Dissipative quantum systems

Start date: 01.01.1998
Funded by: Universität Augsburg
Local head of project: Gert-Ludwig Ingold
Publications: Publication list


System and environment

Real physical systems are never completely decoupled from their environment, they always interact with other degrees of freedom. This coupling can result in an irreversible energy transfer from the system to its environment, a process referred to as dissipation. In addition, the environment exerts a fluctuating force on the system leading to fluctuations of system observables. This effect manifests itself for example in the diffusive motion of a Brownian particle. Finally, in quantum mechanics, the coupling to the environment leads to the phenomenon of decoherence.

In this project, dissipative quantum systems are studied. We consider both, basic conceptual issues as well as specific quantum systems under the influence of environmental degrees of freedom.


Bath of harmonic oscillators

Dissipative quantum systems are often described within in a model where the system is bilinearly coupled to a bath of harmonic oscillators as is schematically indicated in the figure above. Already Magalinskiĭ in 1959 demonstrated for this model that the elimination of the bath degrees of freedom leads to a damped effective equation of motion for the system degree of freedom. Today, this model is often referred to as Caldeira-Leggett model because A. O. Caldeira and A. J. Leggett proved in the eighties of the last century, that the model is also applicable to strong damping. Their results had a significant impact on the study of macroscopic quantum phenomena.

Within this project, dissipative quantum systems are generally described by means of the Caleira-Leggett model and frequently Feynman's path integral method is employed. A detailed discussion of this approach for a broad class of initial conditions for system and bath as well as its application to the damped harmonic oscillator and the free Brownian particle are given in GSI88. This approach was also used in AGI95 in order to determine quantum corrections to dissipative hopping rates across a parabolic barrier.

An introduction to Feynman's path integral method with applications to dissipative quantum systems is given in Ingold02. In contrast, DHIKSZ98 approaches this topic from the point of view of statistical mechanics. Insight into fundamental aspects of quantum Brownian motion can be gained from HI05 while Ingold97 gives a brief introduction into quantum dissipation.

Thermodynamics for nonvanishing coupling to the bath

Nonnegligible between system and environment

In equilibrium thermodynamics one usually assumes an infinitely weak coupling between system and heat bath. However, when quantum effects become important, e.g. for nanosystems, this approximation is often no longer justified. Instead, the system typically is significantly coupled to at least a part of its environment as indicated in the figure. This raises the question of how to define thermodynamic quantities in such a situation. As an example, one way to define the specific heat is based on the expectation value of the system energy. Alternatively, one may start from the reduced partition function. It turns out that the two specific heats obtained in this way differ in general as was demonstrated in HI06. Even the leading high-temperature corrections to the classical result generally do not agree.

Interestingly, for the approach based on the reduced partition function, one finds that the specific heat of a free Brownian particle for specific spectral densities of bath oscillators can become negative at low temperatures (HIT08). In contrast to what one might expect, negative values of the specific heat in this case are not related to a thermodynamic instability. The specific heat obtained from the reduced partition function can rather be understood as change of the specific heat when the system degree of freedom is coupled to the bath degrees of freedom. While the specific heats of the bath as well as of system plus bath are both positive, their difference can well be negative. This can be demonstrated already for a simple model where a free particle is coupled to a single bath oscillator (IHT09). From the perspective of the heat bath, the coupling to the system degree of freedom causes a shift of the bath oscillators to higher frequencies. As a consequence, the density of bath oscillators at low frequencies tends to be suppressed (Ingold11). This suppression is at the origin of the negative specific heat.

A related phenomenon is reentrant classicality. With decreasing temperature a free particle behaves increasingly quantum mechanical. However, quite unexpectedly, at very low temperatures classical behavior is recovered (SIW13). This phenomenon occurs provided the heat bath is sufficiently superohmic. Then, the density of states of the bath is suppressed at low frequencies and the heat bath is no longer capable of rendering the free particle quantum mechanical at low temperatures.

Even though the free Brownian particle may appear to be somewhat academic, these results are also relevant in other contexts. It was shown in ILR09, that a similar scenario as for the free Brownian particle occurs also for the Casimir effect. For two parallel metallic mirrors with electrons described within the Drude model, one finds at low temperatures a negative entropy which initially decreases further as temperature is increased.

Semiclassical methods

Within this project, semiclassical methods have been employed in various respects. The coupling to the environment can induce transitions between energy eigenstates of the system, resulting in a finite level width of these states. As the width of the energy eigenstates of the damped harmonic oscillator is proportional to the quantum number of the state and, for large quantum numbers, to the energy of the state, IJR01 addressed the question whether the quantum number or the energy is the relevant quantity. For one-dimensional potentials which asympototically follow a power law and which may be bounded by a potential wall, a scaling law for the dependence of the level width on the quantum number was drived for large quantum numbers. This scaling behavior was also tested explicitly for a few special cases like the half oscillator, the infinitely deep potential well and the one-dimensional Coulomb problem.

An interesting system from the point of view of semiclassics is the undamped two-dimensional oscillator, which depending on the ratio of the two frequencies offers two different scenarios for the periodic orbits. For the isotropic harmonic oscillator all orbits are periodic while for incommensurate frequencies the only periodic orbits lie along the principal axes. In addition, for commensurate frequencies families of Lissajous trajectories appear. It was shown only in 1995 by M. Brack and S. R. Jain that a semiclassical treatment of the problem leads to the correct eigenenergies. However, they treated the commensurate case as a limiting case of the incommensurate case. In contrast, in DI07 the contribution of the families of periodic orbits was explicitly evaluated for the commensurate case. It was clarified how the different periodic orbits lead to the correct eigenenergies even in this case. Furthermore, this approach allows to analyze particularly well the symmetry properties of the different contributions to the density of states.

Nonclassical trajectories

Another case where semiclassical methods are of interest is the study of dissipative quantum system by means of the Wigner function in phase space. In order to contribute to a better understanding of a generally nonlinear problem, PID10 considered the exactly solvable damped harmonic oscillator in phase space. It could be demonstrated how nonclassical trajectories shown in the figure in red and blue determine the Wigner function. Furthermore, the broadening of the Wigner function with increasing time could be traced back to the nonlocal self interaction of the system coordinate induced by the environment.

Ultrasmall tunnel junctions

Ultrasmall tunnel junctions represent an example where the coupling to environmental degrees of freedom is relevant. The capacitance of these tunnel junctions is very small so that the charging energy associated with the tunneling particle – an electron or, in the case of superconducting junctions, a Cooper pair – becomes important. Then the electromagnetic environment of the tunnel junction plays a crucial role. For a superconducting tunnel junction, a so-called Josephson junction, the current-voltage characteristics depends even directly on the environmental impedance. A detailed introuction into this topic is given in IN92.

At the beginning of this project, ultrasmall Josephson junctions were of particular interest. Leaving the regime of extremely low temperature aside for the moment, one can exactly sum the perturbation series in the Josephson energy and thus obtain the current-voltage characteristics (GIP98). A summary of this work and of previously obtained results is given in GI99. Interestingly, it is also possible to obtain the current-voltage characteristics at zero temperature (IG99, IG00, IG00a). The transition from the Coulomb blockade regime, where the charging energy dominates and Cooper pairs tunnel incoherently, to the Josephson regime, where the tunneling of the Cooper pairs is coherent, can be analyzed particularly well by means of the noise properties. As was shown in GI02, the nature of the tunneling process can be determined by means of the Fano factor.