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.
Institut fu¨r Theoretische Physik, Universita¨t zu K¨oln, De-50937 K¨oln, Germany(Dated: December 16, 2013)Landau-Zener transitions in a Lipkin-Meshkov-Glick model are considered, both as a single systemand in a lattice where each individual site hosts a realization of the corresponding many-bodyproblem. The single site mean-field problem is conceptually different from many other non-linearLandau-Zener models - it does not support any hysteresis and a new phenomenon appears whereadiabaticity totaly breaks down. More precisely, thisbreakdown occurs for isolated classical solutionspropagating on a separatrix in phase space and during the Landau-Zener sweep they get locked toa fixed point which prohibits adiabatic following. Quantum fluctuations tend to destabilize thesesolutions such that for moderate particle numbers the phenomenon is lost. At the quantum levelit is also shown that the non-adiabatic corrections P of the single site problem obey a power-lawdependenceon the Landau-Zenersweep velocity λ in the adiabatic regime, i.e. P ∼ λ
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.
Population transfer in quantum systems has always been an interesting area in physics since the introduction of quantum mechanics. In this paper, transition probabilities for a coupled system consisting of four two - level particles are studied by solving Landau - Zener Hamiltonian. The effects of the first and second nearest neighbors interactions are investigated. Presented results indicate that the second nearest neighbors interactions will decrease the transition probability when the coupling strength for each neighborhood has the same sign. The fast sweep effect on transition probability is also studied here.
We investigate quantum interference effects in a superconducting Cooper-pair box by taking into account the possibility of tunneling processes involving one and two Cooper pairs. The quantum dynamics is analyzed in a framework of three-level model. We compute Landau–Zener probabilities for a linear sweep of the gate charge and investigate Rabi oscillations in a periodically driven three-level system under in- and off-resonance conditions. It was shown that the Landau–Zener probabilities reveal two different patterns: “step”- and “beats”-like behaviors associated with the quantum interference effects. Control on these two regimes is provided by the change of the ratio between two characteristic time scales of the problem. We demonstrate through the analysis of a periodically driven three-level system, that if a direct transition between certain pairs of levels is allowed and fine-tuned to a resonance, the problem is mapped to the two-level Rabi model. If the transition between a pair of levels is forbidden, the off-resonance Rabi oscillations involving second order in tunneling processes are predicted. This effect can be observed by measuring a population difference slowly varying in time between the states of the Cooper-pair box characterized by the same parity.
We investigate quantum interference effects in a superconducting Cooper-pair box by taking into account the \npossibility of tunneling processes involving one and two Cooper pairs. The quantum dynamics is analyzed in a \nframework of three-level model. We compute Landau–Zener probabilities for a linear sweep of the gate charge \nand investigate Rabi oscillations in a periodically driven three-level system under in- and off-resonance conditions. It was shown that the Landau–Zener probabilities reveal two different patterns: “step”- and “beats”-like \nbehaviors associated with the quantum interference effects. Control on these two regimes is provided by change \nof the ratio between two characteristic time scales of the problem. We demonstrate through the analysis of a periodically driven three-level system, that if a direct transition between certain pairs of levels is allowed and finetuned to a resonance, the problem is mapped to the two-level Rabi model. If the transition between pair of levels \nis forbidden, the off-resonance Rabi oscillations involving second order in tunneling processes are predicted. \nThis effect can be observed by measuring a population difference slowly varying in time between the states of \nthe Cooper-pair box characterized by the same parity