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- Homework Statement
- Consider a two-level atom with the state space spanned by the two orthonormal states $$|1>$$ and $$|2>$$, dipole-coupled to an external time dependent driving field E(t). After a unitary transformation to the “rotating frame”, the Hamiltonian reads $$H/\hbar = ∆ |2><2| − f(t)[ |2><1|+|1><2|]$$, where ∆ is the difference between the atomic transition frequency ω0 and the frequency of the nearly monochromatic driving field ω, and f(t) is proportional to the temporal envelope of the driving field: E(t) ∝ f(t) cos(ωt). Suppose that $$f(t) =\frac{ λ }{2 \sqrt{\pi} \tau}(e^{-(\frac{ t + T/2}{\tau})^2}+e^{-(\frac{ t - T/2}{\tau})^2})$$ . This represents two pulses of light of length τ hitting the system at times ∓T/2. Let us assume that the amplitude of the driving field ∝ λ is “very small”, and that the system starts out in the state |1>.

- Relevant Equations
- $$i\hbar c_1(t)=<1|H'|2>c_2(t)$$

$$i\hbar c_2(t)=-f(t)c_2(t)$$

I am assuming this is the interaction picture, so I start with $$|\psi>=c_1(t)|1>+c_2(t)|2>$$. Plugging this into the Schrodinger equation,

I get the equations $$i\hbar c_1(t)=<1|H'|2>c_2(t)$$ and $$i\hbar c_2(t)=<1|H'|2>c_1(t)$$. I am assuming H' (the perturbation) is $$H'= − f(t)[ |2><2|+|1><1|]$$. From there, I get $$i\hbar c_1(t)=-f(t)c_2(t)$$, and $$i\hbar c_2(t)=-f(t)c_2(t)$$. Am I missing anything so far? I just want to make sure I haven't made any faults before continuing my calculation. I know this is supposed to be a derivation of Ramsey Fringes, it that is any help.

I get the equations $$i\hbar c_1(t)=<1|H'|2>c_2(t)$$ and $$i\hbar c_2(t)=<1|H'|2>c_1(t)$$. I am assuming H' (the perturbation) is $$H'= − f(t)[ |2><2|+|1><1|]$$. From there, I get $$i\hbar c_1(t)=-f(t)c_2(t)$$, and $$i\hbar c_2(t)=-f(t)c_2(t)$$. Am I missing anything so far? I just want to make sure I haven't made any faults before continuing my calculation. I know this is supposed to be a derivation of Ramsey Fringes, it that is any help.