ABSTRACT
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)
<Ji = A i c 2 -- A 2 c i , 6,(0) 62 = A3C2 -- A4C1, 62(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).
