The recent developments in the experimental realization of quasi-one-dimensional (1D) systems exhibit many interesting features. These include current quantization which has the potential application for a current standard as well as quantum information and security schemes. In this paper, we investigate the effect of spin-orbit interaction (SOI) on the energy levels of electrons confined to quantum dots on the surface of nanotubes. The radius of a nanotube is a few nanometers and is quasi-1D. The energy levels play a crucial role in determining the electron transport properties.The SOI may arise from the electrostatic confining potential due to gates applied perpendicular to the axis of the nanotube. The quantum computation scheme which we are suggesting consists of a nano-circuit of nanotubes on which electrons are confined within dots. The qubit operation is based on the exchange interaction between a pair of spins occupying states within the quantum dots. We employ a simple model for the electron confinement to obtain the energy eigenstates. Our simplified calculation was able to show that the SOI splits the energy levels which are then used to obtain the exchange energy of a pair of spins with the s-wave Heitler-London approach. We calculate the exchange energy of the entangled electrons on a pair of coaxial and parallel nanotubes as a function of separation between the nanotubes and show that the SOI enhances the entanglement.
The recent developments in the experimental realization of quasi-one-dimensional (1D) systems exhibit many interesting features. These include current quantization which has the potential application for a current standard as well as quantum information and security schemes. In this paper, we investigate the effect of spin-orbit interaction (SOI) on the energy levels of electrons confined to quantum dots on the surface of nanotubes. The radius of a nanotube is a few nanometers and is quasi-1D. The energy levels play a crucial role in determining the electron transport properties.The SOI may arise from the electrostatic confining potential due to gates applied perpendicular to the axis of the nanotube. The quantum computation scheme which we are suggesting consists of a nano-circuit of nanotubes on which electrons are confined within dots. The qubit operation is based on the exchange interaction between a pair of spins occupying states within the quantum dots. We employ a simple model for the electron confinement to obtain the energy eigenstates. Our simplified calculation was able to show that the SOI splits the energy levels which are then used to obtain the exchange energy of a pair of spins with the s-wave Heitler-London approach. We calculate the exchange energy of the entangled electrons on a pair of coaxial and parallel nanotubes as a function of separation between the nanotubes and show that the SOI enhances the entanglement.
We calculate the energy eigenvalues, the spin-split excitation gap (energy separation between the spin-triplet excited state and the spin-singlet ground state) and the concurrence for two interacting electrons captured in a quantum dot (QD) formed by a gigahertz electron pump which is modeled by harmonic confining potentials. We find from our calculations a peak in the QD size dependence of the energy level for the spin-singlet ground state, indicating the effect due to Coulomb blockade. In addition, we observe a local minimum in the QD size dependence of the spin-split excitation gap for a relatively narrow quasi-one-dimensional (1D) channel formed from an etched wire, but a strong positive peak for the spin-split excitation gap in its QD size dependence with a relatively wide 1D channel. From the existence of a robust spin-split excitation gap against both thermal fluctuation due to finite (low) temperatures and the nonadiabatic effect due to fast barrier variations, we predict a spin-entangled electron pair inside the QD with a weak coupling to external leads. An interference-type experiment which employs a gate-controlled electron pump and a beam splitter is proposed to verify this prediction. For the electron pump, a sinusoidal radio-frequency signal is applied to the entrance gate of a two-gated system over a narrow channel etched in a GaAs/AlGaAs heterostructure, where the measured current within the channel shows plateaus at Nef with N = 1, 2,... being the number of captured electrons in a QD and f the frequency of the sinusoidal signal.
Solid state based quantum information processing is focused on physical implementation of all necessary elements of quantum computation and quantum information in solid state systems, mainly due to their scalability compared to e.g. optical systems. Among many proposals to realize these types of devices, such as quantum dots as charge qubits or Josephson junction circuits, we study here one of the most promising candidates, i.e., the spin of an electron confined to a quantum dot as a qubit. Experimentally, it has been shown that the relaxation rate of this two level system can be pushed above few seconds in low magnetic fields. Moreover, using spin echo techniques, \nthe spin dephasing time can be maximized up to milliseconds with the current technology. This long spin decay time is one of the main reasons that make this system desirable for quantum computation and quantum information purposes. \nWe reexamined the recent measurement based proposal called one-way quantum computation which exploits entanglement and local measurements as tools to perform quantum computation on N qubits. Although it was suggested in the original work to entangle the qubits via the nearest neighbor Ising interaction, we investigated how one can generate the so-called cluster states with the Heisenberg interaction. We extended our method to include more general forms of Heisenberg interaction such as asymmetric coupling of adjacent qubits. These forms of couplings, rather than Ising interaction, are more encountered in solid state devices, and therefore make it possible to perform one-way quantum computation with electron spins in quantum dots coupled via exchange interaction to their adjacent spins. \n \nChapters II, III, and IV of my thesis are devoted to the study of the spin-orbit interaction in heterostructure quantum dots and its effect on the spin dynamics. We observed that one can actually use spin-orbit interaction to manipulate the spin state of an electron on time scales much smaller than the spin dephasing time. Specifically, in chapter II, we studied the effect of a nearby functioning quantum point contact (QPC) on the relaxation of the electron spin and show that the charge fluctuations in QPC lead to spin relaxation of the confined electron in the presence