- #1
kelly0303
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Hello! This question is in relation to parity violation (PV) measurements using the optical rotation technique (I can give more details/references about that, but most of it is not relevant for my question). Basically, in a simplified model, they have 2 levels (say of positive parity), g and ##e_1## connected by a magnetic dipole amplitude ##A_{M_1} = <g|M_1|e_1>##. Another level ##e_2## close to ##e_1## (such that we can ignore its effect on g) has negative parity, thus, due to parity violation Hamiltonian, ##H_{PV}##, ##e_1## becomes:
$$|e_1'>=|e_1>+\frac{<e_1|H_{PV}|e_2>}{E_2-E_1}|e_2> = |e_1>+i\eta|e_2>$$
where ##E_1## and ##E_2## are the energies of the ##e_1## and ##e_2## levels (I might have messed up some signs, but that shouldn't matter for my question) and it can be shown that in general, the PV matrix element is always a purely imaginary number, hence ##i\eta = \frac{<e_1|H_{PV}|e_2>}{E_2-E_1}##. Now, in the experiments, people make use of the interference between the M1 transition and the PV effect, in order to amplify the latter one. In the 2D space spanned by g and ##e_1'##, the off diagonal matrix element is:
$$<g|M_1|e_1>+i\eta<g|E_1|e_2> = A_{M_1} + i\eta A_{E_1}$$
and the rate is the square of its modulus. However, in order to get interference i.e. a term proportional to ##\eta A_{M_1}A_{E_1}##, both terms must be either real or imaginary. However, given that ##i\eta## is purely imaginary, this implies, that in order to get the interference ##A_{M_1}## and ##A_{E_1}## should be one purely real and the other one purely imaginary. However, I am not sure I understand why and which is which. Can someone help me figure this out? Thank you!
$$|e_1'>=|e_1>+\frac{<e_1|H_{PV}|e_2>}{E_2-E_1}|e_2> = |e_1>+i\eta|e_2>$$
where ##E_1## and ##E_2## are the energies of the ##e_1## and ##e_2## levels (I might have messed up some signs, but that shouldn't matter for my question) and it can be shown that in general, the PV matrix element is always a purely imaginary number, hence ##i\eta = \frac{<e_1|H_{PV}|e_2>}{E_2-E_1}##. Now, in the experiments, people make use of the interference between the M1 transition and the PV effect, in order to amplify the latter one. In the 2D space spanned by g and ##e_1'##, the off diagonal matrix element is:
$$<g|M_1|e_1>+i\eta<g|E_1|e_2> = A_{M_1} + i\eta A_{E_1}$$
and the rate is the square of its modulus. However, in order to get interference i.e. a term proportional to ##\eta A_{M_1}A_{E_1}##, both terms must be either real or imaginary. However, given that ##i\eta## is purely imaginary, this implies, that in order to get the interference ##A_{M_1}## and ##A_{E_1}## should be one purely real and the other one purely imaginary. However, I am not sure I understand why and which is which. Can someone help me figure this out? Thank you!