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.
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.
An innovative microscopic model with a minimal number of parameters: tunneling splitting gap, external field sweeping velocity, and decoherence rate is used to describe dynamics of the dissipative Landau-Zener transition in the presence of the decoherence. In limiting cases, the derived equation of motion gives rise to the well-known Landau-Zener and Kayanuma formula. In a general case, the description demonstrates a non-monotonic flipping probability with respect to the sweeping velocity, which is also found in some other models. This non-monotony can be explained by considering the competition and timescale of the quantum tunneling, crossing period, and decoherence process. The simplicity and robustness of the theory offer a practical and novel description of the Landau-Zener transition. In addition, it promises an alternative method to the electron paramagnetic resonance in measuring the effective decoherence rate of relevant quantum systems.
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.
Two aspects of the classic two-level Landau-Zener (LZ) problem are considered. First, we address the LZ problem when one or both levels decay, i.e., ${\ensuremath{\varepsilon}}_{j}(t)\ensuremath{\rightarrow}{\ensuremath{\varepsilon}}_{j}(t)\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$. We find that if the system evolves from an initial time $\ensuremath{-}T$ to a final time
+T such that $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|$ is not too large, the LZ survival probability of a state $|j\ensuremath{\rangle}$ can increase with increasing decay rate of the other state $|i\ensuremath{\ne}j\ensuremath{\rangle}$. This surprising result occurs because the decay results in crossing of the two eigenvalues of the instantaneous non-Hermitian Hamiltonian. On the other hand, if $|{\ensuremath{\varepsilon}}_{1}(\ifmmode\pm\else\textpm\fi{}T)\ensuremath{-}{\ensuremath{\varepsilon}}_{2}(\ifmmode\pm\else\textpm\fi{}T)|\ensuremath{\rightarrow}\ensuremath{\infty}$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the probability is independent of the decay rate. These results are based on analytic solutions of the time-dependent Schr\"odinger equations for two cases: (a) the energy levels depend linearly on time, and (b) the energy levels are bounded and of the form ${\ensuremath{\varepsilon}}_{1,2}(t)=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varepsilon}tanh(t/\mathcal{T})$. Second, we study LZ transitions affected by dephasing by formulating the Landau-Zener problem with noise in terms of a Schr\"odinger-Langevin stochastic coupled set of differential equations. The LZ survival probability then becomes a random variable whose probability distribution is shown to behave very differently for long and short dephasing times. We also discuss the combined effects of decay and dephasing on the LZ probability.