We develop a matrix perturbation method for the Lindblad master equation. The first- and second-order corrections are obtained and the method is generalized for higher orders. The perturbation method developed is applied to the problem of a lossy cavity filled with a Kerr medium; the second-order corrections are estimated and compared with the known exact analytic solution. The comparison is done by calculating the

Open systems, that is, systems that interact with an environment, represent an important problem in many branches of physics such as cosmology [

In this work, we show that it is possible to implement a matrix perturbation method on the Lindblad master equation that allows us to determine in a simple and effective form the

The Lindblad master equation, which describes the interaction between a given system and its environment at zero-temperature, is given by [

The formal solution to the master equation is

To get the first-order correction to the nonperturbed solution to the master equation, we expand the exponential in (

The second-order correction to the nonperturbed solution to the master equation may be obtained if we take into account the terms in

The generalization of the method for higher-order corrections can be obtained directly from the results of the first-order and second-order corrections. So, following the same steps that take us to expression (

To demonstrate the accuracy and capability of the method, we obtain the perturbative solution to the master equation of a Kerr medium filling an optical cavity with losses. The exact analytic solution for the master equation in this case is [

The approximate solution for the cavity problem is found using the expression for

A simple and direct form to visualize the evolution of a cavity-Kerr system in phase space is calculating a quasi-probability function. The Husimi

Evolution of the

Evolution of the

As a second way to test the accuracy of the perturbation approximation, we proceed to calculate the mean photon number, which is a relevant physical quantity of the Kerr lossy cavity. Using the exact solution, we obtain

The mean photon number

The figure shows that approximate solutions with different values of

Finally, as another measure of proximity for the solutions, we evaluate the distance between the exact density matrix and the approximated density matrix [

Figure

Plot of

In summary, we can conclude that in the examined case, a lossy cavity filled with a Kerr medium, the matrix perturbative method gives good results. When time grows the results start to differ, but that is not surprising since the real measure of the perturbation is not given by only

The authors declare that they have no competing interests.

B. M. Villegas-Martínez acknowledges CONACYT for support.