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We explore the quantum many-body physics of a three-component Bose-Einstein condensate in an optical lattice driven by laser fields inVand Λ configurations. We obtain exact analytical expressions for the energy spectrum and amplitudes of elementary excitations, and discover symmetries among them. We demonstrate that the applied laser fields induce a gap in the otherwise gapless Bogoliubov spectrum. We find that Landau damping of the collective modes above the energy of the gap is carried by laser-induced roton modes and is considerably suppressed compared to the phonon-mediated damping endemic to undriven scalar condensates

 

We present exact analytical results for the Caputo fractional derivative ofa wide class of elementary functions, including trigonometric andinverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian,quartic Gaussian, Lorentzian, and shifted polynomial functions. Theseresults are especially important for multi-scale physical systems, suchas porous materials, disordered media, and turbulent fluids, in whichtransport is described by fractional partial differential equations. Theexact results for the Caputo fractional derivative are obtained from a singlegeneralized Euler’s integral transform of the generalized hypergeometricfunction with a power-law argument. We present a proof of thegeneralized Euler’s integral transform and directly apply it to theexact evaluation of the Caputo fractional derivative of a broad spectrum offunctions, provided that these functions can be expressed in terms of ageneralized hypergeometric function with a power-law argument. Wedetermine that the Caputo fractional derivative of elementary functions isgiven by the generalized hypergeometric function. Moreover, we show thatin the most general case the final result cannot be reduced toelementary functions, in contrast to both the Liouville-Caputo and Fourier fractionalderivatives. However, we establish that in the infinite limit of theargument of elementary functions, all three definitions of a fractionalderivative - the Coputo, Liouville-Caputo, and Fourier - converge to the same result given by theelementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourierfractional derivatives.