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arXiv:2610.27167 · AppliedMath

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

Conrad Silio, Cristian Staii

quant-phcond-mat.mes-hallmath-phSubmitted Oct 19, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

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. 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. 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.

Comments
24 pages, 4 figures

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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. 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. 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.
We propose and analyze an all-mechanical route to coherent control and quantum-state reconstruction of the fundamental flexural mode of a suspended carbon nanotube (CNT) operated in the anharmonic (Duffing/Kerr). A nearby atomic force microscope (AFM) provides a single, localized actuator that applies calibrated, time-dependent forces to the CNT. In the presence of mechanical anharmonicity this enables spectrally selective control of the lowest vibrational transition and thus supports effective two-level protocols such as Rabi oscillations and Ramsey interferometry. The same actuator also implements phase-space displacements required for Wigner function tomography via displaced-parity sampling, thereby unifying control and tomography without optical heating and without dedicated on-chip microwave drive lines at the CNT resonator. We develop explicit pulse sequences and a master equation framework that connect experimentally accessible signals to energy relaxation and phase coherence times and to parity-based quantum signatures, including negative regions of the Wigner function. The approach is compatible with multiple readout modalities, including direct AFM-based detection and dispersive coupling to superconducting circuitry such as Cooper-pair box, and/or a microwave cavity. Together, these techniques provide complete access to populations, coherence, and parity within a single device architecture. This minimal scheme provides a practical route to all-mechanical quantum control and state-resolved characterization of decoherence in mesoscopic mechanical systems.
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Hybrid carbon nanotube-fullerene architectures provide a controllable platform for studying irreversibility and information flow in structured quantum environments. We analyze entropy generation in a system where paramagnetic endohedral fullerenes, such as N@C60_{60}60​ and P@C60_{60}60​, are encapsulated inside a suspended carbon nanotube (CNT) resonator, with selected multi-level fullerene spin states forming an effective qubit coupled to quantized CNT flexural modes. Building on prior work on fullerene-filled CNTs, spin-phonon control in suspended nanotubes, and phase-space propagators for damped driven oscillators, we develop a hybrid open-system model combining driven quantum Brownian motion of the CNT with an effective Jaynes-Cummings spin-vibrational interaction. The resonator dynamics are represented by a Wigner function whose evolution is written analytically in terms of the initial Wigner distribution and a Gaussian propagator. This phase-space description separates drive-induced displacement, diffusion, and damping, and connects these processes directly to entropy flow. The coupled spin-mechanical dynamics are embedded in a Lindblad master equation including mechanical damping, spin relaxation, pure dephasing, and thermally activated excitation. Within this framework we derive the entropy balance, identify entropy flux and non-negative entropy production, and examine how spin-vibrational hybridization redistributes irreversibility between coherent exchange and dissipative channels. We show that magnetic-gradient-enhanced spin-phonon coupling, resonant driving, and moderate thermal occupation produce crossovers between oscillator-dominated and spin-dominated entropy-production regimes. The framework provides a basis for using CNT-PEF hybrids as nanoscale platforms to study nonequilibrium quantum thermodynamics, decoherence, and information loss in vibrational environments.
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We establish that the exact quantum dynamics of a Brownian particle in the Caldeira-Leggett model, with at most quadratic external potential, can be mapped, at any temperature, onto a classical, non-Markovian stochastic process in phase space. Starting from a correlated thermal equilibrium state between the particle and bath, we demonstrate that this correspondence is exact for quadratic potentials under arbitrary quantum state preparations of the particle itself. Our approach allows to consider arbitrary initial quantum states - including highly non-classical superpositions - which are incorporated via their Wigner functions, which serve as statistical weights for trajectory ensembles. Furthermore, the formalism naturally accommodates external manipulations and measurements modeled by preparation functions acting at arbitrary times, enabling the simulation of complex driven-dissipative quantum protocols. For more general, smooth potentials, we identify a natural small parameter: the density matrix becomes strongly quasidiagonal in the coordinate representation, with its off-diagonal width shrinking as the bath's spectral cutoff increases, suggesting a controlled parameter for a possible approximation.
This article discusses the numerical result predicted by the quantum Langevin equation of the generalized diffusion function of a Brownian particle immersed in an Ohmic quantum bath of harmonic oscillators. The time dependence of the standard deviation of the reduced Wigner function of the system, obtained by integrating the whole function in the momentum space, is determined as well. The complexity of the equations leads to resort to a much simpler calculation based in the position correlation function. They are done for the three possible regimes of the system, namely, periodic, aperiodic, and overdamped. It is found in the periodic case that the generalized diffusion is a discontinuous function exhibiting negative values during short time periods of time. This counterintuitive result, found theoretically in other systems and waiting for its experimental confirmation, can be perfectly explained in the framework of the quantum Langevin equation. Its oscillatory behavior is primordially due to the response to the external field while its quantum origin contributes only in its magnitude. The results are compared to those of the continuum limit which exhibits similar behavior.