Bistability and displacement fluctuations in a quantum nanomechanical oscillator

Remarkable features have been predicted for the mechanical fluctuations at the bistability transition of a classical oscillator coupled capacitively to a quantum dot [Micchi et al., Phys. Rev. Lett. 115, 206802 (2015)]. These results have been obtained in the regime $\ensuremath{\hbar}{\ensuremath{\omega}}₀\ensuremath{\ll}{k}_BT\ensuremath{\ll}\ensuremath{\hbar}\mathrm{\ensuremath{\Gamma}}$, where ${\ensuremath{\omega}}₀, T$, and $\mathrm{\ensuremath{\Gamma}}$ are the mechanical resonating frequency, the temperature, and the tunneling rate, respectively. A similar behavior could be expected in the quantum regime of $\ensuremath{\hbar}\mathrm{\ensuremath{\Gamma}}\ensuremath{\ll}{k}_BT\ensuremath{\ll}\ensuremath{\hbar}{\ensuremath{\omega}}₀$. We thus calculate the energy- and displacement-fluctuation spectra and study their behavior as a function of the electromechanical coupling constant when the system enters the Frank-Condon regime. We find that in analogy with the classical case, the energy-fluctuation spectrum and the displacement spectrum widths show a maximum for values of the coupling constant at which a mechanical bistability is established.