(a) A single gain (n″ < 0) or lossy (n″ > 0) sphere in air impinged by a plane wave propagating along z direction. The corresponding optical torques and rotation speeds of the sphere as a function of ka are plotted in panels (b) and (c), respectively. Here k is the wavenumber and a is the radius of the sphere. For illustration purpose, Γ_z and ω denoted by green lines are multiplied by 0.1 in panels (b) and (c). (d) Absolute value of optical torque versus | n″ | is shown for a gain sphere with ka = 5.1 pointed out by the black arrow in panel (b). The dashed vertical green line is inserted to highlight the paradoxical symmetry between amplification and absorption at gain threshold.