Immersing carbon nanotubes in cold atomic gases

We investigate the sympathetic relaxation of a free-standing, vibrating carbon nanotube that is mounted on an atom chip and is immersed in a cloud of ultracold atoms. Gas atoms colliding with the nanotube excite phonons via a Casimir-Polder potential. We use Fermi's golden rule to estimate the relaxation rates for relevant experimental parameters and develop a fully dynamic theory of relaxation for the multimode phononic field embedded in a thermal atomic reservoir. Based on currently available experimental data, we identify the relaxation rates as a function of atom density and temperature that are required for sympathetic ground-state cooling of carbon nanotubes.

Figure 1

Dimensionless Fourier amplitudes $V_n\left(\overline{q}\right)$ of the Casimir-Polder potential vs the dimensionless wave number $\overline{q}$ for different inverse power laws: $n=3$ (short-dashed orange line), $n=4$ (long-dashed blue line), $n=5$ (solid red line), $n=6$ (dot-dashed green line), and $n=7$ (dotted black line).

Figure 2

Excitation rate ${\Gamma }_0^v$ vs scaled temperature $T/T_{\mathrm{BEC}}\left(n\right)$ for different densities $n$ of the atomic cloud: $n={10}^{12}{\text{cm}}^{-3}$ (crosses), $5\times {10}^{12}{\text{cm}}^{-3}$ (squares), ${10}^{13}{\text{cm}}^{-3}$ (circles), $5\times {10}^{13}{\text{cm}}^{-3}$ (diamonds), and ${10}^{14}{\text{cm}}^{-3}$ (stars). The horizontal line marks the frequency ${\omega }_0=2\pi \times 398$ kHz of the lowest phononic mode and represents the boundary between the overdamped (above) and the underdamped (below) oscillations of the carbon nanotube.

Figure 3

Fugacity $\eta$ as a function of scaled temperature $T/T_{\mathrm{BEC}}$ (solid line), fraction of particles in the ground state $n_0/N$ (dash-dotted line), and fraction of particles in the excited states $\tilde{N}/N$ (dashed line).