Chaotic Quantum Dots Research Articles

Splitting of Andreev levels in a Josephson junction by spin-orbit coupling

We consider the effect of spin-orbit coupling on the energy levels of a single-channel Josephson junction below the superconducting gap. We investigate quantitatively the level splitting arising from the combined effect of spin-orbit coupling and the time-reversal symmetry breaking by the phase difference between the superconductors. Using the scattering matrix approach, we establish a simple connection between the quantum mechanical time delay matrix and the effective Hamiltonian for the level splitting. As an application, we calculate the distribution of level splittings for an ensemble of chaotic Josephson junctions. The distribution falls off as a power law for large splittings, unlike the exponentially decaying splitting distribution given by the Wigner surmise---which applies for normal chaotic quantum dots with spin-orbit coupling in the case that the time-reversal symmetry breaking is due to a magnetic field.

Scrambling of Hartree-Fock levels as a universal Brownian-motion process

We study scrambling of the Hartree-Fock single-particle levels and wave functions as electrons are added to an almost-isolated diffusive or chaotic quantum dot with electron-electron interactions. We use the generic framework of the induced two-body ensembles, where the randomness of the two-body interaction matrix elements is induced by the randomness of the eigenfunctions of the chaotic or diffusive single-particle Hamiltonian. After an appropriate scaling of the number of added electrons, the scrambling behaviors of both the Hartree-Fock levels and wave functions are each described by a universal function. These functions can be derived from a parametric random-matrix process of the Brownian-motion type. An exception to this universality occurs when an empty level gets filled, in which case scrambling is delayed by one electron. An explanation of these results is given in terms of a generalized Koopmans' approach.

Rectification and nonlinear transport in chaotic dots and rings

We investigate the nonlinear current-voltage characteristic of mesoscopic conductors and the current generated through rectification of an alternating external bias. To leading order in applied voltages both the nonlinear and the rectified current are quadratic. This current response can be described in terms of second order conductance coefficients and for a generic mesoscopic conductor they fluctuate randomly from sample to sample. Due to Coulomb interactions the symmetry of transport under magnetic field inversion is broken in a two-terminal setup. Therefore, we consider both the symmetric and antisymmetric nonlinear conductances separately. We treat interactions self-consistently taking into account nearby gates. The nonlinear current is determined by different combinations of second order conductances depending on the way external voltages are varied away from an equilibrium reference point (bias mode). We discuss the role of the bias mode and circuit asymmetry in recent experiments. In a photovoltaic experiment the alternating perturbations are rectified, and the fluctuations of the nonlinear conductance are shown to decrease with frequency. Their asymptotical behavior strongly depends on the bias mode and in general the antisymmetric conductance is suppressed stronger than the symmetric conductance. We next investigate nonlinear transport and rectification in chaotic rings. To this extent we develop a model which combines a chaotic quantum dot and a ballistic arm to enclose an Aharonov-Bohm flux. In the linear two-probe conductance the phase of the Aharonov-Bohm oscillation is pinned while in nonlinear transport phase rigidity is lost. We discuss the shape of the mesoscopic distribution of the phase and determine the phase fluctuations.

Mesoscopic Spin Hall Effect

We investigate the spin Hall effect in ballistic chaotic quantum dots with spin-orbit coupling. We show that a longitudinal charge current can generate a pure transverse spin current. While this transverse spin current is generically nonzero for a fixed sample, we show that when the spin-orbit coupling time is short compared to the mean dwell time inside the dot, it fluctuates universally from sample to sample or upon variation of the chemical potential with a vanishing average.

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.

Weak localization in arrays of metallic quantum dots: Combined scattering matrix formalism and nonlinearσmodel

Combining scattering-matrix formalism with nonlinear $\ensuremath{\sigma}$-model technique we analyze weak localization effects in arrays of chaotic quantum dots connected via barriers with arbitrary distribution of channel transmissions. With the aid of our approach we evaluate magnetoconductance of two arbitrarily connected quantum dots as well as of $N\ifmmode\times\else\texttimes\fi{}M$ arrays of identical quantum dots. Our results can be directly used for a quantitative decription of magnetoconductance measurements in structures containing metallic quantum dots.

