EVOLUTION EQUATIONFOR TWO LEVEL SYSTEMS INTERACTING WITH PUMP AND RELAXATION MECHANISMS*) J. Luczka Institute of Physics, Silesian University, 40-007 Katowice, Poland An integro-differential (non-Markovian) equation of motion for two level systems in contact with one mode of the bosonfield and driven by an external alternating field, the frequency and amplitude of which can be modulated, is derived. The con- nections with other equations are performed. Two cases of the evolution equation are discussed. 1. PRELIMINARIES Let us takeintoconsideration a two level system (a two state atom or spin one-half fermion) coupled to a single mode of the boson field and driven by an external alternating field. The ~Hamiltonian of such a (i) H(t) = Ho (2) = + = (3) H,(t) = h(f(t) + o b ) S+ system is chosen in the form(cf. ref.[1]) + "1(0 hco(S; + b + b) + h(f*(t) + 9b+)S- where S i, i = z, +, -, are operators of a spinS = 1/2, b + and b are the Bose opera- tors, the function f(t) characterizes the alternating (incoherent) fieldthat pumps energy to the system, g is the coupling constant of the system-Bose field interaction. One can mention two kinds of systems described by the Hamiltonian (1)-(3). These are magnetic systems [2] (e.g., the system of paramagnetic spins S = 1/2 interacting with a lattice and with an alternating transverse magnetic field) and optical systems [3]. The function f(t) can be written as (4) f(t) = (1/2) co,(t)exp(-iO(t)) ; ,p(t) = f2t+ q~(t) which allows us to investigate the reaction of the system on the external field with the modulated amplitude cot(t) and the modulated frequency f2(t) = ~t(t)/Ot. The evolution equation for the density operator of the system is proposed to be [4] i (5) dQ(t) + _ [H(t), 0(03 = -e[0(t) - or(t)] . dt h Let us assume that the field mode (boson mode)isin a chaotic (thermal)state [5]. From this assumption and from the form of the Hamiltonian (1)-(3) it is deduced *) This work was supported in part by the Polish Academy of Sciences. 1150 Czech. J. Phys. ~ 34 [19841