  We develop and study quantum and semi-classical models of Rydberg-atom
spectroscopy in amplitude-modulated optical lattices. Both initial- and
target-state Rydberg atoms are trapped in the lattice. Unlike in any other
spectroscopic scheme, the modulation-induced ponderomotive coupling between the
Rydberg states is spatially periodic and perfectly phase-locked to the lattice
trapping potentials. This leads to a novel type of sub-Doppler mechanism, which
we explain in detail. In our exact quantum model, we solve the time-dependent
Schr\"odinger equation in the product space of center-of-mass (COM) momentum
states and the internal-state space. We also develop a perturbative model based
on the band structure in the lattice and Fermi's golden rule, as well as a
semi-classical trajectory model in which the COM is treated classically and the
internal-state dynamics quantum-mechanically. In all models we obtain the
spectrum of the target Rydberg-state population versus the lattice modulation
frequency, averaged over the initial thermal COM momentum distribution of the
atoms. We investigate the quantum-classical correspondence of the problem in
several parameter regimes and exhibit spectral features that arise from
vibrational COM coherences and rotary-echo effects. Applications in
Rydberg-atom spectroscopy are discussed.
