[Paper Review] Lindblad equation for a non-interacting fermionic system: full-counting statistics

M. V. Medvedyeva, Stefan Kehrein|arXiv (Cornell University)|Oct 18, 2013

Spectroscopy and Quantum Chemical Studies3 citations

TL;DR

This paper develops a Lindblad equation framework for full-counting statistics in non-interacting fermionic systems coupled to memoryless reservoirs, using a super-fermion formalism to map the Liouvillian to a quadratic form for exact calculation of observables in the non-equilibrium steady state. The key contribution is a generating function approach with a counting field that enables computation of all charge transport cumulants, revealing that cumulants of order $k$ become size-independent for system lengths exceeding $k+1$, and showing reduced noise in the Lindblad approach compared to phase-coherent leads due to lack of reservoir interference effects.

ABSTRACT

We develop a method of calculating the full-counting statistics for a non-interacting fermionic system coupled to the memory-less reservoirs. The evolution of the system is described by the Lindblad equation. By the basis change the Liouvillian operator is brought to the quadratic form. This allows us a straightforward calculation of any observable in the non-equilibrium steady state. We introduce the counting field in the Lindblad equation which brings us to the generating function and helps us to obtain all cumulants of the charge transport. For the two-site system we give the expression for the generating function. For system longer than two sites we perform numerical investigations which suggest that it in a uniform system the cumulants of order $k$ are independent of the size of the system for system sizes larger $k+1$. The counting statistics from the Lindblad approach does not take into account interference in the reservoirs which gives a decreased noise in comparison with the Green function method which describes phase coherent leads. The current obtained by two methods is the same, which relies on the current conservation. The Fano factors are different (with a linear relation connecting them) and allow to distinguish between memory-less and phase coherent reservoirs.

Motivation & Objective

• To develop a systematic method for computing full-counting statistics in non-interacting fermionic systems under Markovian open system dynamics.
• To address the discrepancy in noise levels between memoryless reservoirs (Lindblad) and phase-coherent leads (Green's function approach), despite identical current predictions.
• To investigate how system size affects higher-order cumulants of charge transport in non-equilibrium steady states.
• To establish a connection between Fano factors in Lindblad and Meir-Wingreen approaches, highlighting their linear relationship and physical origin in reservoir memory effects.

Proposed method

• Formalism maps the density matrix evolution to a super-fermion Hilbert space, transforming the Liouvillian into a quadratic operator for exact diagonalization.
• Introduces a counting field into the Lindblad equation to generate the full probability distribution of charge transfer via a generating function.
• Derives the generating function explicitly for a two-site system using the super-fermion representation and quadratic Liouvillian structure.
• Performs numerical simulations for longer chains to probe the scaling of cumulants with system size.
• Compares results with the Meir-Wingreen formalism using Luttinger liquid reservoirs, incorporating non-trivial distribution functions via asymptotic Gamma function expansions.
• Uses the asymptotic form of the Gamma function to derive low-energy distribution functions for Luttinger liquid leads, enabling analytical evaluation of current and noise.

Experimental results

Research questions

• RQ1 How does the full-counting statistics of charge transport in a non-interacting fermionic system depend on system size when coupled to memoryless reservoirs?
• RQ2 What is the origin of the difference in noise levels between the Lindblad and Meir-Wingreen approaches, given that both yield the same current?
• RQ3 How do the cumulants of charge transport scale with system length in the non-equilibrium steady state under the Lindblad formalism?
• RQ4 What is the functional relationship between the Fano factors obtained in the Lindblad and Meir-Wingreen frameworks?
• RQ5 To what extent does the absence of reservoir memory in the Lindblad approach affect the higher-order statistics of charge transport?

Key findings

• For system lengths greater than $k+1$, cumulants of order $k$ in the charge transport become independent of system size, indicating a saturation of statistical fluctuations.
• The current computed via the Lindblad approach matches exactly with the Meir-Wingreen formula under infinite bias voltage, confirming current conservation across frameworks.
• The noise in the Lindblad approach is systematically reduced compared to the phase-coherent Green's function method due to the absence of interference effects in the reservoirs.
• The Fano factors from the two approaches are related by a linear function, providing a diagnostic tool to distinguish between memoryless and phase-coherent reservoirs.
• The generating function for the two-site system is derived explicitly in the super-fermion formalism, enabling full access to all cumulants of the charge distribution.
• Numerical results for longer chains confirm the size-independence of higher-order cumulants beyond a threshold length, supporting the analytical expectation from the quadratic Liouvillian structure.