We propose to measure the dispersions of Fermi polaron and molecule by engineering spin-orbit coupling (SOC) on the impurity, which induces spin flip with finite momentum transfer. The polaron dispersion can be probed at small SOC momentum from the linear response of impurity spin. For molecule, we show that it can be prepared through an adiabatic steady-state evolution when setting SOC momentum as the Fermi momentum of majority bath. By gradually reducing SOC strength to zero, the steady state smoothly evolves to a molecular state with directional symmetry breaking. The corresponding dispersion can then be probed experimentally through the pair momentum distribution of molecules at finite density. Our scheme reveals a fundamental momentum difference between Fermi polaron and molecule, thereby offering a clear physical picture for their first-order transition in single-impurity system.

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• Polarons
• Spin-orbit coupling
• Fermi gases
• Variational approach