Gert-Ludwig Ingold

Quantum Brownian motion: The functional integral approach

Abstract The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.

Charge Tunneling Rates in Ultrasmall Junctions

With the advances of microfabrication techniques in recent years it has become possible to fabricate tunnel junctions of increasingly smaller dimensions and thereby decreasing capacitance C. Nowadays one can study tunnel junctions in a regime where the charging energy E c = e2/2C is larger than the thermal energy k B T. Then charging effects play an important role and this has been the subject of a by now large body of both theoretical and experimental work.

Fundamental aspects of quantum Brownian motion

With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral approach. We discuss the consequences of the time-reversal symmetry for an open dissipative quantum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise.

Finite quantum dissipation: the challenge of obtaining specific heat

We consider a free particle coupled with finite strength to a bath and investigate the evaluation of its specific heat. A harmonic oscillator bath of Drude type with cutoff frequency !D is employed to model an ohmic friction force with dissipation strength . Two scenarios for obtaining specific heat are presented. The first one uses the measurement of the kinetic energy of the free particle, while the second one is based on the reduced partition function. Both descriptions yield results which are consistent with the Third Law of thermodynamics. Nevertheless, the two methods produce different results that disagree even in their leading quantum corrections at high temperatures. We also consider the regime where the cutoff frequency is smaller than the friction strength, i.e. !D 1.

Path Integrals and Their Application to Dissipative Quantum Systems

The coupling of a system to its environment is a recurrent subject in this collection of lecture notes. The consequences of such a coupling are threefold. First of all, energy may irreversibly be transferred from the system to the environment thereby giving rise to the phenomenon of dissipation. In addition, the fluctuating force exerted by the environment on the system causes fluctuations of the system degree of freedom which manifest itself for example as Brownian motion. While these two effects occur both for classical as well as quantum systems, there exists a third phenomenon which is specific to the quantum world. As a consequence of the entanglement between system and environmental degrees of freedom a coherent superposition of quantum states may be destroyed in a process referred to as decoherence. This effect is of major concern if one wants to implement a quantum computer. Therefore, decoherence is discussed in detail in Chap. 5.

Specific heat anomalies of open quantum systems

The evaluation of the specific heat of an open damped quantum system is a subtle issue. One possible route is based on the thermodynamic partition function which is the ratio of the partition functions of system plus bath and of the bath alone. For the free damped particle it has been shown, however, that the ensuing specific heat may become negative for appropriately chosen environments. Being an open system this quantity then naturally must be interpreted as the change in the specific heat obtained as the difference between the specific heat of the heat bath coupled to the system degrees of freedom and the specific heat of the bath alone. While this difference may become negative, the involved specific heats themselves are always positive; thus, the known thermodynamic stability criteria are perfectly guaranteed. For a damped quantum harmonic oscillator, instead of negative values, under appropriate conditions one can observe a dip in the difference of specific heats as a function of temperature. Stylized minimal models containing a single oscillator heat bath are employed to elucidate the occurrence of the anomalous temperature dependence of the corresponding specific heat values. Moreover, we comment on the consequences for the interpretation of the density of states based on the thermal partition function.

Incoherent charge transport through molecular wires: interplay of Coulomb interaction and wire population

The influence of Coulomb interaction on the electron transport through molecular wires is studied in the regime of incoherent tunneling. In the limit of strong Coulomb repulsion, the current for spinless electrons is determined. It is shown that the voltage profile along the wire crucially influences the dependence of the current on the wire length. Upon inclusion of the spin degree of freedom one finds a blocking effect which depends both on the interaction strength and on the population of the wire. For finite Coulomb interaction, the temperature dependence of the blocking is studied and it is shown that several regimes with different blocking strength may exist.

On the electrostatic potential profile in biased molecular wires

The potential profile across a biased molecular junction is calculated within the framework of a simple Thomas–Fermi-type screening model. In particular, the relationship between this profile and the lateral molecular cross section is examined. We find that a transition from a linear potential profile to a potential that drops mainly near the molecule-metal contacts occurs with increasing cross-section width, in agreement with numerical quantum calculations.

Surface plasmon in metallic nanoparticles: Renormalization effects due to electron-hole excitations

Received 15 May 2006; revised manuscript received 25 August 2006; published 26 October 2006 The electronic environment causes decoherence and dissipation of the collective surface plasmon excitation in metallic nanoparticles. We show that the coupling to the electronic environment influences the width and the position of the surface plasmon resonance. A redshift with respect to the classical Mie frequency appears in addition to the one caused by the spill out of the electronic density outside the nanoparticle. We characterize the spill-out effect by means of a semiclassical expansion and obtain its dependence on temperature and the size of the nanoparticle. We demonstrate that both, the spill-out and the environment-induced shift are necessary to explain the experimentally observed frequencies and confirm our findings by time-dependent local density approximation calculations of the resonance frequency. The size and temperature dependence of the environmental influence results in a qualitative agreement with pump-probe spectroscopic measurements of the differential light transmission.

Single electron tunneling rates in multijunction circuits

The rate of electron tunneling through normal metal tunnel junctions is calculated for the case of ultrasmall junction capacitances. The so-called Coulomb blockade of electron tunneling at low temperatures is shown to be strongly affected by the external electrical circuit. Under the common experimental condition of a low impedance environment the Coulomb blockade is suppressed for single tunnel junctions. However, a Coulomb gap structure emerges for junctions embedded in a high impedance environment. For a double junction setup a Coulomb blockade of tunneling arises even for low impedance environments due to the charge quantization on the metallic island between the junctions. An approach using circuit analysis is presented which allows to reduce the calculation of tunneling rates in multijunction circuits to those of a single junction in series with an effective capacitance. The range of validity of the socalled local rule and global rule rates is clarified. It is found that the tunneling rate tends towards the global rule rate as the number of junctions is increased. Some specific results are given for a one-dimensional array of tunnel junctions.