Spin-Boson Model
In this tutorial, we will use Positive P to study the behaviour of some examples of driven dissipative Jaynes-Cummings models and Jaynes-Cummings-Hubbard lattice models.
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Introduction
While there are many different models for coupled systems of spins and bosons depending on the exact physical arrangement and approximations being used, here we will focus on the Jaynes-Cummings interaction, where each qubit (Spin operators $\hat{S}^{X,Y,Z}_j = \frac{1}{2}\sigma^{X,Y,Z}_j$, $\hat{S}^\pm_j = \hat{S}^X_j \pm i\hat{S}^Y_j$, with $\sigma^{X,Y,Z}_j$ the usual $2\times2$ Pauli matrices) is coupled to a local bosonic mode with annihilation operator $\hat{a}_j$. The local Hamiltonian for each site is given by
\[\hat{H}^{JC}_j = \omega_C\hat{a}^\dagger_j\hat{a}_j +\omega_0\hat{S}_j^Z +g\left(\hat{a}_j\hat{S}_j^+ +\hat{a}^\dagger_j\hat{S}_j^-\right) +h\hat{S}_j^X +F\left(\hat{a}_j+\hat{a}^\dagger_j\right)\, ,\]with $\omega_C$ the boson energy, $\omega_0$ the qubit energy, $g$ the the strength of the Jaynes-Cummings interaction, and we have included coherent driving on both the spins $h$ and bosons $F$. For the case of many-site Jaynes-Cummings-Hubbard lattices, connected sites $(j,j’)$ are coupled by a bosonic hopping $J_{j,j’}$, leading to a total Hamiltonian
\[\hat{H} = \sum_j \hat{H}^{JC}_j -\sum_{j,j'} \left(J_{j,j'}\hat{a}^\dagger_j\hat{a}_{j'} + J^*_{j,j'}\hat{a}^\dagger_{j'}\hat{a}_j \right) \, .\]Including dissipation, the state of the system evolves according to the master equation
\[\frac{\partial\hat{\rho}}{\partial t} = -i\left[\hat{H}, \hat{\rho}\right] + \sum_j \left(\frac{\gamma}{2}D[\hat{a}_j] + \frac{\Gamma_{\downarrow}}{2}D[\hat{S}^-_j] + \frac{\Gamma_{\phi}}{2}D[\hat{S}^Z_j] \right)\, ,\]where $D[\hat{X}] = 2\hat{X}\hat{\rho}\hat{X}^\dagger - \hat{X}^\dagger\hat{X}\hat{\rho} - \hat{\rho}\hat{X}^\dagger\hat{X}$ are Lindbladian dissipation terms, for which we consider three kinds of dissipation: single particle bosonic dissipation $\gamma$, spin decay $\Gamma_\downarrow$, and spin dephasing $\Gamma_\phi$.
Single site
To be completed.
Jaynes-Cummings-Hubbard lattices
To be completed.