arXiv subjects

Z. Kaufmann

Publications and source records attributed to Z. Kaufmann.

14 recordsLinked to original sources

arXiv2007

Magnetic focusing in normal-superconductor hybrid systems: a semiclassical analysis

We study a transverse electron-hole focusing effect in a normal-superconductor system. The spectrum of the quasiparticles is calculated both quantum mechanically and in semiclassical approximation, showing an excellent agreement. A semiclassical conductance formula is derived which takes into account the effect of electron-like as well as hole-like quasiparticles. At low magnetic field the semiclassical conductance shows characteristic oscillations due to the Andreev reflection, while at higher fields it goes to zero. These findigs are in line with the results of previous quantum calculations and with the expectations based on the classical dynamics of the quasiparticles.

cond-mat.mes-hall↗

arXiv2006

Quantized invariant tori in Andreev billiards of mixed phase space

Comparing the results of exact quantum calculations and those obtained from the EBK-like quantization scheme of Silvestrov et al [Phys. Rev. Lett. 90, 116801 (2003)] we show that the spectrum of Andreev billiards of mixed phase space can basically be decomposed into a regular and an irregular part, similarly to normal billiards. We provide the first numerical confirmation of the validity of this quantization scheme for individual eigenstates and discuss its accuracy.

cond-mat.mes-hall↗

arXiv2005

Quantum-classical correspondence in the wavefunctions of Andreev billiards

We present a classical and quantum mechanical study of an Andreev billiard with a chaotic normal dot. We demonstrate that in general the classical dynamics of these normal-superconductor hybrid systems is mixed, thereby indicating the limitations of a widely used retracing approximation. We show that the mixed classical dynamics gives rise to a wealth of wavefunction phenomena, including periodic orbit scarring and localization of the wavefunction onto other classical phase space objects such as intermittent regions and quantized tori.

cond-mat.mes-hall↗

arXiv2004

Andreev bound states for cake shape superconducting-normal systems

The energy spectrum of cake shape normal - superconducting systems is calculated by solving the Bogoliubov-de Gennes equation. We take into account the mismatch in the effective masses and Fermi energies of the normal and superconducting regions as well as the potential barrier at the interface. In the case of a perfect interface and without mismatch, the energy levels are treated by semi-classics. Analytical expressions for the density of states and its integral, the step function, are derived and compared with that obtained from exact numerics. We find a very good agreement between the two calculations. It is shown that the spectrum possesses an energy gap and the density of states is singular at the edge of the gap. The effect of the mismatch and the potential barrier on the gap is also investigated.

cond-mat.supr-con↗

arXiv2003

Effective Random Matrix Theory description of chaotic Andreev billiards

An effective random matrix theory description is developed for the universal gap fluctuations and the ensemble averaged density of states of chaotic Andreev billiards for finite Ehrenfest time. It yields a very good agreement with the numerical calculation for Sinai-Andreev billiards. A systematic linear decrease of the mean field gap with increasing Ehrenfest time $τ_E$ is observed but its derivative with respect to $τ_E$ is in between two competing theoretical predictions and close to that of the recent numerical calculations for Andreev map. The exponential tail of the density of states is interpreted semi-classically.

cond-mat.mes-hall↗

arXiv2003

A logarithmic contribution to the density of states of rectangular Andreev billiards

We demonstrate that the exact quantum mechanical calculations are in good agreement with the semiclassical predictions for rectangular Andreev billiards and therefore for a large number of open channels it is sufficient to investigate the Bohr-Sommerfeld approximation of the density of states. We present exact calculations of the classical path length distribution $P(s)$ which is a non-differentiable function of $s$, but whose integral is a smooth function with logarithmically dependent asymptotic behavior. Consequently, the density of states of rectangular Andreev billiards has two contributions on the scale of the Thouless energy: one which is well-known and it is proportional to the energy, and the other which shows a logarithmic energy dependence. It is shown that the prefactors of both contributions depend on the geometry of the billiards but they have universal limiting values when the width of the superconductor tends to zero.

cond-mat.mes-hall↗

arXiv2002

Proximity-induced sub-gaps in Andreev billiards

We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cut-off in the path length distribution $P(s)$ will possess an energy gap on the scale of the Thouless energy. An exact quantum mechanical calculation for different Andreev billiards gives good agreement with the semi-classical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semi-classical Bohr-Sommerfeld approximation of the density of states, we derive a simple formula for the energy gap. We show that the energy gap, in units of Thouless energy, may exceed the value predicted earlier from random matrix theory for chaotic billiards.

