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Let suppose we have N number of particles in two level system. .Effective number of cobosons in ground state is $ <n_0>$ that can be written as $<\hat{n}_0>= Tr [\hat{n_0}\rho ]$ where $\hat{n}_0$ is number operator of particle while $\rho$ is density operator. $<\hat{n}_0>= \sum\limits_{n=0}^{N}$ $ e^{-(E_0 )n/kT} e^{-(N-n)E_1/kT}/Z . \rho$
where $Z = \sum\limits_{n=0}^{N}$ $e^{-(E_0 )n/kT} e^{-(N-n)E_1/kT}$

For multilevel system like 3D harmonic trap

$<\hat{N}_m>= \sum\limits_{n=0}^{\infty}$ $ e^{-(E_m -\mu)n/kT}/Z_m . \rho$
where $Z_m =(1- e^{-(E_m -\mu)/kT} )^{-1}$. My question is that why for two level system they considered both energy level $E_0$ and $E_1$ but for multilevel they consider specific energy level $E_m$ in which we want to find effective number of particles.

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  • $\begingroup$ where did you find this from? I don't really understand what you're saying, so maybe looking at the original reference may help. $\endgroup$ – SuperCiocia Dec 13 '18 at 0:38

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