Quantized dynamics in a square-wave-driven coherent single-electron source

Abstract

Through analytic derivation within the nonequilibrium Green’s function (NEGF) formalism, we present a comprehensive study of a quantum-coherent single-electron emitter. This emitter is based on a quantum \mathrm{R}\mathrm{C} circuit driven by periodic square-wave potentials with temporal period T and angular frequency \omega =2\pi /T. Our theoretical results reveal three key characteristics of the single-electron emitter: (i) the characteristic time \alpha of the exponentially decaying current J(t)(\propto {\mathrm{e}}^{-t/\alpha }) matches the intrinsic coherence time \tau of the quantum dot; (ii) the Fourier spectrum of the current J(m\omega ) carries information of both external driving amplitude U and internal energy levels {ϵ}_d of the quantum dot; (iii) the quantization of fundamental Fourier component |J_1|=2ef (f=1/T) accompanies the half-quantized relaxation resistance R_q=h/(2e^2), featuring quantized dynamics in AC transport through the single-electron emitter. These findings offers new perspectives into the dynamic properties of quantum-coherent AC transport.

Graphical abstract

Keywords

nonequilibrium Green’s function

1 Introduction

The quantum hierarchy in electrical transport manifests through three cardinal quantization paradigms: (i) Conductance quantization, emerging from discrete electronic states in ballistic channels [1, 2], forming the basis of Landauer−Büttiker formalism. (ii) Flux quantization [3], such as fluxoid \mathrm{\Phi }=h/e in Aharonov−Bohm effect or superconducting fluxoid \mathrm{\Phi }=h/(2e). (iii) Quantum Hall hierarchy, exhibiting \nu-quantized plateaus through Landau levels filling in two-dimensional electron gases (2DEGs) [4, 5], establishing the von Klitzing constant R_K=h/e^2 as resistance metrology’s cornerstone [6–8]. While traditional DC transport regimes have unveiled foundational quantum phenomena, the exploration of time-resolved quantum transport reveals novel phenomena absent in DC regimes as well. A striking example is the half-quantized relaxation resistance R_q=h/(2e^2) which is originated from the frequency-dependent admittance [9] and confirmed through single-electron capacitance spectroscopy [10]. It establishes fundamental constraints on frequency-dependent impedance parameters (capacitance C, resistance R and inductance L) [11]. On the other hand, if frequency and amplitude are sufficiently high, coherent control of single electrons [12] can also be achieved. This can be realized through single-electron pumps [13–17] or single-electron sources [18–20]. The single-electron pump have significantly advanced the development of quantum current standards [21–26], achieving uncertainties of less than 1\mathrm{p}\mathrm{p}\mathrm{m}.

Single-electron source enables on-demand electron emission with temporal precision comparable to photon manipulation in optical systems. This capability forms the foundation for emerging quantum technologies including quantum sensing [27, 28], electron motion capture [29], solid-state flying qubit control [30], electron control in quantum optics [31–33]. Four primary methodologies have been developed for single-electron source: (i) Tunable-barriers pumps through utilizing time-dependent gate voltages [16, 34–36]; (ii) Single-electron transfer through Coulomb interaction [37, 38]; (iii) Itinerant quantum dots obtained using surface acoustic waves caused by piezoelectric effect [39–42]; (iv) Leviton generation, i.e., the creation of minimal-excitation wave packets through Lorentzian voltage pulses [43–46]. These sources have been demonstrated across diverse material systems, including two-dimensional graphene [47], molecular junctions [48], and silicon-based nanostructures [14, 29]. For quantum information applications, given the critical importance of temporal precision in electron emission, the paradigmatic single-electron source employs alternating current [10, 18, 49], where radio-frequency voltage drives induce single-electron emission/absorption during alternating half-cycles of the capacitive charging/discharging sequence. Several theoretical approaches have been employed to solve the dynamic response of quantum system. For harmonic driving sources, the Floquet scattering matrix formalism [45, 50–52] provides a non-perturbed description of time-periodic transport phenomena in low frequency. The quantum master equation approach [16] is effective for modeling dissipative dynamics in weakly coupled systems. The nonequilibrium Green’s function (NEGF) techniques [53–62] and the scattering matrix [63–65] also provide various perturbative treatment of dynamic response in energy representation.

