Classical breathers and quantum coherent states in discrete NLS systems

We present a comparison of quantum and “semiclassical” trajectories of coherent states that correspond to classical breather solutions of finite discrete nonlinear Schrödinger (DNLS) lattices. The main goal is to explain earlier numerical observations of recurrent return to the vicinity of initial coherent states corresponding to stable breathers that are also spatially localized. This effect can be considered as a quantum manifestation of classical spatial localization. We show that these phenomena are encoded in a simple expression for the distance between the quantum and semiclassical states that involves the basic frequencies of the classical and quantum systems, as well as the breather amplitude and quantum spectral decomposition of the system. A corollary is that recurrence phenomena are robust under perturbation of the initial conditions for stable breathers.     

Semiclassical theory of integrable and rough Andreev billiards

We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding light onto the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magnetic fields on the proximity effect in rough Andreev billiards is discussed and an analytical formula is derived. The semiclassical theory provides an interpretation for the suppression of the proximity effect in the presence of magnetic fields as a coherence effect of time reversed trajectories. It is shown to be in good agreement with quantum mechanical calculations. Received 21 August 1999 and Received in final form 21 March 2001     

Beyond Quantum Mechanics: Insights from the Work of Martin Gutzwiller

A complete quantum solution provides all possible knowledge of a system, whereas semiclassical theory provides at best approximate solutions in a limited region. Nevertheless, semiclassical methods based on the work of Martin Gutzwiller can provide stunning physical insights in regimes where quantum solutions are opaque. Furthermore, they can provide a unique bridge between the quantum and classical worlds. We illustrate these ideas with an account of a theoretical and experimental attack on the paradigm problem of the hydrogen atom in strong magnetic and electric fields.     

Quantum corrections to the conductivity and periodic orbits: Integrable systems

We extend the recently developed semiclassical theory for the conductivity to periodic semiconductor structures whose classical dynamics is integrable. We find that the conductivity of integrable systems exhibits quantum oscillations as function of magnetic field and Fermi energy which are closely related to both the Shubnikov-de Haas oscillations and the oscillations observed in the conductivity of antidot lattices. A general expression for the quantum oscillations is derived which is analogous to the Berry-Tabor formula for the spectral density of integrable systems.     

Analytical results for the electronic shell structure in mesoscopic systems: Balian-Bloch type theory for spheres,discs, rings and anti-dot lattices

The electronic shell structure resulting from the interference of closed orbital paths is determined for mesoscopic systems like spherical clusters, discs and rings by extending the semiclassical theory of Balian and Bloch. Analytical results for the shell structure in the density of states are obtained. Thus, the dependence of the shell structure on dimension, size and geometry and potential of the mesoscopic system and on an external magnetic field can be studied systematically. Comparison of the semiclassical results and those of quantum mechanical calculations permits analysis of typical quantum mechanical effects and shows the validity of the semiclassical theory. Our results should stimulate new experiments, can be used to calculate oscillations in the binding-energy, ionization-potential, and can be applied to analyze oscillations in the electronic density of states of quantum dot systems like anti-dot lattices.     

Mesoscopic one-way channels for quantum state transfer via the quantum Hall effect

We show that the one-way channel formalism of quantum optics has a physical realization in electronic systems. In particular, we show that magnetic edge states form unidirectional quantum channels capable of coherently transporting electronic quantum information. Using the equivalence between one-way photonic channels and magnetic edge states, we adapt a proposal for quantum state transfer to mesoscopic systems using edge states as a quantum channel, and show that it is feasible with reasonable experimental parameters. We discuss how this protocol may be used to transfer information encoded in number, charge, or spin states of quantum dots, so it may prove useful for transferring quantum information between parts of a solid-state quantum computer.     

Eigenstates in quantum dots confined by an inhomogeneous magnetic field

We study the eigenstates in quantum dots in which electrons are confined by the application of an inhomogeneous perpendicular magnetic field, focusing on the effect that the specific details of the shape of confining field has on determining these states. In contrast to the edge state picture established in studies on circular dots, we find that dots with more irregular geometries show a more complicated behavior in the interior of the dot. In particular, we find that certain states show indications of having their amplitude enhanced along particular classical periodic orbits in the interior, a phenomenon known as ‘scarring’.     

Nonlinear quantum shock waves in fractional quantum Hall edge states

Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.     

Quantum-chaotic scattering effects in semiconductor microstructures

We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations-a sensitivity of the conductance to either Fermi energy or magnetic field-and weak-localization-a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak-localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak-localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal-approximation theory in describing the magnitude of these quantum transport effects.     

