Mathematical Background

QuantumFCS.jl computes zero-frequency current cumulants for time-independent Lindblad master equations. The system dynamics are written as

\[\frac{d\rho}{dt} = -i[H,\rho] + \sum_{k=1}^r \mathcal{D}[J_k]\rho \equiv \mathcal{L}\rho,\]

with dissipator

\[\mathcal{D}[A]\rho = A\rho A^\dagger - \frac{1}{2}\{A^\dagger A,\rho\}.\]

Here, H is the Hamiltonian, J = (J_1,\ldots,J_r) is the list of jump operators, and L is the Liouvillian superoperator. The recursive construction assumes that $\mathcal{L}$ has a unique stationary state $\rho_{\rm ss}$, normalised by $\operatorname{tr}\rho_{\rm ss}=1$.

Monitored jumps and weights

A stochastic current is defined by choosing a list of monitored jumps

\[\mathcal{M} = (M_1,\ldots,M_p), \qquad M_l \in \{J_1,\ldots,J_r\},\]

and a matching list of weights

\[\nu = (\nu_1,\ldots,\nu_p).\]

In the package, these are the inputs mJ and nu. The weights set both the sign convention and the units of the current: they can be dimensionless values such as $\pm 1$ for particle or photon counts, charges for electric currents, or energy quanta for heat currents.

Let $N_l(t)$ be the number of jumps through monitored channel $M_l$ in the interval $[0,t]$. The integrated current is

\[N(t) = \sum_{l=1}^p \nu_l N_l(t) = \int_0^t dt'\, I(t'),\]

which implicitly defines the stochastic current $I(t)$.

Tilted Liouvillian

To generate the statistics of $N(t)$, introduce the counting-field-resolved density matrix

\[\rho_\chi(t) = \sum_{n\in\mathcal{N}} e^{i\chi n}\rho(n;t),\]

where $\operatorname{tr}\rho(n;t)$ is the probability that the integrated current has value $n$. It evolves under the tilted Liouvillian

\[\mathcal{L}_\chi \rho_\chi = \mathcal{L}\rho_\chi + \delta\mathcal{L}_\chi\rho_\chi,\]

with

\[\delta\mathcal{L}_\chi\rho = \sum_{l=1}^p \left(e^{i\chi\nu_l}-1\right)M_l\rho M_l^\dagger.\]

This sign convention matches the definition of $\rho_\chi$ above and the package implementation. The integrated-current cumulant generating function is

\[C_N(\chi;t)=\log\operatorname{tr}\rho_\chi(t),\]

and the long-time current cumulants are

\[c_n \equiv \langle\!\langle I^n\rangle\!\rangle = \left.(-i\partial_\chi)^n \lim_{t\to\infty}\frac{C_N(\chi;t)}{t}\right|_{\chi=0}.\]

Recursive cumulant scheme

The package computes these cumulants without differentiating the dominant eigenvalue of $\mathcal{L}_\chi$ numerically. In vectorised notation, $|A\rangle\rangle$ denotes the vector form of an operator and $\langle\langle 1|$ is the trace functional. Define

\[\mathcal{L}^{(n)} = \left.(-i\partial_\chi)^n \mathcal{L}_\chi\right|_{\chi=0} = \sum_{l=1}^p \nu_l^n M_l^* \otimes M_l.\]

Starting from $|\sigma_0\rangle\rangle = |\rho_{\rm ss}\rangle\rangle$, the cumulants and auxiliary pseudo-states are generated recursively as

\[\langle\!\langle I^n\rangle\!\rangle = \sum_{m=1}^n {n\choose m} \langle\!\langle 1|\mathcal{L}^{(m)}|\sigma_{n-m}\rangle\!\rangle,\]

\[|\sigma_n\rangle\rangle = \mathcal{L}^+ \sum_{m=1}^n {n\choose m} \left(\langle\!\langle I^m\rangle\!\rangle\mathbb{I} - \mathcal{L}^{(m)}\right) |\sigma_{n-m}\rangle\rangle.\]

Here, $\mathcal{L}^+$ is the Drazin inverse of the Liouvillian on the subspace orthogonal to the stationary state. Numerically, QuantumFCS.jl prepares a solver for applying $\mathcal{L}^+$ and reuses that solver across cumulant orders; see Drazin solvers.

Mapping to package inputs

The mathematical objects above map directly to the main package interface:

• (H, J) or a preassembled L define the Liouvillian $\mathcal{L}$.
• rho_ss supplies the stationary state $\rho_{\rm ss}$.
• mJ is the monitored-jump list $\mathcal{M}$.
• nu is the weight list $\nu$.
• nC is the number of cumulants to return.

These inputs can be passed directly to fcscumulants_recursive, or bundled in a LindbladFCS problem.

References

• FCS in Lindblad master equations: Potts (2024). "Quantum Thermodynamics" arXiv

• FCS, recursive scheme, vectorization, Drazin inverse: Landi et al. (2024). "Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics", PRX Quantum 5, 020201 (2024) arXiv

• Detailed exposition of the recursive scheme: Flindt et al. (2010). "Counting statistics of transport through Coulomb blockade nanostructures: High-order cumulants and non-Markovian effects" Phys. Rev. B 82, 155407 arXiv