Universal Quantum Signatures of Chaos in Ballistic Transport

Abstract

The conductance of a ballistic quantum dot (having chaotic classical dynamics and being coupled by ballistic point contacts to two electron reservoirs) is computed on the single assumption that its scattering matrix is a member of Dyson's circular ensemble. General formulas are obtained for the mean and variance of transport properties in the orthogonal (β = 1), unitary (β = 2), and symplectic (β = 4) symmetry class. Applications include universal conductance fluctuations, weak localization, sub-Poissonian shot noise, and normal-metal-superconductor junctions. The complete distribution P (g) of the conductance g is computed for the case that the coupling to the reservoirs occurs via two quantum point contacts with a single transmitted channel. The result P (g) ∝ g −1+β/2 is qualitatively different in the three symmetry classes. The search for signatures of chaotic behavior in quantum mechanical systems [1] has recently been extended to semiconductor nanostructures known as "quantum dots" [2,3]. A quantum dot is essentially a mesoscopic electron billiard, consisting of a ballistic cavity connected by two small holes to two electron reservoirs. An electron which is injected through one of the holes will either return through the same hole, with probability R, or be transmitted through the other hole, with probability T . Classically, the uniform (ergodic) exploration of the boundaries yields T = R, if the two holes are of the same size and sufficiently small that direct transmission (without boundary reflections) can be ignored.For a closed quantum dot (without holes), it is well known that one of the quantum signatures of its classically chaotic character consists in the Wigner-Dyson distribution of the energy levels [4,5]. The WignerDyson distribution was originally derived by randommatrix theory (RMT), and is characterized by a repulsion of nearby levels which depends only on the symmetry of the Hamiltonian. The quantum analog of the ergodic exploration of the dot boundaries by the classical trajectories consists in the Porter-Thomas distribution of the eigenfunctions, as confirmed by numerical studies of their amplitude distribution at the boundaries [6]. The quantum dot with holes is an open, rather than a closed system. Just as the Wigner-Dyson distribution describes the Hamiltonian H of the closed system, Dyson's circular ensemble [7] provides the statistical properties of the scattering matrix S of the open system. To have spectral or scattering properties given by the universal RMT description for H or S can be actually regarded as a precise definition of the somewhat vague concept of "quantum chaos". To what extent a real ballistic cavity is close to this precise universal limit is the subject of the theory of quantum billiards [5,8].Assuming this definition of a quantum chaotic system -and this is our only assumption -we will calculate the statistics of the transmission and reflection eigenvalues of the quantum dot, and hence its transport properties. This allows us to determine the universal quantum si...

Shot noise in mesoscopic conductors

“…12a). The description of transport properties of open chaotic cavities based on the random matrix theory was proposed independently by Baranger and Mello [96] and Jalabert, Pichard, and Beenakker [97]. They assumed that the scattering matrix of the chaotic cavity is a member of 24 In the case when the cavity is open, i.e.…”

Conductance distribution of a quantum dot with nonideal single-channel leads

“…Recently, it was found that a "quantum dot" has a qualitatively different conductance distribution. [2][3][4] A quantum dot is a small confined region, having a large level spacing compared to the thermal energy, which is weakly coupled by point contacts to two electron reservoirs. The classical motion within the dot is assumed to be ballistic and chaotic.…”

“…(1) was found to be in good agreement with numerical simulations of transmission through a chaotic billiard connected to ideal leads having a single propagating mode. 4 (The case β = 4 was not considered in Ref. 4.)…”

A variance formula related to a quantum conductance problem

“…Relating S to an evolution operator or a Hamiltonian, one may think that S * = S. Assuming that S is a random matrix from the Circular Orthogonal Ensemble (COE, β = 1), the Circular Unitary Ensemble (CUE, β = 2) or the Circular Symplectic Ensemble (CSE, β = 4), it is shown in [2,4,18,24,34] that…”

“…It is worthwhile to mention the difference and a connection between our model behind formula (1.6) and the traditional models behind formula (1.4): our model t is based on a truncation part of an Haar-invariant matrix from O(n)(β = 1), U (n)(β = 2) or Sp(n)(β = 4). The t in the traditional model (see, e.g., [2,4,5,18,24,34]) comes from a block of the Circular Orthogonal Ensemble (β = 1), the Circular Unitary Ensemble (β = 2) or the Circular Symplectic Ensemble (β = 4). However, for β = 2, the Haar-invariant matrix from U (n) and the n × n Circular Unitary Ensemble have the same probability distribution, see [28].…”