Author Correction: Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity

Correction to: Nature Communications 10.1038/s41467-018-02854-4; published online 26 January 2018

The original version of this Article contained an error in the last sentence of the second paragraph of the ‘Raman-like coupling between well-separated resonators’ section of the Results, which incorrectly read ‘On the contrary, when the detuning Δ₁₂ is lowered to ~180 kHz, a distinct avoided level crossing between modes R₁ and R₃ is observed, as shown inside the dashed circle in Fig. 2b.’ The correct version states ‘Δ₁₂/2π’ in place of ‘Δ₁₂.’

Additionally, the first sentence of the ‘Theory of three-mode coupling’ section of the Methods originally incorrectly read ‘We describe this three-mode system with the Hamiltonian (ħ = 4).’ The correct version states ‘ħ = 1’ instead of ‘ħ = 4’.

Equation 8 was originally missing a factor of ‘a₂’ from the second term, and incorrectly read:

$$a_{\mathrm{\Delta }\pm }=\frac{\left({\mathrm{\Omega }}_{12}{a}_1\pm \left({\omega }_{\Delta 0}\mp \Delta \right)+{\mathrm{\Omega }}_{23}{a}_3\right)}{\sqrt{2{\omega }_{\mathrm{\Delta }0}\left({\omega }_{\mathrm{\Delta }0}\mp \Delta \right)}}.$$

The correct form of Equation 8 is:

$$a_{\mathrm{\Delta }\pm }=\frac{\left({\mathrm{\Omega }}_{12}{a}_1\pm \left({\omega }_{\Delta 0}\mp \Delta \right)a_2+{\mathrm{\Omega }}_{23}{a}_3\right)}{\sqrt{2{\omega }_{\mathrm{\Delta }0}\left({\omega }_{\mathrm{\Delta }0}\mp \Delta \right)}}.$$

The eleventh and twelfth sentences of the second paragraph of the ‘Theory of three-mode coupling’ section of the Methods originally incorrectly read ‘The nearly degenerate modes can be viewed as a hybridization of α₁ and α₃ with an effective coupling ${\mathrm{\Omega }}_{13}\left({\mathrm{\alpha }}_1^{†}{\mathrm{\alpha }}_3+{\mathrm{\alpha }}_3^{†}{\alpha }_1\right)\mathrm{∕}2$. The magnitude of this effective coupling is equal to the frequency splitting between the nearly degenerate modes, with ${\mathrm{\Omega }}_3=\left({\mathrm{\Omega }}_{12}^2+{\mathrm{\Omega }}_{23}^2\right)\mathrm{∕}2\mathrm{\Delta }$.’ These sentences have been replaced with ‘The nearly degenerate modes can be viewed as a hybridization of α₁ and α₃ with an effective splitting $\left({\mathrm{\Omega }}_{12}^2+{\mathrm{\Omega }}_{23}^2\right)\mathrm{∕}2\mathrm{\Delta }$.’

In Equation 10, the term on the left-hand side was originally incorrectly given as ‘M’. The corrected version changes this to ‘M_eff’.

Finally, the original version of this Article contained an error in Fig. 3b, in which the units on the y-axis were incorrectly given as ‘MHz’, rather than the correct ‘kHz’.

This has been corrected in both the PDF and HTML versions of the Article.