In a nutshell: MPSDynamics.jl for open quantum systems

MPSDynamics.jl was originally developed to perform tensor network simulations of open quantum systems by solving the Schrödinger equation for the closed {System + Environment}. The key idea to construct the simulations is to reformulate the open quantum system Hamiltonian as a one-dimensional many-body problem with nearest-neighbor interactions. The dynamics are then efficiently simulated with Tensor Networks methods.

Sketch of the different ingredients behind MPSDynamics

The starting point is a system linearly coupled to a bosonic environment, where the coupling between system and environment is specified by the spectral density function $J(\omega)$: the prototypical example being The Spin-Boson Model. At finite temperature, the initial state of the environment will be a thermal state (mixed), requiring density matrix formalism and thus making the problem more complex. To circumvent this issue, we consider instead an extended environment, with coupling to the system characterized by a thermalized spectral density function $J(\omega,\beta)$. The two environments induce the same reduced dynamics on the system, enabling us to deal with pure states only[1]. The vacuum state (pure) of the extended environment corresponds to the thermal state of the original environment.

To exploit the computational efficiency of the matrix product states (MPS) description of pure states, we apply the so called Chain-Mapping of bosonic environments —a unitary transformation dependent on $J(\omega,\beta)$— to the Spin-Boson Hamiltonian, mapping it on a chain Hamiltonian with nearest-neighbor interactions. This enables efficient representation of the full {System + Environment} state and its time evolution using the Time-Dependent Variational Principle.

To go further, you will find in this documentation:

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