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European Conference on Modelling and Simulation (ECMS)

Modeling And Control Of Berry Phase In Quantum Dots

Sanjay Prabhakar, Roderick Melnik, Ali Sebetci

Published May 27, 2014• 0 citations
Publisher

Abstract

We study numerically the Berry phase in semiconductor quantum dots (QDs) that is induced by letting the dots to move adiabatically in a closed loop in the 2D plane along the circular trajectory. We show that the Berry phase is highly sensitive to the Rashba and Dresselhaus spin-orbit lengths. Based on the Finite Element Method, we solve the Schrodinger equation and investigate the evolution of the spin dynamics during the adiabatic transport of the QDs in the 2D plane along circular trajectory. Results of numerical simulations are discussed in detail, indicating that this work might be used for the realization of solid state quantum information processing.

DOI
10.7148/2014-0166

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We study numerically the Berry phase in semiconductor quantum dots (QDs) that is induced by letting the dots to move adiabatically in a closed loop in the 2D plane along the circular trajectory. We show that the Berry phase is highly sensitive to the Rashba and Dresselhaus spin-orbit lengths. Based on the Finite Element Method, we solve the Schrodinger equation and investigate the evolution of the spin dynamics during the adiabatic transport of the QDs in the 2D plane along circular trajectory. Results of numerical simulations are discussed in detail, indicating that this work might be used for the realization of solid state quantum information processing.
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.
We consider a system with spin-orbit coupling and derive equations of motion which include the effects of Berry curvatures. We apply these equations to investigate the dynamics of particles with equal Rashba-Dresselhaus spin-orbit coupling in one dimension. In our derivation, the adiabatic transformation is performed first and leads to quantum Heisenberg equations of motion for momentum and position operators. These equations explicitly contain position-space, momentum-space, and phase-space Berry curvature terms. Subsequently, we perform the semiclassical approximation and obtain the semiclassical equations of motion. Taking the low-Berry-curvature limit results in equations that can be directly compared to previous results for the motion of wave packets. Finally, we show that in the semiclassical regime, the effective mass of the equal Rashba-Dresselhaus spin-orbit-coupled system can be viewed as a direct effect of the phase-space Berry curvature.
Tuning spin-orbit coupling is essential in controlling both spin and charge in confined semiconductor nanostructures, yet it is rarely a truly controllable parameter. Here, we show control over the spin-orbit Hamiltonian in quantum dots and the resulting quantum states by tailoring the confinement potential with atomic-scale precision. Using scanning tunnelling microscopy and spectroscopy, we pattern individual Cs ions into designer quantum dot structures on the surface of indium antimonide, in which electrons from a two-dimensional electron gas are confined with chosen in-plane electric-field gradients. We then quantify the atomic level structure, both spatially resolving the orbital character of the electronic states and their magnetic-field evolution. We demonstrate that the level structure, including the induced zero-field splitting, can be tailored by the designed geometry of the local electric fields. These effects can be described using a Hamiltonian that allows consistent treatment of the confinement-induced spin-orbit coupling beyond the conventional Bychkov-Rashba description. This Hamiltonian is derived from a multiband k.p model and takes the energy dependence of the relevant physical parameters into account. Such precise control of spin-orbit coupling in semiconductor quantum dots is relevant to quantum and spintronic technologies.
Semiconductor quantum dots (QDs) are ideal candidates for quantum information processing, due to their strong resemblance to atomic systems. This paper will show that an optically driven approach is an attractive scheme for quantum computing with QDs. Results will be presented from the experiments with the optical Bloch vector qubit in neutral QDs, demonstrating phase‐locked single qubit rotations and two‐qubit universal control‐NOT logic gates. We will conclude by discussing how the optical approach can be extended to control the electron spin qubit in charged QDs.
We propose a method to simulate adiabatically driven two-level quantum systems and detect the Berry phase induced in the cyclic evolution by observing the classical orbit motion of a charged particle under a time-varying magnetic field. Analogy between the dynamics of these two species of systems is revealed and the correspondence relationship is established by virtue of their equations of motion, i.e., the von Neumman equation of the quantum binary system and the Newton-Lorentz equation of the charged particle. In particular, we show how the complex phase information of the quantum system, say, the Berry phase shift acquired by its wave function during the loop evolution, can be visually captured by examining the spatial motion of the classical particle.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.