin
S=N/2. The mean-field limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ corresponds to the classical limit $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ for the effective spin. It is shown that the ferromagnetic coupling
J>0 tends to suppress the LZS transitions. For $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. For
J<0 LZS transitions are boosted and
P=0 for a set of finite values of the sweep rate for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.
Motivated by the experimental observation of resonant tunneling in systems of half-integer spin, such as V15 and Mn4, we study the mechanism of the adiabatic change of the magnetization in systems with time-reversal symmetry. Although tunneling between the degenerate ground states is suppressed in a simple Kramers doublet, some types of perturbational interactions cause an avoided level crossing structure in the energy level structures when the system possesses additional degeneracy of the ground state. We show that a nonadiabatic transition governed by the Landau-Zener-Stückelberg mechanism occurs in many cases, and also find more general cases of adiabatic change of the magnetization in which LZS mechanism cannot be applied directly.
We present a comprehensive analysis of the Landau-Zener tunneling of a nonlinear three-level system in a linearly sweeping external field. We find the presence of nonzero tunneling probability in the adiabatic limit (i.e., very slowly sweeping field) even for the situation that the nonlinear term is very small and the energy levels keep the same topological structure as that of the linear case. In particular, the tunneling is irregular with showing an unresolved sensitivity on the sweeping rate. For the case of fast-sweeping fields, we derive an analytic expression for the tunneling probability with stationary phase approximation and show that the nonlinearity can dramatically influence the tunneling probability when the nonlinear ``internal field'' resonate with the external field. We also discuss the asymmetry of the tunneling probability induced by the nonlinearity. Physics behind the above phenomena is revealed and possible application of our model to triple-well trapped Bose-Einstein condensate is discussed.
We will discuss on the quantum dynamics on the following three cases. (1) First, we will study structure of adiabatic energy levels for a triangle magnetic cluster with the Dzyaloshinsky-Moriya (DM) interaction. Generally, in single molecular magnets, the magnetic energy levels are discrete and we can find various interesting magnetic behavior when we sweep the external magnetic field. For the dynamics, the adiabatic energy level structure, in particular, avoided level structures at the energy level crossings, plays an important role. The mechanism to open the gap is attributed to some interactions which do not commute with the magnetization. We will discuss dependence of the dynamical magnetization process under the field cycling on the angle between the DM interaction vector and the external magnetic fields.[1,2] We also discuss a hidden conserved quantity due to the chiral symmetry.[3] The dynamical DM effect will be also discussed. (2) We will study the quantum dynamics of one-dimensional transverse Ising model which exhibits a quantum phase transition at zero temperature. There, we find a metastable state and a kind of spinodal decomposition phenomena in the sense of the quantum dynamics. We also refer to the cases of very fast sweep where some size independent collective motions are observed.[4] (3) Finally, we will also study an itinerant electron system in which the Mott insulator-Nagaoka ferromagnetism transition. There, we study the adiabatic change from the Mott insulator to the Nagaoka ferromagnetic state under removal of an electron from the lattice. We also study the magnetization process as a function of the magnetic field.
We study the Landau-Zener-Stueckelberg (LZS) effect for a two-level system with a time-dependent nonlinear bias field (the sweep function)
W(t). Our main concern is to investigate the influence of the nonlinearity of
W(t) on the probability P to remain in the initial state. The dimensionless quantity $\ensuremath{\varepsilon}=\ensuremath{\pi}{\ensuremath{\Delta}}^{2}/(2\ensuremath{\Elzxh}v)$ depends on the coupling $\ensuremath{\Delta}$ of both levels and on the sweep rate
v. For fast sweep rates, i.e., \ensuremath{\varepsilon}\ensuremath{\ll}1, and monotonic, analytic sweep functions linearizable in the vicinity of the resonance we find the transition probability $1\ensuremath{-}P\ensuremath{\cong}\ensuremath{\varepsilon}(1+a),$ where
a>0 is the correction to the LSZ result due to the nonlinearity of the sweep. Further increase of the sweep rate with nonlinearity fixed brings the system into the nonlinear-sweep regime characterized by $1\ensuremath{-}P\ensuremath{\cong}{\ensuremath{\varepsilon}}^{\ensuremath{\gamma}}$ with \ensuremath{\gamma}\ensuremath{\ne}1, depending on the type of sweep function. In the case of slow sweep rates, i.e., \ensuremath{\varepsilon}\ensuremath{\gg}1, an interesting interference phenomenon occurs. For analytic
W(t) the probability ${P=P}_{0}{e}^{\ensuremath{-}\ensuremath{\eta}}$ is determined by the singularities of $\sqrt{{\ensuremath{\Delta}}^{2}{+W}^{2}(t)}$ in the upper complex plane of t. If
W(t) is close to linear, there is only one singularity, which leads to the LZS result ${P=e}^{\ensuremath{-}\ensuremath{\varepsilon}}$ with important corrections to the exponent due to nonlinearity. However, for, e.g., $W(t)\ensuremath{\propto}{t}^{3}$ there is a pair of singularities in the upper complex plane. Interference of their contributions leads to oscillations of the prefactor
P0 that depends on the sweep rate through \ensuremath{\varepsilon} and turns to zero at some \ensuremath{\varepsilon}. Measurements of the oscillation period and of the exponential factor would allow one to determine \ensuremath{\Delta}, independently.