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Find the eigenvalues of Liouvillian

  1. Aug 16, 2011 #1
    The master equation of the damped harmonic oscillator is
    [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.
     
  2. jcsd
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