Dissipative optomechanical coupling between a single-wall carbon nanotube and a high-$Q$ microcavity

We theoretically investigate optomechanical coupling between a single-wall carbon nanotube and a high-$Q$ whispering gallery microcavity. The anisotropy in the refractive index of the nanotube results in polarization-sensitive optomechanical coupling. Moreover, the band structure of the nanotube leads to a strong dependence of the coupling strength on wavelength. In particular, strong and pure dissipative coupling can be achieved at the absorption resonance frequency of the nanotube, and the single-photon coupling strength can exceed $10\text{kHz}$. This strong coupling induces dissipative cooling of the nanotube oscillation, and ground-state cooling can be achieved.

Figure 1

Schematic of the optomechanical system. Inset: Illustration of the CNT lattice. Yellow (light gray) and blue (dark gray) dots denote sublattices $A$ and $B$, respectively.

Figure 2

Real (black solid lines) and imaginary (blue dashed lines) parts of (a) ${\chi }_{\mathrm{xx}}\left(\omega \right)$ and (b) ${\chi }_{\mathrm{zz}}\left(\omega \right)$. The parameters are ${\gamma }_0=2.89\text{eV}$ and $\tau ={10}^{-13}\text{s}$.

Figure 3

Frequency shift (black solid line) and extra linewidth (blue dashed line) of the optical mode coupled to SWNTs with chiral index $(7,0)$, $(8,0)$, $(10,0)$, respectively. The length of the SWNTs is $1\mu \mathrm{m}$. The distance between the SWNT and the microcavity is 30 nm.

Figure 4

(a) Extra linewidth and (b) static coupling strength induced by distance-dependent SWNT absorption. SWNT chiral index $(8,0)$, length $L=1\mu$m. Microcavity major diameter $D=20\mu \text{m}$, minor diameter $d=2.7\mu \text{m}$. Inset: Illustration of the distance $x$.

Figure 5

Force noise spectral density for pure dissipative or dispersive coupling. The black solid line corresponds to pure dispersive coupling with $\delta =-{\omega }_{\mathrm{m}},{\kappa }_{\mathrm{c}}={\omega }_{\mathrm{m}}$, the short-dashed blue curve to pure dissipative coupling with $\delta =-{\omega }_{\mathrm{m}},{\kappa }_{\mathrm{c}}={\omega }_{\mathrm{m}}$, and the long-dashed red curve to pure dissipative coupling with $\delta =-3{\omega }_{\mathrm{m}},{\kappa }_{\mathrm{c}}={\omega }_{\mathrm{m}}$. Each spectrum has been scaled by its maximum.

Figure 6

Optical damping as a function of detuning $\delta$. The blue (${\gamma }_{\mathrm{o}}>0$) and pink (${\gamma }_{\mathrm{o}}<0$) areas correspond to cooling and heating, respectively. Input power $P=5\text{pW}$ and other parameters are $g_{\kappa }=1.77\times {10}^{-4}$, $[\zeta ,{\kappa }_{\mathrm{c}},{\omega }_{\mathrm{m}},{\gamma }_{\mathrm{m}},{\kappa }_{\omega }]=[7.5\text{MHz},1\text{MHz},2\pi \times 12\text{MHz},2\pi \times 80\text{Hz},2.58\times {10}^{18}\text{Hz}]$.

Figure 7

Steady-state phonon number as a function of detuning $\delta$ and pump power.