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arXiv:cond-mat/0501089 · Physical Review B

Many-body Landau-Zener effect at fast sweep

Dmitry A. Garanin, R. Schilling

Published May 26, 2005• 24 citations
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Abstract

The asymptotic staying probability $P$ in the Landau-Zener effect with interaction is analytically investigated at fast sweep $\ensuremath{\epsilon}=\ensuremath{\pi}{\ensuremath{\Delta}}^{2}∕(2\ensuremath{\hbar}v)⪡1$. We have rigorously calculated the value of ${I}_{0}$ in the expansion $P\ensuremath{\cong}1\ensuremath{-}\ensuremath{\epsilon}+{\ensuremath{\epsilon}}^{2}∕2+{\ensuremath{\epsilon}}^{2}{I}_{0}$ for arbitrary couplings and relative resonance shifts of individual tunneling particles. The results essentially differ from those of the mean-field approximation. It is shown that strong long-range interactions such as dipole-dipole interaction generate huge values of ${I}_{0}$ because a flip of one particle strongly influences many others. However, in the presence of strong static disorder, making the resonances for individual particles shifted with respect to each other, the influence of interactions is strongly reduced. In molecular magnets the main source of static disorder is the coupling to nuclear spins. Our calculations using the actual shape of the ${\mathrm{Fe}}_{8}$ crystal studied in the Landau-Zener experiments [Wernsdorfer et al., Europhys. Lett. 50, 552 (2000)] yield a value of ${I}_{0}$ that is in good agreement with the value extracted from the experimental data.

DOI
10.1103/physrevb.71.184414

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The asymptotic staying probability PPP in the Landau-Zener effect with interaction is analytically investigated at fast sweep \ensuremathϵ=\ensuremathπ\ensuremathΔ2∕(2\ensuremathℏv)⪡1\ensuremath{\epsilon}=\ensuremath{\pi}{\ensuremath{\Delta}}^{2}∕(2\ensuremath{\hbar}v)⪡1\ensuremathϵ=\ensuremathπ\ensuremathΔ2∕(2\ensuremathℏv)⪡1. We have rigorously calculated the value of I0{I}_{0}I0​ in the expansion P\ensuremath≅1\ensuremath−\ensuremathϵ+\ensuremathϵ2∕2+\ensuremathϵ2I0P\ensuremath{\cong}1\ensuremath{-}\ensuremath{\epsilon}+{\ensuremath{\epsilon}}^{2}∕2+{\ensuremath{\epsilon}}^{2}{I}_{0}P\ensuremath≅1\ensuremath−\ensuremathϵ+\ensuremathϵ2∕2+\ensuremathϵ2I0​ for arbitrary couplings and relative resonance shifts of individual tunneling particles. The results essentially differ from those of the mean-field approximation. It is shown that strong long-range interactions such as dipole-dipole interaction generate huge values of I0{I}_{0}I0​ because a flip of one particle strongly influences many others. However, in the presence of strong static disorder, making the resonances for individual particles shifted with respect to each other, the influence of interactions is strongly reduced. In molecular magnets the main source of static disorder is the coupling to nuclear spins. Our calculations using the actual shape of the Fe8{\mathrm{Fe}}_{8}Fe8​ crystal studied in the Landau-Zener experiments [Wernsdorfer et al., Europhys. Lett. 50, 552 (2000)] yield a value of I0{I}_{0}I0​ that is in good agreement with the value extracted from the experimental data.
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.S=N/2.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>0J>0J>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<0J<0J<0 LZS transitions are boosted and P=0P=0P=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.
The study of transition probabilities in few-body systems and their long-range interactions using the Landau-Zener method could be helpful for solving a wide range of problems in fields of quantum simulations, Rydberg blockade, quantum gates, and dipole transition in Rydberg atoms. Here, the transition probabilities for a four-particle system in a square-shaped lattice are studied by solving the many-body Landau-Zener Hamiltonian in the next-nearest-neighbor approximation. It is observed that the complete transition occurs for the anti-ferromagnetic coupling while it is limited to a constant probability for the ferromagnetic coupling. These probabilities suppress when the magnetic field's energy rate is increased. We demonstrate that in the presence of the next-nearest-neighbor's interactions, the final probabilities’ behavior is different. Considering the next-nearest-neighbor's interactions, it is possible to achieve the complete transition in imperfect anti-ferromagnetic regions. At the same time, the complete transition does not occur in perfect anti-ferromagnetic regions for certain values of the sweeping rate. These results emphasize that it is necessary to consider the next-nearest-neighbors’ interactions in a many-body system. Furthermore, the implementation of this Hamiltonian in the dipole-dipole and van der Waals interaction reveals that a steeper decrease for dipole-dipole interaction.
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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).W(t).W(t). Our main concern is to investigate the influence of the nonlinearity of W(t)W(t)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.v.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>0a>0a>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)W(t)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)W(t)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{P}_{0}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.
We study stationary and dynamical properties of the many-body Landau-Zener dynamics of a Bose quantum fluid confined in two coupled one-dimensional chains by using a many-body generalization recently reported Chen et al. [Nature Phys. 7, 61 (2011)], within the decoupling approximation and the one-level band scheme. The energy spectrum shows evidence of the structure of the avoided level crossings as a function of the onsite interparticle interaction strength. On the dynamical side, a phase diagram of the transfer efficiency across ground-state and inverse sweeps is presented. A totally different scenario with respect to the original single-particle Landau-Zener scheme is found for ground-state sweeps, in which a breakdown of the adiabatic region emerges as the sweep rate decreases. On the contrary, the transfer efficiency across inverse sweeps reveals consistent results with the single-particle Landau-Zener predictions. In the strong-coupling regime, we find that there is a critical value of the onsite interaction for which the transfer of particles starts to vanish independently of the sweep rate. Our results are in qualitative agreement with those of the experimental counterpart.