Table 1 Fluctuation-dissipation theorem for time-varying systems: First and second-order current density correlations

\(\begin{array}{l}\hskip4pt{\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}^{{{{\dagger}}} }({{{{{{{\boldsymbol{\rho }}}}}}}};\omega )\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{1}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} };{\omega }^{{\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}} \,=\, {\omega }^{{\prime} }\left[\frac{{\mu }_{0}}{\pi }\right]{\int}_{{{{{{{{\mathcal{V}}}}}}}}}{d}^{3}{{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime\prime} }\int\,d{\omega }^{{\prime\prime} }{\omega }^{{\prime\prime} }\Delta \tilde{\chi }({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{\omega }^{{\prime} }-{\omega }^{{\prime\prime} }){{{{{{{\bf{G}}}}}}}}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime\prime} },{\omega }^{{\prime\prime} }){\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}^{{{{\dagger}}} }({{{{{{{\boldsymbol{\rho }}}}}}}};\omega )\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime\prime} };{\omega }^{{\prime\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}};\\ {\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}^{{{{\dagger}}} }({{{{{{{\boldsymbol{\rho }}}}}}}};\omega )\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{2}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} };{\omega }^{{\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}}=\, {\omega }^{{\prime} }\left[\frac{{\mu }_{0}}{\pi }\right]{\int}_{{{{{{{{\mathcal{V}}}}}}}}}{d}^{3}{{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime\prime} }\int\,d{\omega }^{{\prime\prime} }{\omega }^{{\prime\prime} }\Delta \tilde{\chi }({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{\omega }^{{\prime} }-{\omega }^{{\prime\prime} }){{{{{{{\bf{G}}}}}}}}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime\prime} },{\omega }^{{\prime\prime} }){\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}^{{{{\dagger}}} }({{{{{{{\boldsymbol{\rho }}}}}}}};\omega )\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{1}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime\prime} };{\omega }^{{\prime\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}}\\ \hskip5pt {\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{1}^{{{{\dagger}}} }({{{{{{{\boldsymbol{\rho }}}}}}}};\omega )\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{1}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} };{\omega }^{{\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}} \,=\, \omega {\omega }^{{\prime} }{\left[\frac{{\mu }_{0}}{\pi }\right]}^{2}{\iint }_{{{{{{{{\mathcal{V}}}}}}}}}{d}^{3}\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}}{d}^{3}{\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}}}^{{\prime} }\iint d\tilde{\omega }d{\tilde{\omega }}^{{\prime} }\tilde{\omega }{\tilde{\omega }}^{{\prime} }\Delta {\tilde{\chi }}^{*}({{{{{{{\boldsymbol{\rho }}}}}}}},\omega -\tilde{\omega })\Delta \tilde{\chi }({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{\omega }^{{\prime} }-{\tilde{\omega }}^{{\prime} })\hfill\\ \left[{{{{{{{{\bf{G}}}}}}}}}^{*}({{{{{{{\boldsymbol{\rho }}}}}}}},\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}},\tilde{\omega }){{{{{{{\bf{G}}}}}}}}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}}}^{{\prime} },{\tilde{\omega }}^{{\prime} }){\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}^{{{{\dagger}}} }(\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}};\tilde{\omega })\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}({\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}}}^{{\prime} };{\tilde{\omega }}^{{\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}}+{{{{{{{\bf{G}}}}}}}}({{{{{{{\boldsymbol{\rho }}}}}}}},\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}},\tilde{\omega }){{{{{{{{\bf{G}}}}}}}}}^{*}({{{{{{{{\boldsymbol{\rho }}}}}}}}}^{{\prime} },{\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}}}^{{\prime} },{\tilde{\omega }}^{{\prime} }){\langle {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}(\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}};\tilde{\omega })\cdot {\hat{{{{{{{{\bf{j}}}}}}}}}}_{0}^{{{{\dagger}}} }({\tilde{{{{{{{{\boldsymbol{\rho }}}}}}}}}}^{{\prime} };{\tilde{\omega }}^{{\prime} })\rangle }_{{{{{{{{\rm{th}}}}}}}}}\right]\hskip-175pt\end{array}\)