P13: Geometric pumping in driven non-Markovian open systems
Within the RTG, we found  that the geometric contribution to the response of external observables to the driving of an open system has its physical origin in the freedom to recali-brate the detector registering the response. When explicitly including an ideal detector in the description, one can derive, using standard considerations, either of two prominent formalisms for geometric pumping: the adiabatic-response (AR) and the full counting statistics (FCS). We have shown  that this works for Markovian dynamics of open systems with strong interactions, when restricting the comparison of the approaches to the first moment (which is equivalent to the first cumulant).
In this project, we make the decisive generalization to arbitrary non-Markovian open systems, all moments and cumulants of the detector output, both in real-time and Laplace-frequency formulations. This allows us to systematically study slow- and fast-driving dynamics (corresponding to adiabatic, weakly non-adiabatic and antiadiabatic driving). On the one hand, using our encompassing ideal-detector approach, we can systematically extend the two prominent formalisms for pumping (slow driving) without losing track of their exact equivalence, which exists if no coupling or driving expansion is made. In particular, the AR approach has a non-Markovian extension, as already demonstrated . We will extend its geometric (gauge-invariant) formulation, as derived in Refs.  and , to all moments. On the other hand, the geometric formulation of the FCS  approach concerns ar-bitrary cumulants, but has so far been restricted to driven Markovian systems. We will develop the general non-Markovian formulation following up prior work . Our goal is to fully develop both approaches consistently on the level of physical approximations and geometric structure (gauge freedom, connection, and curvature). This is mutually beneficial: on the one hand, the FCS is closely tied with the ideal-detector model and the geometry of its physical calibration. On the other hand, the slow-driving expansion requires a systematic account of memory contributions which is less transparent in the FCS approach and which was com-pletely ignored in the geometric Markovian FCS . This poses no problem in the AR approach  for which, however, the meter calibration of higher moments seems less accessible. We will compare the slow-driving expansions for higher moments obtained from these approaches which automatically give different approximation schemes due to the non-linear relation between moments (AR) and cumulants (FCS).
We will investigate how memory effects renormalize the geometric dynamics for slow driving for concrete model systems. Applications, that we have in mind, are driven quantum-optical setups (polaritons with repulsive Kerr interaction, qubits with attractive interaction) and nano-electric circuits with tuneable interaction that we have studied in the first funding period of the RTG.
Finally, within the general ideal-detector framework, we will investigate the universal geometric phase for open systems, i.e., a geometric phase that does not rely on slow driving. For closed systems it was realized soon  after Berry that his celebrated phase is actually a special case  of such a more general non-adiabatic geometric phase. For Markovian open systems, this universal geometric phase has been discussed . Our aim is to establish a more concrete relation with computational schemes such as Floquet – studied within the RTG – and exact adiabatic iteration  and go beyond the Markovian approximation. The universal geometric phase is of interest for a topological classification of periodically driven, strongly interacting, non-equilibrium open systems analogous to materials with real-space periodicity.
We note that this project complements the RTRG approach developed in project P4 which actually corresponds to an AR formulation. The present project supplements this with the FCS formulation which relies on a different type of expansion (cumulants) and leads to different approximation schemes. Furthermore, the mere application of Floquet theory as in project P4 also does not address the gauge-invariance and geometric nature of driven responses (pumping): one has to apply Floquet to the combined system and measurement device as we learned in the first funding period .
If successful, the results of projects P4 and P13 can be brought together into a framework that enhances both computational power and physical transparency. The projects P7, P12, and P13 have a common core but each define a well separated PhD project.
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