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I'm currently doing some work in Quantum Optics and now have a question about the transition dipole moment.

We were told in class that the off-diagonal elements of this matrix, namely $\langle n|d|m\rangle$, gives us information about allowed and forbidden transitions of our defined system. For example if $$\langle 1|d|2\rangle\neq 0$$ then we could say that transitions from $|1\rangle$ to $|2\rangle$ and vice versa are possible.

In my work I've now come across a transition dipole moment for the Hamiltonian of a RF-SQUID qubit with a bias flux of $\Phi = \Phi_0/0.4$ which looks like the following

Transition dipole moment

I have some trouble interpreting the diagonal elements of this matrix since a transition from $|1\rangle$ to $|1\rangle$ for example doesn't make much sense to me. I'm kind of more used to matrices of this kind where the diagonal elements are all zero, for example the Harmonic oscillator. Would really appreciate it if someone could shed some light on this.

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The diagonal elements of the Hamiltonian give you information about the energy of a certain configuration. Not about the transitions.

For example, let's consider the Pairing Hamiltonian used in the BCS theory of superconductivity. Its form in second quantization is:

$$\hat{H} = \sum_{p} \varepsilon_p \hat{a}_p^\dagger\hat{a}_p - g \sum_{i}\sum_{j|j \neq i} \hat{a}_i^{\dagger} \hat{a}_j$$

This Hamiltonian describe a physical system with particles in different levels p and with transition coeficient -g. The coeficients $\varepsilon_p$ are the energies associated with the ocupation of the levels p.

Its matricial form for the case of 2 levels can be writen as:

$$ \[ \left( \begin{array}{cc} 0&0&0&0\\ 0&\varepsilon_0 & -g &0\\ 0& -g & \varepsilon_1&0\\ 0&0&0&\varepsilon_0 + \varepsilon_1 \end{array} \right) % \left| \begin{array}{cc} |00\rangle \\ |01\rangle \\ |10\rangle \\ |11\rangle \\ \end{array} \right| \] $$

The states $|b_1 b_0\rangle$ denote the possible configurations of the sysmtem and the value of the bit $b_i$ denote the occupation of the levels. For example, the state $|10\rangle$ corresponds to 1 particle in the level 1 and the level 0 empty.

You can see directly that the energy of the state $|10\rangle$ correspond to the matrix element $\langle 10 | H | 10 \rangle = \varepsilon_1$. Likewise, the energy of both levels occupaid is given by $\langle 11 | H | 11 \rangle = \varepsilon_0 + \varepsilon_1$.

A more graphical view of the problem can be seen as follows:

schematic_pairing

In this case there are 5 levels and 3 particles in the levels 1, 3 and 4.The energy of this particular configuration is given by $\langle 010110|H|010110 \rangle = \varepsilon_1 + \varepsilon_3 + \varepsilon_4$.

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