Amplitude Noise Reduction in a Nano-Mechanical Oscillator

We study the quantum properties of a nano-mechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force 0 F close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nano-mechanical oscillator are to be obtained from the Hamiltonian of the form H = a⁺a + β(a⁺ + a) / 4 . This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms a⁺² a², (a²a)² and a⁺⁴ + a⁴ - (a⁺² a² + a² a⁺² ) previously employed in quantum optics to study squeezing. Here we solve the Schrodinger equation numerically and show that in-phase quadrature gets squeezed for both vacuum and coherent states. The squeezing can be controlled by bringing F₀ close to or far from the critical value F_c. We further study the effect of the transverse driving force on the squeezing in nano-mechanical oscillator.