Weak localization correction to the density of transmission eigenvalues in the presence of magnetic field and spin-orbit coupling for a chaotic quantum dot

We calculated the weak localization correction to the density of the transmission eigenvalues in the case of chaotic quantum dots in the framework of Random Matrix Theory including the parametric dependence on the magnetic field and spin-orbit coupling. The result is interpreted in terms of spin singlet and triplet Cooperon modes of conventional diagrammatic perturbation theory. As simple applications, we obtained the weak localization correction to the conductance, the shot noise power and the third cumulant of the distribution of the transmitted charge.

Degradation of electron-hole entanglement by spin-orbit coupling

Electron-hole pairs produced by tunneling in a degenerate electron gas lose their spin entanglement by spin-orbit coupling, which transforms the fully entangled Bell state into a partially entangled mixed density matrix of the electron and hole spins. We calculate the dependence of the entanglement (quantified by the concurrence) on the spin-orbit coupling time ${\ensuremath{\tau}}_{\mathrm{so}}$ and on the diffusion time (or dwell time) ${\ensuremath{\tau}}_{\text{dwell}}$ of electrons and holes in the conductors (with conductances $⪢{e}^{2}∕h$) at the two sides of the tunnel barrier (with conductance $⪡{e}^{2}∕h$). The entanglement disappears when the ratio ${\ensuremath{\tau}}_{\text{dwell}}∕{\ensuremath{\tau}}_{\mathrm{so}}$ exceeds a critical value of order unity. The results depend on the type of conductor (disordered wire or chaotic quantum dot), but they are independent of other microscopic parameters (number of channels, level spacing). Our analytical treatment relies on an ``isotropy approximation'' (no preferential basis in spin space), which allows us to express the concurrence entirely in terms of spin correlators. We test this approximation for the case of chaotic dynamics with a computer simulation (using the spin-kicked rotator) and find good agreement.

Photovoltaic Current Response of Mesoscopic Conductors to Quantized Cavity Modes

We extend the analysis of the effects of electromagnetic (EM) fields on mesoscopic conductors to include the effects of field quantization, motivated by recent experiments on circuit QED. We show that in general there is a photovoltaic (PV) current induced by quantized cavity modes at zero bias across the conductor. This current depends on the average photon occupation number and vanishes identically when it is equal to the average number of thermal electron-hole pairs. We analyze in detail the case of a chaotic quantum dot at temperature Te in contact with a thermal EM field at temperature Tf, calculating the rms size of the PV current as a function of the temperature difference, finding an effect approximately pA.

Spin-orbit interaction in quantum dots in the presence of exchange correlations: An approach based on a good-spin basis of the universal Hamiltonian

We discuss the problem of spin-orbit interaction in a two-dimensional chaotic or diffusive quantum dot in the presence of exchange correlations. Spin-orbit scattering breaks spin rotation invariance, and in the crossover regime between different symmetries of the spin-orbit coupling, the problem has no closed solution. A conventional choice of a many-particle basis in a numerical diagonalization is the set of Slater determinants built from the single-particle eigenstates of the one-body Hamiltonian (including the spin-orbit terms). We develop a different approach based on the use of a good-spin many-particle basis that is composed of the eigenstates of the universal Hamiltonian in the absence of spin-orbit scattering. We introduce a complete labeling of this good-spin basis and use angular momentum algebra to calculate in closed form the matrix elements of the spin-orbit interaction in this basis. Spin properties, such as the ground-state spin distribution and the spin excitation function, can be directly calculated in this basis. Our approach is not limited to the spin-orbit coupling problem but can be applied to any perturbation that is added to the universal Hamiltonian.

Entanglement production in chaotic quantum dots subject to spin-orbit coupling

We study numerically the production of orbital and spin entangled states in chaotic quantum dots for non-interacting electrons. The introduction of spin-orbit coupling permit us to identify signatures of time-reversal symmetry correlations in the entanglement production previously unnoticed, resembling weak-(anti)localization quantum corrections to the conductance. We find the entanglement to be strongly dependent on spin-orbit coupling, showing universal features for broken time-reversal and spin-rotation symmetries.

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.

Asymmetry of Nonlinear Transport and Electron Interactions in Quantum Dots

The symmetry properties of transport beyond the linear regime in chaotic quantum dots are investigated experimentally. A component of differential conductance that is antisymmetric in both applied source-drain bias V and magnetic field B, absent in linear transport, is found to exhibit mesoscopic fluctuations around a zero average. Typical values of this component allow a measurement of the electron interaction strength.

