## Dynamics and thermalization of a fermion in a fermionic bath embedded in a bosonic bath.

##### Abstract

We consider a model consisting of a finite number of quantum dots each of which
confines a spinless electron. We zoom in on a single quantum dot containing an
electron of interest treating it as being strongly coupled to the surrounding finite
bath of electrons. The fermionic bath is embedded in a bosonic Markovian bath.
The master equation for the fermion of interest interacting with the fermionic bath
is derived. Based on the master equation for this system, the reduced dynamics
and thermalization of the spinless electron is studied.
We start with a description of the Hamiltonian of the entire system which we
call total Hamiltonian. Because the electron of interest is strongly coupled to the
surrounding fermionic bath, we treat the Hamiltonian consisting of the electron
of interest, the fermionic bath and the interaction between them as the system
Hamiltonian. Then using techniques of linear algebra, we diagolize the system
Hamiltonian making it appear in what we are calling quasi-fermionic picture. After
this, we take the diagonalized system Hamiltonian back to the total Hamiltonian.
We then use this total Hamiltonian to switch to the interaction picture. Since
the general expression of the Markovian quantum master equation is in terms of
the interaction Hamiltonian, we now substitute our interaction Hamiltonian into
it and begin from there to derive the quantum master equation of our system.
In the next step, we solve the derived quantum master equation casting the solution
in Kraus representation. Using the explicit form of the Kraus operators and initial
conditions, the density matrix of the reduced system is obtained in the quasifermionic
picture. We then transform to the original fermionic picture and trace
out the fermionic bath coming out with the density matrix of the electron of
interest.
We then check the normalization of the density matrix of the electron of interest
by calculating the trace and then use it to calculate the mean number of fermions.
The mean number of fermions is then plotted against time for different coupling
strengths and varying numbers of fermions in the fermionic bath to visually check
the dynamics and thermalization of the fermion of interest.