Description and contextualization of the subject

Este curso sobre Materia Cuántica explora conceptos clave en gases cuánticos ultrafríos y temas en la intersección de la teoría de la información cuántica y la materia cuántica, con un enfoque en el entrelazamiento, la computación cuántica, las redes tensoriales y los sistemas cuánticos de muchos cuerpos. También revisa plataformas experimentales para la simulación cuántica y fenómenos como la criticidad cuántica y los campos gauge emergentes. A través de discusiones teóricas rigurosas y análisis de investigaciones de vanguardia, se abordarán tanto desarrollos fundamentales como contemporáneos en el campo.

La primera parte cubre los gases cuánticos, la teoría de Gross-Pitaevskii, las excitaciones colectivas, las fluctuaciones cuánticas, las gotas cuánticas, los condensados dipolares y los átomos ultrafríos en redes ópticas. La segunda parte se centra en la materia cuántica y la información cuántica, abordando las redes tensoriales, las plataformas de simulación y la criticidad cuántica. Una tercera parte está dedicada a los estados de Majorana, las fases topológicas y sus aplicaciones para la computación cuántica topológica.

Learning outcomes of the subject

Knowledge or content:

RCO1. Demonstrate the ability to explain the fundamental principles of the quantum world, both at a basic and technical level.

RCO2. Have a basic knowledge of the relevant literature in quantum mechanics and be capable of effectively reading and understanding research articles.

RCO3. Be able to initiate the development of original ideas and applications within the context of quantum physics research.

RCO4. Possess the capacity for independent research, synthesis, and be able to present in a clear and structured way complex issues related to the various areas of quantum mechanics addressed in this Master¿s program.

RCO5. Under supervision, demonstrate the ability to write and defend original work that meets the quality standards required for publication in high-impact indexed journals.

RCO6. Be able to identify opportunities for innovation and technology transfer in the field of quantum science and technology.

RCO7. Know the basic literature and demonstrate the ability to solve standard problems in the field of Quantum Information and Computation.

RCO11. Know the basic literature and demonstrate the ability to solve standard problems in the field of Condensed Matter Physics.

RCO12. Know the basic literature and demonstrate the ability to solve standard problems in the field of Quantum Technologies.



Competencies:

RC1. Possess and understand knowledge that provides a basis or opportunity for developing and/or applying original ideas, often in a research context.

RC2. Apply acquired knowledge and problem-solving skills in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their field of study.

RC3. Demonstrate the ability to integrate knowledge and address the complexity of formulating judgments based on incomplete or limited information, including reflection on social and ethical responsibilities linked to the application of their knowledge and judgments.

RC4. Communicate conclusions, as well as the underlying knowledge and rationale, clearly and unambiguously to both specialized and non-specialized audiences.

RC5. Possess learning skills that enable continued study in a largely self-directed or autonomous manner.



Abilities or skills:



RHE1. Demonstrate proficiency in using tools for bibliographic resource searches.

RHE2. Exhibit critical capacity to read research articles and incorporate their findings into one¿s own work.

RHE3. Write and present original work in one of the official languages and in English.

RHE4. Communicate scientific concepts and results clearly and effectively to both specialized and non-specialized audiences, through presentations and publications.

RHE5. Demonstrate the ability for autonomous learning and staying current with scientific and technological advances.





RHT1. Understand and apply the fundamental principles of quantum mechanics to analyze and solve problems in basic research in quantum science.

RHT2. Understand and apply the fundamental principles of quantum mechanics to analyze and solve problems in quantum technology.

RHT3. Effectively integrate into a fundamental or applied research project involving quantum aspects, and solve problems in multidisciplinary environments.

RHT4. Evaluate and select appropriate tools and techniques for research in fundamental physics.

RHT5. Evaluar y seleccionar las herramientas y técnicas adecuadas para la obtención de aplicaciones tecnológicas basadas en la física cuántica.

