quantum-transport-statistics-framework

name	quantum-transport-statistics-framework
description	Exact framework for computing heat, energy, and particle transport statistics in quadratic quantum systems coupled to Gaussian reservoirs — combines full counting statistics with non-Markovian master equations. Use when: analyzing quantum transport in mesoscopic systems, computing full counting statistics for particle/heat currents, studying non-Markovian open quantum systems, evaluating transport between quantum reservoirs, or modeling quantum thermodynamic engines.
license	Complete terms in LICENSE.txt
metadata	{"arxiv_id":"2602.21190","published":"2026-02-21","authors":"Guglielmo Pellitteri, Vittorio Giovannetti, Vasco Cavina","tags":["quantum","transport","statistics","full-counting","non-markovian","open-systems","thermodynamics","gaussian"]}

Quantum Transport Statistics Framework

Exact computational framework for evaluating heat, energy, and particle transport statistics between Gaussian reservoirs mediated by quadratic quantum systems.

Core Methodology

Full Counting Statistics (FCS) Setup

For a quadratic system coupled to M Gaussian reservoirs:

1. Hamiltonian: H = H_S + Σ_α H_R^α + Σ_α H_{SR}^α

   • H_S = ½ Ψ† h Ψ (quadratic system)
   • H_R^α = Σ_k ε_{αk} c_{αk}† c_{αk} (Gaussian reservoir α)
   • H_{SR}^α = Σ_k (t_{αk} Ψ† c_{αk} + h.c.) (linear coupling)

2. Counting field: introduce χ_α for each reservoir to track transferred particles/energy

   • Modified Hamiltonian: H(χ) = e^{iχN/2} H e^{-iχN/2}

3. Cumulant generating function: G(χ, t) = ⟨e^{iχQ(t)}⟩ where Q = accumulated current

4. Levitov-Lesovik formula: for non-interacting systems, G(χ, t) = det[1 + T(f_L - f_R)(e^{iχ} - 1)] where T = transmission matrix, f = Fermi functions

Non-Markovian Master Equation Approach

For systems where Markov approximation fails:

1. Derive time-convolutionless (TCL) master equation:
   ∂_t ρ(t) = ℒ(t)[ρ(t)] where ℒ(t) = Σ_n ℒ_n(t) (Born expansion)

2. Counting-field modified Liouvillian:
   ℒ(χ, t) = ℒ_0(t) + Σ_α (e^{iχ_α} - 1) J_α(t) + (e^{-iχ_α} - 1) J_α†(t)

3. Cumulant generating function:
   ln G(χ, t) = λ_0(χ, t) · t where λ_0 is dominant eigenvalue of ℒ(χ, t)

4. Current moments:
   ⟨I^n⟩ = (-i∂_χ)^n ln G(χ, t)|_{χ=0}

Efficient Computational Method

The framework introduces an algorithm that:

1. Reduces the full counting statistics to solving a set of coupled differential equations
2. Exploits Gaussianity: only first and second moments needed (Wick's theorem)
3. Computational cost: O(N³) for N system modes (vs O(4^N) for general states)
4. Handles arbitrary time-dependent driving and multi-reservoir setups

Key Results

Current Statistics

For steady-state transport:

• Average current: ⟨I⟩ = Tr[T(E)·(f_L(E) - f_R(E))] (Landauer formula generalization)
• Noise (variance): S = ∫ dE Tr[T(E)(1-T(E))(f_L-f_R)² + T(E)(f_L(1-f_L)+f_R(1-f_R))]
• Skewness: higher cumulants encode interaction effects and non-Gaussianity

Thermal Transport

For heat current between reservoirs at temperatures T_L, T_R:

• Fourier's law emerges in the diffusive limit
• Ballistic transport: heat current independent of system size
• Quantum thermal rectification possible with asymmetric couplings

Usage Patterns

Pattern 1: Steady-state transport

Set up Hamiltonian, compute transmission T(E), evaluate Landauer-type integrals.

Pattern 2: Time-dependent driving

Use the non-Markovian master equation with time-dependent counting fields.

Pattern 3: Full distribution

Compute all cumulants via ∂χ^n ln G(χ)|{χ=0} for complete current statistics.

Error Handling

Non-Gaussian Reservoirs

• The framework assumes Gaussian reservoirs; for non-Gaussian (e.g., spin baths), use polaron transformation or reaction coordinate mapping
• For strongly coupled reservoirs, include system-reservoir correlations via reaction coordinate

Numerical Stability

• For long times, the TCL expansion may diverge — switch to time-convolution (Nakajima-Zwanzig) form
• Use adaptive time-stepping for stiff differential equations
• Verify complete positivity of the reduced dynamics