**1. Introduction**

According to the rapid development of experimental techniques, the downsizing of devices has been accelerated to extend possibilities to control single-electron current [1], spin-polarized current [2,3] and even thermal transport [4]. The trend is based on the aim to construct electronics with low energy consumption, quantum information processing, as well as quantum metrology [1,5].

Quantum pumping phenomena have attracted intensive attention, since they show the controllability of quantum transfer to extend the possibility of nano fabrications. The first proposal of quantum pumping was given by Thouless to transport electrons between two environments [6,7]. Its essential point is to adiabatically or slowly modulate the potential, which is described with the superposition of two standing waves in an out-of phase way [6–8]. Since the work of Thouless, the "adiabatic" change or "slow" modulation of parameters has played a central role in theoretical treatments of quantum pumping phenomena. However, the fast development of experimental techniques after the first experimental study on electron pumping with a quantum dot [9] requires us to investigate conditions on transferring quanta quicker and more precisely. In the present review, we classify the meaning of "quick" or "slow" in quantum pumping and show a standardized theoretical treatment—called full counting statistics (FCS)—to attain the purpose.

The physical situations referred to by the same term "adiabaticity" are roughly divided into three categories: (1) slow change in potential to allow the application of the adiabatic approximations to wave functions associated with transported particles [6,7]; (2) slow and small change of parameter(s) such as the chemical potential and voltage to allow a linear expansion of the scattering matrix [10–17] or Green functions [18–26] associated with transported electron charge or spin; and (3) slow change of parameters compared with the relaxation time of the relevant system with using FCS [27–32]. Different from the former two treatments, the third succeeds in including explicitly the finite relaxation time in adiabaticity. Because isolating any quantum system from its surroundings is impossible, considering relaxation phenomena is indispensable in implementing quantum pumping systems. With further developments of experimental techniques in mind, removing the adiabatic condition in this third instance is necessary. Before going further, let us provide a quick review of the conventional studies on pumping phenomena from the view point of the above summarized classifications of the adiabaticity.

Sinitsyn and Nemenman treated the classical two-state stochastic system [27] using FCS to represent the pumped quantity with a geometrical phase, first expounded in Reference [33]. The relationship between finite relaxation time and adiabaticity in the context of open quantum systems was first discussed by Ren, Hänggi, and Li [28] by extending the FCS approach in Reference [27]. Considering a two-level system coupled to two environments, they found that the pumping of energy quanta occurs under out-of-phase conditions and sufficiently slow (adiabatic) temperature modulations of the environments even if the bias is averaged out during a period. The power of the FCS approach can be found in the further application to electron charge pumping through single or double quantum dot(s) coupled to two leads [29–31]. They found that the sufficiently slow and out-of-phase modulations of the chemical potential can induce electron charge pumping, which is represented by a geometrical formula. In these instances, the condition for a sufficiently slow (adiabatic) environmental modulation means that the relevant two-level system approaches steady state sufficiently quickly.

In many varieties of quantum pumping phenomena, the generation of spin polarized electron current (spin current) by periodic parameter modulation has attracted a grea<sup>t</sup> deal of attention because of its promising applications in spintronics. Referred to as spin pumping, much effort has been made to develop its protocols. Conventionally, the protocols fall roughly into three classes: those using (i) a precession of magnetization in a magnetic material attached to a normal metal [12,13,19–26,34–36]; (ii) a periodic modulation of parameters such as gate voltages and/or tunneling amplitudes in a system consisting of quantum dots subjected to a magnetic field and normal metal leads [14,32,37–40]; and (iii) a periodic modulation of strength of magnetization in addition to parameters in a system consisting of quantum dots attached to a magnetic lead and/or normal metal leads [34,35,41,42]. Among these protocols, those using precession of magnetization—protocol (i)—have attracted intensive studies because it can generate pure spin current in a simple ferromagnet/normal metal heterojunction [3]. So far, the protocol has been mostly studied in situations where the precession of the magnetization is sufficiently slow, which is called adiabatic pumping. It was first proposed by Tserkovnyak et al. [12,13] based on the scattering theory of adiabatic quantum pumping given by Brouwer [11]. Its alternative formalisms based on Green's function [19,20,22–26,36] have also been proposed by several authors. In these studies, the adiabatic contribution to the spin current generation has been obtained as a linear response to the precession, which corresponds to adiabaticity No. (2), with an implicit assumption of an infinite relaxation time. There are a few studies addressing adiabatic spin pumping with a finite relaxation time [34,35], where a slow modulation means smallness of the precession frequency comparing with the tunneling rate.

In the present article, we intend to review our recent studies on the role of nonadiabaticity with a finite relaxation time in quantum pumping of energy quanta and electron spins. For the purpose, we rely on adiabaticity condition No. (3), where the adiabaticity means a slow modulation of parameters compared to the relaxation time of the relevant system—defining the relaxation time of the relevant system as *τr*, we find that the condition for a slow modulation requires *τr* to be much shorter than the period of the temperature modulation, which is written equivalently in terms of the modulation frequency Ω, *τ*<sup>−</sup><sup>1</sup> *r* Ω. Thus, we consider the nonadiabatic regime up to *τ*<sup>−</sup><sup>1</sup> *r* - Ω in the following. As a formulation of nonadiabatic pumping, we present our extension of the FCS approach to quantum pumping toward the nonadiabatic regime. By applying the formulation to the pumping phenomena of energy quanta and electron spin, we find the following features: For the former, we demonstrate that nonadiabaticity yields a contribution to the pumped quantity in addition to the terms such as dynamical and geometrical phase terms which were obtained under adiabatic conditions. For the latter, surprisingly, we show that there are no contributions under the adiabatic condition and nonadiabaticity is an essential feature.

In the rest of the paper, we present in Section 2 our formulation describing the pumped quantity based on FCS. We discuss quantum heat pumping in Section 3 and spin pumping in Section 4, followed by a discussion and conclusion in Section 5.
