Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts
91
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider wave propagation in a complex structure coupled to a finite number $N$ of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, related to the energy (or frequency) derivative of the scattering matrix $mathcal{S}$. We develop a random matrix approach to study the statistical properties of the symmetrised Wigner-Smith time-delay matrix $mathcal{Q}_s=-mathrm{i}hbar,mathcal{S}^{-1/2}big(partial_varepsilonmathcal{S}big),mathcal{S}^{-1/2}$, and obtain the joint distribution of $mathcal{S}$ and $mathcal{Q}_s$ for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) $0<Tleq1$. We derive two representations of the distribution of $mathcal{Q}_s$ in terms of matrix integrals specified by the Dyson symmetry index $beta=1,2,4$ (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay $tau_mathrm{W}=(1/N),mathrm{tr}big{mathcal{Q}_sbig}$, which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution $mathscr{P}_{N,beta}(tau)$ of the Wigner time delay in the weak coupling limit $NTll1$ and identify three different asymptotic regimes.
rate research
Read More
We consider a multichannel wire with a disordered region of length $L$ and a reflecting boundary. The reflection of a wave of frequency $omega$ is described by the scattering matrix $mathcal{S}(omega)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix $mathcal{Q}=-mathrm{i}, mathcal{S}^daggerpartial_omegamathcal{S}$ is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, $mathcal{S}=mathrm{e}^{2mathrm{i}kL}mathcal{U}_Lmathcal{U}_R$ (with $mathcal{U}_L=mathcal{U}_R^mathrm{T}$ in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix: $widetilde{mathcal{Q}} =mathcal{U}_R,mathcal{Q},mathcal{U}_R^dagger = (2L/v),mathbf{1}_N -mathrm{i},mathcal{U}_L^daggerpartial_omegabig(mathcal{U}_Lmathcal{U}_Rbig),mathcal{U}_R^dagger$, where $k$ is the wave vector and $v$ the group velocity. We demonstrate that $widetilde{mathcal{Q}}$ can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, $Ltoinfty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for $mathcal{Q}$s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length $L$, the exponential functional representation is used to calculate the first moments $langlemathrm{tr}(mathcal{Q})rangle$, $langlemathrm{tr}(mathcal{Q}^2)rangle$ and $langlebig[mathrm{tr}(mathcal{Q})big]^2rangle$. Finally we derive a partial differential equation for the resolvent $g(z;L)=lim_{Ntoinfty}(1/N),mathrm{tr}big{big( z,mathbf{1}_N - N,mathcal{Q}big)^{-1}big}$ in the large $N$ limit.
The Painleve transcendents discovered at the turn of the XX century by pure mathematical reasoning, have later made their surprising appearance -- much in the way of Wigners miracle of appropriateness -- in various problems of theoretical physics. The notable examples include the two-dimensional Ising model, one-dimensional impenetrable Bose gas, corner and polynuclear growth models, one dimensional directed polymers, string theory, two dimensional quantum gravity, and spectral distributions of random matrices. In the present contribution, ideas of integrability are utilized to advocate emergence of an one-dimensional Toda Lattice and the fifth Painleve transcendent in the paradigmatic problem of conductance fluctuations in quantum chaotic cavities coupled to the external world via ballistic point contacts. Specifically, the cumulants of the Landauer conductance of a cavity with broken time-reversal symmetry are proven to be furnished by the coefficients of a Taylor-expanded Painleve V function. Further, the relevance of the fifth Painleve transcendent for a closely related problem of sample-to-sample fluctuations of the noise power is discussed. Finally, it is demonstrated that inclusion of tunneling effects inherent in realistic point contacts does not destroy the integrability: in this case, conductance fluctuations are shown to be governed by a two-dimensional Toda Lattice.
We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assumption that the Landauer-Butticker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of Random Matrix Theory (RMT). The starting points are the finite-n formulae that we recently discovered (Mezzadri and Simm, J. Math. Phys. 52 (2011), 103511). Our analysis includes all the symmetry classes beta=1,2,4; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer (J. Math. Phys. 37 (1996), 4986-5018) and Altland and Zirnbauer (Phys. Rev. B. 55 (1997), 1142-1161). Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. (J. Phys. A.: Math. Theor. 41 (2008), 365102) and Berkolaiko and Kuipers (J. Phys. A: Math. Theor. 43 (2010), 035101 and New J. Phys. 13 (2011), 063020). Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.
We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes beta=1,2,4 of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for beta=1,4, thus proving a number of conjectures of Khoruzhenko et al. (Phys. Rev. B, Vol. 80 (2009), 125301). We derive differential equations that characterize the cumulant generating functions for all beta=1,2,4. Furthermore, we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painleve III transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit n -> infinity. Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.
We propose a new version of Wigner-Weyl calculus for tight-binding lattice models. It allows to express various physical quantities through Weyl symbols of operators and Greens functions. In particular, Hall conductivity in the presence of varying and arbitrarily strong magnetic field is represented using the proposed formalism as a topological invariant.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
