We consider an optomechanical system comprising a single cavity mode and a dense spectrum of acoustic modes and solve for the quantum dynamics of initial cavity mode Fock (i.e., photon number) superposition states and thermal acoustic states. The optomechanical interaction results in dephasing without damping and bears some analogy to gravitational decoherence. For a cavity mode locally coupled to a one-dimensional (1D) elastic string-like environment or two-dimensional (2D) elastic membrane-like environment, we find that the dephasing dynamics depends respectively on the string length and membrane area--a consequence of an infrared divergence in the limit of an infinite-sized string or membrane. On the other hand, for a cavity mode locally coupled to a three-dimensional (3D) bulk elastic solid, the dephasing dynamics is independent of the solid volume (i.e., is infrared finite), but dependent on the local geometry of the coupled cavity--a consequence of an ultraviolet divergence in the limit of a pointlike coupled cavity. We consider as possible respective realizations for the cavity-coupled-1D and 2D acoustic environments, an LC oscillator capacitively coupled to a partially metallized strip and a cavity light mode interacting via light pressure with a membrane.
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