Floquet NEGF for Driven Quantum Systems

• Floquet NEGF is a formalism that extends NEGF by incorporating periodic driving to analyze steady-state transport, noise, and response in quantum devices.
• It systematically includes many-body interactions using approaches such as the Hubbard, FLEX, and GD approximations to model complex correlation effects.
• The framework decomposes observables into Floquet sidebands, revealing quantized transport phenomena and topological signatures in driven systems.

Floquet Nonequilibrium Green’s Function (Floquet NEGF) is a formalism that extends the nonequilibrium Green’s function (NEGF) approach to quantum systems subject to time-periodic (Floquet) driving. It enables rigorous calculation of steady-state transport, noise, and response properties in driven open systems, fully incorporating quantum coherence, many-body interactions (at various levels of approximation), and the structure of Floquet sidebands. The Floquet NEGF framework is applicable to a wide range of systems—quantum dots, molecular junctions, mesoscopic rings, periodically driven dielectrics, and topological phases—creating a unified methodology for treating nonequilibrium, periodically modulated quantum systems.

1. Mathematical Formulation of Floquet NEGF

The core of Floquet NEGF is the reformulation of the two-time Kadanoff–Baym equations for Green's functions in the presence of a time-periodic Hamiltonian $h(t)=h(t+T)$ with period $T=2\pi/\omega$. Two principal expansions are widely used:

a) Matrix-Like (“M-like”) Formulation:

The Green’s function is expanded in both time arguments:

$$G^r(t, t') = \sum_{n,m=-N}^N \int_{0}^{\omega} \frac{d\mathcal{E}}{2\pi} e^{-i(\mathcal{E}+n\omega)t} g^r_{nm}(\mathcal{E}) e^{i(\mathcal{E}+m\omega)t'}$$

The resulting Dyson equation in Floquet space is:

$$\Bigl[\mathcal{E} I^F - h^F - \bar\Sigma^{r, F}\Bigr] G^{r, F}(\mathcal{E}) = I^F$$

where $h^F$ and $\bar\Sigma^{r,F}$ are block matrices of size $(2N+1)M \times (2N+1)M$, $M$ being the number of orbitals (Mosallanejad et al., 2023, Balabanov, 2018).

b) Vector-Like (“V-like”) Formulation:

A two-step procedure, Fourier transforming first in the relative time and then expanding the remaining time argument in discrete Floquet harmonics, leads to an alternate block structure for the Dyson equation, where the Floquet expansion applies after a first partial Fourier transform (Mosallanejad et al., 2023).

In both cases, all observables—density, current, susceptibility—are encoded in the Floquet Green’s function and its “lesser” counterpart, which obeys a Keldysh equation:

$$G^{<,F}(\mathcal{E}) = G^{r,F}(\mathcal{E})\, \bar\Sigma^{<,F}(\mathcal{E})\, G^{a,F}(\mathcal{E})$$

where self-energies encode lead coupling and interactions as appropriate.

2. Inclusion of Many-Body Interactions

Floquet NEGF supports the systematic inclusion of interactions at various levels:

• Hubbard Ansatz:

A self-consistent NEGF equation-of-motion truncation enables treatment of on-site Hubbard-like interaction terms on quantum dots. The resulting coupled Dyson equations in Floquet space for one- and two-particle GFs are solved either self-consistently or with a static occupation approximation (Mosallanejad et al., 2023).

• Fluctuation-Exchange (FLEX) Approximation:

For multi-site or strongly correlated systems, the self-energy in Floquet space can be built from FLEX diagrams, including bubble, $GW$, and $T$-matrix contributions, all expanded into the Floquet index structure. Full self-consistency is imposed at the matrix level (Honeychurch et al., 2023).

• Electron-Phonon Coupling (GD Approximation):

NEGF with phonon–electron interactions (self-consistent Born approximation) extends to Floquet space, allowing calculation of inelastic transport with photon–assisted and phonon–assisted sidebands, and explicit computation of the phonon occupancy under drive (Honeychurch et al., 2022).

A summary of these techniques and their relation to quantum master equation approaches is given in (Mosallanejad et al., 2023), where quantitative agreement is found in non-interacting and weakly interacting regimes, but NEGF uniquely captures lifetime broadening and interaction-induced features essential for strong correlations.

