# Open Quantum Systems: Liouville-space Field Theory

## Lecture details

The lecture will be given by Professor Wegewijs.

**Prerequisites**

quantum mechanics and statistical mechanics

second quantization + Wick theorem

suitable for students in the first Master semester

**Topics**

The course develops quantum-field theoretical techniques for addressing open quantum systems out of equilibrium within a density-operator approach. Examples of such systems range from quantum dots realized in nanoscale devices (atoms, molecules, heterostructures) to quantum-information processing setups (e.g., readout-devices). Such open quantum systems can be strongly coupled to their environments and can be subject to external time-dependent driving, resulting, e.g., in pumping phenomena. This requires a systematic account of higher-order coupling effects and memory effects beyond the standard Born-Markov approximation. A general framework for performing such calculations is set up and is illustrated, focusing on transport problems involving quantum dots coupled to normal metal electrodes or ferromagnets.

The density operator approach has wide applicability and is used in particular by researchers working on quantum transport through strongly interacting systems as well as quantum information. A central notion is that of Liouville space, the space of physical mixed states of a quantum system coupled to (possibly competing) environments. The Liouville-space formalism that is presented provides a physically elegant and a technically efficient way of "importing" the usual techniques of quantum mechanics and field theory (which apply only to wave vectors) such that they also apply to density operators.

Basic topics:

- Liouville space formalism: physical density operators, superoperators, bra-ket formalism in Liouville space

- Quantum field operators for mixed states: second quantization and Wick-theorem in Liouville space

- Generalized quantum master equations for the density operator

- Symmetries for open quantum systems and superselection rules

- Diagrammatic perturbation theory time-evolution kernels: real-time and frequency space formulations, "Born-Markov" expansion, adiabatic expansion for time-dependently driven systems

- Transport of charge, spin and energy through quantum dots

Advanced topics introduced:

- Counting-statistics: calculation of distribution of 2-point measurement outcomes from a "counting-field" density operator, time-reversal and nonequilibrium fluctuation relations that quantify violation of thermodynamical laws

- Transient time-dependent decay after a parameter "quench"

- Adiabatic pumping of charge, spin and energy

- Renormalization of perturbation theory: exact resummation of "infinite-temperature" contributions leads to huge simplifications

- Renormalization group flow: real-time RG for nonequilibrium to access low-temperature and strong coupling to reservoirs

**Related lecture**

There is a complimentary parallel lecture "Functional renormalization group for nonequilibrium transport through mesoscopic systems" by Dr. Jakobs which is based on a Green's function approach: there the starting point is an expansion in the many-particle interactions around an exact 1-particle solution. In the density operator approach discussed in my course one instead expands in the 1-particle interaction around an exact many-particle solution.

**Subsequent lecture**

There will be a another lecture on "Open Quantum Systems" by Professor Schoeller (semester not yet known) focusing more on real-time renormalization group calculations of the difficult low-temperature, strong coupling regimes of, e.g., the Kondo and spin-boson model.

**First lecture: Thursday, April 9th: 14.15 – 15.45 Modulbau 2, Room 116**

**Note: The regular lecture + exercise class time and place will be optimally rescheduled based on the participants preferences.**

**Please contact **
Maarten Wegewijs
** if you cannot make it to the first lecture**.

For the time schedule, you can also refer to the university calendar:

Time | Room | Start |
---|---|---|

Thurs. 2.15pm -3.45pm | 4273 (MBP2 116) | 09.04.2015 |