IL NUOVO CIMENTO VOL. 107 B, N. 5 Maggio 1992 Squeezing in Quantum Parametric Chain (*). V. V. DODONOV(1), O. V. MAN'KO(2) and V. I. MAN'KO(3) (1) Moscow Physicotechnical Institute - Moscow Region 140160, Zhukovsky, Gagarina~ 16 (2) Institute of Nuclear Research of the Academy of Science of the USSR Moscow 117312, 60th October Universary Prospec~ 7a (3) Lebedev Physics Institute - 117333, Moscow, Leninsky Prospect, 53 (ricevuto il 15 Marzo 1991; approvato il 29 Ottobre 1991) Summary. -- The coupled parametric quantum oscillators are discussed. The ground, coherent states and integrals of motion for the parametric chain with time-dependent frequencies are constructed. The possibility of controlling in principle the dispersions of coordinates and momenta is suggested. The squeezing coefficients in coherent states are found. PACS 03.65.Bz - Foundations, theory of measurement, miscellaneous theories. PACS 03.70 - Theory of quantized fields. 1. - Introduction. The aim of this paper is to discuss the integrals of motion and coherent-states properties of the time-dependent discrete string consisting of quantum interacting parametric oscillators and to find the squeezing coefficients. Some solutions to the stationary chain may be found in[l,2]. We follow the procedure suggested for a nonstationary chain in [3]. A different model of the chain was suggested in [4]. The model of the coupled harmonic oscillators was discussed in [3, 5, 6]. The methods of time-dependent quantum quadratic and linear integrals of motion [7-10], the methods of continuous representation [11-14] and the methods of coherent and correlated states are used in this paper and in [3-6]. The term and the idea of coherent states were introduced by Glauber in 1963 [15], when he considered electromagnetic-field oscillators and studied their statistical properties. The coherent states of the quantized electromagnetic-field oscillator turned out to coincide with Gaussian wave packets in the coordinate representation studied for the quantum harmonic oscillator by SchrSdinger[16]. It was suggested (see, for example[17-19]), that the coherent states of arbitrary quantum systems could be constructed by means of the integrals of motion of these systems. (*) The authors of this paper have agreed to not receive the proofs for correction. 513