Chaotic Cavities Research Articles

Voltage and dephasing probes in mesoscopic conductors: A study of full-counting statistics

Voltage and dephasing probes introduce incoherent inelastic and incoherent quasi-elastic scattering into a coherent mesoscopic conductor. We discuss in detail the concepts of voltage and dephasing probes and develop a full counting statistics approach to investigate their effect on the transport statistics. The formalism is applied to several experimentally relevant examples. A comparison of different probe models and with procedures like phase averaging over an appropriate phase distribution shows that there is a perfect equivalence between the models for the case of one single-channel probe. Interestingly, the appropriate phase distribution function is found to be uniform. A uniform distribution is provided by a chaotic cavity with a long dwell time. The dwell time of a chaotic cavity plays a role similar to the charge response time of a voltage or dephasing probe. For multi-channel or multiple probes the transport statistics of voltage and dephasing probes differs and the equivalence with phase averaging is similarly lost.

Effect of symmetry class transitions on the shot noise in chaotic quantum dots

Using the random matrix theory (RMT) approach, we calculated the weak localization correction to the shot noise power in a chaotic cavity as a function of magnetic field and spin-orbit coupling. We found a remarkably simple relation between the weak localization correction to the conductance and to the shot noise power, that depends only on the channel number asymmetry of the cavity. In the special case of an orthogonal-unitary crossover, our result coincides with the prediction of Braun et. al [J. Phys. A: Math. Gen. {\bf 39}, L159-L165 (2006)], illustrating the equivalence of the semiclassical method to RMT.

Semiclassical approach to chaotic quantum transport

We describe a semiclassical method to calculate universal transport properties of chaotic cavities. While the energy-averaged conductance turns out governed by pairs of entrance-to-exit trajectories, the conductance variance, shot noise and other related quantities require trajectory quadruplets; simple diagrammatic rules allow us to find the contributions of these pairs and quadruplets. Both pure symmetry classes and the crossover due to an external magnetic field are considered.

Spectral form factor near the Ehrenfest time

We calculate the Ehrenfest-time dependence of the leading quantum correction to the spectral form factor of a ballistic chaotic cavity using periodic orbit theory. For the case of broken time-reversal symmetry, when the quantum correction to the form factor involves two small-angle encounters of classical trajectories, our result differs from that previously obtained using field-theoretic methods [Tian and Larkin, Phys. Rev. B 70, 035305 (2004)]. While we believe that the existing field-theoretic calculation is technically flawed, the question whether the field theoretic and periodic-orbit approaches agree when more than one small-angle encounter of classical orbits is involved remains unanswered.

Universal properties of two-port scattering, impedance, and admittance matrices of wave-chaotic systems

Statistical fluctuations in the eigenvalues of the scattering, impedance, and admittance matrices of two-port wave-chaotic systems are studied experimentally using a chaotic microwave cavity. These fluctuations are universal in that their properties are dependent only upon the degree of loss in the cavity. We remove the direct processes introduced by the nonideally coupled driving ports through a matrix normalization process that involves the radiation-impedance matrix of the two driving ports. We find good agreement between the experimentally obtained marginal probability density functions (PDFs) of the eigenvalues of the normalized impedance, admittance, and scattering matrix and those from random matrix theory (RMT). We also experimentally study the evolution of the joint PDF of the eigenphases of the normalized scattering matrix as a function of loss. Experimental agreement with the theory by Brouwer and Beenakker for the joint PDF of the magnitude of the eigenvalues of the normalized scattering matrix is also shown.

Towards a semiclassical justification of the effective random matrix theory for transport through ballistic chaotic quantum dots

The scattering matrix $S$ of a ballistic chaotic cavity is the direct sum of a classical and a quantum part, which describe the scattering of channels with typical dwell time smaller and larger than the Ehrenfest time, respectively. According to the effective random matrix theory of Silvestrov, Goorden, and Beenakker [Phys. Rev. Lett. 90, 116801 (2003)], statistical averages involving the quantum-mechanical scattering matrix are given by random matrix theory. While this effective random matrix theory is known not to be applicable for quantum interference corrections to transport, which appear to subleading order in the number of scattering channels $N$, it is believed to correctly describe quantum transport to leading order in $N$. We here partially verify this belief, by comparing the predictions of the effective random matrix theory for the ensemble averages of polynomial functions of $S$ and ${S}^{\ifmmode\dagger\else\textdagger\fi{}}$ of degree 2, 4, and 6 to a semiclassical calculation.

