where the Hamiltonian H ~ ~ B S = e i ( H S + H B ) t H B S e − − i ( H S + H B ) t is explicitly time dependent. Also, according to the interaction picture, χ χ ~ ~ = U B S ( t , t 0 ) χ χ U B S † † ( t , t 0 ) , where U B S = U 0 † † U ( t , t 0 ) . This equation can be integrated directly to give