Derivation of the Keldysh Lagrangian from a Lindblad Master Equation
This document presents a derivation of the Keldysh Lagrangian for a driven–dissipative Kerr oscillator following.
1. Starting Point: The Lindblad Master Equation
The Lindblad master equation for the density matrix evolution is given by
\[\partial_t \rho = -i\,[H,\rho] + \kappa\Bigl(2a\,\rho\,a^\dagger - \{a^\dagger a,\rho\}\Bigr),\]where the Hamiltonian is
\[H = \delta\,a^\dagger a + \chi\,a^\dagger a^\dagger aa + i\varepsilon\,(a^\dagger - a).\]The Lindblad operator can be expressed as
\[L = \sqrt{2\kappa}\,a,\]so that the dissipative part is in the standard Lindblad form.
2. Matrix Element of the Liouvillian Superoperator
In the coherent-state representation, we work with forward ($a_+$) and backward ($a_-$) contours. Denoting the fields by $a_+$ and $a_-$ we define the matrix element of the Liouvillian superoperator as
\[L_{\rm super}(a_+^*,a_+,a_-^*,a_-) = -i\Bigl[H(a_+^*,a_+) - H(a_-^*,a_-)\Bigr] + \kappa\Bigl[2\,a_+a_-^* - a_+^*a_+ - a_-^*a_-\Bigr].\]Here, the Hamiltonian evaluated on each contour is
\[H(a_\pm^*,a_\pm) = \delta\,a_\pm^*a_\pm + \chi\,a_\pm^{*2}a_\pm^2 + i\varepsilon\,(a_\pm^* - a_\pm).\]3. Constructing the Keldysh Lagrangian
Following the recipe, the full Keldysh Lagrangian is written as
\[L_K = a_+^*\,i\partial_t a_+ - a_-^*\,i\partial_t a_- - i\,L_{\rm super}(a_+^*,a_+,a_-^*,a_-).\]Substituting the expression for $L_{\rm super}$, we have
\[\begin{aligned} L_K &= a_+^*\,i\partial_t a_+ - a_-^*\,i\partial_t a_- \\ &\quad {} - i\Bigl\{-i\Bigl[H(a_+^*,a_+) - H(a_-^*,a_-)\Bigr] + \kappa\Bigl[2\,a_+a_-^* - a_+^*a_+ - a_-^*a_-\Bigr]\Bigr\} \\ &= a_+^*\,i\partial_t a_+ - a_-^*\,i\partial_t a_- + \Bigl[H(a_+^*,a_+) - H(a_-^*,a_-)\Bigr] \\ &\quad {} - i\,\kappa\Bigl[2\,a_+a_-^* - a_+^*a_+ - a_-^*a_-\Bigr]. \end{aligned}\]Inserting the explicit form of the Hamiltonian difference
\[\begin{aligned} H(a_+^*,a_+) - H(a_-^*,a_-) &= \delta\,\Bigl(a_+^*a_+ - a_-^*a_-\Bigr) + \chi\,\Bigl(a_+^{*2}a_+^2 - a_-^{*2}a_-^2\Bigr) \\ &\quad {} + i\varepsilon\Bigl[(a_+^* - a_+) - (a_-^* - a_-)\Bigr], \end{aligned}\]the final form of the Keldysh Lagrangian becomes
\[\boxed{ \begin{aligned} L_K =\; & a_+^*\,i\partial_t a_+ - a_-^*\,i\partial_t a_- \\ & - \delta\,\Bigl(a_+^*a_+ - a_-^*a_-\Bigr) - \chi\,\Bigl(a_+^{*2}a_+^2 - a_-^{*2}a_-^2\Bigr) \\ & - i\varepsilon\,\Bigl[(a_+^* - a_+) - (a_-^* - a_-)\Bigr] \\ & - i\,\kappa\Bigl[2\,a_+a_-^* - a_+^*a_+ - a_-^*a_-\Bigr]. \end{aligned} }\]