In this work, we present a theoretical investigation for a quantum-coherent single-electron emitter using the NEGF formalism. Based on a quantum \mathrm{R}\mathrm{C} circuit setup [66] featuring a single-electron reservoir [67] driven by periodic square-wave potentials, we reproduce the existing experimental findings [18], i.e., the coherent emission-absorption cycles and the quantized alternating current. Remarkably, our results achieve superior current quantization precision beyond existing benchmarks [20, 68], exhibiting enhanced operational robustness in open quantum environments. It demonstrates that square-wave signals can also effectively drive single-electron emission, providing an alternative to established approaches based on smooth AC driving potentials [20, 59] or transient square waves [19, 69]. Employing NEGF techniques, we systematically derive the time-resolved spectral function A(t,ϵ), thereby enabling rigorous characterization of the non-stationary AC current J(t). Frequency-domain analysis through Fourier decomposition yields essential harmonic coefficients J(m\omega ), providing information into the dynamic response of the quantum system. These findings include: (i) the characteristic time \alpha of the exponentially decaying current J(t)(\propto {\mathrm{e}}^{-t/\alpha }) matches the intrinsic coherence time \tau of the quantum dot; (ii) the decaying current oscillates with a frequency proportional to the external driving amplitude U, which is also supported by the Fourier component of the current J(\mathrm{\Omega }) occurring precisely at \mathrm{\Omega }=U; (iii) quantized fundamental Fourier component |J_1|=2ef is directly related to the half-quantized relaxation resistance R_q=h/(2e^2), revealing the quantum dynamics of AC transport through this single-electron emitter. These theoretical results derived from the quantum \mathrm{R}\mathrm{C} circuit provide new insights for understanding AC transport phenomena.

The rest of this paper is organized as follows: Section 2 introduces the system Hamiltonian and establishes the theoretical framework for analyzing alternating current under periodic square wave voltage signals. In Section 3, we perform numerical calculations and discuss the physical interpretations of the obtained results. Section 4 extends our analysis beyond the conventional wideband regime, exploring more generalized operational conditions. Finally, a comprehensive summary is shown in Section 5.

2 Theoretical formalism

Our analysis begins with the quantum \mathrm{R}\mathrm{C} circuit configuration illustrated in Fig.1(a). The system comprises a semi-infinite lead and a quantum dot with gate-tunable energy levels {ϵ}_d. The quantum dot is electrically coupled to a semi-infinite lead through quantum point contact technology [10, 18]. This setup employs a single-lead design as documented in Refs. [19, 55, 67], where the lead is subjected to a time-periodic external potential U(t)=U(t+T). The quantum dot’s energy spectrum can be precisely controlled via gate voltage V_g modulation, following established methodologies in Refs. [70–72]. The periodic square wave

is selected as the driving potential. The configuration of U(t) is shown in Fig.1(b). The abrupt transition edges provide optimal temporal resolution for investigating AC response dynamics. Within each oscillation cycle, two distinct operational phases emerge. During the first half-period, electron injection occurs from the external terminal into the quantum dot, while retrieval processes dominate the latter half-period [Fig.1(c) and (d)]. This bidirectional charge transfer mechanism manifests as alternating-polarity current pulses in the temporal domain, thereby giving rise to temporally resolved current signatures.

The total Hamiltonian is constituted from three components: the Hamiltonian of the isolated quantum dot H_d, the Hamiltonian of the isolated semi-infinite lead H_l, and their coupling term H_t. It is expressed as H=H_l+H_d+H_t where

\begin{array}{rl}{H}_d& =\sum _n{ϵ}_{d,n}{d}_n^{†}{d}_n,\\ H_l& =\sum _k[{ϵ}_k+U(t)]c_k^{†}{c}_k,\\ H_t& =\sum _{k,n}{t}_k{c}_k^{†}{d}_n+h.c..\end{array}

Here, d_n (d_n^{†}) represents particle annihilation (creation) operator for n-th quantum level of quantum dot. The intrinsic quantum energy levels of the central quantum dot, denoted by {ϵ}_{d,n}, can be adjusted through gate voltage V_g. c_k (c_k^{†}) is the annihilation (creation) operator for electrons with momentum state k in the semi-infinite lead. The quantum dot is coupled to an external Fermi reservoir through semi-infinite lead. The Fermi level of Fermi reservior is modulated by the external excitation U(t). Performing a unitary transformation [73]

P(t)=\exp \left[{\displaystyle \sum _k{\displaystyle \frac{\mathrm{i}}{\hslash }}{\displaystyle {\int }_0^t\mathrm{d}\tau U(\tau )c_k^{†}{c}_k}}\right],

H_l becomes time independent, which is convenient for current calculation. Finally, the transformed Hamiltonian becomes

\tilde{H}={\displaystyle \sum _{k,n}{ϵ}_k{c}_k^{†}{c}_k+{ϵ}_n{d}_n^{†}{d}_n+\left[t_k(t)c_k^{†}{d}_n+h.c.\right],}

where the time dependence of H_l is effectively transferred to the coupling terms H_t with t_k(t)=t_k{\mathrm{e}}^{\frac{\mathrm{i}e}{\hslash }{\int }_0^t\mathrm{d}\tau U(\tau )}.

