Variational Problem for Overlapping Laser Pulses

placed beside a problem number indicates that the problem was submitted without solution. Proposers and solvers whose solutions are published will receive 5 reprints of the corresponding problem section. Other solvers will receive just one reprint provided a self-addressed stamped (U.S.A. or Canada) envelope is enclosed Proposers and solvers desiring acknowledgment of their contributions should include a self-addressed stamped postcard. Transfer of an atom or molecule from its ground state to any desired higher state is often accomphshed by applying a succession of laser pulses, because use of only one laser pulse may be difficuh or impossible. What shape should the laser pulses have? In a simplified model of this process, the atom or molecule is a three-state quantum system driven by the oscillating electric fields of two laser pulses that overlap in time. We solve the time-dependent Schrodinger equation 0 Ai{t) 0 \laA A,{t) 0 A2{t) fl2 , 0 A2{t) 0 l\a^l where a^, aj, fla are probability amplitudes. We assume that Ai{t) and Ajit) are real; these two arbitrary functions are proportional to the amplitudes ofthe two applied oscillating fields. Since we demand complete transfer of the occupation probability from state one to state three, we have the following initial and final conditions: (2) |ail=l, 02 = 03 = 0 for t=-oo, «1 = 02 = 0, |a3l = l for t =-\-co. There are many sets of functions a^, 02, 03, ^i(0, ^2(0 that satisfy (1) and (2). We cannot use this simple model to find the minimum energy or minimum time for transfer from state one to state three, because Ai{t) and A2{t) could be replaced by (1 /k)Ai{kt) and (1 /k)A2{kt), where k is any positive constant. However, some calculations for this model involve eigenvalues of the matrix that appears in (1). We consider the time integral of an eigenvalue or a difference of eigenvalues; we want to minimize /sqrt= r [{A,{t)]^+[A2{t)fY'ut.