Spectroscopic properties of large open quantum-chaotic cavities with and without separated time scales

The spectroscopic properties of an open large Bunimovich cavity are studied numerically in the framework of the effective Hamiltonian formalism. The cavity is opened by attaching two leads to it in four different ways. In some cases, the transmission takes place via standing waves with an intensity that closely follows the profile of the resonances. In other cases, short-lived and long-lived resonance states coexist. The short-lived states cause traveling waves in the transmission while the long-lived ones generate superposed fluctuations. The traveling waves oscillate as a function of energy. They are not localized in the interior of the large chaotic cavity. In all considered cases, the phase rigidity fluctuates with energy. It is mostly near to its maximum value and agrees well with the theoretical value for the two-channel case.

Shot Noise in Semiclassical Chaotic Cavities

We construct a trajectory-based semiclassical theory of shot noise in clean chaotic cavities. In the universal regime of vanishing Ehrenfest time tau(E), we reproduce the random matrix theory result and show that the Fano factor is exponentially suppressed as tau(E) increases. We demonstrate how our theory preserves the unitarity of the scattering matrix even in the regime of finite tau(E). We discuss the range of validity of our semiclassical approach and point out subtleties relevant to the recent semiclassical treatment of shot noise in the universal regime by Braun et al. (cond-mat/0511292).

Characterization of fluctuations of impedance and scattering matrices in wave chaotic scattering

In wave chaotic scattering, statistical fluctuations of the scattering matrix S and the impedance matrix Z depend both on universal properties and on nonuniversal details of how the scatterer is coupled to external channels. This paper considers the impedance and scattering variance ratios, Xi(z) and Xi(s), where Xi(z) = Var[Z(ij)]/{Var[Z(ii)]Var[Z(jj)]}1/2, Xi(s) = Var[S(ij)]/{Var[S(ii)]Var[S(jj)]}1/2, and Var[.] denotes variance. Xi(z) is shown to be a universal function of distributed losses within the scatterer. That is, Xi(z) is independent of nonuniversal coupling details. This contrasts with s for which universality applies only in the large loss limit. Explicit results are given for Xi(z) for time reversal symmetric and broken time reversal symmetric systems. Experimental tests of the theory are presented using data taken from scattering measurements on a chaotic microwave cavity.

Mesoscopic Fluctuations of Nonlinear Conductance of Chaotic Quantum Dots

The nonlinear dc conductance of a two-terminal chaotic cavity is investigated. The fluctuations of the conductance (anti)symmetric with respect to magnetic-flux inversion through multichannel cavities are found analytically for arbitrary temperature, magnetic field, and interaction strength. For few-channel dots the effect of dephasing is investigated numerically. A comparison with recent experimental data is provided.

Statistical study of the conductance and shot noise in open quantum-chaotic cavities: Contribution from whispering gallery modes

In the past, a maximum-entropy model was introduced and applied to the study of statistical scattering by chaotic cavities, when short paths may play an important role in the scattering process. In particular, the validity of the model was investigated in relation with the statistical properties of the conductance in open chaotic cavities. In this article we investigate further the validity of the maximum-entropy model, by comparing the theoretical predictions with the results of computer simulations, in which the Schroedinger equation is solved numerically inside the cavity for one and two open channels in the leads; we analyze, in addition to the conductance, the zero-frequency limit of the shot-noise power spectrum. We also obtain theoretical results for the ensemble average of this last quantity, for the orthogonal and unitary cases of the circular ensemble and an arbitrary number of channels. Generally speaking, the agreement between theory and numerics is good. In some of the cavities that we study, short paths consist of whispering gallery modes, which were excluded in previous studies. These cavities turn out to be all the more interesting, as it is in relation with them that we found certain systematic discrepancies in the comparison with theory. We give evidence that it is the lack of stationarity inside the energy interval that is analyzed, and hence the lack of ergodicity that gives rise to the discrepancies. Indeed, the agreement between theory and numerical simulations is improved when the energy interval is reduced to a point and the statistics is then collected over an ensemble. It thus appears that the maximum-entropy model is valid beyond the domain where it was originally derived. An understanding of this situation is still lacking at the present moment.

Semiclassical prediction for shot noise in chaotic cavities

We show that in clean chaotic cavities the power of shot noise takes a universal form. Our predictions go beyond previous results from random-matrix theory, in covering the experimentally relevant case of few channels. Following a semiclassical approach we evaluate the contributions of quadruplets of classical trajectories to shot noise. Our approach can be extended to a variety of transport phenomena as illustrated for the crossover between symmetry classes in the presence of a weak magnetic field.

Shot noise in chaotic cavities with an arbitrary number of open channels

Using the random matrix approach, we calculate analytically the average shot-noise power in a chaotic cavity at an arbitrary number of propagating modes (channels) in each of the two attached leads. A simple relationship between this quantity, the average conductance and the conductance variance is found. The dependence of the Fano factor on the channel number is considered in detail.

