Spin-Boson Model

In a Nutshell

The spin-boson model studies the dynamics of a two-state system interacting with a continuous bosonic bath. Such systems are ubiquitous in nature. Examples include state decay of atoms, certain types of chemical reactions, and exciton transport in photosynthesis.

The great applicability of the spin-boson theory is based on the idea that the bath does not need to be microscopically bosonic, but needs only to behave like one. If an interaction with a bath is mediated by an operator \(Y\), then this coupling operator can be replaced by the linear bosonic operator \(X\) of a fictitious (non-interacting) bosonic bath, if:

the coupling is weaker than the bath memory decay, or
the coupling is strong but the statistics of \(Y\) are approximately Gaussian.

The corresponding spectral functions must then match:

\[ \int_{-\infty}^\infty e^{i\omega t} \langle X(t) X(0)\rangle_0 dt = \int_{-\infty}^\infty e^{i\omega t} \langle Y(t) Y(0)\rangle_0 dt \nobreakspace . \]

This is imposed over some relevant frequency range \(\omega_\textrm{min}<\omega<\omega_\textrm{max}\) and "smoothing" \(d\omega\), corresponding to the shortest and longest relevant timescales in the problem. Here the subscript-0 refers to an average during free-evolution.

In general, solving the spin-boson model with numerical methods becomes more difficult when increasing the system-bath couplings as well as the spectral structure.

Hamiltonian, Spectral Function, and Spectral Density

The Hamiltonian of the "traditional" spin-boson problem is of the form:

\[ H = \frac{\Delta}{2}\sigma_z + \frac{1}{2}\sigma_x\left[ \epsilon + \sum_i (g_i a_i + g_i^* a^\dagger_i)\right] + \sum_i \omega_i a^\dagger_i a_i \nobreakspace , \]

with possible (trivial) exchange between \(\sigma_z\) and \(\sigma_x\). Here, the two-level system (spin) has a level-splitting \(\Delta\) and a transverse coupling to a bath position operator:

\[ X=\sum_i g_i a_i + g_i^* a_i^\dagger \nobreakspace . \]

A central role is played by the (free evolution) bath spectral function:

\[ S(\omega) = \int e^{i\omega t}\left< X(t)X(0)\right> _0 dt \nobreakspace , \] as it fully defines the effect of the bath on the system. In thermal equilibrium, we have:

\[ S(\omega) = \frac{2\pi J(\omega)}{1-\exp\left( -\frac{\omega}{k_{\rm B}T } \right)} \nobreakspace . \] where the spectral density is defined as:

\[ J(\omega) = \sum_i |g_i|^2 \delta(\omega-\omega_i) \quad \omega >0 \nobreakspace , \]

thereby connecting the spectral density to the couplings and frequencies in the Hamiltonian.

Representing a Finite-Temperature Bath by a Zero-Temperature Bath

At zero temperature, the spectral function and the density differ only by a factor. It follows that the couplings in the Hamiltonian define the spectral function.

It is possible to represent the spectral function at \(T>0\) by a spectral function of some other system at \(T=0\). This requires the introduction of negative bosonic frequencies. We also define \(T=0\) as the situation where all boson modes are empty. The corresponding new Hamiltonian couplings are then defined as:

\[ \sum_i |g_i|^2 \delta(\omega-\omega_i) \equiv \frac{S(\omega)}{2\pi} . \] We use this description of spin-boson systems due to its simplicity, and since it is the description that the final coarse-grained system also corresponds to.

Other Forms of the (single) Spin-Boson Hamiltonians

We can perform a generalization of the "standard" spin-boson model and include several baths coupling via different directions, each of them have their own coupling polarization (X, Y, or Z),

\[ \begin{align} H &= \frac{\Delta}{2}\sigma_z + \frac{1}{2}\sigma_x X + \frac{1}{2}\sigma_y Y + \frac{1}{2}\sigma_z Z \\ &+ \sum_i \omega_{Xi} a^\dagger_{Xi} a_{Xi}+ \sum_i \omega_{Yi} a^\dagger_{Yi} a_{Yi}+ \sum_i \omega_{Zi} a^\dagger_{Zi} a_{Zi} \end{align} \] Here, the spin is coupled to three independent baths.

A model that emerges when mapping electron-transport models to the spin-boson theory is:

\[ \begin{align} H &= \frac{\Delta}{2}\sigma_z + \frac{1}{2}\sigma^+\left( \sum_i g_i (a_{1i} + a^\dagger_{2i})\right) + \frac{1}{2}\sigma^-\left( \sum_i g_i (a^\dagger_{1i} + a_{2i})\right) \\ &+ \sum_i \omega_i \left( a^\dagger_{1i} a_{1i} + a^\dagger_{2i} a_{2i} \right) \nobreakspace . \end{align} \]

Here we have two sets of bosonic environments with identical parameters. This is a special model since it allows the system-qubit damping to be mapped to the bath spectral density.

Generalization to the Multi-Spin Boson Model

The above models can be straightforwardly generalized to multi-spin systems. An example is the following Hamiltonian:

\[ \begin{align} H &= \sum_{j \in \textrm{system}} \frac{\Delta}{2} \sigma^j_z + \sum_{jk \in \textrm{system}} \frac{1}{2} \left( g_{jk} \sigma_j^+ \sigma_k^- + g_{jk} \sigma_j^- \sigma_k^+ \right) \\ &+ \sum_{j \in \textrm{system}} \sum_{sm \in \textrm{bath}} \frac{1}{2} \sigma^z_j \left[ g_{jsm} a_s(\omega_m) + g_{jsm}^* a^\dagger_s(\omega_m)\right] + \sum_{sm} \omega_m a^\dagger_s(\omega_m) a_s(\omega_m) \end{align} \] This is used to describe exciton transport in photosynthesis. The system spins describe the molecular vibrations of the excitons and the bosonic modes. In principle, all excitons couple to the same bath. However, such cross-couplings are often neglected. In that case, each exciton has its own bath and we can set \(g_{ism}=0\), if \(i\neq s\).

Spectral function, spectral density, and cross-correlations are generalized analogously. In the case of multi-spin boson model, the spectral function is defined as:

\[ S_{ij}(\omega) = \int e^{i\omega t}\langle X_i(t)X_j(0)\rangle dt \nobreakspace , \] where:

\[ X_i = \sum_{sm} \left[g_{ism} a_s(\omega_m) + g_{ism}^* a^\dagger_s(\omega_m)\right] \nobreakspace , \]

and \(i, j\) refer to system spins. These functions fully define the effect of the bosonic bath on the system. In thermal equilibrium,

\[ S_{ij}(\omega) = \frac{2\pi J_{ij}(\omega)}{1-\exp\left( -\frac{\omega}{k_{\rm B}T } \right)} , , \] where the spectral density is generalized to:

\[ J_{ij}(\omega) = \sum_{sm} g_{ism}^* g_{jsm} \delta(\omega-\omega_m) \quad \omega >0 \] We see that cross-correlations between two spins, elements \(J_{ij}(\omega)\) with \(i\neq j\), can be only finite if different spins couple to the same bath mode.