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Register a new userDensity of proper delay times in chaotic and integrable quantum billiards
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Added by
Michael G. A. Crawford
Publication date
2001
fields
Physics
and research's language is
English
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We calculate the density P(tau) of the eigenvalues of the Wigner-Smith time delay matrix for two-dimensional rectangular and circular billiards with one opening. For long times, the density of these so-called proper delay times decays algebraically, in contradistinction to chaotic quantum billiards for which P(tau) exhibits a long-time cut-off.
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By an inductive reasoning, and based on recent results of the joint moments of proper delay times of open chaotic systems for ideal coupling to leads, we obtain a general expression for the distribution of the partial delay times for an arbitrary number of channels and any symmetry. This distribution was not completely known for all symmetry classes. Our theoretical distribution is verified by random matrix theory simulations of ballistic chaotic cavities.
This article reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor $T_{alpha beta}(x,y)$ for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as $psi = u+iv$. With the assumption that $u$ and $v$ are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components $T_{alpha beta}$. The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schroedinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogues. Hence we report on microwave measurements for an open 2D cavity and how the quantum stress tensor analogue is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for $T_{alpha beta}(x,y)$ is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.
In this work, we perform a statistical study on Dirac Billiards in the extreme quantum limit (a single open channel on the leads). Our numerical analysis uses a large ensemble of random matrices and demonstrates the preponderant role of dephasing mechanisms in such chaotic billiards. Physical implementations of these billiards range from quantum dots of graphene to topological insulators structures. We show, in particular, that the role of finite crossover fields between the universal symmetries quickly leaves the conductance to the asymptotic limit of unitary ensembles. Furthermore, we show that the dephasing mechanisms strikingly lead Dirac billiards from the extreme quantum regime to the semiclassical Gaussian regime.
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 $tau_E$ is observed but its derivative with respect to $tau_E$ 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.
Resonant systems emerge as weakly nonlinear approximations to problems with highly resonant linearized perturbations. Examples include nonlinear Schroedinger equations in harmonic potentials and nonlinear dynamics in Anti-de Sitter spacetime. The classical dynamics within this class of systems can be very rich, ranging from fully integrable to chaotic as one changes the values of the mode coupling coefficients. Here, we initiate a study of quantum infinite-dimensional resonant systems, which are mathematically a highly special case of two-body interaction Hamiltonians (extensively researched in condensed matter, nuclear and high-energy physics). Despite the complexity of the corresponding classical dynamics, the quantum version turns out to be remarkably simple: the Hamiltonian is block-diagonal in the Fock basis, with all blocks of varying finite sizes. Being solvable in terms of diagonalizing finite numerical matrices, these systems are thus arguably the simplest interacting quantum field theories known to man. We demonstrate how to perform the diagonalization in practice, and study both numerical patterns emerging for the integrable cases, and the spectral statistics, which efficiently distinguishes the special integrable cases from generic (chaotic) points in the parameter space. We discuss a range of potential applications in view of the computational simplicity and dynamical richness of quantum resonant systems.
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