Finite Frequency Quantum Noise in an Interacting Mesoscopic Conductor

We present a quantum calculation based on scattering theory of the frequency-dependent noise of current in an interacting chaotic cavity. We include interactions of the electron system via long range Coulomb forces between the conductor and a gate with capacitance C. We obtain explicit results exhibiting the two time scales of the problem, the cavity's dwell time tau(D) and the RC time tau(C) of the cavity in relation to the gate. The noise shows peculiarities at frequencies of the order and exceeding the inverse charge relaxation time tau(-1) = tau(D)(-1) + tau(C)(-1).

Aspects of the Scattering and Impedance Properties of Chaotic Microwave Cavities

We consider the statistics of the impedance Z of a chaotic microwave cavity coupled to a single port. We remove the non-universal effects of the coupling from the experimental Z data using the radiation impedance obtained directly from the experiments. We thus obtain the normalized impedance whose probability density function is predicted to be universal in that it depends only on the loss (quality factor) of the cavity. We find that impedance fluctuations decrease with increasing loss. The results apply to scattering measurements on any wave chaotic system.

Charge fluctuations in open chaotic cavities

We present a discussion of the charge response and the charge fluctuations of mesoscopic chaotic cavities in terms of a generalized Wigner-Smith matrix. The Wigner-Smith matrix is well known in investigations of time-delay of quantum scattering. It is expressed in terms of the scattering matrix and its derivatives with energy. We consider a similar matrix but instead of an energy derivative we investigate the derivative with regard to the electric potential. The resulting matrix is then the operator of charge. If this charge operator is combined with a self-consistent treatment of Coulomb interaction, the charge operator determines the capacitance of the system, the non-dissipative ac-linear response, the RC-time with a novel charge relaxation resistance, and in the presence of transport a resistance that governs the displacement currents induced into a nearby conductor. In particular these capacitances and resistances determine the relaxation rate and dephasing rate of a nearby qubit (a double quantum dot). We discuss the role of screening of mesoscopic chaotic detectors. Coulomb interaction effects in quantum pumping and in photon assisted electron-hole shot noise are treated similarly. For the latter we present novel results for chaotic cavities with non-ideal leads.

Interpolation formula for the reflection coefficient distribution of absorbing chaotic cavities in the presence of time reversal symmetry

We propose an interpolation formula for the distribution of the reflection coefficient in the presence of time reversal symmetry for chaotic cavities with absorption. This is done assuming a similar functional form as that when time reversal invariance is absent. The interpolation formula reduces to the analytical expressions for the strong and weak absorption limits. Our proposal is compared to the quite complicated exact result existing in the literature.

Chaos, coherence, and the double-slit experiment

We investigate the influence that classical dynamics has on interference patterns in coherence experiments. We calculate the time-integrated probability current through an absorbing screen and the conductance through a doubly connected ballistic cavity, both in an Aharonov-Bohm geometry with forward scattering only. We show how interference fringes in the probability current generically disappear in the case of a chaotic system with small openings, and how they may persist in the case of an integrable cavity. Simultaneously, the typical, sample dependent amplitude of the flux-sensitive g(phi) part of the conductance survives in all cases, and becomes universal in the case of a chaotic cavity. In the presence of dephasing by fluctuations of the electric potential in one arm of the Aharonov-Bohm loop, we find an exponential damping of the flux-dependent part of the conductance, g(phi) proportional exp[-tauL/tauphi, in term of the traversal time tauL through the arm and the dephasing time tauphi. This extends previous works on dephasing in ballistic systems to the case of many conducting channels.

Full counting statistics of Andreev scattering in an asymmetric chaotic cavity

We study the charge transport statistics in coherent two-terminal double junctions within the framework of the circuit theory of mesoscopic transport. We obtain the general solution of the circuit-theory matrix equations for the Green's function of a chaotic cavity between arbitrary contacts. As an example we discuss the full counting statistics and the first three cumulants for an open asymmetric cavity between a superconductor and a normal-metal lead at temperatures and voltages below the superconducting gap. The third cumulant shows a characteristic sign change as a function of the asymmetry of the two quantum point contacts, which is related to the properties of the Andreev reflection eigenvalue distribution.

Semiclassical theory of weak antilocalization and spin relaxation in ballistic quantum dots

We develop a semiclassical theory for spin-dependent quantum transport in ballistic quantum dots. The theory is based on the semiclassical Landauer formula, that we generalize to include spin-orbit and Zeeman interaction. Within this approach, the orbital degrees of freedom are treated semiclassically, while the spin dynamics is computed quantum mechanically. Employing this method, we calculate the quantum correction to the conductance in quantum dots with Rashba and Dresselhaus spin-orbit interaction. We find a strong sensitivity of the quantum correction to the underlying classical dynamics of the system. In particular, a suppression of weak antilocalization in integrable systems is observed. These results are attributed to the qualitatively different types of spin relaxation in integrable and chaotic quantum cavities.