Excess conductance of a spin-filtering quantum dot

The conductance G of a pair of single-channel point contacts in series, one of which is a spin filter, increases from 1/2 to 2/3 x e^2/h with more and more spin-flip scattering. This excess conductance was observed in a quantum dot by Zumbuhl et al., and proposed as a measure for the spin relaxation time T_1. Here we present a quantum mechanical theory for the effect in a chaotic quantum dot (mean level spacing Delta, dephasing time tau_phi, charging energy e^2/C), in order to answer the question whether T_1 can be determined independently of tau_phi and C. We find that this is possible in a time-reversal-symmetry-breaking magnetic field, when the average conductance follows closely the formula <G>=(2e^2/h)(T_1+h/Delta)/(4T_1+3h/Delta).

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.

Transport in chaotic quantum dots: Effects of spatial symmetries which interchange the leads

We investigate the effect of spatial symmetries on phase coherent electronic transport through chaotic quantum dots. For systems which have a spatial symmetry that interchanges the source and drain leads, we find in the framework of random matrix theory that the density of the transmission eigenvalues is indepedent of the number of channels N in the leads. As a consequence, the weak localization correction to the conductance vanishes in these systems, and the shot noise suppression factor F is independent of N. We confirm this prediction by means of numerical calculations for stadium billiards with various lead geometries. These calculations also uncover transport signatures of partially preserved symmetries.

Voltage probe model of spin decay in a chaotic quantum dot with applications to spin-flip noise and entanglement production

The voltage probe model is a model of incoherent scattering in quantum transport. Here we use this model to study the effect of spin-flip scattering on electrical conduction through a quantum dot with chaotic dynamics. The spin decay rate gamma is quantified by the correlation of spin-up and spin-down current fluctuations (spin-flip noise). The resulting decoherence reduces the ability of the quantum dot to produce spin-entangled electron-hole pairs. For gamma greater than a critical value gamma_c, the entanglement production rate vanishes identically. The statistical distribution P(gamma_c) of the critical decay rate in an ensemble of chaotic quantum dots is calculated using the methods of random-matrix theory. For small gamma_c this distribution is proportional to gamma_c^(-1+beta/2), depending on the presence (beta=1) or absence (beta=2) of time-reversal symmetry. To make contact with experimental observables, we derive a one-to-one relationship between the entanglement production rate and the spin-resolved shot noise, under the assumption that the density matrix is isotropic in the spin degrees of freedom. Unlike the Bell inequality, this relationship holds for both pure and mixed states. In the tunneling regime, the electron-hole pairs are entangled if and only if the correlator of parallel spin currents is at least twice larger than the correlator of antiparallel spin currents.

Quantum-to-classical crossover for Andreev billiards in a magnetic field

We extend the existing quasiclassical theory for the superconducting proximity effect in a chaotic quantum dot, to include a time-reversal-symmetry breaking magnetic field. Random-matrix theory (RMT) breaks down once the Ehrenfest time $\tau_E$ becomes longer than the mean time $\tau_D$ between Andreev reflections. As a consequence, the critical field at which the excitation gap closes drops below the RMT prediction as $\tau_E/\tau_D$ is increased. Our quasiclassical results are supported by comparison with a fully quantum mechanical simulation of a stroboscopic model (the Andreev kicked rotator).

Ballistic chaotic quantum dots with interactions: A numerical study of the Robnik-Berry billiard

In previous work we have found a regime in ballistic quantum dots where interelectron interactions can be treated asymptotically exactly as the Thouless number $g$ of the dot becomes very large. However, this work depends on some assumptions concerning the renormalization group and various properties of the dot obeying Random Matrix Theory predictions at scales of the order of the Thouless energy. In this work we test the validity of those assumptions by considering a particular ballistic dot, the Robnik-Berry billiard, numerically. We find that almost all of our predictions based on the earlier work are borne out, with the exception of fluctuations of certain matrix elements of interaction operators. We conclude that, at least in the Robnik-Berry billiard, one can trust the results of our previous work at a qualitative and semi-quantitative level.

Microwave control of transport through a chaotic mesoscopic dot

We establish analogy between a microwave ionization of Rydberg atoms and a charge transport through a chaotic quantum dot induced by a monochromatic field in a regime with a potential barrier between dot contacts. We show that the quantum coherence leads to dynamical localization of electron excitation in number of photons absorbed inside the dot. The theory developed determines the dependence of localization length on dot and microwave parameters showing that the microwave power can switch the dot between metallic and insulating regimes.