Quantum and thermal fluctuations of a thin elastic plate

Abstract

We consider a Hamiltonian description of the vibrations of a clamped, elastic circular plate. The Hamiltonian of this system features a potential energy with two distinct contributions: one that depends on the local mean curvature of the plate, and a second that depends on its Gaussian curvature. We quantize this model using a complete, orthonormal set of eigenfunctions for the clamped, vibrating plate. The resulting quanta are the flexural phonons of the thin circular plate. As an application, we use this quantized description to calculate the fluctuations in displacement of the plate for arbitrary temperature. We compare the fluctuation profile with that from an elastic membrane under tension. At low temperature, we find that while both profiles have a circular ring of local maxima, the ring in the membrane profile is much more pronounced and sharper. We also note that with increasing temperature the plate profile develops two additional rings of extrema.

arXiv comment12 pages, 4 figures, 2 tables

References in corpus (15)

• Impermeable Atomic Membranes from Graphene Sheets
• Entangled massive mechanical oscillators
• Creation and control of multi-phonon Fock states in a bulk acoustic wave resonator
• Cavity optomechanics with stoichiometric SiN films
• Control of Material Damping in High-Q Membrane Microresonators
• Integration of fiber coupled high-Q silicon nitride microdisks with atom chips
• Bending modes, anharmonic effects and thermal expansion coefficient in single layer and multilayer graphene
• Quantum Electromechanics on Silicon Nitride Nanomembranes
• Phonon number sensitive electromechanics
• Control of Recoil Losses in Nanomechanical SiN Membrane Resonators
• Specific heat measurement of thin suspended SiN membrane from 8 K to 300 K using the 3$\omega$-V$\ddot{o}$lklein method
• Quantum Sticking of Atoms on Membranes
• Orthogonality Catastrophe in Quantum Sticking
• Infrared problem in quantum acoustodynamics at finite temperature
• Infrared Problem in Cold Atom Quantum Physisorption on 2D Materials