By 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:
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)
Counting field: introduce χ_α for each reservoir to track transferred particles/energy
- Modified Hamiltonian: H(χ) = e^{iχN/2} H e^{-iχN/2}
Cumulant generating function: G(χ, t) = ⟨e^{iχQ(t)}⟩ where Q = accumulated current
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:
- Reduces the full counting statistics to solving a set of coupled differential equations
- Exploits Gaussianity: only first and second moments needed (Wick's theorem)
- Computational cost: O(N³) for N system modes (vs O(4^N) for general states)
- 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