Parametric excitation and generation of nonclassical states in linear media V V Dodonov
V.V. Dodonov, V.I. Man'ko
The single-mode case is equivalent to the model of a one-dimensional generalized oscillator with the time-dependent Hamiltonian H(t) = i [fi(t)p2 + v(t)x2 + p(t) [xp + px)] + ci(t)p + C2(t)x, (1) where the quadrature operators x and p satisfy the canonical commutation relation [x,p] = ih. The same Hamiltonian can also be written in terms of the bosonic annihilation and creation operators as H = \h [ D 0 ( a f a + aat) + DYa2 + D*tf2] +h(ga + tfg*) , (2) 154 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES 2N 2N •qB(t)q + C(t)q. (3) (5) PROPAGATORS AND INTEGRALS OF MOTION 155 156 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES SINGLE MODE: QUADRATURE REPRESENTATION 157 Ai = Mp-A2/*, Ai(0) A 2 = A i i / — A2P, A 2(0) A 3 - A 3 p - A4/1, A 3(0) A 4 = A 3z/ - A 4 p , A 4(0) Putting expressions (21) into equation (20) we arrive at the set of first-order ordinary differential equations and initial conditions 1 (22) 0 (23) 0 (24) 1 (25) : 0 (26) : 0 (27) (the dot means the time derivative). An immediate consequence of equations (22)- (25) and the initial conditions is the identity Its physical meaning is very simple: the commutation relations between the operators £ and p are the same as between X and P, [x , p] = [X , P] = ih, i.e., transformation (21) is canonical (and the evolution operator is unitary).
CRC Press eBooks•Mar 13, 2003•1 citation