Exceptional Points in Random-Defect Phonon Lasers

Intrinsic defects in optomechanical devices are generally viewed to be detrimental for achieving coherent amplification of phonons, and great care has thus been exercised in fabricating devices and materials with no (or a minimal number of) defects. Contrary to this view, here we show that, by surpassing an exceptional point (EP), both the mechanical gain and the phonon number can be enhanced despite increasing defect losses. This counterintuitive effect, well described by an effective non-Hermitian phonon-defect model, provides a mechanical analog of the loss-induced purely optical lasing. This opens the way to operating random-defect phonon devices at EPs.

Figure 1

Schematic illustrations of the random-defect phonon laser. (a) The top resonator, coupled to a tapered fiber, supports a mechanical mode $b$ [3] and contains a defect-induced TLS. (b) The optical mode $a_1$, which is coupled to mode $a_2$ with strength $J$, interacts with $b$, which, in turn, is coupled to the TLS with the strength $g_d$. The TLS decay rate is denoted by ${\gamma }_q$, and ${\gamma }_m^{\prime }={\gamma }_m-G_0$ is the effective mechanical damping rate, while $G_0$ is the mechanical gain.

Figure 2

(a) The mechanical gains $G_0$ (without defect) and $G$ (with defect) as a function of the optical detuning $\mathrm{\Delta }$. (b) $G$ as a function of the defect loss ${\gamma }_q$. Here, for simplicity, we use ${\gamma }_q/\gamma =1$, $g_d=1\mathrm{MHz}$ in (a), $\mathrm{\Delta }=0.5{\omega }_m$ in (b), and $J=0.5{\omega }_m$ and $P_l=10\mu \mathrm{W}$ in both (a) and (b).

Figure 3

(a) Mechanical gain $G$ in a defect-assisted phonon laser versus the pump-cavity detuning $\mathrm{\Delta }$. (b) The threshold power $P_{\mathrm{th}}$ of the defect-assisted phonon laser versus the TLS decay rate ${\gamma }_q$. The parameters used here are (a) ${\gamma }_q=\gamma$, $g_d=1\mathrm{MHz}$ and (b) $J=0.5{\omega }_m$, $\mathrm{\Delta }=0.5{\omega }_m$. We choose ${\omega }_q={\omega }_m$ and $P_l=10\mu \mathrm{W}$ in both (a) and (b).

Figure 4

Supermode spectrum of the reduced system, i.e., the (a) real and (b) imaginary parts of the eigenvalues of ${\mathcal{H}}_{\mathrm{eff}}$, with the experimentally accessible values of the tunable parameters $J=0.5{\omega }_m$, ${\gamma }_q/\gamma =1$, and $P_l=7\mu \mathrm{W}$ (the threshold value as measured in the experiment [3]).

Figure 5

Mechanical gain $G$ as a function of the TLS loss rate ${\gamma }_q$, for different values of $\mathrm{\Delta }$. The parameters used here are $J=0.5{\omega }_m$, ${\omega }_q/{\omega }_m=1$, and $P_l=10\mu \mathrm{W}$.

Figure 6

Stimulated emitted phonon number. The phonon number $N_b$ versus (a) the pump power $P_l$ and (b) the damping rate ${\gamma }_q$, with ${\omega }_q/{\omega }_m=1$. Also, we take the achievable values (a) $P_l=10\mu \mathrm{W}$ and (b) ${\gamma }_q/\gamma =1$, $g_d=1\mathrm{MHz}$.