Reconstructing effective Hamiltonians of condensed matter systems directly from experimental data is challenging because of the intricate relationship between Hamiltonian parameters and observables. Here, we reconstruct an effective three-band electron-hole Hamiltonian in bulk GaAs based on high-order sideband generation (HSG) induced by quasicontinuous near-infrared (NIR) and terahertz (THz) lasers. We perform polarimetry of high-order sidebands while varying the wavelength and polarization of the NIR laser, as well as the strength of the THz field (27 to 64 kV/cm at 447 GHz). An analytic model is derived to incorporate the effects of both dephasing and quantum fluctuations around the semiclassical electron-hole recollision pathways. Surprisingly, the contribution of quantum fluctuations to the decay of sideband intensity with increasing sideband order is comparable to the contribution of dephasing. Assuming that the exciton reduced mass and the single parameter that defines the hole Bloch wave functions in bulk GaAs are known from previous experiments, we simultaneously and unambiguously determine through Bloch-wave interferometry the following: the effective Hamiltonian parameter that determines the electron-hole reduced masses, $\xi =0.125\pm 0.011$; the band gap of GaAs at 30 K, $E_{\mathrm{g}}=1.530\pm 0.001\mathrm{eV}$; and two dephasing constants associated with two electron-hole species, ${\mathrm{\Gamma }}_{\text{E-HH}}=10.4\pm 0.2\mathrm{meV}$ and ${\mathrm{\Gamma }}_{\text{E-LH}}=7.6\pm 0.2\mathrm{meV}$. We demonstrate that full Hamiltonian reconstruction can be achieved by combining HSG measurements with absorbance spectroscopy. Unexpectedly, we find that the extracted band gap of GaAs is about 10 meV larger than the value inferred from previous absorbance measurements. Quantum-kinetic analysis suggests that, in the HSG experiments, the electron-hole energy may be renormalized through Fröhlich interaction that is modulated by the strong THz fields. We also show that the energy threshold for optical-phonon emission can be suppressed by applying a strong THz field, leading to nearly constant dephasing rates. Our work provides an opportunity to explore possible modification of polaronic effects under strong THz fields. We show that polarimetry of high-order sidebands, together with interband absorption measurements, symmetry analysis, and analytic theory, can potentially enable reconstruction of effective Hamiltonians for other direct-gap insulators and semiconductors, whose bulk is not directly accessible to surface-sensitive techniques such as angle-resolved photoemission spectroscopy (ARPES).

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• Electron-phonon coupling
• Electronic structure
• Optical coherence
• Quantum kinetic theory
• Strong-field-induced spectra
• High-order harmonic spectroscopy
• Monte Carlo methods
• Nonperturbative methods