3. Observables and Mode Decomposition

Given the Floquet Green's functions, time-periodic expectation values (occupancy, current, noise spectra) decompose naturally into Floquet sidebands. The general structure for any observable $\mathcal{O}$ is:

$$\overline{\mathcal{O}} = \sum_{n=-N}^N \int_{0}^{\omega} \frac{d\mathcal{E}}{2\pi} \operatorname{Tr}[\cdots]$$

For the time-averaged dot occupancy and lead currents in a quantum dot setup (Mosallanejad et al., 2023):

$$\overline{n} = -i \sum_n \int_{0}^{\omega} \frac{d\mathcal{E}}{2\pi} \operatorname{Tr}[g^<_{nn}(\mathcal{E})]$$

$$\overline{I}_l = \sum_n \int_{0}^{\omega} \frac{d\mathcal{E}}{2\pi} 2\,\operatorname{Re} \operatorname{Tr}[g^r_{nn}\,\bar\Sigma_l^<(\mathcal{E}+n\omega) + g^<_{nn}\,\bar\Sigma_l^a(\mathcal{E}+n\omega)]$$

In the non-interacting limit, the Landauer–Floquet form is recovered:

$$\overline{I} = \int_0^\omega \frac{d\mathcal{E}}{2\pi} \mathrm{Tr}[\Gamma_L^F\,G^{r,F}(\mathcal{E})\,\Gamma_R^F\,G^{a,F}(\mathcal{E})]\, [f_L(\mathcal{E})-f_R(\mathcal{E})]$$

For voltage-noise or current fluctuations, the noise spectrum $S_{ii}(\omega)$ can be formulated in terms of convolution of greater and lesser Floquet GFs, enabling detection of topological bound states via characteristic spectral peaks (Rodriguez-Vega et al., 2018).

4. Numerical Strategy and Implementation

Floquet NEGF requires truncation of the infinite Floquet (sideband/photon) indices to a finite cutoff $N$; convergence is ensured when the truncation energy $N\hbar\omega$ exceeds all device energy scales. The necessary block-matrix inversions scale as $(2N+1)M \times (2N+1)M$, with practical $N\sim 3-10$ for moderate driving and up to $N\sim 50$ for complex spectra (Mosallanejad et al., 2023, Honeychurch et al., 2023). Numerical schemes exploit block-banded structure, sparse solvers, and contour integration for observables. The reduced-zone scheme ensures charge conservation at any truncation (Chen et al., 2012). For multi-terminal or disordered systems, ensemble averages and parallelization can be straightforwardly incorporated (Rodriguez-Vega et al., 2018).

5. Applications: Transport, Response, and Topological Signatures

Floquet NEGF has been deployed in the following contexts:

• Electronic transport in driven nanostructures:

Quantized steps (“Floquet plateaus”) and photon-assisted tunneling in quantum dots, molecular junctions, and driven molecules are captured, including the effects of electron–electron and electron–phonon interactions (Mosallanejad et al., 2023, Honeychurch et al., 2022).

• Spin phenomena:

Circularly polarized periodic driving can generate spin current even without intrinsic spin–orbit coupling (Mosallanejad et al., 2023, Chen et al., 2012).

• Topological phases:

Voltage-noise spectra computed via Floquet NEGF reveal both regular and anomalous Floquet topological bound states through robust spectral peaks at characteristic drive harmonics (Rodriguez-Vega et al., 2018).

• Driven dielectrics and photon emission:

Floquet NEGF for time-modulated permittivity quantifies thermal emission and the opening of sidebands in intensity spectra, with finite, non-divergent output and modest Floquet enhancement (Ren et al., 10 Oct 2025).

• Susceptibility and Floquet engineering:

Floquet NEGF/Keldysh enables the calculation of spin and charge susceptibilities, capturing dynamical localization, Fermi surface deformation, and multi-peak structures arising from Floquet band crossings (Ono et al., 2018).

6. Comparison to Other Methods and Limitations

The NEGF approach, particularly with Floquet generalization, encompasses broader physics than Floquet Redfield or quantum master equation approaches, especially regarding finite lifetime broadening, spectral weight transfer, and non-Markovian effects (Mosallanejad et al., 2023). In the weakly interacting or noninteracting limit, numerical and analytical results from NEGF, master equations, and frequency-domain methods are fully consistent for plateau locations, while NEGF remains uniquely suited for treating strong coupling, correlated regimes, and noise spectra.

Limitations arise in truncation-induced spectral artifacts (remedied by convergence tests), and the formalism assumes periodic long-time steady states are reachable. Mean-field or self-consistent diagrammatic treatments (e.g., FLEX, Born) are necessary for strong correlations, with further extensions required for nonperturbative regimes.

7. Outlook and Physical Implications

Floquet NEGF underpins quantitative design of periodically driven quantum devices, molecular junctions under light, optically modulated dielectric media, and topological quantum matter. The capability to recover charge conservation, to treat a wide class of interactions, and to yield direct algorithmic correspondences to static NEGF supports its application to both theoretical modeling and experimental interpretation. A plausible implication is the utility of Floquet NEGF for optimizing energy transfer, thermal emission, and dynamical control in engineered quantum systems subjected to temporal modulation (Mosallanejad et al., 2023, Ren et al., 10 Oct 2025, Honeychurch et al., 2023).