Semiclassical approach to orbital magnetism of interacting diffusive quantum systems

We study interaction effects on the orbital magnetism of diffusive mesoscopic quantum systems. By combining many-body perturbation theory with semiclassical techniques, we show that the interaction contribution to the ensemble-averaged quantum thermodynamic potential can be reduced to an essentially classical operator. We compute the magnetic response of disordered rings and dots for diffusive classical dynamics. Our semiclassical approach reproduces the results of previous diagrammatic quantum calculations.     

Quantum to classical correspondence for the Fermi-acceleration model

A quantum particle which is confined to the interior of a box with infinitely high but periodically oscillating walls can have an unusual semiclassical limit: For the special case of a one-dimensional linear wall motion we show that the semiclassical domain corresponds to a classical motion in phase space where the initial momentum depends on the particle's position in the box. Another result is that quantum states which correspond to classical cycle-1 fixed points have maximum stability against the boundary induced perturbation (caused by the moving wall). Higher cycle-n fixed points are calculated by numerical bookkeeping up to n = 20. The classical motion is marginally stable. We show how a slight change in the boundary condition will lead to chaotic motion.     

Semiclassical theory for two-anyon system

The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm type interactions between the anyons are dealt with by the modified WKB method of Friedrich and Trost. For s-wave bound state problems in which the choice of the boundary condition at short distance gives rise to an additional ambiguity, a suitable generalization of the latter method is required to develop a consistent WKB approach. We here show how the related self-adjoint extension parameter affects the semiclassical quantization condition for energy levels. For some simple cases admitting exact answers, we verify that our semiclassical formulas in fact provide highly accurate results over a broad quantum number range.     

Duality in Two Capacitively Coupled Layered Arrays of Ultrasmall Josephson Junctions

We consider the problem of two capacitively coupled Josephson junction arrays made of ultrasmall junctions. Each one of the arrays can be in the semiclassical or quantum regimes, depending on its physical parameter values. The former case is dominated by a Cooper-pair superfluid, while the quantum case is dominated by dynamic vortices leading to an insulating behavior. We first consider the limit when both arrays are in the semiclassical limit, and next the case when one array is quantum and the other semiclassical. We present WKB and mean-field theory results for the critical temperature of each array when both are in the semiclassical limit. When one array is in the semiclassical regime and the other one is in the quantum-fluctuations-dominated regime, we derive a duality transformation between the charge and vortex-dominated arrays that involves a gauge vector field which is proportional to the site coupling capacitance between the arrays. The system considered here has been fabricated and we make some predictions as to possible experimentally measurable quantities that could be compared with theory.     

Black holes and beyond

The black hole information paradox forces us into a strange situation: we must find a way to break the semiclassical approximation in a domain where no quantum gravity effects would normally be expected. Traditional quantizations of gravity do not exhibit any such breakdown, and this forces us into a difficult corner: either we must give up quantum mechanics or we must accept the existence of troublesome ‘remnants’. In string theory, however, the fundamental quanta are extended objects, and it turns out that the bound states of such objects acquire a size that grows with the number of quanta in the bound state. The interior of the black hole gets completely altered to a ‘fuzzball’ structure, and information is able to escape in radiation from the hole. The semiclassical approximation can break at macroscopic scales due to the large entropy of the hole: the measure in the path integral competes with the classical action, instead of giving a subleading correction. Putting this picture of black hole microstates together with ideas about entangled states leads to a natural set of conjectures on many long-standing questions in gravity: the significance of Rindler and de Sitter entropies, the notion of black hole complementarity, and the fate of an observer falling into a black hole.     

二维椭圆量子台球中的谱分析

研究了二维椭圆台球中的量子谱和经典轨道之间的对应关系.为尝试求解没有解析波函数和本征能量又不能分离变量的体系，采用了定态展开方法(expansion method for stationary states，简称EMSS)得到尽可能精确的数值解，这是闭合轨道理论被推广到计算开轨道的情况.比较了傅里叶变换谱和经典轨道，发现量子谱的峰位置与经典轨道的长度在可分辨的范围内符合得很好，这是半经典理论为经典与量子力学的联系提供桥梁作用的又一个例子. 关键词： 椭圆量子台球 定态展开方法 闭合轨道理论 量子谱     

Semiclassical theory of spin-orbit interactions using spin coherent states

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller's trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome.