Quantum Transport through Ballistic Cavities: Soft vs Hard Quantum Chaos

Abstract

We study transport through a two-dimensional billiard attached to two infinite leads by numerically calculating the Landauer conductance and the Wigner time delay. In the generic case of a mixed phase space we find a power law distribution of resonance widths and a power law dependence of conductance increments apparently reflecting the classical dwell time exponent, in striking difference to the case of a fully chaotic phase space. Surprisingly, these power laws appear on energy scales below the mean level spacing, in contrast to semiclassical expectations.PACS numbers: 05.45.+b,05.60.Gg,72.20.Dp,73.23.Ad Advances in the fabrication of semiconductor heterostructures and metal films have made it possible to produce two dimensional nanostructures with a very low amount of disorder [1]. At low temperatures, scattering of the electrons happens mostly at edges of the structures with the electrons moving ballistically between collisions with the boundary. Theoretical and experimental investigations have shown that the spectral and transport properties of such quantum coherent cavities, commonly called "billiards", depend strongly on the nature of their classical dynamics. In particular, integrable and chaotic systems were found to behave quite differently [2,3].Generic billiards are neither integrable nor ergodic [4], but have a mixed phase space with regions of regular as well as chaotic dynamics [5]. Their dynamics is much richer than in either of the extreme cases, as phase space has a hierarchical structure at the boundary of regular and chaotic motion. In particular, this leads to a trapping of chaotic trajectories close to regular regions with a probability P (t) ∼ t −β for t > t 0 , to be trapped longer than a time t, with t 0 of the order of a few traversal times [6]. The exponent β > 1 depends on system and parameters with typically β ≈ 1.5 [6]. This power-law trapping in mixed systems is in contrast to the typical exponentially decaying staying probability of fully chaotic systems (see Fig. 1).Recently, it was shown semiclassically employing the diagonal approximation that the variance of conductance increments (for a small dc bias voltage) over small energy intervals ∆E grows as [7,8] for mixed systems if β < 2. This is in strong contrast to an increase as (∆E) 2 in the case of fully chaotic systems [3]. The semiclassical approximation requires ∆E to be larger than the mean level spacing ∆, corresponding to the picture that quantum mechanics can follow the classical power law trapping at most until the Heisenberg time. In the semiclassical approximation the graph of g vs. E has the statistical properties of fractional Brownian motion with a fractal dimension. Fractal conductance fluctuations have indeed been found in experiments on gold wires [10] and semiconductor nanostructures [11] and numerically for the quantum separatrix map [12]. In this Letter, we numerically study quantum transport through a simple cavity, the cosine billiard [13] (see insets of Fig. 1). Although we observe completely different...

New Class of Eigenstates in Generic Hamiltonian Systems

“…It was semiclassically derived that these fluctuations should have a fractal dimension D = 2 − γ/2 [7], which was confirmed in gold nanowires [8], semiconductor nanostructures [9], and numerics [10]. Quite recently, a second type of conductance fluctuations in mixed systems has been discovered numerically [11,12], namely isolated resonances. There the classical exponent γ seems to appear in the scaling of the variance of conductance increments, surprisingly, on scales below the mean level spacing, what is not understood so far.…”

Unitary-stochastic matrix ensembles and spectral statistics

“…Quantum networks have been used with great success to model quantum phenomena observed in disordered metals and mesoscopic systems (Shapiro 1982, Chalker andCoddington 1988); typical behaviour found in extended, diffusive systems such as localisation -delocalisation transitions (Freche et al 1999), multifractal properties of wavefunctions at the transition point Metzler 1995, Huckestein andKlesse 1999), transport properties (Pascaud and Montambaux 1999, Huckestein et al 2000) and the statistical properties of quantum spectra (Klesse and Metzler 1997) have been studied on graphs in the limit of infinite network size. Recently, Kottos andSmilansky (1997, 1999) proposed to study quantum spectra of non-diffusive graphs with only relatively few vertices or nodes.…”

Nonperiodic echoes from mushroom billiard hats

“…In particular the spectral behavior and properties of the wave functions of quantum systems, whose classical analogue is chaotic [18,19,20,21,22,23], integrable [24,25] or intermediate [26,27,28] can be investigated experimentally with such systems. In scattering systems similar matters have been discussed mainly for systems that are chaotic and hyperbolic [12,29], or mixed [30,31,32,33,34]. The role of parabolic manifolds has also received considerable attention [7,35].…”