# P13: Quantum dots with time-dependent parameters

## Project Description

The real-time dynamics of locally correlated quantum dots which are coupled to leads continuous to reveal new and interesting many-body physics (see e.g. Ref. (1) and references therein). In this project we focus on protocols in which the dot parameters, e.g. the level energies or level-lead couplings depend on time. They are relevant for dot based quantum information science and quantum control theory as well as for possible applications in nanoelectronics and metrology. In addition, this type of physics is expected to come within reach in cold atomic

gases as well. An example for such protocols are abrupt parameter changes, so-called quantum quenches. Several promising methods to tackle time-dependent problems in locally correlated electron systems have recently been developed. Here we will use time-dependent numerical renormalization group (TDNRG)^{(2)} and functional renormalization group (FRG) ^{(3)}.

Within this project we aim at answering three different types of questions.

Firstly, we address fundamental issues about the reliability of TDNRG and FRG. The NRG approach is based on using so-called Wilson chains with logarithmically decaying hopping matrix elements as leads ^{(4)}. It was argued ^{(5)} that this prevents that the leads act as thermal reservoirs---energy is trapped in the dot region. By comparison with the real-time dynamics resulting out of an abrupt parameter change on the dot (or repeated abrupt changes) obtained using FRG we will thoroughly investigate this question. In FRG bothWilson chains as well as standard leads can be implemented. As discussed recently^{(6),(7)} one way out might be the use of hybrid leads (see also Ref. (8)). Here we will further investigate this and search for other possibilities. In the current truncation scheme used in FRG for real-time evolution, terms of second order in the two-particle interaction and higher are only partially included. While the consequences of this at large times have been investigated ^{(3),(9)} it is less clear how reliable the approximate approach is on short to transient times. By a comparison to TDNRG results, which are expected to be highly accurate on these time scales we will address this question.

Secondly, we will work on an extension of the TDNRG approach to changes of the dot parameters continuous in time (e.g. ``quenches'' linear in time). Here the dynamics obtained by FRG in which such type of parameter variations do not present an additional challenge ^{(10),(11)} can be used for benchmarking. Up to now only single abrupt parameter changes^{(2)} or repeated abrupt changes ^{(6),(7)} have been studied using TDNRG. This relates the present project to P8 and P14 in which periodically driven quantum dots are considered. Time-dependent variations of the parameters are also considered in P12 considering closed one-dimensional systems.

Thirdly, we will investigate in detail the non-equilibrium physics for selected protocols. An abrupt quench leads e.g. to oscillations in the relaxation dynamics with frequencies which are well understood in the limits of small^{(12),(13)} and strong ^{(6),(7)} local two-particle interactions. The reasoning in those two limits is very different raising the interesting question of whether and how the explanations are (continuously) connected when considering intermediate interactions. It was recently shown using FRG ^{(11),(14)} that quenching from attractive to repulsive interactions leads to unexpected effects in the dynamics which can be traced back to the non-Markovian memory of the system. We will investigate to what extent this physics can be found in TDNRG and whether it survives if the quenches become continuous. The FRG based approximation is computationally very efficient and can be used to study a variety of protocols rather quickly but is controlled for weak (to intermediate) local correlations. In contrast the TDNRG applies to both strong and weak local correlations, particularly at intermediate to short times. Hence, we plan to use these complementary approaches to search for new correlation physics within different protocols. Besides the already mentioned overlaps with other projects the present one is linked to P10, P11, and P16 in which the equilibrium as well as non-equilibrium physics of quantum dots is investigated.

(1) S. Andergassen, V. Meden, H. Schoeller, J. Splettstoesser and M. R. Wegewijs,

Nanotechnology **21**, 272001 (2010)

(2) F. B. Anders and A. Schiller, Phys. Rev. B **74**, 245113 (2006)

(3) D. M. Kennes, S. G. Jakobs, C. Karrasch and V. Meden Phys. Rev. B **85**, 085113 (2012)

(4) R. Bulla, T. A. Costi and Th. Pruschke, Rev. Mod. Phys.** 80**, 395 (2008)

(5) A. Rosch, Eur. Phys. J. B **85**, 6 (2012)

(6) E. Eidelstein, A. Schiller, F. Güttge and F. B. Anders, Phys. Rev. B **85**, 075118 (2012)

(7) F. Güttge, F. B. Anders, U. Schollwöck, E. Eidelstein and A. Schiller, Phys. Rev. B **87**, 115115 (2013)

(8) A. Branschädel, G. Schneider and P. Schmitteckert, Annalen der Physik **522**, 657 (2010)

(9) D. Kennes and V. Meden, Phys. Rev. B **87**, 075130 (2013)

(10) D. M. Kennes and V. Meden, Phys. Rev. B **85**, 245101 (2012)

(11) D. M. Kennes, O. Kashuba, M. Pletyukhov, H. Schoeller and V. Meden, Phys. Rev. Lett.** 110**, 100405 (2013)

(12) C. Karrasch, S. Andergassen, M. Pletyukhov, D. Schuricht, L. Borda, V. Meden and H. Schoeller,

Europhys. Lett. **90**, 30003 (2010)

(13) S. Andergassen, M. Pletyukhov, D. Schuricht, H. Schoeller and L. Borda, Phys. Rev. B **83**, 205103 (2011)

(14) O. Kashuba, D. M. Kennes, M. Pletyukhov, V. Meden and H. Schoeller, arXiv:1307.3191