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arXiv:cond-mat/0612693 · Physical Review Letters

Effects of Exchange Symmetry on Full Counting Statistics

Fabian Hassler, Gordey B. Lesovik, G. Blatter

Published Aug 16, 2007• 13 citations
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Abstract

We study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange for the situation where the scattering depends on energy. We find that, in addition to the expected sensitivity of the noise and higher cumulants, the exchange symmetry has a huge effect on the average transmitted charge; for equal-spin exchange-correlated electrons, the average transmitted charge can be orders of magnitude larger than the corresponding value for independent electrons.

DOI
10.1103/physrevlett.99.076804

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We study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange for the situation where the scattering depends on energy. We find that, in addition to the expected sensitivity of the noise and higher cumulants, the exchange symmetry has a huge effect on the average transmitted charge; for equal-spin exchange-correlated electrons, the average transmitted charge can be orders of magnitude larger than the corresponding value for independent electrons.
Traditionally, (charge) transport phenomena describe the average current flow in response to a constant applied driving force. Additional information is provided by current-current correlations or noise, telling about the granularity, the statistical properties, and the interactions between the particles. The complete characterization of transport is given by the full counting statistics, quantifying the probability that n particles pass the device during a time window t. The desire for a complete characterization of transport in a mesoscopic device goes together with the development of new single-particle sources and the availability of quantum bits as measurement devices. In this thesis, we will describe how to combine these three elements, full counting statistics, single-particle sources, and qubit measurement devices and how these concepts relate to the description of charge transport of noninteracting electrons in a mesoscopic device. In the work on full counting statistics, we develop a new wave-packet formalism that provides access to an entire class of interesting results. The wavepacket formalism is derived from an insight regarding the equivalence between the fidelity of a quantum system and the generating function of full counting statistics; furthermore, we show how qubits can be used to measure both quantities, fidelity and full counting statistics, exploiting the induced ‘decoherence’ as a signal; thereby, we transfer the measurement of the fidelity and of the full counting statistics from the realm of a ‘Gedanken’ experiment to a practical proposal realizable with today’s qubit technology. Subsequently, we study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange, for the situation where the scattering depends on energy. We make use of our firstquantized wave-packet formalism and find interesting results. While it is expected that the noise as well as higher-order correlators will be sensitive to the exchange symmetry, we find the astounding result that exchange can hugely enhance (or suppress) the average charge already. The effect is a consequence of the symmetry-induced reshuffling of weight in the momentum distribution of the
The counting statistics (CS) for charges passing through a coherent conductor comprise the most general quantity that characterizes electronic transport. The CS depend not only on the transport properties of the conductor, but also on the correlations among particles which compose the incident beam. In this paper we present general results for the CS of entangled electron pairs traversing a beam splitter, and we show that the probability that Q charges have passed is not binomial, as in the uncorrelated case, but is symmetric with respect to the average transferred charge. We furthermore consider the joint probability for transmitted charges of a given spin, and we show that the signature of entanglement distinctly appears in a correlation which is not present for a nonentangled case.
We demonstrate that the exchange of two identical particles can be effected by a rotation of a coordinate system defined by the two particles through an angle π. It is then a consequence of the intimate connection between the generator of a space rotation and the total angular momentum operator, that the effect of the particle exchange on the ket corresponding to the physical system is dependent upon the integer or half-integer character of the particle spins. From this it can be concluded that the connection between spin and statistics that derives from the symmetry of the identical particle system under particle exchange is explained (in the most fundamental way) by the equivalence between the exchange of two identical particles and a rotation in configuration space.
An analysis of the Wigner function for identical particles is presented. Four situations have been considered. (i) The first is scattering process between two indistinguishable particles described by a minimum uncertainty wave packets showing the exchange and correlation effects in Wigner phase space. (ii) An equilibrium ensemble of NNN particles in a one-dimensional box and in a one-dimensional harmonic potential is considered second, showing that the reduced one-particle Wigner function, as a function of the energy defined in the Wigner phase space, tends to the Fermi-Dirac or to the Bose-Einstein distribution function, depending on the considered statistics. (iii) The third situation is reduced one-particle transport equation for the Wigner function, in the case of interacting particles, showing the need for the two-particle reduced Wigner function within the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy scheme. (iv) Finally, the electron-phonon interaction in the two-particle case is considered, showing coparticipation of two electrons in the interaction with the phonon bath.
Unless constrained by symmetry, measurement of an observable on an ensemble of identical quantum systems returns a distribution of values which are encoded in the full counting statistics. While the mean value of this distribution is important for determining certain properties of a system, the full distribution can also exhibit universal behavior. In this paper we study the full counting statistics of particle number in one-dimensional interacting Bose and Fermi gases which have been quenched far from equilibrium. In particular, we consider the time evolution of the Lieb-Liniger and Gaudin-Yang models quenched from a Bose-Einstein condensate initial state and calculate the full counting statistics of the particle number within a subsystem. We show that the scaled cumulants of the charge in the initial state and at long times are simply related and, in particular, the latter are independent of the model parameters. Using the quasiparticle picture we obtain the full time evolution of the cumulants and find that although their end points are fixed, the finite-time dynamics depends strongly on the model parameters. We go on to construct the scaled cumulant generating functions and from this determine the limiting charge probability distributions at long time which are shown to exhibit distinct, nontrivial, and non-Gaussian fluctuations and large deviations. Published by the American Physical Society 2024
We show that a ``pileup'' effect occurring for a train of noninteracting identical particles incident on the same side of a resonance scatterer leads to significant interference effects, different from those observed in Hong-Ou-Mandel experiments. These include characteristic changes in the overall transmission rate and full counting statistics, as well as ``bunching'' and ``antibunching'' behavior in the all-particle transmission channel. With several resonances involved, pseudoresonant driving of the two-level system in the barrier may also result in a sharp enhancement of scattering probabilities for certain values of temporal delay between the particles.