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arXiv:2509.01858 · Physical review. B./Physical review. B

Quantum tomography of suspended carbon nanotubes

Jialiang Chang, Nicholas Pietrzak, Cristian Staii

Published Apr 1, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/sh37-qczd

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•Jan 1, 2021•0 citations
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.
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.
Mechanical oscillators have been demonstrated with very high quality factors over a wide range of frequencies.They also couple to a wide variety of fields and forces, making them ideal as sensors.The realization of a mechanically based quantum bit could therefore provide an important new platform for quantum computation and sensing.Here, we show that by coupling one of the flexural modes of a suspended carbon nanotube to the charge states of a double quantum dot defined in the nanotube, it is possible to induce sufficient anharmonicity in the mechanical oscillator so that the coupled system can be used as a mechanical quantum bit.However, these results can only be achieved when the device enters the ultrastrong coupling regime.We discuss the conditions for the anharmonicity to appear, and we show that the Hamiltonian can be mapped onto an anharmonic oscillator, allowing us to work out the energy level structure and find how decoherence from the quantum dot and the mechanical oscillator is inherited by the qubit.Remarkably, the dephasing due to the quantum dot is expected to be reduced by several orders of magnitude in the coupled system.We outline qubit control, readout protocols, the realization of a CNOT gate by coupling two qubits to a microwave cavity, and finally how the qubit can be used as a static-force quantum sensor.
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.
The quantum behaviour of mechanical resonators is a new and emerging field driven by recent experiments reaching the quantum ground state. The high frequency, small mass, and large quality-factor of carbon nanotube resonators make them attractive for quantum nanomechanical applications. A common element in experiments achieving the resonator ground state is a second quantum system, such as coherent photons or a superconducting device, coupled to the resonators motion. For nanotubes, however, this is a challenge due to their small size. Here, we couple a carbon nanoelectromechanical (NEMS) device to a superconducting circuit. Suspended carbon nanotubes act as both superconducting junctions and moving elements in a Superconducting Quantum Interference Device (SQUID). We observe a strong modulation of the flux through the SQUID from displacements of the nanotube. Incorporating this SQUID into superconducting resonators and qubits should enable the detection and manipulation of nanotube mechanical quantum states at the single-phonon level.
This thesis deals with the implementation and demonstration of an optomechanical system that is composed of a superconducting microwave resonator and a suspended carbon nanotube (CNT). This is of particular interest, as CNTS - being macromolecules - have outstanding electronic and mechanical properties. Single electron effects combined with large mechanical resonance frequencies, quality factors, and zero-point motion amplitudes make possible promising applications in the field of quantum information processing or fundamental research, e.g. the synthesis of arbitrary quantum states. Besides the assembly of a high frequency measurement setup for cryogenic temperatures in the millikelvin range, introductory chapters discuss electronic and mechanical characteristics of carbon nanotubes. Here, two different fabrication techniques are described, the overgrowth and the fork transfer process. The physics of superconducting microwave resonators in coplanar waveguide geometry is introduced. Their high frequency behaviour at low temperatures is described by a transition from Mattis-Bardeen to two-level system theory. This serves as a means to investigate the high frequency material properties of a molybdenum/rhenium alloy, which is particularly suitable for the overgrowth process. Device design adjustments and considerations are described that were necessary for the integration of CNTs into hybrid geometries using the fork transfer process. Through this, the (previously demonstrated) electronic coupling of a carbon nanotube to a microwave resonator could be successfully experimentally demonstrated. There, the transmission of the resonator is sensitive to single electron tunneling processes in the CNT. The central point of this thesis is the introduction to the field of dispersive optomechanics. Here, the interaction of a vibrating carbon nanotube and a microwave field is investigated. A mechanism based on the quantum capacitance enhances this otherwise negligible coupling by orders of magnitude. Through an optomechanically induced transparency experiment, this coupling was experimentally quantified. A comparison to the beforehand deduced theory shows a very good agreement. In conclusion, subsequent prospective experiments towards optomechanical cooling and motion manipulation are explained and preliminary experimental data are discussed.