Researchers have developed a theoretical formulation to describe the behavior of electrons in crystalline lattices when driven by a single-mode Gaussian quantum light source. This framework, based on Floquet Green's functions, allows for the analysis of non-interacting, single-band electronic systems, where the light source is considered external and is not modified by many-electron polarization, although an active electronic probe can condition the source's evolution through Peierls coupling.

The proposed methodology addresses Peierls coupling non-perturbatively within the prescribed-source model, recovering classical Floquet theory in the appropriate limit. The key lies in the two time arguments of a Green's function sharing a single source history, with the lesser and greater components obtained by convolving a shared-history four-endpoint kernel with continuous bath kernels before the final source trace. The bath's canonical anticommutation relations provide an equal-time covariance that seeds the retarded and advanced single-leg propagations.

Numerical calculations on a minimal one-dimensional model have revealed significant quantum-source corrections, even with finite coupling, extending beyond predictions from classical drives. Spectral reconstruction and occupancy redistribution were observed for squeezed vacuum and squeezed coherent sources. The squeezing parameter and phase of the quantum light offer additional control knobs, beyond classical amplitude modulation, for manipulating both the sideband structure and the occupied weight of electronic states. This work establishes a theoretical framework for quantum Floquet engineering in condensed matter using externally prescribed quantum light sources.

This advance is fundamental for quantum Floquet engineering, a technique that seeks to manipulate material properties through the periodic application of external fields. By introducing quantum control of light, new avenues are opened for designing and controlling the electronic properties of materials, which could have implications for the development of new quantum devices and the understanding of transport phenomena in out-of-equilibrium systems.