Prerequisites

• Part II Advanced Quantum Physics (ladder operators)

• Part II Thermal and Statistical Physics (Bose & Fermi statistics, Phase transitions)

• Part III Atomic and Optical Physics

While all concepts will be briefly reviewed, basic knowledge of Bose and Fermi statistics and the formalism of second quantization (ladder operators) from Part II Thermal and Statistical Physics and Part II Advanced Quantum Physics is required, as is the content of the Part III Atomic and Optical Physics course. Familiarity with the Bloch wave formalism from Part II Quantum Condensed Matter is helpful but not required.

Learning Outcomes and Assessment

Quantum simulators use tailored quantum systems - such as ultracold atoms, trapped ions, or superconducting qubits - to experimentally investigate fundamental models of complex quantum systems.  They operate in a fashion broadly similar to wind tunnels for aerodynamics research and can be used to study fascinating effects in condensed matter physics, high energy physics, relativistic and black hole physics, and other contexts.

This course will mostly focus on non-relativistic many-body quantum systems that are, due to their exponentially large Hilbert spaces, impossible to model classically even on the best supercomputers.  Our main example will be interacting electrons in a solid and their emergent phenomena such as magnetism, (High-Tc) superconductivity and other transport phenomena, and topological phases.

Synopsis

Introduction - The challenge of many-body physics

• Exponential scaling of Hilbert space, Limits for exact solutions
• What are Quantum Simulations?
• Examples of emergent phenomena: Mott insulators, Superfluidity & Superconductivity, Magnetism, Strongly correlated systems, Quantum Hall effect, Topology, Spin liquids

Synthetic many-body systems

• Ultracold atoms
• Rydberg atoms in tweezer arrays
• Others: Trapped ions, Superconducting qubits,  NV centers,...

Phase Transitions and Quantum Fluids

• Bose condensation, Superfluidity, and long-range order.

Optical Lattices

• Implementation
• Bloch waves, Wannier functions, tight-binding limit, Hubbard models.

Bosons in lattices

• Bose Hubbard model
• Bosonic Mott insulators
• Quantum phase transition / Mean-field phase diagram / Local density approximation
• Measuring structure factors
• Quantum gas microscopes
• Higgs mode, connection to relativistic theories (not examinable).

Quantum Thermodynamics

• Closed system quantum thermodynamics
• Negative absolute temperatures
• Measuring entropy
• Critical temperatures / critical entropy scales.

Quantum Magnetism and Fermi- Hubbard model

• Band mapping techniques
• Fermionic Mott insulators
• Heisenberg antiferromagnet
• Connection to high-Tc superconductors
• Quantum Magnetism in Tweezer arrays

Out-of-Equilibrium dynamics & Thermalization in closed quantum systems

• Eigenstate thermalization hypothesis
• Measuring entanglement
• Alternatives to thermalization: Integrable systems, Many-body localization, Quantum many-body scars
• Transport properties / (anomalous) diffusion
• Light-cone effects and correlation measurements
• Relativistic theories, Dirac equation and Klein tunneling (not examinable)
• Structure Formation, Kibble-Zurek Mechanism
• Driven-dissipative systems, reservoir engineering.

Artificial Gauge fields and Topology

• Hall effect, geometric phases, topological invariants
• Artificial gauge fields
• Periodic modulations and Floquet theory
• Hall conductance and topological insulators
• Thouless charge and spin pumps
• Non-Abelian gauge fields.