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Progress of Theoretical Physics Supplement

Quantum Mechanical Transitions in a Dissipative Environment

Seiji Miyashita, Keiji Saito, Hans A. De Raedt

Published Jan 1, 2000• 1 citation
Publisher

Abstract

We have constructed an algorithm to treat quantum mechanical dynamics, such as the nonadiabatic transitions in a dissipative environment, by making use of the formula of reduced density matrix, i.e., Quantum Master Equation (QME). We applied the method to investigate dynamics of magnetization of nanoscale magnets, such as Mn12 and Fe8, in a time dependent field. Due to the discreteness of the energy levels in small systems the resonating tunneling phenomena play an important role. At relatively high temperatures the excited levels contribute to the transition and the transition rate depends significantly on the temperature. On the other hand, at very low temperatures the magnetization process does not depend on the temperature. Even at such a low temperature, the effect of environments is still relevant. We demonstrate such a process with the method of QME. We also analyze temperature dependent phenomena of hysteresis of the nanoscale magnets.

DOI
10.1143/ptps.138.501

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We have constructed an algorithm to treat quantum mechanical dynamics, such as the nonadiabatic transitions in a dissipative environment, by making use of the formula of reduced density matrix, i.e., Quantum Master Equation (QME). We applied the method to investigate dynamics of magnetization of nanoscale magnets, such as Mn12 and Fe8, in a time dependent field. Due to the discreteness of the energy levels in small systems the resonating tunneling phenomena play an important role. At relatively high temperatures the excited levels contribute to the transition and the transition rate depends significantly on the temperature. On the other hand, at very low temperatures the magnetization process does not depend on the temperature. Even at such a low temperature, the effect of environments is still relevant. We demonstrate such a process with the method of QME. We also analyze temperature dependent phenomena of hysteresis of the nanoscale magnets.
A new approach to the magnetic hysteresis loop and magnetization relaxation of molecule magnet, Mn12, is presented. The method, so called "quasi-thermal equilibrium (QTE) + erase phase (EP)" (QTE + EP), leads the dissipation in state evolution. The method gives an method to approach an irreversible quantum dynamic tunnelling and time evolution of system and straightly leads the hysteresis loops. The QTE can work well for the double-well system and is able to be extended to the multi-wells quantum system. The EP is based on the physics of random interaction with the environment that results the phase de-coherence or say "re-initial" of the phase of quantum state during the time evolution. The stairs in the magnetic loops of Mn12 we have obtained quantitatively show the quantum resonant tunnelling. The relaxation of magnetization from initial lowest state is calculated for different temperatures and well shows the exponential behavior. The theoretical results well agree with the experiments.
We will discuss on the quantum dynamics on the following three cases. (1) First, we will study structure of adiabatic energy levels for a triangle magnetic cluster with the Dzyaloshinsky-Moriya (DM) interaction. Generally, in single molecular magnets, the magnetic energy levels are discrete and we can find various interesting magnetic behavior when we sweep the external magnetic field. For the dynamics, the adiabatic energy level structure, in particular, avoided level structures at the energy level crossings, plays an important role. The mechanism to open the gap is attributed to some interactions which do not commute with the magnetization. We will discuss dependence of the dynamical magnetization process under the field cycling on the angle between the DM interaction vector and the external magnetic fields.[1,2] We also discuss a hidden conserved quantity due to the chiral symmetry.[3] The dynamical DM effect will be also discussed. (2) We will study the quantum dynamics of one-dimensional transverse Ising model which exhibits a quantum phase transition at zero temperature. There, we find a metastable state and a kind of spinodal decomposition phenomena in the sense of the quantum dynamics. We also refer to the cases of very fast sweep where some size independent collective motions are observed.[4] (3) Finally, we will also study an itinerant electron system in which the Mott insulator-Nagaoka ferromagnetism transition. There, we study the adiabatic change from the Mott insulator to the Nagaoka ferromagnetic state under removal of an electron from the lattice. We also study the magnetization process as a function of the magnetic field.
We investigate the dynamic evolution and thermodynamic process of a driven quantum system immersed in a finite-temperature heat bath. A Born-Markovian quantum master equation is formally derived for the time-dependent system with discrete energy levels. This quantum master equation can be applied to situations with a broad range of driving speeds and bath temperatures and thus be used to study the finite-time quantum thermodynamics even when nonadiabatic transition and dissipation coexist. The dissipative Landau-Zener model is analyzed as an example. The population evolution and transition probability of the model reveal the importance of the competition between driving and dissipation beyond the adiabatic regime. Moreover, local maxima of irreversible entropy production occur at intermediate sweep velocity and finite temperature, which the low-dissipation model cannot describe.
An analytical study of the dissipative Landau-Zener model is presented. The model where two energy levels at constant speed are brought to cross is a standard model used to describe a large variety of phenomena. In many cases of interest the presence of coupling of the two-state system to an environment is of importance, the accounting for which shall be done here from first principles. Analytical results for the excitation transition from the ground state of the two-state system at large negative times to the excited state at large positive times (as well as the opposite, decay, transition) are obtained in terms of the speed by which the two levels approach each other, the energy gap between the adiabatic energies, the coupling strength to the environment, and its temperature. For the excitation transition we find the following results: In the slow-passage limit of small sweeping speed it is shown that adiabaticity is limited to low temperatures and the quantitative adiabatic criterion is established. Particularly, at zero temperature there is no influence of the environment on the transition probability as a consequence of a compensation property shown to be peculiar to the linear-sweep model. The transition is at low temperatures due to quantum tunneling and, with an increase in temperature, an intermediate region appears where the transition is dominated by thermally assisted transitions across the energy gap before finally at high temperatures a saturated regime is reached with equal population of the levels. In contrast to the dependence on the temperature the dependence of the transition probability as a function of coupling strength is nonmonotonic with maximum influence at intermediate strength. In the fast-passage limit with rapid sweep speed there is no influence of the environment on the transition probability. For the decay transition the adiabatic limit does not exist for the linear-sweep model for physically relevant spectra of the environment and the decay transition is dominated by spontaneous emission, except in the fast-passage limit and the high-temperature limit where the decay transition probability equals the excitation probability.
A new perturbative quantum master equation is presented for the dissipative dynamics of a subsystem inside a nanosystem with a constant and finite total energy. The degrees of freedom besides those of the subsystem play the role of the environment. Our equation rules the time evolution of the distribution functions of the subsystem populations and coherences over the energy of the environment. Thanks to these distribution functions, we can take into account the effects of the conservation of the total energy. This equation is more general than the standard perturbative equations used for describing a subsystem interacting with an environment (such as the Redfield or Cohen-Tannoudji equations) because these latter equations can be deduced from it in the limit of an infinitely large environment. We numerically apply this equation to the spin-GORM model for the interaction of a two-level subsystem with an environment described by random matrices. We compare our equation with the exact von Neumann equation of the total system and show its superiority compared to the Redfield equation (in the Markovian and non-Markovian cases).