Hyperfine-Resolved Polarizabilities in Alkali Optical Traps

Optical trapping of alkali atoms such as rubidium (Rb) and cesium (Cs) underpins many modern quantum technologies, including atomic clocks, quantum information processing, and precision spectroscopy. While J-dependent polarizabilities and magic wavelengths of the $5S_{1/2}\rightarrow5P_{1/2,3/2}$ transition in Rb and the $6S_{1/2}\rightarrow6P_{1/2,3/2}$ transition in Cs are well studied, significant gaps remain in understanding hyperfine (F-dependent) polarizabilities and vector contributions. These effects are crucial for hyperfine qubits, where differential light shifts introduce decoherence, and for optical clocks, where vector polarizability can limit accuracy at the $10^{-16}$ level. We employ a relativistic all-order (AO) single–double (SD) method to calculate highly accurate dipole matrix elements and hyperfine-dependent wavefunctions. Both static ($\omega = 0$) and dynamic polarizabilities near the D1 and D2 lines are analyzed to identify magic wavelengths. For Rb $5S_{1/2}(F=1,2)$ and Cs $6S_{1/2}(F=3,4)$ ground states, we present complete sets of scalar, vector, and tensor polarizabilities. The dominant contributions arise from $5P$ and $6P$ states, while core and tail terms are small but non-negligible. Our calculations reveal strong cancellations in tensor components, large resonance-driven variations in dynamic polarizabilities, and precise magic wavelengths for both species. Vector components, particularly in Cs, were found to significantly alter trapping conditions. This work establishes a comprehensive framework for F-dependent polarizability calculations in alkali atoms. By combining relativistic all-order methods with full vector light-shift treatment, we provide benchmark data enabling precision optical trapping, state-insensitive magic wavelengths, and improved control for quantum simulation, metrology, and quantum information applications.