In this paper, we show the relation between
sp(4,R), the Lie algebra of the symplectic group, and the elements of
Sp(4,R). We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.
We propose a novel scheme for generating mechanical squeezed states based on the combined mechanism of a two-tone driving and a squeezed vacuum field. This innovative approach achieves a remarkable improvement in mechanical squeezing performance across the entire range of red/blue detuning ratios. Our study reveals that the squeezed vacuum field not only induces position squeezing of the mechanical oscillator but also facilitates momentum squeezing through phase matching. Moreover, the total squeezing degree exhibits nonlinear enhancement with the increasing of squeezing parameter
r. The mechanical squeezed state exhibits a
2π-periodic dependence in relation to the squeezing phase
θ, offering experimental implementation with a high degree of operational flexibility. Notably, the scheme exhibits strong robustness against cavity dissipation and environmental thermal noise, substantially relaxing the strict parameter-matching requirements inherent in conventional approaches.