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## Main Question or Discussion Point

Assume there is a two level system, two eigenstates are written as

[tex]|\psi_1\rangle = \cos\theta |1, g\rangle + \sin\theta |0, e\rangle[/tex]

and

[tex]|\psi_2\rangle = -\sin\theta |1, g\rangle + \cos\theta |0, e\rangle[/tex]

For the density operator of the system is written as

[tex]\rho = \frac{1}{2}|\psi_1\rangle\langle \psi_1| + \frac{1}{2}|\psi_2\rangle\langle \psi_2| = \frac{1}{2}|1, g\rangle\langle 1, g| + \frac{1}{2}|0, e\rangle\langle 0, e|[/tex]

where [tex]g[/tex] stands for ground state, [tex]e[/tex] stands for excited state, 0 and 1 stands for the number of photon.

If the inital state of the system is in [tex]|0, e\rangle[/tex], what's the probability of transition from [tex]|0, e\rangle \to |1, g\rangle[/tex] ? I am quite confuse how to use density operator to find the probability, shoud it be

[tex]\langle1, g|\rho|0, e\rangle[/tex]

or

[tex]\left|\langle1, g|\rho|0, e\rangle\right|^2[/tex] ?

[tex]|\psi_1\rangle = \cos\theta |1, g\rangle + \sin\theta |0, e\rangle[/tex]

and

[tex]|\psi_2\rangle = -\sin\theta |1, g\rangle + \cos\theta |0, e\rangle[/tex]

For the density operator of the system is written as

[tex]\rho = \frac{1}{2}|\psi_1\rangle\langle \psi_1| + \frac{1}{2}|\psi_2\rangle\langle \psi_2| = \frac{1}{2}|1, g\rangle\langle 1, g| + \frac{1}{2}|0, e\rangle\langle 0, e|[/tex]

where [tex]g[/tex] stands for ground state, [tex]e[/tex] stands for excited state, 0 and 1 stands for the number of photon.

If the inital state of the system is in [tex]|0, e\rangle[/tex], what's the probability of transition from [tex]|0, e\rangle \to |1, g\rangle[/tex] ? I am quite confuse how to use density operator to find the probability, shoud it be

[tex]\langle1, g|\rho|0, e\rangle[/tex]

or

[tex]\left|\langle1, g|\rho|0, e\rangle\right|^2[/tex] ?