Quantum Tomography of Suspended Carbon Nanotubes

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.

Figures (3)

• Figure 1: Schematic of the proposed experimental setup. A doubly clamped single-walled carbon nanotube (CNT) of length $L=100\text{--}1000\,\mathrm{nm}$ is suspended over a $200\,\mathrm{nm}$ trench and driven by localized forces from a nearby atomic force microscope (AFM) tip. This experimental configuration provides a fully mechanical platform for implementing Ramsey interferometry and Wigner-function tomography of a nanomechanical mode under dilution refrigerator conditions ($T\sim 10\,\mathrm{mK}$, $P<10^{-7}\,\mathrm{Torr}$). Also indicated schematically is a Cooper-pair box patterned on the $\mathrm{SiO_2}/\mathrm{Si}$ substrate, illustrating one possible dispersive-coupling readout scheme for the CNT.
• Figure 2: (a) Excited-state population $P_1(t)$ of a driven CNT in the effective two-level description for three representative relaxation strengths relative to the control rate: $\Gamma_1/\Omega_R=0.1$ (weak relaxation; blue), $\Gamma_1/\Omega_R=0.5$ (intermediate relaxation; red), and $\Gamma_1/\Omega_R=0.9$ (strong relaxation; green). The Rabi frequency is $\Omega_R=0.75MHz$ for all curves. Dotted lines are fits to Eq. \ref{['eq:P1']} used to extract $T_1$. (b) Ramsey population $P_1^{\mathrm{R}}(\tau)$ versus delay time, for detuning $\Delta=300kHz$ in two CNT devices of different lengths: $L=800nm$ (blue) and $L=250nm$ (red). Dotted lines are fits to Eq. \ref{['eq:PR']} used to extract $T_2$.
• Figure 3: Time evolution of the calculated Wigner function $W(\alpha)$. For all panels, $T_1 = 5/\omega_{0}$. Top row: 3D surfaces showing $W(\alpha)$ at $t=0$ (a), $t=T_1$ (b), and $t=5T_1$ (c). Bottom row: corresponding 2D plots at $t=0$ (d), $t=T_1$ (e), and $t=5T_1$ (f). A single color scale is shown on the right, applying to all six panels.