The system’s configuration employs a single-electron reservoir setup inherently eliminating conventional conduction current pathways. Within this framework, our analysis focuses exclusively on the displacement current component. Adopting the rigorous NEGF formalism, the displacement current flowing through the semi-infinite lead is formulated as

(1)

Here, we have introduced the integral abbreviation

where is retarded or less Green’s function originated from n-th energy level of central quantum dot, and is self energy from the semi-infinite lead. In the presence of time-dependent external excitation U(t), self energy is

(2){\mathrm{\Sigma }}^{\gamma }(t_1,t)={\mathrm{e}}^{\mathrm{i}{\int }_t^{t_1}d\tau U(\tau )}{\mathrm{\Sigma }}^{\gamma ,0}(t_1-t),

where and {\mathrm{\Sigma }}_{\alpha }^{\gamma ,0} is the self energy when U(t)=0, which are expressed as

(3)

where we have introduced the linewidth function

\mathrm{\Gamma }(ϵ)=2\pi {\displaystyle \sum _k\delta (ϵ-{ϵ}_k)|t_k{|}^2=2\pi \rho (ϵ)|t(ϵ){|}^2}

and the Fermi distribution f(ϵ). In the wide-band limit (WBL), \rho (ϵ) and t(ϵ) are energy independent, and \mathrm{\Gamma }=2\pi \rho |t{|}^2, which is also energy independent. Then {\mathrm{\Sigma }}^{r/a,0}(t_1-t)=\mp \frac{\mathrm{i}}{2}\delta (t_1-t)\mathrm{\Gamma }. For notational simplicity, we henceforth set the elementary charge e and the reduced Planck constant \hslash to unity, i.e., e=\hslash =1. Subsequently, by introducing the spectral function [74]

(4)A_n(t,ϵ)={\displaystyle {\int }_{-\mathrm{\infty }}^t\mathrm{d}{t}_1{G}_n^r(t,t_1){\mathrm{e}}^{\mathrm{i}\epsilon \left(t-t_1\right)}{\mathrm{e}}^{\mathrm{i}{\int }_t^{t_1}U(\tau )\mathrm{d}\tau }}

and considering the WBL, Eq. (1) becomes

(5)

with

The first and second term in Eq. (5) are corresponding to the current flowing into and out of quantum dot, respectively. Once the spectral function A(t,ϵ) is ready, the calculation of the current J(t) is straightforward.

Now, we proceed to solve the spectral function A(t,ϵ). Given that {\mathrm{\Sigma }}^{r/a,0}(t_1-t)=\mp \frac{\mathrm{i}}{2}\delta (t_1-t)\mathrm{\Gamma }, from Eq. (2) we have {\mathrm{\Sigma }}^{r/a}={\mathrm{\Sigma }}^{r/a,0}. This implies that the external excitation U(t) exerts no effect on the Green’s function G_n^r. Consequently, we have

(6)G_n^r(t-t_1)={\displaystyle \int {\displaystyle \frac{\mathrm{d}E}{2\pi }}{\mathrm{e}}^{-\mathrm{i}E(t-t_1)}{G}_n^r(E)}

with G_n^r(E)=1/(E-{\widetilde{ϵ}}_n). Here, the complex-valued energy {\widetilde{ϵ}}_n={ϵ}_{d,n}-\mathrm{i}\mathrm{\Gamma }/2 represents the broadened energy level of the quantum dot system. The parameter \mathrm{\Gamma } quantitatively describes the spectral width of this broadened state. Substituting Eq. (6) into Eq. (4), we have

A_n(t,ϵ)={\displaystyle {\int }_{-\mathrm{\infty }}^t\mathrm{d}{t}_1{\displaystyle \int {\displaystyle \frac{\mathrm{d}E}{2\pi }}{G}_n^r(E){\mathrm{e}}^{\mathrm{i}(ϵ-E)\left(t-t_1\right)}{\mathrm{e}}^{\mathrm{i}{\int }_t^{t_1}U(\tau )\mathrm{d}\tau }.}}

