**2. General Formalism**

We formulate a model of quantum pumping under periodic modulation of a parameter applying the FCS based on two-point projective measurements [43,44]. Let us consider a system consisting of a relevant system (S) and an environment (E) described by the Hamiltonian

$$H = H\_0 + H\_{\rm SE} \tag{1}$$

where *H*0 ≡ *H*S + *H*E and *H*SE is the system–environment interaction. The FCS provides the statistical average of the net amount of a physical quantity, such as energy and particle number, exchanged between the system and the environment during a certain time interval. It is based on a joint probability of outcomes of two successive projective measurements of an observable of the environment *Q* corresponding to the exchanged quantity. The measurement scheme is—at *t* = *ti*, we perform a projective measurement of *Q* to obtain a measurement outcome *qti* ; for *ti* ≤ *t* ≤ *ti*+1, the system undergoes a unitary time evolution through an interaction between the system and the environment; and at *t* = *ti*+1, we perform a second projective measurement of *Q* to obtain another outcome *qti*+<sup>1</sup> . The joint probability for the measurement scheme is given by

$$P[q\_{t\_{i+1}}, q\_{t\_i}] \equiv \text{Tr}[P\_{q\_{t\_{i+1}}} \mathcal{U}(t\_{i+1}, t\_i) P\_{q\_{t\_j}} \mathcal{W}(t\_i) P\_{q\_{t\_j}} \mathcal{U}^\dagger(t\_{i+1}, t\_i) P\_{q\_{t\_{i+1}}}],\tag{2}$$

where Tr denotes the trace taken over the total system, *Pqt* ≡ |*qtqt*| the projective measurement of *Q* at *t*, *<sup>U</sup>*(*<sup>t</sup>*, *ti*) the unitary time evolution operator for the total system, and *W*(*ti*) the initial condition of the total system (see Note [45]). The net amount of exchanged quantity during the time interval *δt* ≡ *ti*+<sup>1</sup> − *ti* is then given by Δ*qi* ≡ *qti*+<sup>1</sup> − *qti* , where its sign is chosen to be positive when the physical quantity is transferred from the system to the environment. The statistics of Δ*qi* is contained in its probability distribution function

$$P(\Delta q\_i) \equiv \sum\_{q\_{t\_{i+1}}, q\_{t\_i}} \delta(\Delta q\_i - (q\_{t\_{i+1}} - q\_{t\_i})) P[q\_{t\_{i+1}}, q\_{t\_i}]\_\prime \tag{3}$$

The moments of Δ*qi*are provided by the moment generating function,

$$Z(\lambda) \equiv \int\_{-\infty}^{\infty} P(\Delta q\_i) e^{i\lambda \Delta q\_i} d\Delta q\_i. \tag{4}$$

where *λ* is the counting field associated with *Q*, for example, the mean value is computed from

$$
\langle \Delta q\_i \rangle = \frac{\partial Z(\lambda)}{\partial (i\lambda)} \Big|\_{\lambda=0} \,. \tag{5}
$$

Our next task is to describe the time evolution of *<sup>Z</sup>*(*λ*). Using the Definition (3) and introducing the modified evolution operator *<sup>U</sup>λ*(*<sup>t</sup>*, *ti*) ≡ *<sup>e</sup>i<sup>λ</sup>QU*(*<sup>t</sup>*, *ti*)*e*<sup>−</sup>*iλQ*, the moment generating function *Z*(*λ*) is expressed as

$$Z(\lambda) = \text{Tr}[\mathcal{W}^{(\lambda)}(t\_{i+1})],\tag{6}$$

with

$$\mathcal{W}^{(\lambda)}(t) \equiv \mathcal{U}\_{\lambda/2}(t, t\_i) \bar{\mathcal{W}}(t\_i) \mathcal{U}\_{-\lambda/2}^\dagger(t, t\_i),\tag{7}$$

where *W*¯ (*ti*) ≡ ∑*qt i Pqt i W*(*ti*)*Pqt i* is the diagonal part of *<sup>W</sup>*(*ti*). For *λ* = 0, *<sup>W</sup>*(*<sup>λ</sup>*=<sup>0</sup>)(*t*) reduces to the usual reduced density matrix of the total system. By taking the time derivative of *<sup>W</sup>*(*λ*)(*t*), we obtain a modified Liouville–von Neumann equation

$$i\frac{\partial}{\partial t}\mathcal{W}^{(\lambda)}(t) = \mathcal{L}^{(\lambda)}\mathcal{W}^{(\lambda)}(t),\tag{8}$$

with a modified Liouvillian <sup>L</sup>(*λ*)*W*(*λ*)(*t*) ≡ *h*¯ <sup>−</sup><sup>1</sup>[*<sup>H</sup>*, *<sup>W</sup>*(*λ*)(*t*)]*<sup>λ</sup>*, where [*<sup>A</sup>*, *<sup>B</sup>*]*λ* ≡ *A*(*λ*)*B* − *BA*(−*<sup>λ</sup>*), and *A*(*λ*) ≡ *eiλQ*/2*Ae*−*iλQ*/2. In Appendix B, we explain the connection between the formalism of the FCS based on two-point measurements and the formalism by Sinitsyn and Nemenman in Reference [27].

