We set up a general density-operator approach to geometric steady-state pumping through slowly driven open quantum systems. This approach applies to strongly interacting systems that are weakly coupled to multiple reservoirs at high temperature, illustrated by an Anderson quantum dot. Pumping gives rise to a nonadiabatic geometric phase that can be described by a framework originally developed for classical dissipative systems by Landsberg. This geometric phase is accumulated by the transported observable (charge, spin, energy) and not by the quantum state. It thus differs radically from the adiabatic Berry-Simon phase, even when generalizing it to mixed states, following Sarandy and Lidar.As a key feature, our geometric formulation of pumping stays close to a direct physical intuition (i) by tying gauge transformations to calibration of the meter registering the transported observable and (ii) by deriving a geometric connection from a driving-frequency expansion of the current. Furthermore, our approach provides a systematic and efficient way to compute the geometric pumping of various observables, including charge, spin, energy and heat. These insights seem to be generalizable beyond the present paper's working assumptions (e.g., Born-Markov limit) to more general opensystem evolutions involving memory and strong-coupling effects due to low temperature reservoirs as well.Our geometric curvature formula reveals a general experimental scheme for performing geometric transport spectroscopy that enhances standard nonlinear spectroscopies based on measurements for static parameters. We indicate measurement strategies for separating the useful geometric pumping contribution to transport from nongeometric effects.A large part of the paper is devoted to an explicit comparison with the Sinitsyn-Nemenmann fullcounting statistics (FCS) approach to geometric pumping, restricting attention to the first moments of the pumped observable. Covering all key aspects, gauge freedom, pumping connection, curvature, and gap condition, we argue that our approach is physically more transparent and, importantly, simpler for practical calculations. In particular, this comparison allows us to clarify how in the FCS approach an "adiabatic" approximation leads to a manifestly nonadiabatic result involving a finite retardation time of the response to parameter driving. PACS numbers: 73.63.Kv, 05.60.Gg, 72.10.Bg 03.65.VfTABLE I. Comparison of geometric density-operator approaches relevant to this paper. Approach Prior works Present work (I) Adiabatic state Adiabatic mixed-state geometric phase 37,38 Zero geometric phase for adiabatic steady-state [Sec. III A] evolution (ASE) Mixed-state adiabatic-response correction 37,38 Zero geometric phase for nonadiabatic state [Sec. III A] Gauge freedom related to eigenvector rescaling 37,38 Restriction of gauge freedom by normalization and hermiticity [Sec. III A] (II) Full counting Geometric part of generating function 40,44,72-75 Restriction of gauge freedom by real-valuedness observable [Sec. V C 3] ...

We analyze time-dependent transport through a quantum dot with electron-electron interaction that is statically tunable to both repulsive and attractive regimes, or even dynamically driven. Motivated by the recent experimental realization [Hamo et. al, Nature 535, 395 (2016)] of such a system in a static double quantum dot we compute the geometric pumping of charge in the limit of weak tunneling, high temperature and slow driving. We analyze the pumping responses for all pairs of driving parameters (gate voltage, bias voltage, tunnel coupling, electron-electron interaction). We show that the responses are analytically related when these different driving protocols are governed by the same pumping mechanism, which is characterized by effective driving parameters that differ from the experimental ones. For static attractive interaction we find a characteristic pumping resonance despite the 'attractive Coulomb blockade' that hinders stationary transport. Moreover, we identify a pumping mechanism that is unique to driving of the interaction. Finally, although a single-dot model with orbital pseudo-spin describes most of the physics of the mentioned experimental setup, it is crucial to account for the additional (real-)spin degeneracies of the double dot and the associated electron-hole symmetry breaking. This is necessary because the pumping response is more sensitive than DC transport measurements and detects this difference through pronounced qualitative effects.