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Physical Review B

Statistics and fluctuation theorem for boson and fermion transport through mesoscopic junctions

Upendra Harbola, Massimiliano Esposito, Shaul Mukamel

Published Aug 8, 2007• 36 citations
Publisher

Abstract

The statistical properties of quantum particles moving between two heat reservoirs at different temperatures are studied by solving the quantum master equation using a generating function technique. Bosons and fermions satisfy the same fluctuation theorem: Logarithm of the ratio of the probabilities of forward and backward transports reaches a constant value at long times, with an asymptotic $\ensuremath{\sim}1∕t$ correction. Non-Poissonian transfer statistics (bunching for bosons and antibunching for fermions) are examined using the Mandel parameter. These come primarily from the tail of the distribution corresponding to transfer of large number of particles, $k⪢⟨k⟩$.

DOI
10.1103/physrevb.76.085408

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The statistical properties of quantum particles moving between two heat reservoirs at different temperatures are studied by solving the quantum master equation using a generating function technique. Bosons and fermions satisfy the same fluctuation theorem: Logarithm of the ratio of the probabilities of forward and backward transports reaches a constant value at long times, with an asymptotic \ensuremath∼1∕t\ensuremath{\sim}1∕t\ensuremath∼1∕t correction. Non-Poissonian transfer statistics (bunching for bosons and antibunching for fermions) are examined using the Mandel parameter. These come primarily from the tail of the distribution corresponding to transfer of large number of particles, k⪢⟨k⟩k⪢⟨k⟩k⪢⟨k⟩.
We demonstrate that the probability distribution of the net number of electrons passing through a quantum system in a junction obeys a steady-state fluctuation theorem (FT) which can be tested experimentally by the full counting statistics (FCS) of electrons crossing the lead-system interface. The FCS is calculated using a many-body quantum master equation combined with a Liouville space generating function formalism. For a model of two coupled quantum dots, we show that the FT becomes valid for long binning times, and provide an estimate for the finite-time deviations. The Mandel (or Fano) parameter associated with the incoming or outgoing electron transfers shows sub-Poissonian (antibunching) statistics.
Fluctuation theorems (FTs), which describe some universal properties of nonequilibrium fluctuations, are examined from a quantum perspective and derived by introducing a two-point measurement on the system. FTs for closed and open systems driven out of equilibrium by an external time-dependent force, and for open systems maintained in a nonequilibrium steady state by nonequilibrium boundary conditions, are derived from a unified approach. Applications to fermion and boson transport in quantum junctions are discussed. Quantum master equations and Green's functions techniques for computing the energy and particle statistics are presented.
Equilibrium distribution functions are obtained for boson and fermion ensembles with a limited number of particles. It is shown that the number-of-particle distribution functions in different quantum states are statistically dependent; this dependence disappears only for a large number of particles in the ensemble. The distributions are transformed into the Boltzmann distribution at a high temperature and into the Bose-Einstein and Fermi-Dirac distributions for a large number of particles in the ensemble.
A fundamental pillar of quantum mechanics concerns indistinguishable quantum particles. In three dimensions they may be classified into fermions or bosons, having, respectively, antisymmetric or symmetric wave functions under particle exchange. One of the numerous manifestations of this quantum statistics is the tendency of fermions (bosons) to antibunch (bunch). In a two-particle scattering experiment with two possible outgoing channels [C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987)], the probability of the two particles to arrive each at a different terminal is enhanced (with respect to classical particles) for fermions, and reduced for bosons. Here we show that by entangling the particles with an external degree of freedom, we can engineer quantum statistical transmutation, which may cause fermions to bunch. Our analysis may have consequences on the observed fractional statistics of anyons, including non-Abelian statistics, with serious implications for quantum computing operations in the presence of external degrees of freedom.
The waiting time distribution w(τ)w(τ)w(τ), i.e. the probability for a delay τττ between two subsequent transition (`jumps') of particles, is a statistical tool in (quantum) transport. Using generalized Master equations for systems coupled to external particle reservoirs, one can establish relations between w(τ)w(τ)w(τ) and other statistical transport quantities such as the noise spectrum and the Full Counting Statistics. It turns out that w(τ)w(τ)w(τ) usually contains additional information on system parameters and properties such as quantum coherence, the number of internal states, or the entropy of the current channels that participate in transport.
Abstract This paper investigates the dynamics of current and quantum transport factor in a bosonic system consisting of a central system interacting with two reservoirs at different temperatures. We derive a master equation describing the time evolution of the density matrix of the system, accounting for the interactions and energy transfer between the components. We quantify the current, representing the flow of bosons through the system and analyse its dependence on the system's parameters and temperatures of the thermal reservoirs. In the steady state regime, we derived an expression for the quantum transport factor of the energy transfer process. Our analysis show that quantum effects, such as the dependence on temperature can significantly impact this factor. In particular, we observe that the transport factor of the quantum system is greater than the corresponding factor when the temperature goes to infinity, where the factor has an identical form with the Carnot efficiency of an ideal heat engine. We then derived the Fokker-Planck equation to find out the Glauber-Sudarsan P-representation. In the steady state of the equation, the probability distribution comes out to be in Gaussian form. We then calculated the entropic force for this probability distribution which gives the Hooke's law in the steady state, in agreement with the fact that our system is a harmonic oscillator.