Correction to: Nature Communications https://doi.org/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:

$$a_{{\mathrm{\Delta }} \pm } = \frac{{({\mathrm{\Omega }}_{12}a_1 \pm \left( {\omega _{\Delta 0} \mp \Delta } \right) + {\mathrm{\Omega }}_{23}a_3)}}{{\sqrt {2\omega _{{\mathrm{\Delta }}0}(\omega _{{\mathrm{\Delta }}0} \mp \Delta )} }}.$$

The correct form of Equation 8 is:

$$a_{{\mathrm{\Delta }} \pm } = \frac{{({\mathrm{\Omega }}_{12}a_1 \pm \left( {\omega _{\Delta 0} \mp \Delta } \right)a_2 + {\mathrm{\Omega }}_{23}a_3)}}{{\sqrt {2\omega _{{\mathrm{\Delta }}0}(\omega _{{\mathrm{\Delta }}0} \mp \Delta )} }}.$$

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 \({\mathrm{\Omega }}_{13}\left( {{\mathrm{\alpha }}_1^\dagger {\mathrm{\alpha }}_3 + {\mathrm{\alpha }}_3^\dagger \alpha _1} \right){\mathrm{/}}2\). The magnitude of this effective coupling is equal to the frequency splitting between the nearly degenerate modes, with \({\mathrm{\Omega }}_3 = ({\mathrm{\Omega }}_{12}^2 + {\mathrm{\Omega }}_{23}^2){\mathrm{/}}2{\mathrm{\Delta }}\).’ These sentences have been replaced with ‘The nearly degenerate modes can be viewed as a hybridization of α1 and α3 with an effective splitting \(({\mathrm{\Omega }}_{12}^2 + {\mathrm{\Omega }}_{23}^2){\mathrm{/}}2{\mathrm{\Delta }}\).’

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

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.