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

Michael. Mwalaba

Published Jul 7, 2015• 0 citations
OpenAlex

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.

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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.
We consider a central fermion strongly interacting with a surrounding mesoscopic bath of fermions which is weakly coupled to a Markovian bath of fermions. The master equation of the system consisting of the central fermion and the mesoscopic bath is derived, and based on this master equation, the reduced dynamics and thermalization of the central fermion are investigated. As probes for studying the dynamics and thermalization, the mean number of the central fermion and the current density are calculated. The observed dynamics reveal that strong coupling strength induces non-Markovian behavior and that in the long time limit thermalization is attained leaving the central fermion in a thermal state that is described by the corresponding Gibbs distribution. Upon attaining the thermal state, the central fermion is found to be entangled with the collective mode of the surrounding mesoscopic bath in the high-temperature regime, whereas in the low-temperature regime, it is found to undergo sudden death of entanglement.
We solve the fermionic master equation for a thermal bath to obtain its explicit Kraus operator solutions via the fermionic state approach. The normalization condition of the Kraus operators is proved. The matrix representation for these solutions is obtained, which is incongruous with the result in the book completed by Nielsen and Chuang [ Quantum Computation and Quantum Information , Cambridge University Press, 2000]. As especial cases, we also present the Kraus operator solutions to master equations for describing the amplitude-decay model and the diffusion process at finite temperature.
Coupling to a thermal bath leads to thermalisation and decoherence of stored quantum information. For a system of Gaussian fermions, the fermionic analog of linear or Gaussian optics, these dynamics can be elegantly and efficiently described by evolution of the system's covariance matrix. Taking both system and bath to be Gaussian fermionic, we observe that thermalization and decoherence occurs at a rate that is independent of the bath temperature. Furthermore, we also consider a weak coupling regime where the dynamics are Markovian. We present a microscopic derivation of Markovian master equations entirely in the language of covariance matrices, where temperature independence remains manifest. This is radically different from behaviour seen in other scenarios, such as when fermions interact with a bosonic bath. Our analysis applies to many Majorana fermion systems that have been heralded as very robust, topologically protected, qubits. In these systems, it has been claimed that thermal decoherence can be exponentially suppressed by reducing temperature, but we find Gaussian decoherence cannot be cooled away.
Employing quadratic fermionic and bosonic Hamiltonians for collective and internal subsystems with a linear rotating-wave-approximation coupling, we studied the role of heat-bath statistics on the dynamics of the collective motion. The master equations for the collective occupation number derived directly from the quadratic Hamiltonians and within the Non-Markovian Langevin approach are discussed and their solutions are obtained. Because of the different nature of the heat-bath statistics, the path to equilibrium or the relaxation time is affected as shown in the numerical calculations.
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath to the models in which the system operators commute with the fermionic bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"odinger equation (SSE), called the fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the stochastic average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model, and the NNN-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.
We develop analytical tools and numerical methods for time evolving the total density matrix of the finite-size Anderson model. The model is composed of two finite metal grains, each prepared in canonical states of differing chemical potential and connected through a single electronic level (quantum dot or impurity). Coulomb interactions are either excluded all together or allowed on the dot only. We extend this basic model to emulate decohering and inelastic scattering processes for the dot electrons with the probe technique. Three methods, originally developed to treat impurity dynamics, are augmented to yield global system dynamics: the quantum Langevin equation method, the well-known fermionic trace formula and an iterative path integral approach. The latter accommodates interactions on the dot in a numerically exact fashion. We apply the developed techniques to two open topics in nonequilibrium many-body physics. (i) We explore the role of many-body electron–electron repulsion effects on the dynamics of the system. Results, obtained using exact path integral simulations, are compared with mean-field quantum Langevin equation predictions. (ii) We analyze aspects of quantum equilibration and thermalization in large quantum systems using the probe technique, mimicking elastic-dephasing effects and inelastic interactions on the dot. Here, unitary simulations based on the fermionic trace formula are accompanied by quantum Langevin equation calculations.