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(I will not use the “hat” (\(\hat{}\)) for the operators unless it is strictly necessary)

The Rabi Hamiltonian for 2 qubit is \[\label{eqn:H2R} H_{2q}=\hbar \omega_0{a^{\dagger}}a + \hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z + \hbar(\Gamma_1\sigma^{(1)}_x + \Gamma_2\sigma^{(2)}_x)({a^{\dagger}}+ a)\].

Using the following displacement operator properties \[\begin{split} &{D^{\dagger}(\alpha)}aD(\alpha)=a + \alpha \\ &{D^{\dagger}(\alpha)}{a^{\dagger}}D(\alpha)={a^{\dagger}}+ \alpha^*\\ &{D^{\dagger}(\alpha)}D(\alpha)=D(\alpha){D^{\dagger}(\alpha)}={\mathbb{1}}\end{split}\]

the Hamiltonian is easily rewritten in term of the displacement operator \[H_{2q}=\hbar\omega_0 D^{\dagger}(\omega^{-1}_0 \hat{\gamma}^{\dagger}){a^{\dagger}}a D(\omega^{-1}_0 \hat{\gamma}) + \hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z - \omega^{-2}\hat{\gamma}^{\dagger}\hat{\gamma}\]

where \[\hat{\gamma}=\Gamma_1\sigma^{(1)}_x + \Gamma_2\sigma^{(2)}_x\].

Taking \(\Gamma_i\in\mathbb{R}\) and remembering \(\sigma_x^{\dagger}=\sigma_x\) one has \(\hat{\gamma}^{\dagger}=\hat{\gamma}\).

The eigenstates of Hamiltonian (\ref{eqn:H2R}) are the displaced Fock states....: \[|N_{nm}nm>={D^{\dagger}(\omega^{-1}\gamma_{nm})}|N>|n>|m>\] with \(N\in\mathbb{N}\), \(n,m=\{+,-\}\) and \(\gamma_{nm}=n\Gamma_1+m\Gamma_2\) is the eigenvalue of the operator \(\hat{\gamma}\) on the 2-qubits bases \(\{|nm>\}=\{|++>,|+->,|-+>,|-->\}\).

Negletting the costant therm of the Hamiltonian it becomes

\[H_{2q}=\hbar\omega_0 D^{\dagger}(\omega^{-1}_0 \hat{\gamma}^{\dagger}){a^{\dagger}}a D(\omega^{-1}_0 \hat{\gamma}) + \hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z - 2\hbar\frac{\Gamma_1\Gamma_2}{\omega^2}\sigma^{(1)}_x\otimes\sigma^{(2)}_x\]

Calculate now the matrix’s elements in the bases \(\{|N_{nm}nm\}\) in order to find later the eigenenergies of the system \[\begin{split} &<M_{st}st|H_1|N_{nm}nm>=\epsilon_N\delta_{sn}\delta_{tm}<M_{st}|N_{nm}>\\ &<M_{st}st|H_2|N_{nm}nm>=\epsilon_1(1-\delta_{sn})\delta_{tm}<M_{st}|N_{\bar{n}m}>+\epsilon_2\delta_{sn}(1-\delta_{tm})<M_{st}|N_{n\bar{m}}>\\ &<M_{st}st|H_3|N_{nm}nm>=-\epsilon_{12}(nm)\delta_{sn}\delta_{tm}<M_{st}|N_{nm}> \end{split}\] where \(\bar{n}\) is the negation of \(n\), \(H_1=\hbar\omega_0 D^{\dagger}(\omega^{-1}_0 \hat{\gamma}){a^{\dagger}}a D(\omega^{-1}_0 \hat{\gamma})\), \(H_2=\hbar \frac{\omega_1}{2}\sigma^{(1)}_z + \hbar \frac{\omega_2}{2}\sigma^{(2)}_z\), \(H_3=- 2\hbar\frac{\Gamma_1\Gamma_2}{\omega^2}\sigma^{(1)}_x\otimes\sigma^{(2)}_x\), \(\epsilon_N=\hbar\omega_0 N\), \(\epsilon_1=\hbar \frac{\omega_1}{2}\), \(\epsilon_2= \hbar \frac{\omega_2}{2}\) and \(\epsilon_{12}=2\hbar\frac{\Gamma_1\Gamma_2}{\omega^2}\). One can easily identify two diagonal terms and an off-diagonal one.


  1. CMS/CERN. A New Boson with a Mass of 125 GeV Observed with the CMS Experiment at the Large Hadron Collider. Science 338, 1569-1575 (2012). Link

  2. Barry R Holstein. The mysterious disappearance of Ettore Majorana. J. Phys.: Conf. Ser. 173, 012019 IOP Publishing, 2009. Link