- #1
lutherblissett
Dear all,
I am encoutering some difficulties while calculating the Hamiltonian after the transformation to the interaction picture. I am following the tutorial by Sasura and Buzek:
https://arxiv.org/abs/quant-ph/0112041
Previous:
I already know that the Hamiltonian for the j-th ion is given by two terms:
[tex] \hat{H} = \hat{H}_{0j} + \hat{V}_j [/tex]
where the two terms are the free Hamiltonian and the interaction hamiltonian
[tex] \hat{H}_{0j} = \dfrac{1}{2} \hbar \omega_0 \sigma_{zj} + \nu \hat{a}^\dagger \hat{a} [/tex]
[tex] \hat{V}_j = -q_e\left [ \left (\mathbf{r}_{eg}\right )_j \hat{\sigma}_{+j} + \left (\mathbf{r}_{eg}\right )^*_j \hat{\sigma}_{-j} \right ] \cdot \dfrac{E_0 \varepsilon }{2} \left \{ e^{ -i \left [ \omega_L t - \eta_j \left ( \hat{a}^\dagger + \hat{a} \right ) + \phi_j \right ] } + \text{ h.c.} \right \} [/tex]
I don't understand how one can obtain the transformed Hamiltonian [itex] \hat{\mathcal{H}} = \hat{U}^\dagger_0 \hat{V} \hat{U}_0 [/itex] where [itex] \hat{U}_0 = exp \left \{ - \dfrac{i}{\hbar} \hat{H}_0 t \right \} [/itex].
Could someone explain me the steps to obtain it, please? I will be very grateful :D
thank you in advance
LB
I am encoutering some difficulties while calculating the Hamiltonian after the transformation to the interaction picture. I am following the tutorial by Sasura and Buzek:
https://arxiv.org/abs/quant-ph/0112041
Previous:
I already know that the Hamiltonian for the j-th ion is given by two terms:
[tex] \hat{H} = \hat{H}_{0j} + \hat{V}_j [/tex]
where the two terms are the free Hamiltonian and the interaction hamiltonian
[tex] \hat{H}_{0j} = \dfrac{1}{2} \hbar \omega_0 \sigma_{zj} + \nu \hat{a}^\dagger \hat{a} [/tex]
[tex] \hat{V}_j = -q_e\left [ \left (\mathbf{r}_{eg}\right )_j \hat{\sigma}_{+j} + \left (\mathbf{r}_{eg}\right )^*_j \hat{\sigma}_{-j} \right ] \cdot \dfrac{E_0 \varepsilon }{2} \left \{ e^{ -i \left [ \omega_L t - \eta_j \left ( \hat{a}^\dagger + \hat{a} \right ) + \phi_j \right ] } + \text{ h.c.} \right \} [/tex]
I don't understand how one can obtain the transformed Hamiltonian [itex] \hat{\mathcal{H}} = \hat{U}^\dagger_0 \hat{V} \hat{U}_0 [/itex] where [itex] \hat{U}_0 = exp \left \{ - \dfrac{i}{\hbar} \hat{H}_0 t \right \} [/itex].
Could someone explain me the steps to obtain it, please? I will be very grateful :D
thank you in advance
LB
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