cond-mat.mes-hall↗

arXiv2001

Comparison of averages of flows and maps

It is shown that in transient chaos there is no direct relation between averages in a continuos time dynamical system (flow) and averages using the analogous discrete system defined by the corresponding Poincare map. In contrast to permanent chaos, results obtained from the Poincare map can even be qualitatively incorrect. The reason is that the return time between intersections on the Poincare surface becomes relevant. However, after introducing a true-time Poincare map, quantities known from the usual Poincare map, such as conditionally invariant measure and natural measure, can be generalized to this case. Escape rates and averages, e.g. Liapunov exponents and drifts can be determined correctly using these novel measures. Significant differences become evident when we compare with results obtained from the usual Poincare map.

nlin.CD↗

arXiv2000

Generalized multibaker maps for open dissipative systems

Generalized multibaker maps are introduced to model dissipative systems which are spatially extended only in certain directions and escape of particles is allowed in other ones. Effects of nonlinearity are investigated by varying a control parameter. Emphasis is put on the appearance of the critical state representing the borderline of transient chaos, where anomalous behavior sets in. The investigations extend to the conditionally invariant and the related natural measures and to transient diffusion in normal and critical states as well. Permanent chaos is also considered as a special case.

chao-dyn↗

arXiv1999

Critical states of transient chaos

One-dimensional maps exhibiting transient chaos and defined on two preimages of the unit interval [0,1] are investigated. It is shown that such maps have continuously many conditionally invariant measures $μ_σ$ scaling at the fixed point at x=0 as $x^σ$, but smooth elsewhere. Here $σ$ should be smaller than a critical value $σ_{c}$ that is related to the spectral properties of the Frobenius-Perron operator. The corresponding natural measures are proven to be entirely concentrated on the fixed point.

chao-dyn↗

arXiv1997

New approach to the correlation spectrum near intermittency: a quantum mechanical analogy

The correlation spectrum of fully developed one-dimensional mappings are studied near and at a weakly intermittent situation. Using a suitable infinite matrix representation, the eigenvalue equation of the Frobenius-Perron operator is approximately reduced to the radial Schro"dinger equation of the hydrogen atom. Corrections are calculated by quantum mechanical perturbation theory. Analytical expressions for the spectral properties and correlation functions are derived and checked numerically. Compared to our previous works, the accuracy of the present results is significantly higher owing to the controlled and systematic approximation scheme.

chao-dyn↗

arXiv1997

Relationships among coefficients in deterministic and stochastic transient diffusion

Systems are studied in which transport is possible due to large extension with open boundaries in certain directions but the particles responsible for transport can disappear from it by leaving it in other directions, by chemical reaction or by adsorption. The connection of the total escape rate, the rate of the disappearance and the diffusion constant is investigated. It leads to the observation that the diffusion coefficient defined by $< x^2>$ is in general different from the one present in the effective Fokker-Planck equation. The result makes it possible to generalize the Gaspard-Nicolis formula for deterministic systems to this transient case.

chao-dyn↗

arXiv1997

Diffusion in normal and critical transient chaos

In this paper we investigate deterministic diffusion in systems which are spatially extended in certain directions but are restricted in size and open in other directions, consequently particles can escape. We introduce besides the diffusion coefficient D on the chaotic repeller a coefficient ${\hat D}$ which measures the broadening of the distribution of trajectories during the transient chaotic motion. Both coefficients are explicitly computed for one-dimensional models, and they are found to be different in most cases. We show furthermore that a jump develops in both of the coefficients for most of the initial distributions when we approach the critical borderline where the escape rate equals the Liapunov exponent of a periodic orbit.

chao-dyn↗

arXiv1995

Eigenvalue spectrum of the Frobenius-Perron operator near intermittency

The spectral properties of the Frobenius-Perron operator of one-dimensional maps are studied when approaching a weakly intermittent situation. Numerical investigation of a particular family of maps shows that the spectrum becomes extremely dense and the eigenfunctions become concentrated in the vicinity of the intermittent fixed point. Analytical considerations generalize the results to a broader class of maps near and at weak intermittency and show that one branch of the map is dominant in determination of the spectrum. Explicit approximate expressions are derived for both the eigenvalues and the eigenfunctions and are compared with the numerical results.

chao-dyn↗