Correction to: Nature Photonics https://doi.org/10.1038/s41566-021-00771-5, published online 4 March 2021.

In the PDF version of this Article originally published, equation (1) was incorrect as:

$$\begin{array}{l}\frac{\partial }{{\partial t}}\left( {R_xx_yy} \right) = \frac{c}{{n_{\mathrm{g}}}}\left[ { - i\delta + \frac{{{\it{\Gamma}} g\left( N \right) - \alpha _{{\mathrm{in}}}}}{2}} \right]\left( {R_xx_yy} \right)\\ - \frac{c}{{n_{\mathrm{g}}}}\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\partial R_x/\partial x} \\ { - \partial S_x/\partial x} \\ {\partial R_y/\partial y} \end{array}} \\ { - \partial S_y/\partial y} \end{array}} \right) - \gamma \left( {R_xx_yy} \right) + \frac{{ic}}{{n_{\mathrm{g}}}}C\left( {R_xx_yy} \right) + \left( {f_12_34} \right)\end{array}$$

It should have read:

$$\begin{array}{l}\frac{\partial }{{\partial t}}\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {R_x} \\ {S_x} \\ {R_y} \end{array}} \\ {S_y} \end{array}} \right) = \frac{c}{{n_{\mathrm{g}}}}\left[ { - i\delta + \frac{{{\it{\Gamma}} g\left( N \right) - \alpha _{{\mathrm{in}}}}}{2}} \right]\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {R_x} \\ {S_x} \\ {R_y} \end{array}} \\ {S_y} \end{array}} \right)\\ - \frac{c}{{n_{\mathrm{g}}}}\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\partial R_x/\partial x} \\ { - \partial S_x/\partial x} \\ {\partial R_y/\partial y} \end{array}} \\ { - \partial S_y/\partial y} \end{array}} \right) - \gamma \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {R_x} \\ {S_x} \\ {R_y} \end{array}} \\ {S_y} \end{array}} \right) + \frac{{ic}}{{n_{\mathrm{g}}}}C\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {R_x} \\ {S_x} \\ {R_y} \end{array}} \\ {S_y} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {f_1} \\ {f_2} \\ {f_3} \end{array}} \\ {f_4} \end{array}} \right)\end{array}$$

This has now been corrected.