Joachim Ankerhold

Strong Friction Limit in Quantum Mechanics: The Quantum Smoluchowski Equation

For a quantum system coupled to a heat bath environment the strong friction limit is studied starting from the exact path integral formulation. Generalizing the classical Smoluchowski limit to low temperatures, a time evolution equation for the position distribution is derived and the strong role of quantum fluctuations in this limit is revealed.

Optimal control of open quantum systems: cooperative effects of driving and dissipation.

We investigate the optimal control of open quantum systems, in particular, the mutual influence of driving and dissipation. A stochastic approach to open-system control is developed, using a generalized version of Krotovs iterative algorithm, with no need for Markovian or rotating-wave approximations. The application to a harmonic degree of freedom reveals cooperative effects of driving and dissipation that a standard Markovian treatment cannot capture. Remarkably, control can modify the open-system dynamics to the point where the entropy change turns negative, thus achieving cooling of translational motion without any reliance on internal degrees of freedom.

IS THE DIRECT OBSERVATION OF ELECTRONIC COHERENCE IN ELECTRON TRANSFER REACTIONS POSSIBLE

The observability of electronic coherence in electron transfer reactions is discussed. We show that under appropriate circumstances large-amplitude oscillations can be found in the electronic occupation probabilities. The initial preparation of the system is of crucial importance for this effect, and we discuss conditions under which experiments detecting electronic coherence should be feasible. The Feynman–Vernon influence functional formalism is extended to examine more general and experimentally relevant initial preparations. Analytical expressions and path integral quantum dynamics simulations were developed to study the effects of various initial preparations on the observability of electronic coherence.

A study of the semiclassical initial value representation at short times

The short time dynamics of the semiclassical initial value separation are studied analytically for a one dimensional system. We find that at short times the approximation introduces spurious errors that depend on ℏ and result from the anharmonic part of the potential. This is in contrast to classical mechanics which gives the first three initial time derivatives of a coordinate dependent operator exactly. Consideration of a model system shows, though, that the error introduced is not very large and that for times which are longer than a typical period of classical motion, semiclassical initial value representation propagation is superior to classical time propagation.

Non-Markovian Dissipative Semiclassical Dynamics

The exact stochastic decomposition of non-Markovian dissipative quantum dynamics is combined with the time-dependent semiclassical initial value formalism. It is shown that even in the challenging regime of moderate friction and low temperatures, where non-Markovian effects are substantial, this approach allows for the accurate description of dissipative dynamics in anharmonic potentials over many oscillation periods until thermalization is reached. The problem of convergence of the stochastic average at long times, which plagues full quantum mechanical implementations, is avoided through a joint sampling of the stochastic noise and the semiclassical phase-space distribution.

Persistence of Coherent Quantum Dynamics at Strong Dissipation

The quantum dynamics of a two-state system coupled to a bosonic reservoir with sub-Ohmic spectral density is investigated for strong friction. Numerically exact path integral Monte Carlo methods reveal that a changeover from coherent to incoherent relaxation does not occur for a broad class of spectral distributions. In nonequilibrium coherences associated with substantial system-reservoir entanglement exist even when strong dissipation forces the thermodynamic state of the system to behave almost classically. This may be of relevance for current experiments with nanoscale devices.

Quantum Brownian motion with large friction.

Quantum Brownian motion in the strong friction limit is studied based on the exact path integral formulation of dissipative systems. In this limit the time-nonlocal reduced dynamics can be cast into an effective equation of motion, the quantum Smoluchowski equation. For strongly condensed phase environments it plays a similar role as master equations in the weak coupling range. Applications for chemical, mesoscopic, and soft matter systems are discussed and reveal the substantial role of quantum fluctuations.

Low temperature electron transfer in strongly condensed phase

Electron transfer coupled to a collective vibronic degree of freedom is studied in strongly condensed phase and at lower temperatures where quantum fluctuations are essential. Based on an exact representation of the reduced density matrix of the electronic + reaction coordinate compound in terms of path integrals, recent findings on the overdamped limit in quantum dissipative systems are employed. This allows us to give a consistent generalization of the well-known Zusman equations to the quantum domain. Detailed conditions for the range of validity are specified. Using the Wigner transform these results are also extended to the quantum dynamics in full phase space. As an important application electronic transfer rates are derived that comprise adiabatic and nonadiabatic processes in the low temperature regime including nuclear tunneling. Accurate agreement with precise quantum Monte Carlo data is observed.

Path-integral Monte Carlo simulations for electronic dynamics on molecular chains. I. Sequential hopping and super exchange

An improved real-time quantum Monte Carlo procedure is presented and applied to describe the electronic transfer dynamics along molecular chains. The model consists of discrete electronic sites coupled to a thermal environment which is integrated out exactly within the path integral formulation. The approach is numerically exact and its results reduce to known analytical findings (Marcus theory, golden rule) in proper limits. Special attention is paid to the role of superexchange and sequential hopping at lower temperatures in symmetric donor-bridge-acceptor systems. In contrast to previous approximate studies, superexchange turns out to play a significant role only for extremely high-lying bridges where the transfer is basically frozen or for extremely low temperatures where for weaker dissipation a description in terms of rate constants is no longer feasible. For bridges with increasing length an algebraic decrease of the yield is found for short as well as for long bridges. The approach can be extended to electronic systems with more complicated topologies including impurities and in presence of external time-dependent forces.

Quantum Smoluchowski equation

The strong friction limit of the Caldeira-Leggett model for quantum Brownian motion is analyzed. In this Smoluchowski limit the density operator of the Brownian particle is essentially diagonal in coordinate, and the diagonal element satisfies the classical Smoluchowski equation. This result cannot be obtained from the master equations that have been proposed for quantum Brownian motion, whether or not these equations are of Lindblad form.