E. Räsänen

Effects of geometry and impurities on quantum rings in magnetic fields

We investigate the effects of impurities and changing ring geometry on the energetics of quantum rings under different magnetic field strengths. We show that as the magnetic field and/or the electron number are/is increased, both the quasiperiodic Aharonov-Bohm oscillations and various magnetic phases become insensitive to whether the ring is circular or square in shape. This is in qualitative agreement with experiments. However, we also find that the Aharonov-Bohm oscillation can be greatly phase-shifted by only a few impurities and can be completely obliterated by a high level of impurity density. In the many-electron calculations we use a recently developed fourth-order imaginary time projection algorithm that can exactly compute the density matrix of a free-electron in a uniform magnetic field.

Optimal laser control of double quantum dots

Coherent single-electron control in a realistic semiconductor double quantum dot is studied theoretically. Using optimal-control theory we show that the energy spectrum of a two-dimensional double quantum dot has a fully controllable transition line. We find that optimized picosecond laser pulses generate population transfer at significantly higher fidelities (> 0.99) than conventional sinusoidal pulses. Finally we design a robust and fast charge switch driven by optimal pulses that are within reach of terahertz laser technology. Double quantum dots (DQDs), i.e., coupled twodimensional (2D) electron traps, have been under recent and extensive studies both experimentally 1,2,3 and theoretically. 4,7,8 The main interest in DQDs arises from their potential for solid-state quantum computation that could be achieved in principle by rapidly switching voltages of electrostatic gates. The gates permit to tune at will the system geometry and hence the electronic properties of DQDs. Coherent manipulation of a single charge 2 and coupled spins 1 has already been achieved, and recently a coherence time of ∼ 200 ns was obtained for a well isolated silicon DQD. 3 Theoretical studies on single-electron transport inside the DQD driven by linear switches and linearly polarized continuous waves (CWs) were reported very recently. 4 In the latter case the transport is rather sensitive to possible anharmonicity of the potential and limited to uncoupled dots far apart from each other. Electron control in DQDs has been studied also using genetic algorithms 5 as well as rotating-wave and resonant approximations leading to a reduction to a threelevel system. 6 To the best of our knowledge, however, a general N-level control scheme by using direct external electric fields has not been introduced for 2D-DQDs until now. In this paper we discuss the controllability criteria for single-electron states of DQDs by means of external laser pulses. We show that at certain interdot distances some of the single-electron states allow full population transfer from the ground state to those states. We apply quantum optimal-control theory (OCT) 9 which yields the optimal laser pulses for predefined transitions. We obtain high occupations (& 99%) of the target states in a realistic DQD in a few picoseconds, which is well in the coherent regime. If the initial and final states are chosen to have full localization of the electron in one or the other dot, this scheme enables rapid and controlled transport which is not sensitive to the interdot distance or to the inevitable anharmonicities in the confining potential.

Femtosecond laser pulse shaping for enhanced ionization

We demonstrate how the shape of femtosecond laser pulses can be tailored in order to obtain maximal ionization of atoms or molecules. For that purpose, we have overlayed a direct- optimization scheme on top of a fully unconstrained computation of the three-dimensional time- dependent Schrodinger equation. The procedure looks for pulses that maintain the same total length and integrated intensity or fluence as a given pulse that serves as an initial guess. It allows, however, for changes in frequencies —within a certain, predefined range— and overall shape, leading to enhanced ionization. We illustrate the scheme by calculating ionization yields for the H + molecule when irradiated with short (≈ 5 fs), high-intensity laser pulses. The magnitude of the obtained enhancement, as well as the shape of the solution optimal field depend strongly on the constrains imposed on the search space. In particular, when only small frequencies are allowed, the solution merely increases the peak intensity through temporal compression, as expected from a simple tunneling picture. If larger frequencies are allowed the structure of the solution field is more complicated. Copyright c EPLA, 2009

Orbital-free energy functional for electrons in two dimensions

We derive a nonempirical, orbital-free density functional for the total energy of interacting electrons in two dimensions. The functional consists of a local formula for the interaction energy, where we follow the lines introduced by Parr for three-dimensional systems [R. G. Parr, J. Phys. Chem. 92, 3060 (1988)], and the Thomas-Fermi approximation for the kinetic energy. The freedom from orbitals and from the Hartree integral makes the proposed approximation numerically highly efficient. The total energies obtained for confined two-dimensional systems are in a good agreement with the standard local-density approximation within density-functional theory and considerably more accurate than the Thomas-Fermi approximation.