of spin-orbit interactiom and an applied magnetic field. We also addressed the relation of this rate to the microscopic parameters of the system and found some geometrical dependence of the spin relaxation time on the orientation of the QPC on the substrate. Moreover, we showed in chapters III and IV that the spin-orbit interaction can play a positive role, in order to rotate the spin around the Bloch sphere. We considered different mechanisms, particularly, Electron Dipole Spin Resonance (EDSR) and holonomic unitary gates in quantum dots. We verified that these mechanisms of spin manipulation can be realized in solid state systems with the state of the art semiconductor technologies. \n \nChapter V of the thesis covers a slightly different topic and focuses on the role of the Coulomb interaction in electronic transport. There, we reviewed the non-analytic corrections to the Fermi liquid behavior and their consequences on the momentum occupation number of the electrons in a two dimensional electron gas (2DEG). As an example, we calculated the tunneling rate from an interacting electron reservoir onto a quantum dot and compared our result to the corresponding case for electron tunneling between bilayer 2DEGs. Moreover, within RPA approximation, we found that the electron-plasmon coupling leads to a quadratic frequency dependence of the electron self-energy at low frequencies at the Fermi surface. This correction suppresses the same order corrections due to the particle-hole bubble. \n \n \n
In the last years, the electron spin in quantum dots has become a most promising candidate for the realization of qubits, the basic element for quantum computation. In quantum dot spin qubits the exchange interaction implements a fundamental two qubit gate, which originates in the Coulomb interaction and the Pauli principle. Compared to single qubit gates, the exchange based gates are much faster and easier to control locally, motivating the solely exchange energy based quantum computation. The control is based on the exponential sensitivity of the exchange energy on the interparticle distance. Manipulation then can proceed, for example, shifting the single particle states electrically or compressing them magnetically. The practical manipulation schemes require quantitative knowledge of the exchange energy. The spin-orbit interaction plays an important role in this nano-system. Indeed the usual exchange interaction changes its magnitude, while a new term, called anisotropic exchange, appears. In this thesis we propose an accurate effective spin Hamiltonian for modeling spin-orbit-induced spin dynamics in the presence of the magnetic field. We have also performed numerically exact calculations of the isotropic and anisotropic exchange coupling in realistic two-electron GaAs coupled quantum dots in the presence of both Dresselhaus and Bychkov-Rashba spin-orbit interactions. The numerics allow us to establish the goodness of our model and to see the limits of previous statements. Furthermore we find that in zero magnetic field the second-order spin-orbit effects are absent at all interdot couplings.
With the help of the spin-orbit interaction, we propose a scheme to perform holonomic single-qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective non-Abelian gauge potential whose structure depends on the type of the spin-orbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius R) on the two dimensional electron gas (2DEG) and show that all single-qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate two-qubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary N qubits. This proposal shows a way of realizing holonomic quantum computers in solid-state systems.
The properties of an electron pair confined in a coupled quantum dot (QD) system are currently being intensively investigated both experimentally and theoretically. Indeed the manipulation of the two-electron spin states is paramount for the realization of controllable entanglement so as to produce the fundamental quantum gate operations in a solid state quantum computer. The works usually concentrate on the properties of a two-electron system without going into detailed analysis of the contributions from various single-particle states or the effects produced by the constituent single QDs. Zeeman splitting is neglected as it affects only the magnitude of the exchange energy but not the single-particle state mixing. In summary, after numerically diagonalizing the two-electron Schrodinger equation in the model confinement potential of both single and laterally coupled double quantum dots, we computed the contributions from the various single-particle states to the final two-particle wave function.
Entanglement is a valuable resource for quantum computing and information technologies. A promising way to obtain spin-entangled electrons is the splitting of Cooper pairs residing in superconductors. This is realized by a nanodevice called Cooper pair splitter. A Cooper pair splitter is realized by tunnel-coupling a superconductor to two quantum dots, which are each further tunnelcoupled to separate terminal leads. This setup enables the extraction and splitting of Cooper pairs from the superconductor into the terminals via the quantum dots. The quantum dots are commonly fabricated from semiconducting materials like InAs and InSb, that can manifest spin-orbit interaction. This thesis studies the effect of spin-orbit interaction on the spin state of the electron pair in the Cooper pair splitter. Activating the spin-orbit effects in a Cooper pair splitter requires a static magnetic field to be applied to the quantum dots. Together with the gate electrodes controlling the electrostatic energy of the quantum dots, the external magnetic field provides a handle for addressing different two-particle spin states. The spin-orbit interaction can be activated by various combinations of the gate voltage and Zeeman magnetic field, that create resonances between different states of the system. The most interesting coherent evolution, that can be invoked in this way, involves the simultaneous activation of the two spin-polarized triplet states. This gives a rise to an entangled Bell state, (|↑↑>+|↓↓>) / √2, that moreover exhibits spin blockade and thus stabilizes entanglement in the Cooper pair splitter. Secondly, the spin-orbit interaction can be used to bypass suppressed Cooper pair tunneling at a high superconductor-quantum dots detuning. Introducing a large Zeeman magnetic field allows Cooper pairs to tunnel in to the dots given that their spin state flips to the triplet |↓↓>. Although this regime lacks spin-entanglement, it is of interest for creating triplet Cooper pairs and spin-polarized superconducting currents.