10. Spin Dynamics in spin and phonon baths

This is a module for studying the quantum dynamics of spins, in the presence of both spin and phonon baths..

10.1. Hamiltonian

The package allows one to simulate the dynamics of a central spin or multiple central spins interacting with spin and/or honon bath through the following Hamiltonian:

\[\hat H = \hat H_S + \hat H_{SB} + \hat H_{B}\]

Where \(\hat H_S\) is the Hamiltonian of the central spin(s). \(\hat H_{SB}=\hat H_{SP} + \hat H_{SN}\) denotes interactions between central spin(s) and spin/phonon bath. \(\hat H_B=\hat H_{P} + \hat H_{N}\) are intrinsic spin/phonon bath interactions. For a single central spin, this corresponds to the following Hamiltonian:

\[\begin{split}&\hat H_S = \mathbf{SDS} + \mathbf{B\gamma}_{S}\mathbf{S} \\ &\hat H_{SN} = \sum_i \mathbf{S}\mathbf{A}_i\mathbf{I}_i \\ &\hat H_{SP} = \sum_{\alpha} \mathbf{B} g^S_\alpha \mathbf{S} Q_\alpha \\ &\hat H_{P} = \sum_{\alpha} \Omega_\alpha b^{+}_\alpha b_\alpha \\ &\hat H_{N} = \sum_i{\mathbf{I}_i\mathbf{P}_i \mathbf{I}_i + \mathbf{B}\mathbf{\gamma}_i\mathbf{I}_i} + \sum_{i<j} \mathbf{I}_i\mathbf{J}_{ij}\mathbf{I}_j\end{split}\]

Where \(\mathbf{S}=(\hat{S}_x, \hat{S}_y, \hat{S}_z)\) are the spin operators of the central spin, \(\mathbf{I}=(\hat{I}_x, \hat{I}_y, \hat{I}_z)\) are the bath spin operators, \(Q_\alpha=(\hat{b}^{+} + \hat b_\alpha)\) are the bath phonon operators, and \(\mathbf{B}=(B_x,B_y,B_z)\) is an external applied magnetic field.

If several central spins are considered, the central spin Hamiltonian is modified as following:

\[\hat H_S = \sum_i (\mathbf{S_i D_i S_i} + \mathbf{B\gamma}_{S_i}\mathbf{S_i} + \sum_{i<j}\mathbf{S_i K_{ij} S_j})\]

And the spin-bath Hamiltonians become:

\[\begin{split}&\hat H_{SN} = \sum_{i,j} \mathbf{S}_i \mathbf{A}_{ij} \mathbf{I}_j, \\ &\hat H_{SP} = \sum_{i\alpha}[ \mathbf{B}g^S_{i\alpha}\mathbf{S} + \sum_{ij,\alpha} \mathbf{S}_i \mathbf{g}^K_{ij,\alpha} \mathbf{S}_j] Q_\alpha.\end{split}\]

The interactions are described by the following tensors that are either required to be input by user or can be generated by the package itself.

\(\mathbf{D}\) (\(\mathbf{P}\)) is the self-interaction tensor of the central spin (bath spin). For the electron spin, the tensor corresponds to the zero-field splitting (ZFS) tensor. For nuclear spins corresponds to the quadrupole interactions tensor.
\(\mathbf{\gamma}_i\) is the magnetic field interaction tensor of the \(i\)-spin describing the interaction of the spin and the external magnetic field \(B\).
\(\mathbf{A}\) is the interaction tensor between central and bath spins. In the case of the nuclear spin bath, it corresponds to the hyperfine couplings.
\(\mathbf{J}\) is the interaction tensor between bath spins.
\(\mathbf{K}\) is the interaction tensor between central spins.
\(\mathbf{g}^K\) is the phonon-mediated interaction tensor between central spins \(\frac{d\mathbf{K}_i}{dQ_\alpha}\).
\(\mathbf{g}^S\) is the phonon-mediated interaction tensor between central spins and external field \(\frac{d\gamma_i}{dQ_\alpha}\).

10. Spin Dynamics in spin and phonon baths

10.1. Hamiltonian

10.2. Implementation

10.2.1. Spin system