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arXiv:1009.1428 · Physical Review E

Derivation of quantum master equation with counting fields by monitoring a probe

Takaaki Monnai

Published Nov 10, 2010• 3 citations
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Abstract

We show a microscopic derivation of a quantum master equation with counting terms which describes the electron statistics. A localized spin behaves as a probe whose precession angle monitors the net electron current by the magnetic moment interaction. The probe Hamiltonian is proportional to the current and is determined self-consistently for a model of a quantum dot. Then it turns out that the quantum master equation for the spin precession contains the counting terms. As an application, we show the fluctuation theorem for the electron current.

DOI
10.1103/physreve.82.051113

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We show a microscopic derivation of a quantum master equation with counting terms which describes the electron statistics. A localized spin behaves as a probe whose precession angle monitors the net electron current by the magnetic moment interaction. The probe Hamiltonian is proportional to the current and is determined self-consistently for a model of a quantum dot. Then it turns out that the quantum master equation for the spin precession contains the counting terms. As an application, we show the fluctuation theorem for the electron current.
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The operation of a source of entangled electron spins, based on a superconductor and two quantum dots in parallel, is described in detail with the help of quantum master equations. These are derived including the main parasitic processes in a fully consistent and nonperturbative way, starting from a microscopic Hamiltonian. The average current is calculated, including the contribution of entangled and nonentangled pairs. The constraints on the operation of the device are illustrated by a calculation of the various charge state probabilities.
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A theory of electron counting statistics in quantum transport is presented. It involves an idealized scheme of current measurement using a spin 1/2 coupled to the current so that it precesses at the rate proportional to the current. Within such an approach, counting charge without breaking the circuit is possible. As an application, we derive the counting statistics in a single channel conductor at finite temperature and bias. For a perfectly transmitting channel the counting distribution is gaussian, both for zero-point fluctuations and at finite temperature. At constant bias and low temperature the distribution is binomial, i.e., it arises from Bernoulli statistics. Another application considered is the noise due to short current pulses that involve few electrons. We find the time-dependence of the driving potential that produces coherent noise-minimizing current pulses, and display analogies of such current stateswith quantum-mechanical coherent states.
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We study fluctuations of electric current in a quantum resistor and derive a general quantum-mechanical formula for the distribution of transmitted charge. For that we introduce a scheme of current measurement that involves a spin 1/21/21/2 coupled to the current so that it precesses at the rate proportional to the current. Our approach allows the study of charge transfer without breaking the circuit. We analyze a single channel conductor and derive electron counting statistics for arbitrary relation between temperature and voltage. For a perfectly transmitting channel the counting distribution is gaussian, both for zero-point fluctuations and at finite temperature. At constant voltage and low temperature the distribution is binomial, i.e., it arises from Bernoulli statistics.