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theuselessone

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## Homework Statement

I've been reading through

*Quantum Optics*by Scully and Zubairy and have been stuck in section 6.2 getting from the definition of the atomic inversion given by equation (6.2.20, pg 199)

[itex]W(t)=\sum_{n}\left[|c_{a,n}(t)|^2-|c_{b,n}(t)|^2\right][/itex]

to the atomic inversion for the Jaynes-Cummings model Hamiltonian between a single-mode quantized field and a single two-level atom given by (6.2.21, pg 200)

[itex]W(t)=\sum_{n=0}^{\infty}\rho_{nn}(0)\left[\frac{\Delta^2}{\Omega_n^2}+\frac{4g^2(n+1)}{\Omega_n^2}\cos(\Omega_n t)\right][/itex]

where [itex]|c_n(0)|^2=\rho_{nn}(0)[/itex] is the initial probability of n-photons. The coefficient [itex]|c_{a,n}(t)|^2[/itex] is the probability of being in atomic state [itex]\left|a\right\rangle[/itex] and photonic state [itex]\left|n\right\rangle[/itex] (similarly for [itex]|c_{b,n}(t)|^2[/itex]). In the book, they solve for the case of the atom initially in the excited state [itex]\left|a\right\rangle[/itex] such that [itex]c_{a,n}(0)=c_n(0)[/itex] and [itex]c_{b,n+1}(0)=0[/itex].

## Homework Equations

Computing [itex]|c_{a,n}(t)|^2[/itex] and [itex]|c_{b,n}(t)|^2[/itex] using equations (6.2.16) and (6.2.17), respectively (or reading ahead to 6.2.18), I get

[itex]|c_{a,n}(t)|^2=\rho_{nn}(0)\left[\cos^2(\frac{\Omega_n t}{2})+\frac{\Delta^2}{\Omega_n^2}\sin^2(\frac{\Omega_n t}{2})\right][/itex]

and

[itex]|c_{b,n}(t)|^2=\rho_{n-1,n-1}(0)(\frac{4g^2n}{\Omega_{n-1}^2})\sin^2(\frac{\Omega_{n-1}t}{2})[/itex]

## The Attempt at a Solution

Plugging these into (6.2.20), I find

[tex]

\begin{align}

W(t)&=\sum_{n=0}^\infty\rho_{nn}(0)\left[\cos^2(\frac{\Omega_n t}{2})+\frac{\Delta^2}{\Omega_n^2}\sin^2(\frac{\Omega_n t}{2})\right]-\sum_{n=0}^\infty\rho_{n-1,n-1}(0)(\frac{4g^2n}{\Omega_{n-1}^2})\sin^2(\frac{\Omega_{n-1}t}{2}) \\

&=\sum_{n=0}^\infty\rho_{nn}(0)\left[\cos^2(\frac{\Omega_n t}{2})+\frac{\Delta^2}{\Omega_n^2}\sin^2(\frac{ \Omega_n t}{2})\right]-\sum_{n=-1}^\infty\rho_{nn}(0)(\frac{4g^2(n+1)}{\Omega_{n}^2})\sin^2(\frac{\Omega_{n}t}{2})

\end{align}

[/tex]

And that's where I get stuck... So far, the only justification I've come up with is that [itex]\rho_{nn}(0)=0[/itex] for [itex]n<0[/itex] since there shouldn't be less than zero photons. I'm able to get the desired equation (6.2.21) if the second sum runs from [itex]n=0[/itex] to [itex]n=\infty[/itex] instead of starting from [itex]n=-1[/itex].

Thanks in advance for any help.

Note: It's showing \Omeg[itex]a_n t[/itex] instead of [itex]\Omega_n t[/itex] for me in the last equation above.

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