Emergent Bell Phase in an Electro-Nanomechanical Quantum Simulator

TL;DR

Two parallel suspended carbon nanotubes with four quantum dots each are proposed as a quasi-2D electro-nanomechanical quantum simulator to study strongly correlated electron-phonon systems. The model includes electron hopping $t$, on-site repulsion $U$, inter-tube Coulomb interaction $V$, and a spectrum of phonon modes with frequencies $\omega_{\mu}$ coupled through $g_{i\mu}$; applying the Lang-Firsov transformation yields an effective attractive interaction $\hat{H}_{\tilde{U}}$ that competes with $U$, enabling tractable analysis. Numerical results reveal three electronic ground-state regimes—Mott, Bell, and Paired—along with an intermediate, highly entangled state; the Bell phase exhibits maximal electronic entanglement across tubes and nonzero mutual information in the phonons, while the negativity remains zero, indicating classical phonon correlations. The study shows the Bell phase persists under finite tunneling $t$ and inter-tube coupling $V$, and argues that the proposed platform is within reach of current CNT fabrication and gating techniques for quantum simulation of strongly correlated materials.

Abstract

Suspended carbon nanotubes hosting electrostatically defined quantum dots allow for exceptionally strong and tunable electromechanical coupling as well as mechanical modes that can reach the quantum ground state of motion simply by cryogenic cooling. This makes them a unique platform for quantum simulation of electron-phonon coupling. Here, we propose an experimentally realisable setup with two such carbon nanotubes in parallel, each hosting four quantum dots. Our system not only exhibits phonon-mediated electron-electron attraction, but also supports a robust, maximally entangled Bell phase at mesoscopic scales shared across the subsystems. These features highlight its potential as a simulator of strongly correlated quantum systems.

Figures (4)

• Figure 1: Sketch of the proposed quasi 2-D setup. Each CNT, labeled $\mathcal{A}$ and $\mathcal{B}$, hosts four quantum dots at half filling. The electronic states are capacitively coupled to the vibrational modes of the carbon nanotubes via gate electrodes located at the bottom of the trench. This results in configuration-dependent displacements ($\Delta x$). Coulomb interactions (indicated by red arrows) occur between opposing occupied QDs on different CNTs. We assume $D > d$, i.e. the distance between neighbouring QDs on a single tube is greater than the distance between the tubes. Depending on the Hamiltonian parameters, various electronic configurations can emerge, including Mott insulating, Paired and Intermediate states.
• Figure 2: The phase spectrum of the two-tube system with intertube coupling strength $V=0.02$. The figures correspond to the following observables: (a) the electronic double occupancies in single tube, (b) the phonon number in a single tube, (c) the average electronic charge correlations in a single tube, (d) the variance of the phononic annihilation operator in a single tube, (e) the mutual information between phonons of different tubes, and (f) the entanglement entropy between subsystem $\mathcal{A}$ and $\mathcal{B}$. The entropic quantities were computed with a logarithm of base e.
• Figure 3: Phase diagram dependence on the inter-tube coupling $V$. We display the number of electronic double occupations as a function of the inter-tube coupling: (a1) $V/U = 0.01$, (a2) $V/U = 0.02$, and (a3) $V/U=0.04$. The transition from the Mott to Bell phase occurs at the same critical value regardless of the value of $V/U$, while the transition from Bell to Paired phase shifts to larger values of $\lambda$ as $V/U$ increases. Notice that the Bell phase, characterised by a non-integer double occupancy, grows linearly with the inter-tube coupling strength.
• Figure 4: The effect of the rescaling. The phase spectrum of the one-dimensional setup is displayed for a realistic system (left) and a system of low phonon numbers (right), exemplified on the phonon number (top), the average charge correlations (middle) and the variance of the phononic annihilation operator (bottom).