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arXiv:0811.0010 · Physical Review B

Incoherent Landau-Zener-Stückelberg transitions in single-molecule magnets

Avinash Vijayaraghavan, Anupam Garg

Published Mar 20, 2009• 16 citations
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Abstract

It is shown that in experiments on single molecule magnets in which transitions between two lowest spins states are induced by sweeping the applied magnetic field along the easy axis; the transitions are fully incoherent. Nuclear spins and the dipolar coupling of molecular spins are identified as the main sources of decoherence, and the form of the decoherence is calculated. The Landau-Zener-St\"uckelberg (LZS) process is examined in light of this decoherence, and it is shown that the correct formula for the spin-flip probability is better given by a more recent formula of Kayanuma than that of LZS. The two formulas are shown to be identical in the limit of rapid sweeps. An approximate way of incorporating the molecular spin dipole field into the rate equations for this process is developed.

DOI
10.1103/physrevb.79.104423

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It is shown that in experiments on single molecule magnets in which transitions between two lowest spins states are induced by sweeping the applied magnetic field along the easy axis; the transitions are fully incoherent. Nuclear spins and the dipolar coupling of molecular spins are identified as the main sources of decoherence, and the form of the decoherence is calculated. The Landau-Zener-St"uckelberg (LZS) process is examined in light of this decoherence, and it is shown that the correct formula for the spin-flip probability is better given by a more recent formula of Kayanuma than that of LZS. The two formulas are shown to be identical in the limit of rapid sweeps. An approximate way of incorporating the molecular spin dipole field into the rate equations for this process is developed.
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.
The asymptotic staying probability PPP 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 I0{I}_{0}I0​ 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 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.
Landau-Zener (LZ) transitions of a two-level system (e.g., electronic spin in molecular magnets) coupled to one or many environmental spins (e.g., nuclear spins) are studied. For rather general interactions the LZ problem is reduced to that of a Landau-Zener grid. It is shown analytically that environmental spins initially in their ground state do not influence the staying probability PPP. This changes if they are prepared in a statistical ensemble. For a more specific model with environmental spins in a transverse field, LZ transitions are studied in the case of well-separated resonances in the LZ grid. The full evolution of the system is described as a succession of elementary transitions at avoided crossings and free evolution between them. If the environmental spins are strongly coupled to the central spin, their effect on PPP is weak. In other cases LZ transitions are strongly suppressed and PPP is decreasing very slowly with the sweep-rate parameter $\ensuremath{\epsilon}\ensuremath{\propto}1/v$, vvv being the energy sweep rate.
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.
Decoherence processes in crystals of molecular magnets are prototypical for interacting electronic spin systems. We analyze the Landau-Zener dynamics of the archetypical TbPc2{\mathrm{TbPc}}_{2}TbPc2​ complex diluted in a diamagnetic monocrystal. The dependence of the tunneling probability on the field sweep rate is evaluated in the framework of the recently proposed master equation in which the decoherence processes are described through a phenomenological Lindblad operator. Thus, we showcase low-temperature magnetic measurements that complement resonant techniques in determining small tunnel splittings and dephasing times.
We monitor the Landau-Zener dynamics of a single-ion magnet inserted into a spin-transistor geometry. For increasing field-sweep rates, the spin reversal probability shows increasing deviations from that of a closed system. In the low-conductance limit, such deviations are shown to result from a dephasing process. In particular, the observed behaviors are successfully simulated by means of an adiabatic master equation, with time averaged dephasing (Lindblad) operators. The time average is tentatively interpreted in terms of the finite time resolution of the continuous measurement.