The master equation of the damped harmonic oscillator is(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

\frac{d}{dt}\rho_S(t)

=

-i\omega_0

[a^\dagger a,\rho_S(t)]

+

\gamma_0(\bar n+1)

\{

a\rho_S(t) a^\dagger

-\frac{1}{2}

a^\dagger a \rho_S(t)

-\frac{1}{2}

\rho_S(t) a^\dagger a

\}

+

\gamma_0\bar n

\{

a^\dagger

\rho_S(t)

a

-\frac{1}{2}

a a^\dagger \rho_S(t)

-\frac{1}{2}

\rho_S(t)

a

a^\dagger

\}

\equiv

\mathcal{L}\rho_S(t).

[/itex]

Is there any method to analytically find out some eigenvalues, like the smallest three eigenvalues, of the liouvillian?

I am an engineering graduate, but now a rookie in physics.

Thanks for any advices.

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# Find the eigenvalues of Liouvillian

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