Temary

Ultracold Quantum Gases

Introduction. General overview of the field. Condensation of bosons in harmonic traps: trapped bosons at finite temperature; finite size effects; role of dimensionality; effects of interactions.

Gross-Pitaevskii theory. Many-body Hamiltonian, Gross-Pitaevskii equation. Hydrodynamic formulation and scaling solutions. Collective excitations, Bogoliubov theory. Landau criterion for superfluidity, energetic and dynamical instabilities. Quantum fluctuation and beyond mean-field corrections. Quantum droplets. Dipolar condensates, supersolids.

Ultracold atoms in optical lattices. General results for periodic systems: semiclassical equation of motion,Bloch oscillations. Tight binding regime. Discrete non-linear Schrödinger equation. Bose-Hubbard Hamiltonian, superfluid-Mott insulator transition.



Quantum Matter and Quantum Information

Entanglement, Quantum Computation, and Tensor Networks. The role of entanglement in quantum systems. Quantum computation and quantum circuits. Tensor networks for quantum many-body states. Matrix Product States. Efficient representations of quantum systems via entanglement entropy. Application of tensor networks in quantum simulations.

Quantum Simulation of Many-Body Systems. Introduction to quantum simulators: definition and purpose. Experimental platforms: ultracold atoms, trapped ions, photonic systems. Optical lattices and Hubbard models. Quantum spin models. Strongly correlated systems and emergent phenomena. Comparison of analog vs. digital quantum simulation.

Quantum Criticality and Gauge Fields. Quantum criticality and phase transitions at zero temperature. Landau theory vs. topological order. Gauge theories in condensed matter. Emergence of gauge fields in quantum spin liquids. U(1) and Z_2 gauge theories. The relation between quantum entanglement and quantum criticality.



Majorana Bound States and Topological Quantum Computing.

Introduction to Majorana Bound States and Topological Phases. Overview of Majorana Fermions. Topological Phases of Matter: Introduction to topological insulators and superconductors; Berry phase, Chern numbers, and topological invariants; Symmetry protection; Bulk-boundary correspondence. Majorana Bound States in Topological Superconductors. The Kitaev chain as a 1D topological superconductor.

Mathematical and Experimental Framework. Majorana Bound States: Hamiltonians for topological superconductors; Majorana operators and their properties; Non-Abelian statistics and braiding Majorana states. Experimental Realizations: Semiconductor-superconductor nanowires; Vortex bound states in 2D superconductors; Signatures of MBS in experiments. Challenges in Experiment and Theory.

Majorana Bound States in Quantum Computing. Topological Quantum Computing: Topological qubits; Braiding of Majorana fermions as quantum gates; Non-Abelian anyons and their use in quantum computation; encoding and manipulating Majorana qubits. Recent Advances and Future Directions.

Bibliography

Basic bibliography

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity, Oxford (2016).

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge (2008).

M. Lewenstein, A. Sanpera, V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems, Oxford University Press (2012).

S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2011).

X.-G. Wen, Quantum Field Theory of Many-body Systems, Oxford University Press (2004).

J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75 076501 (2012).

B. A. Bernevig and T. L. Hughes,Topological Insulators and Topological Superconductors, Princeton University Press (2013).

Journals

F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).

I. Bloch et al., Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).

T. Lahaye et al., The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72, 126401 (2009).



R. P. Feynman, Simulating Physics with Computers, Int. J. Theor. Phys. 21, 467-488 (1982).

R. Orús, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Ann. Phys. 349, 117-158 (2014).

I. Bloch, J. Dalibard, and S. Nascimbène, Quantum Simulations with Ultracold Quantum Gases, Nat. Phys. 8, 267-276 (2012).

I.M. Georgescu, S. Ashhab, and F. Nori, Quantum Simulation, Rev. Mod. Phys. 86, 153-185 (2014).

X.-G. Wen, Colloquium: Topological Orders in Rigid States, Rev. Mod. Phys. 89, 041004 (2017).