Quantum Fluctuations and Wigner Dynamics of a Carbon Nanotube Oscillator Driven by an Atomic Force Microscope

Suspended carbon nanotubes provide a useful platform for studying controlled nanomechanical motion in regimes where environmental fluctuations, memory effects, and quantum coherence can all influence the dynamics. We model the fundamental flexural mode of a carbon nanotube driven by an atomic force microscope as a harmonic quantum Brownian oscillator subject to a prescribed classical force and coupled linearly to a thermal environment. Starting from the generalized Langevin equation, we formulate the reduced dynamics directly in phase space, where the Wigner function evolves through a Gaussian propagator determined by dissipative linear transport, a deterministic force-induced displacement, and bath-induced fluctuations. For a Drude memory kernel, the retarded response is expressed through the three roots of a characteristic cubic equation. In the classical-noise limit, the three independent entries of the finite-time covariance matrix reduce to explicit finite sums over these response poles, eliminating numerical double-time covariance integration. We extend the construction to the full quantum fluctuation spectrum, obtaining a thermal frequency series representation that retains zero-point and low-temperature fluctuations. The asymptotic classical covariance further yields the uncertainty-based consistency criterion kBT≥ℏω0/2, which identifies regimes in which the classical equipartition covariance is incompatible with a physical quantum state. The pole construction is also generalized to finite sums of exponential memory kernels, providing an analytic treatment of environments with multiple relaxation times. For an initial two-packet superposition, we examine interference attenuation and the integrated negative part of the Wigner function, which provides a phase space measure of nonclassicality. A deterministic atomic force microscope drive translates the Wigner distribution without changing its integrated negative part, whereas environmental fluctuations suppress nonclassical interference. These results provide an explicit analytical framework for finite-memory quantum Brownian motion of bound oscillators and a tractable phase space description of driven carbon nanotube dynamics relevant to future quantum state preparation and quantum tomography experiments.