Biblioteca Allievi della Scuola Superiore di Catania

Contents:

Dissertation note: Tesi di diploma di 1° livello per la Classe delle Scienze Sperimentali Diploma di 1° livello Scuola Superiore di Catania, Catania, Italy 2014 A.A. 2012/2013 Abstract: "In the last twenty years, ultracold atoms have emerged as a flexible platform for the realization of the so called quantum simulator, a concept envisaged by Feynman in 1982. A fundamental issue has been the possibility of creating gauge fields for the (neutral) atoms. The main reason is the simulation of charged particles in U(1) electromagnetic fields, but there are many other important examples of gauge fields associated to higher groups, e.g. SU(N). In this last case the fields can be non-abelian, in the sense that their different components do not commute between themselves. Now, starting from the pioneering work of Dum and Olshanii for the creation of magnetic fields, it was realized that gauge fields could appear in the center of mass Hamiltonian as a consequence of the adiabatic following of some internal (electronic) states. This mechanism, already known in the context of molecular physics, can be explained in terms of the Berry phase acquired by the internal state after the cyclic adiabatic evolution of the center of mass coordinates. Depending on the number N of levels the internal subsystem lives in, a U(N) or SU(N) field will appear. In the case of ultracold gases, a careful coupling of internal levels with lasers is essential. In the present work we start with a brief chapter on the main properties and definition of gauge elds. Then we derive the Berry phase underlining its geometrical meaning and show how the Born-Oppenheimer approximation, in the general case of N internal levels, can give rise to effective adiabatic gauge fields. In the following chapter we see how these concepts translate into the field of ultracold atoms, also from the experimental point of view. In particular we analyze a scheme for the creation of a very simple case of non-abelian gauge potential, an SU(2) configuration that is equivalent, after a gauge transformation to Dresselhaus spin-orbit coupling. In the last chapter we investigate the properties of this potential. We carry out a numerical solution of the time dependent Schrödinger equation, analyzing the ballistic expansion of a single atom and studying the presence of zitterbewegung."
1 Introduction.
2 Gauge fields.
2.1 Non-abelian gauge fields.
3 Geometric phase and adiabatic Hamiltonians.
3.1 Parallel transport.
3.2 Parallel transport in quantum mechanics: Berry phase.
3.3 Some manifestations of the geometrical phase.
3.4 Non-abelian geometric phase.
3.5 Adiabatic effective Hamiltonians.
4 Implementation of artificial gauge fields for ultracold atoms.
4.1 Abelian field configurations.
4.2 Non-Abelian field configurations.
5 A simple example: dynamics of a particle in a SU(2) gauge field.
5.1 Some properties of the potential.
5.2 Results and discussion.
6 Conclusions.
Bibliography.

Location | Call number | Copy number | Status | Date due |
---|---|---|---|---|

Sala B : Armadio Tesi | THS_2014 530.41 P9619 (Browse shelf) | 1 | Available | |

Sala B : Armadio Tesi | THS_2014 530.41 P9619 (Browse shelf) | 2 | Available |

Tesi di diploma di 1° livello per la Classe delle Scienze Sperimentali Diploma di 1° livello Scuola Superiore di Catania, Catania, Italy 2014 A.A. 2012/2013

Includes bibliographical references (58-62 p.) and index.

1 Introduction.

2 Gauge fields.

2.1 Non-abelian gauge fields.

3 Geometric phase and adiabatic Hamiltonians.

3.1 Parallel transport.

3.2 Parallel transport in quantum mechanics: Berry phase.

3.3 Some manifestations of the geometrical phase.

3.4 Non-abelian geometric phase.

3.5 Adiabatic effective Hamiltonians.

4 Implementation of artificial gauge fields for ultracold atoms.

4.1 Abelian field configurations.

4.2 Non-Abelian field configurations.

5 A simple example: dynamics of a particle in a SU(2) gauge field.

5.1 Some properties of the potential.

5.2 Results and discussion.

6 Conclusions.

Bibliography.

"In the last twenty years, ultracold atoms have emerged as a flexible platform for the realization of the so called quantum simulator, a concept envisaged by Feynman in 1982. A fundamental issue has been the possibility of creating gauge fields for the (neutral) atoms. The main reason is the simulation of charged particles in U(1) electromagnetic fields, but there are many other important examples of gauge fields associated to higher groups, e.g. SU(N). In this last case the fields can be non-abelian, in the sense that their different components do not commute between themselves. Now, starting from the pioneering work of Dum and Olshanii for the creation of magnetic fields, it was realized that gauge fields could appear in the center of mass Hamiltonian as a consequence of the adiabatic following of some internal (electronic) states. This mechanism, already known in the context of molecular physics, can be explained in terms of the Berry phase acquired by the internal state after the cyclic adiabatic evolution of the center of mass coordinates. Depending on the number N of levels the internal subsystem lives in, a U(N) or SU(N) field will appear. In the case of ultracold gases, a careful coupling of internal levels with lasers is essential. In the present work we start with a brief chapter on the main properties and definition of gauge elds. Then we derive the Berry phase underlining its geometrical meaning and show how the Born-Oppenheimer approximation, in the general case of N internal levels, can give rise to effective adiabatic gauge fields. In the following chapter we see how these concepts translate into the field of ultracold atoms, also from the experimental point of view. In particular we analyze a scheme for the creation of a very simple case of non-abelian gauge potential, an SU(2) configuration that is equivalent, after a gauge transformation to Dresselhaus spin-orbit coupling. In the last chapter we investigate the properties of this potential. We carry out a numerical solution of the time dependent Schrödinger equation, analyzing the ballistic expansion of a single atom and studying the presence of zitterbewegung."