Multiphonon interactions between nitrogen-vacancy centers and nanomechanical resonators

We investigate the multiphonon interactions in a hybrid system composed of a nitrogen-vacancy center and a mechanical resonator. We show that, through appropriate sideband engineering analogy to the Mollow or Lamb-Dicke dynamics, the enhanced nonlinear interactions can dominate the coupled system. As an example, we show the preparation of nonclassical states of the mechanical motion and explore the quantum correlation of $n$-phonon bundles with the engineered dissipation, based on the large multiphonon coupling strength attainable in this sideband structure. This work may take full advantage of the structure and coherence features of defect centers in diamond and may be useful for quantum information processing.

Figure 1

Schematic of the setup. A sharp magnetic tip is attached at the end of a silicon nanomechanical cantilever of dimensions ($l,w,t$). A single NV center is positioned below the tip at a distance of $h$, and is immersed in a static magnetic field. Additionally, microwave fields are added to drive the defect center.

Figure 2

Level diagrams with respect to the Mollow and Lamb-Dicke sideband engineering schemes. (a) The level diagram of the NV center driven by two microwave fields. The two driving fields have the same detunings and Rabi frequencies. (b) Dressed spin states in the large detuning limit corresponding to (a). The dressed states $|d\rangle$ and $|b\rangle$ are resonantly driven by microwave fields. (c) The level diagram of the NV center driven by a weak microwave field.

Figure 3

(a) Curve of the energy eigenvalues with the pumping $\mathrm{\Omega }$ increasing from 0 to $20\lambda$. The result is calculated from Eq. (9) at ${\mathrm{\Delta }}_a=5\lambda$. This plot only contains the second to the ninth energy eigenvalues. (b) Enlarged view of the spectral region delimited by a square in (a).

Figure 4

(a) Time evolution of the phonon number. (b) Wigner function of the mechanical mode at the steady state given by the density matrix. (c), (d) Wigner function of the mechanical mode given by the wave function of a single trajectory. This two pictures depict a jumping cat at times before and after a phonon damping process at time $t_1$. Here, ${\lambda }^{\left(2\right)}=2\times {10}^{-2}\lambda , {\mathrm{\Omega }}_0=10{\lambda }^{\left(2\right)}, (n_{\text{th}}+1){\gamma }_m=0.5{\lambda }^{\left(2\right)}$, and ${\gamma }_s=0.05{\lambda }^{\left(2\right)}$. The numerical results are based on Eq. (21) and Eq. (22).

Figure 5

Time evolution of the probabilities for the resonator being in certain number states $|n\rangle$, with the initial state $\left|\psi \right(t=0)\rangle =|+,0\rangle$. The relevant dissipation parameters are ${\gamma }_s={10}^{-3}\lambda , {\gamma }_m=5\times {10}^{-5}\lambda$, and $n_{th}=40$. (a) One-phonon resonance, with ${\mathrm{\Delta }}_a=10\lambda$ and $\mathrm{\Omega }=5\lambda$; (b) two-phonon resonance, with ${\mathrm{\Delta }}_a=5\lambda$ and $\mathrm{\Omega }=5\lambda$; (c) three-phonon resonance, with ${\mathrm{\Delta }}_a=5\lambda$ and $\mathrm{\Omega }=7.5\lambda$; (d) four-phonon resonance, with ${\mathrm{\Delta }}_a=3\lambda$ and $\mathrm{\Omega }=6\lambda$.

Figure 6

Second-order phonon correlations at the $N=1$ (dashed, red) and $N=2$ (solid, blue) level at two-phonon resonance, with quadratic coupling strength ${\lambda }^{\left(2\right)}={10}^{-2}\lambda$, dephasing rate ${\gamma }_s={10}^2{\lambda }^{\left(2\right)}$, and damping rate $(n_{\text{th}}+1){\gamma }_m=10{\lambda }^{\left(2\right)}$. (a) Short effective lifetime of the spin qubit with ${\mathrm{\Gamma }}_0=10{\lambda }^{\left(2\right)}$; (b) long effective lifetime of the spin qubit with ${\mathrm{\Gamma }}_0=0.25{\lambda }^{\left(2\right)}$.

Figure 7

Population of the two-phonon state on two-phonon resonance under different noise strength. The detuning is $\mathrm{\Delta }\approx 2\pi \times 250$ MHz and the driving amplitude is ${\mathrm{\Omega }}_x\approx 2\pi \times 25$ MHz, with $\lambda \approx 2\pi \times 100$ kHz, ${\mathrm{\Delta }}_a=8\lambda , \mathrm{\Omega }=7.94\lambda , {\gamma }_s={10}^{-3}\lambda , {\gamma }_m=5\times {10}^{-5}\lambda$, and $n_{\text{th}}=40$.