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Physical Review B

Conduction electrons and the decoherence of impurity-bound electrons in a semiconductor

Kuljit S. Virk, John E. Sipe

Published Oct 18, 2005• 7 citations
Publisher

Abstract

We study the dynamics of impurity-bound electrons interacting with a bath of conduction band electrons in a semiconductor. Only the exchange interaction is considered. We derive master equations for the density matrices of single- and two-qubit systems under the usual Born and Markov approximations. The bath mediated Ruderman-Kittel-Kasuya-Yoshida interaction in the two-qubit case arises naturally. It leads to an energy shift significant only when the ratio $({R}_{T})$ of the interqubit distance to the thermal de Broglie wavelength of the bath electrons is small. This bath mediated interaction also has a profound impact on the decoherence times; the effect decreases monotonically with ${R}_{T}$.

DOI
10.1103/physrevb.72.155312

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We study the dynamics of impurity bound electrons interacting with a bath of conduction band electrons in a semiconductor. Only the exchange interaction is considered. We derive master equations for the density matrices of single and two qubit systems under the usual Born and Markov approximations. The bath mediated RKKY interaction in the two qubit case arises naturally. It leads to an energy shift significant only when the ratio (RTR_{T}RT​) of the inter-qubit distance to the thermal deBroglie wavelength of the bath electrons is small. This bath mediated interaction also has a profound impact on the decoherence times; the effect decreases monotonically with RTR_{T}RT​.
We study theoretically the phonon-induced relaxation and decoherence processes in the hybrid qubit in silicon. A hybrid qubit behaves as a charge qubit when the detuning is close to zero and as a spin qubit for large detuning values. It is realized starting from an electrostatically defined double quantum dot where three electrons are confined and manipulated through only electrical tuning. By employing a three-level effective model for the qubit and describing the environment bath as a series of harmonic oscillators in the thermal equilibrium states, we extract the relaxation and decoherence times as a function of the bath spectral density and of the bath temperature using the Bloch-Redfield theory. For Si quantum dots the energy dispersion is strongly affected by the physics of the valley, i.e., the conduction band minima, so we also included the contribution of the valley excitations in our analysis. Our results offer fundamental information on the system decoherence properties when the unavoidable interaction with the environment is included and temperature effects are considered.
We theoretically study decoherence of two localized spins interacting via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in one-, two-, and three-dimensional electron gas. We derive the kinetic equation for the reduced density matrix of the localized spins and show that energy relaxation caused by singlet-triplet transition is suppressed when the RKKY interaction is ferromagnetic. We also estimate the decoherence time of the system consisting of two quantum dots embedded in a two-dimensional electron gas.
We use the nonequilibrium Green's function formalism and a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons (with and without spin) on the conductance of a quantum wire. We study how the conductance varies with the wire length, the temperature, and the strengths of the impurity and interactions. The numerical results for the dependence of the conductance on the wire length and temperature are compared with the results obtained from a renormalization group analysis based on the Hartree-Fock approximation. For the spin-1∕21∕21∕2 model with a repulsive on-site interaction or the spinless model with an attractive nearest neighbor interaction, we find that the conductance increases with increasing wire length or decreasing temperature. This can be explained using the Born approximation in scattering theory. For a strong impurity, the conductance is significantly different for a repulsive and an attractive impurity; this is due to the existence of a bound state in the latter case. In general, the large density deviations close to the impurity have an appreciable effect on the conductance at short distances which is not captured by the renormalization group equations.
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.