Quantum Effects in a Mechanically Modulated Single-Photon Emitter

Recent observation of quantum emitters in monolayers of hexagonal boron nitride (h-BN) has provided a novel platform for optomechanical experiments where the single-photon emitters can couple to the motion of a freely suspended h-BN membrane. Here, we propose a scheme where the electronic degree of freedom (d.o.f.) of an embedded color center is coupled to the motion of the hosting h-BN resonator via dispersive forces. We show that the coupling of membrane vibrations to the electronic d.o.f. of the emitter can reach the strong regime. By suitable driving of a three-level $\mathrm{\Lambda }$-system composed of two spin d.o.f. in the electronic ground state as well as an isolated excited state of the emitter, a multiple electromagnetically induced transparency spectrum becomes available. The experimental feasibility of the efficient vibrational ground-state cooling of the membrane via quantum interference effects in the two-color drive scheme is numerically confirmed. More interestingly, the emission spectrum of the defect exhibits a frequency comb with frequency spacings as small as the fundamental vibrational mode, which finds applications in high-precision spectroscopy.

Figure 1

(a) Sketch for the proposed setup: The h-BN nanoribbon hosting a color center whose electronic ground and excited states form a $\mathrm{\Lambda }$ system. (b) Normalized dispersion force frequency pull of an h-BN emitter as a function of its distance from the surface; the solid green line corresponds to the average emission energies 1.95 eV while the shade depicts the inhomogenous broadening of the emitters. The blue and red triangles show the distances attained experimentally [20] and assumed in this Letter, respectively. (c) Absolute electromechanical coupling rate for a monolayer h-BN strip versus clamping strain for the parameter values $z=10\mathrm{nm}$, $L=1\mu \mathrm{m}$, $w=10\mathrm{nm}$, and ${\tau }_{eg}=3.2\mathrm{ns}$. For reference, the top axis gives the corresponding mechanical frequency. The dashed red and dotted blue lines separate the ultrastrong coupling region. The green diamond indicates our working point in the rest of Letter.

Figure 2

(a) EIT cooling efficiency as function of probe detuning and the control Rabi frequency. The probe Rabi frequency is set at ${\mathcal{E}}_p=0.1{\mathcal{E}}_c$ and ${\delta }_c=+10\mathrm{\Omega }$. (b) Absorption spectrum of the probe field in the absence (black) and presence (blue) of coupling to the mechanical mode. (c) Variations of the steady-state mechanical occupation number with probe drive detuning (solid green line). The absorption spectrum as well as the phonon assisted transitions are also shown for reference (see the text for a discussion). In (b) and (c) ${\mathcal{E}}_c=14\mathrm{\Omega }$ and ${\overline{N}}_{\mathrm{\Omega }}=210$ corresponding to the temperature $T=0.1\mathrm{K}$.

Figure 3

Electromagnetically induced signal: (a) At $t=10{\tau }_{\mathrm{m}}$ as a function of probe detuning and the strength of coupling to the mechanical mode (${\overline{N}}_{\mathrm{\Omega }}=210$). In (b)–(d) the ideal (${\overline{N}}_{\mathrm{\Omega }}=0$) steady-state EIT signal is illustrated at three different coupling rates: The solid lines are for numerical computation, while the dashed lines indicate analytical expression given in Eq. (4). The parameters used in these plots are ${\mathcal{E}}_c=10{\mathcal{E}}_p=\mathrm{\Omega }$ and ${\delta }_c=0$.

Figure 4

(a) Steady-state RFS of the emitter $S(t\to \infty ,\omega )$ at the two resonance frequencies ${\omega }_{eg}$ and ${\omega }_{eg}-{\mathrm{\Delta }}_0$ for three different coupling rates (${\overline{N}}_{\mathrm{\Omega }}=0$). (b) Time-resolved RFS for $G=5\mathrm{\Omega }$ and ${\overline{N}}_{\mathrm{\Omega }}=210$ at different times with system initialized with the ground state. The ideal steady-state spectrum is also shown for comparison (the thin line). In these plots we have set ${\delta }_c=0$, ${\delta }_p=\mathrm{\Omega }$, and ${\mathcal{E}}_c=10{\mathcal{E}}_p=\mathrm{\Omega }$. The curves are shifted vertically for clarity.