# Open quantum systems

- Introduction
- Driven open quantum systems and Cooper pair pumping
- Quantum driving
- Non-Markovian quantum dynamics
- Conservation laws

**Introduction**

An open quantum system is a quantum system that interacts with another quantum system, known as the environment. It is usually impossible to calculate the time evolution of the total system, consisting of the open quantum system and the environment, exactly. This would also lead to a large amount of redundant information as the knowledge of the detailed behavior of the environment is often not needed. In this situation, it suffices to describe the open quantum system by a differential equation that takes into account the influence of the environment but neglects the details of its time evolution. A paradigmatic example of an open quantum system is a two-level system. The environment is typically chosen to consist of harmonic oscillators.

Essentially all realistic quantum systems are open quantum systems. For this reason, the methods of the theory of open quantum systems have found applications in numerous fields of physics. In our research group, we have used these methods to study, for example, the effects of noise and driving on Cooper pair pumping, the relation between quantum driving and quantum work relations, the non-Markovian dynamics caused by information backflow from the environment, and the conservation laws of open quantum systems.

See also our research pages on quantum information processing for applications of the theory of open quantum systems on quantum devices.

**Driven open quantum systems and Cooper pair pumping**

Describing the combined effect of noise and driving is no easy task when it comes to quantum systems. The typical Redfield approach strictly speaking only applies to non-driven systems rendering its use highly speculative especially for fast driving. However, the control of quantum systems using external fields is a day-to-day practise in microscale engineering where an accurate description of the dynamics is necessary to, for example, mitigate the error in quantum information processing. Our approach to solving this issue is based around the proper selection of the dynamical basis in the near-adiabatic limit [Phys. Rev. Lett. 105, 030401 (2010); Phys. Rev. B 82, 134517 (2010)] and beyond [Phys. Rev. A 82, 062112 (2010); Phys. Rev. B 84, 174507 (2011)]. These efforts have sparked further interest in improving and extending the theory in the research community (See for example [Phys. Rev. B 83, 214508 (2011); Phys. Rev. A 87, 042111 (2013)]).

**Fig.** Charge pumped through a Cooper pair sluice under gate charge noise with respect to (a) the superconducting phase difference over the sluice and (b) the environmental coupling strength.

Our main applications for the accurate theory of driven dissipative dynamics lies in superconducting nanoelectronics. More precisely, the geometric phase evolution for driven systems gives rise to Cooper pair pumping for which we have additionally derived fundamental theoretical [Phys. Rev. B 73, 214523 (2006)] and experimental results [Phys. Rev. Lett. 100, 177201 (2008)]. From these considerations, we propose superconducting realizations of non-Abelian phases [Phys. Rev. B 81, 174506 (2010)] and geometric quantum computing [Phys. Rev. B 83, 214518 (2011)]. In addition, we have theoretically studied and maintain interest in improving the modeling of such superconducting systems related to geometric charge transport [Phys. Rev. A 84, 052103 (2011); Phys. Rev. B 86, 184512 (2012)].

**Fig.** Circuit diagram of the Cooper pair sluice required for the inclusion of a nonvanishing loop inductance.

**Quantum driving**

Typically the theoretical description of driven quantum dynamics arises from the external-field-induced time-dependence of the system Hamiltonian (See for example [Rev. Mod. Phys. 81, 1665 (2009)]). However, this starting point for developing theoretical tools and methods already entails the assumption that the driving is done from outside the dynamical picture. We aim to go beyond such assumption and deconstruct the total system into its composite subparts assigning the task of driving to a specific quantum subsystem. This brings about a natural definition of injected work based on changes in internal and interaction energies which reduces to a previous result [Phys. Rev. B 87, 060508(R) (2013)] when we go to the limit of classical driving. Much work is to be done with this novel approach to driving with regards to its relation to quantum work relations, its practical applicability to modeling physical systems and the design of complex driving protocols from the composite interactions. These are our future research directions in this area.

**Non-Markovian quantum dynamics**

Most theoretical models of open quantum systems rely on several approximations allowing an analytically solvable treatment of their dynamics. The Markov approximation, which neglects any back action of the environment on the open system, is often employed, giving us a memoryless description of the open system dynamics. Although this is the most commonly studied situation, non-Markovian effects often arise in current experimental systems where the high level of control can give rise to scenarios, where e.g. the system-environment coupling is strong, the reservoir is structured or has a finite size, or the temperature is low.

Recent advancements in the definition and quantification of quantum non-Markovianity [Phys. Rev. Lett. 103, 210401 (2009); Phys. Rev. Lett. 101, 150402 (2008); Phys. Rev. Lett. 105, 050403 (2010)] have initiated many essential steps towards the development of a general consistent theory of non-Markovian quantum dynamics as well as achievements in the experimental detection and control of memory effects [Nature Physics 7, 931 (2011); EPL 97, 10002 (2012); Scientific Reports 3, 1781 (2013)]. Yet, many fundamental problems regarding the applicability of some of the most widely applied approximations are still unsolved as studied out by us in Ref. [Phys. Rev. A 88, 052111 (2013)]. At the moment, we concentrate on finding exact open quantum system descriptions of superconducting quantum circuits and on developing reservoir engineering techniques to achieve adjustable coupling schemes that allow to arbitrarily tune the environment and modify the open system dynamics at will.

**Conservation laws**

Conservation laws quantify some of the most fundamental properties of any physical system.For example, the conservation of electric charge is the key in almost all experiments on electron transport in nanoelectronics. However, in theoretical modeling of open quantum systems some simplifying assumptions are typically necessary to obtain a usable picture of the system evolution. Here, one has to be especially careful when treating the fast-oscillating terms in accordance with the so-called secular approximation in an effort to obtain a master equation for the reduced system dynamics. Depending on the nature of the dissipative environment, the secular approximation may lead to a non-conservation of physical quantities in time-evolution. To generally define the principle of such conservation for master equations, we identify a conservation law for operator current in Ref. [Phys. Rev. A 85, 032110 (2012)]. We have already applied these principles to Cooper pair pumping using an artificial flux noise environment [Phys. Rev. B 85, 024527 (2012)] but more fundamental ideas might arise from the nature of the law itself.

**Fig.** Charge pumped through a Cooper pair sluice under gate charge noise with respect to the environmental coupling strength using different master equations. The straight line uses the full master equation and the descending lines use the secular master equation giving the charge through the left and right junctions for a closed path in the Hamiltonian space.