Correlation energy of finite two-dimensional systems: Toward nonempirical and universal modeling

The capability of density-functional theory to deal with the ground-state of strongly correlated low-dimensional systems, such as semiconductor quantum dots, depends on the accuracy of functionals developed for the exchange and correlation energies. Here we extend a successful approximation for the correlation energy of the three dimensional inhomogeneous electron gas, originally introduced by Becke [J. Chem. Phys. {bf 88}, 1053 (1988)], to the two-dimensional case. The approach aims to non-empirical modeling of the correlation-hole functions satisfying a set of exact properties. Furthermore, the electron current and spin are explicitly taken into account. As a result, good performance is obtained in comparison with numerically exact data for quantum dots with varying external magnetic field, and for the homogeneous two-dimensional electron gas, respectively.

Gaussian approximations for the exchange-energy functional of current-carrying states: Applications to two-dimensional systems

Electronic structure calculations are routinely carried out within the framework of density-functional theory, often with great success. For electrons in reduced dimensions, however, there is still a need for better approximations to the exchange-correlation energy functional. Furthermore, the need for properly describing current-carrying states represents an additional challenge for the development of approximate functionals. In order to make progress along these directions, we show that simple and efficient expressions for the exchange energy can be obtained by considering the short-range behavior of the one-body spin-density matrix. Applications to several two-dimensional systems confirm the excellent performance of the derived approximations, and verify the gauge-invariance requirement to be of great importance for dealing with current-carrying states.

Local correlation functional for electrons in two dimensions

We derive a local approximation for the correlation energy in two-dimensional electronic systems. In the derivation we follow the scheme originally developed by Colle and Salvetti for three dimensions, and consider a Gaussian approximation for the pair density. Then, we introduce an ad-hoc modification which better accounts for both the long-range correlation, and the kinetic-energy contribution to the correlation energy. The resulting functional is local, and depends parametrically on the number of electrons in the system. We apply this functional to the homogeneous electron gas and to a set of two-dimensional quantum dots covering a wide range of electron densities and thus various amounts of correlation. In all test cases we find an excellent agreement between our results and the exact correlation energies. Our correlation functional has a form that is simple and straightforward to implement, but broadly outperforms the commonly used local-density approximation.

Spin droplets in confined quantum Hall systems

Two-dimensional semiconductor quantum dots are studied in the filling-factor range 2<v<3. We find both theoretical and experimental evidence of a collective many-body phenomenon, where a fraction of the trapped electrons form an incompressible spin droplet on the highest occupied Landau level. The phenomenon occurs only when the number of electrons in the quantum dot is larger than ~30. We find the onset of the spin-droplet regime at v=5/2. This proposes a finite-geometry alternative to the Moore-Read-type Pfaffian state of the bulk two-dimensional electron gas. Hence, the spin-droplet formation may be related to the observed fragility of the v=5/2 quantum Hall state in narrow quantum point contacts.

Electron localization function for two-dimensional systems

The concept of the electron localization function (ELF) is extended to two-dimensional (2D) electron systems. We show that the topological properties of the ELF in two dimensions are considerably simpler than in molecules studied previously. We compute the ELF and demonstrate its usefulness for various physical 2D systems focusing on semiconductor quantum dots that effectively correspond to a confined 2D electron gas. The ELF visualizes the shell structure of harmonic quantum dots and provides insight into electron bonding in quantum-dot molecules. In external magnetic fields, the ELF is found to be a useful measure of vorticity when analyzing the properties of quantum-Hall droplets. We show that the current-dependent term in the ELF expression is important in magnetic fields.

Pfaffian and fragmented states at v=5/2 in quantum Hall droplets

When a gas of electrons is confined to two dimensions, application of a strong magnetic field may lead to startling phenomena such as emergence of electron pairing. According to a theory this manifests itself as appearance of the fractional quantum Hall effect with a quantized conductivity at an unusual half-integer v=5/2 Landau level filling. Here we show that similar electron pairing may occur in quantum dots where the gas of electrons is trapped by external electric potentials into small quantum Hall droplets. However, we also find theoretical and experimental evidence that, depending on the shape of the external potential, the paired electron state can break down, which leads to a fragmentation of charge and spin densities into incompressible domains. The fragmentation of the quantum Hall states could be an issue in the proposed experiments that aim to probe for non-Abelian quasiparticle characteristics of the v=5/2 quantum Hall state.