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.
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.
Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.
We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schr\"odinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time
tw; we find an exact expression for the excitation probability as a function of
tw. We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.
We study the
S matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].
Considering a temperature dependent two-level quantum system, we have numerically solved the Landau-Zener transition problem.The method includes the incorporation of temperature effect as a thermal noise added Schrödinger equation for the construction of the Hamiltonian.Here, the obtained results which describe the changes in the system including the quantum states and the transition probabilities are investigated and discussed.The results successfully describe the behavior of the transition probabilities by sweeping the temperature.