Geometry, statistics, and asymptotics of quantum pumps

Abstract

We give a pedestrian derivation of a formula of Büttiker et. al. (BPT) relating the adiabatically pumped current to the S matrix and its (time) derivatives. We relate the charge in BPT to Berry's phase and the corresponding Brouwer pumping formula to curvature. As applications we derive explicit formulas for the joint probability density of pumping and conductance when the S matrix is uniformly distributed; and derive a new formula that describes hard pumping when the S matrix is periodic in the driving parameters.PACS numbers: 72.10. Bg, Brouwer [3], and Aleiner et. al. [4], building on results of Büttiker, Pretre and Thomas (BPT) [5], pointed out that adiabatic scattering theory leads to a geometric description of charge transport in mesoscopic quantum pumps. Some of these works, and certainly our own work, was motivated by experimental results of Switkes et. al.[6] on such pumps.In this article we examine the formula of BPT [5], which relates adiabatic charge transport to the S matrix and its (time) derivatives, in the special case of singlechannel scattering. We show that the formula admits a simple interpretation in terms of three basic processes at the Fermi energy. Two of these are dissipative and nonquantized. The third integrates to zero for any cyclic variation in the system.Next, we describe the geometric significance of BPT and relate it to Berry's phase [2]. It follows that the pumping formula of Brouwer [3] can be interpreted as curvature and is formally identical to the adiabatic curvature [2]. In spite of the interesting geometry the topological aspects of pumping are trivial. In particular, we prove that all Chern numbers associated to the Brouwer formula are identically zero.We proceed with two applications. First we give an elementary and explicit derivation of the joint probability density for pumping and conductance. This problem was studied in [3]. Brouwer's results go beyond ours as he also calculates the tails of the distributions and we don't. On the other hand, parts of his results are numerical, and they are certainly not elementary. Finally, we calculate, for the first time, the asymptotics of hard pumping for S matrices that depend periodically on two parameters. If the system traverses a circle of radius R in parameter space, with R large, then the amount of charge transported is order √ R, multiplied by a quasi-periodic (oscillatory) function of R leading to ergodic behavior.We shall use units where e = m =h = 1, so the electron charge is −1 and the quantum of conductance is e 2 /h = 1 2π . The mutual Coulombic interaction of the electrons is disregarded.The BPT formula: Consider a scatterer connected to leads that terminate at electron reservoirs. All the reservoirs are initially at the same chemical potential and at zero temperature. The scatterer is described by its (on-shell) S matrix, which, in the case of n channels is an n × n matrix parameterized by the energy and other parameters associated with the adiabatic driving of the system (e.g. gate voltages and magnetic fields).T...

Transport and Dissipation in Quantum Pumps

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Nonadiabatic charge pumping by oscillating potentials in one dimension: Results for infinite system and finite ring

“…If oscillating potentials are applied to K sites, and their effects add up constructively, the reflection and transmission amplitudes in the first side band would be of the order of K times what one would get if there was pumping at only one site; this is evident from Eqs. (9). The pumped current would therefore be magnified by a factor of K 2 .…”

“…For the case of non-interacting electrons, theoretical studies of this phenomenon have used adiabatic scattering theory [9][10][11][12][13][14][15][16][17], Floquet scattering theory [21][22][23], the nonequilibrium Green function formalism [24][25][26][27], and the equation of motion approach [36,37]. The case of interacting electrons has been studied using a renormalization group method for weak interactions [50], and the method of bosonization for arbitrary interactions [51][52][53][54][55][56][57][58][59][60][61].…”

Electron counting with a two-particle emitter

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