By introducing a projection operator P : *W*(*λ*)(*t*) → TrE[*W*(*λ*)(*t*)] ⊗ *ρ*E, where TrE is the partial trace taken over the environment and *ρ*E is a fixed state of the environment, the equation of motion for the reduced operator of the relevant system *ρ*(*λ*)(*t*) ≡ TrE[*W*(*λ*)(*t*)] can be cast into the form of a time convolutionless (TCL)-type quantum master equation [46–53].

Assuming that the initial state is factorized between system and environment as *W*(*ti*) = *ρ*(*ti*) ⊗ *ρ*E and the fixed state of the environment *ρ*E is the Gibbs state with an inverse temperature *β*, the TCL master equation including the counting field is expressed as

$$
\frac{
\partial
}{
\partial t
} \rho^{(\lambda)}(t) = \mathfrak{J}^{(\lambda)}(t) \rho^{(\lambda)}(t).
\tag{9}
$$

The super-operator *ξ*(*λ*)(*t*) is expanded as a sum of "ordered cumulants" of the interaction Hamiltonian *H*SE up to infinite order. Taking leading terms up to second-order, we have

$$\xi^{(\lambda)}(t)\rho^{(\lambda)}(t) = -\frac{i}{\hbar}[H\_{\text{S}}\rho^{(\lambda)}(t)] - \frac{1}{\hbar^2}\int\_0^t d\tau \text{Tr}\_{\text{E}}[H\_{\text{SE}}, [H\_{\text{SE}}(-\tau), \rho^{(\lambda)}(t)\otimes \rho^{\text{eq}}\_{\text{E}}]\_{\lambda}]\_{\lambda}.\tag{10}$$

Note that the time dependence of the memory kernel reflects the finiteness of the correlation time of the dot–lead interaction, which allows us to describe the non-Markovian dynamics.

To work with the super-operator, it is convenient to introduce its supermatrix representation, where we represent the density matrix *ρ*(*λ*) in vector form and the super-operator *ξ*(*λ*) in matrix form. In this representation, the formal solution of the master equation Equation (9) is expressed as

$$\langle |\rho^{(\lambda)}(t)\rangle \rangle = \mathcal{T}\_{+} \exp\left[\int\_{t\_i}^{t} \Xi^{(\lambda)}(s) ds\right] |\rho^{(\lambda)}(t\_i)\rangle \rangle,\tag{11}$$

where |*ρ*(*λ*)(*t*) represents the vector form of *<sup>ρ</sup>*(*λ*)(*t*), T+ exp the time-ordered exponential, and Ξ(*λ*)(*t*) the supermatrix form of *ξ*(*λ*)(*s*). With the representation, the moment generating function Equation (6) is rewritten as *Z*(*λ*) = TrS[*ρ*(*λ*)(*ti*+<sup>1</sup>)] = 1|*ρ*(*λ*)(*ti*+<sup>1</sup>), where TrS is the partial trace taken over the relevant system and 1| the vector representation of the partial trace TrS. Using the formal solution Equation (11), we recast the expression of the mean value into the form

$$
\langle \Delta q\_i \rangle = \int\_{t\_i}^{t\_{i+1}} f(s)ds \tag{12}
$$

with the inertial flow of the quantity,

$$J(t) \equiv \langle \langle 1 | \left[ \frac{\Im \Xi^{(\lambda)}(t)}{\partial(i\lambda)} \right]\_{\lambda=0} | \rho^{(\lambda=0)}(t) \rangle \rangle. \tag{13}$$

To formulate quantum pumping based on the above framework, we need to accumulate transfers of the physical quantity under a cyclic modulation of system and/or environmental parameters during a period T . For this purpose, we consider a step-like change of the parameters; specifically, dividing the period T into *N* intervals, *ti* ≤ *t* ≤ *ti*+<sup>1</sup> (*i* = 1, 2, ··· , *N*) with *t*1 = 0 and *tN*+<sup>1</sup> = T , fixing a value of the parameters during each interval, and changing the value at each *ti* discretely. With the total density matrix factorized at each *ti*, the mean value as well as the inertial flow of the quantity for each interval are given by Equations (12) and (13), respectively. The time integration of *J*(*t*) over one period T provides the accumulated value of the quantity over one cycle

$$<\langle \Delta q \rangle \equiv \int\_0^\mathcal{T} J(t)dt = \sum\_{i=1}^N \langle \Delta q\_i \rangle,\tag{14}$$

where <sup>Δ</sup>*qi* is the mean value of the net transferred quantity in the *i*th interval.
