Bistability of a slow mechanical oscillator coupled to a laser-driven two-level system

It has been recently proposed that single molecule spectroscopy could be employed to detect the motion of nanomechanical resonators. Estimates of the coupling constant ($g$) between the molecular two-level system and the oscillator indicate that it can reach values much larger than the mechanical resonating pulsation (${\omega }_m$) and the two-level system linewidth ($\mathrm{\Gamma }$). Other experimental realizations of the same system are also approaching this strong-coupling regime. In this paper we investigate the behavior of the system in the limit for slow mechanical oscillator ${\omega }_m\ll \mathrm{\Gamma }$. We find that, for sufficiently large coupling, the system undergoes a bistability reminiscent of that observed in optical cavities coupled to mechanical resonators.

Figure 1

Left: schematics of the system proposed in Ref. [18] and considered in this paper. Right: simplified view of the two-level system coupled to a mechanical oscillator via the Stark effect and to a laser.

Figure 2

Region of static (shaded blue region for negative ${\delta }^{\prime }$) and dynamical (shaded regions for positive ${\delta }^{\prime }$) instabilities in the plane $\lambda -{\delta }^{\prime }$ for $\mathrm{\Omega }=\mathrm{\Gamma }$ and for $Q{\omega }_m/\mathrm{\Gamma }=4$ (leftmost brown region), 2 (gray), and 1 (green). The dashed line corresponds to the value ${\delta }^{\prime }=0$, separatrix between cooling and heating in the case $\gamma =0$ (or $Q=\infty$).

Figure 3

Same as Fig. 2 in the plane $\delta -\lambda$. Due to the bistability the contour of the bistability region can be in coincidence with the dynamically unstable region of the phase space.

Figure 4

Probability of occupation of the excited state ($P_e$ proportional to the luminescence) as a function of the detuning $\delta$ for $\mathrm{\Omega }=\mathrm{\Gamma }$ and $\lambda =0.1,0.5,1,2$, and 3 (from left to right). The bistable behavior begins at $\lambda =2$.

Figure 5

Mapping of the luminescence line from ${\delta }^{\prime }$ to $\delta$. The points $A, B$, and $C$ correspond to cooling, neutral, and heating values, respectively. One can see that for $\delta =2$ both cooling and heating are possible, depending on value of $x$ that determines if the point $A$ or $C$ is actually occupied.

Figure 6

Luminescence: $\lambda =2, {\omega }_m/\mathrm{\Gamma }={10}^{-3}, k_{B}T/{\epsilon }_P=0.01, Q=10, {10}^2, {10}^3$, and ${10}^4$ for the curves from the steepest to the smoothest, respectively. Increasing $Q$ increases the fluctuations and smoothens the line shape. The parameter $Q{\omega }_m/\mathrm{\Gamma }$ entering Eq. (30) takes thus the values ${10}^{-2}, {10}^{-1}$, 1, and 10.

Figure 7

Effective temperature $k_B{T}_{\mathrm{eff}}/{\epsilon }_P$ as a function of the detuning $\delta$ for the same value of the quality factors of Fig. 4. The higher values of the temperature are of course obtained for the highest $Q$.