Integrating over time t_1, we have

(7)\begin{array}{rl}{A}_n(t,ϵ)=& \mathrm{i}\int \frac{\mathrm{d}E}{2\pi }{G}_n^r(E)[\frac{1}{B\pm U}\\ & \pm \frac{2U{\mathrm{e}}^{\mathrm{i}(B\pm U)\overline{t}}}{(B-U)(B+U)}\left(1-{\mathrm{e}}^{\mathrm{i}(B\mp U)T/2}\right)]\end{array}

with B=ϵ-E+\mathrm{i}{0}^{+}. In the WBL, the parameter \mathrm{\Gamma } is energy independent. So the retarded Green’s function G_n^r(E) possesses a single pole at {\widetilde{ϵ}}_n, with an associated residue of unity 1. This analytical structure enables explicit evaluation of the integral in Eq. (7) through systematic application of the Residue Theorem. Through the semi-infinite lead, electrons can escape from quantum dot to the Fermi sea, and the life time of the energy level of quantum dot is \tau \sim 1/\mathrm{\Gamma }. Under radio-frequency driving conditions, where the period T is much greater than the relaxation time \tau (T\gg \tau). It means \mathrm{\Gamma }T\gg 1, so the exponential phase factor {\mathrm{e}}^{\mathrm{i}(ϵ-{ϵ}_{d,n}+\mathrm{i}\mathrm{\Gamma }/2\mp U)T/2} exhibits asymptotic decay to zero. This temporal hierarchy ensures the vanishing of non-stationary contributions in the long-time limit. Consequently, the spectral function is simplified to

(8)A_n(t,ϵ)={\displaystyle \frac{1}{ϵ-{\widetilde{ϵ}}_n\pm U}}\pm {\displaystyle \frac{2U{\mathrm{e}}^{\mathrm{i}(ϵ-{\widetilde{ϵ}}_n\pm U)\overline{t}}}{(ϵ-{\widetilde{ϵ}}_n-U)(ϵ-{\widetilde{ϵ}}_n+U)}},

where “\pm” denote the first and second half cycle and \overline{t} is the time duration within each half-period. While Eq. (8) is formulated within WBL and does not fully account for realistic spectral constraints, it captures essential physical mechanisms governing the system’s nonequilibrium dynamics.

3 Results and discussion

In our calculations, all quantities are expressed in dimensionless arbitrary units. The Fermi level of the free electron reservoir serves as our reference zero energy, with all other energy values measured relative to this baseline. The system is driven by a periodic square wave potential U(t) with radio-frequency \omega applied to the single lead. Unlike conventional harmonic excitations, this abruptly switching square wave not only probes the intrinsic response speed of nanoscale devices, but also enables precise characterization of transient current decay dynamics. Within the single-terminal RC circuit configuration under square wave excitation, coherent electrons shuttle between the lead and quantum dot. This charge transfer manifests as a distinct positive current pulse during the rising edge of the square wave, followed by a negative pulse during the falling edge. Fig.2 specifically illustrates the transient current J(t) profile during the first half-period excitation. While the external drive U(t) theoretically induces an instantaneous Fermi level shift, practical device limitations prevent abrupt Fermi surface modifications. Consequently, the current cannot instantaneously achieve the maximum due to finite temporal response. The switching dynamics exhibit a strong dependence on the U/\mathrm{\Gamma } ratio. Increasing this parameter enhances switching speed through enhanced field-effect dominance. Fig.2(a) reveals three distinct temporal regimes: (i) rapid current ascent from baseline to maximum amplitude, (ii) peak current, and (iii) exponential relaxation to equilibrium. For sufficiently large U values [Fig.2(b)−(d)], the charging process becomes nearly instantaneous, and region (i) is invisible. After reaching maximum quickly, the current begins to decay slowly. The decaying dynamics are governed by quantum dot energy level broadening effects, enabling electrons/holes escape into electron reservoir from the quantum dot. It results in the decay of the J(t) with characteristic time constant \alpha.

The transient current J(t) can be effectively modeled through analogy with classical \mathrm{R}\mathrm{C} circuit behavior, where the exponential relaxation process follows

(9)J_{RC}(t)=\pm {\displaystyle \frac{1}{\alpha }}{\mathrm{e}}^{-\frac{\overline{t}}{\alpha }}

for respective half-periods. The characteristic decay time \alpha corresponds to the 1/e reduction timescale of the maximum current. Fig.2 juxtaposes our theoretical predictions for J(t) (black solid curves) with numerically fitted \mathrm{R}\mathrm{C}-model simulations J_{\mathrm{R}\mathrm{C}}(t) (red dotted curves). The normalization condition \int |J_{\mathrm{R}\mathrm{C}}(t)|\mathrm{d}t=1 per half-period ensures consistent comparison across pulse events. Our results reveal complete current relaxation J(t)\to 0 within t\simeq 30, demonstrating \alpha \ll T where T is the driving period. This temporal scaling justifies our truncated visualization (t\le 40) in Fig.2 and subsequent analyses. Systematic fitting procedures yield the U-dependence of \alpha, revealing a saturation phenomenon: For U>3, \alpha converges to 5.2. This converged value 5.2 reveals the timescale of the current decay for quantum \mathrm{R}\mathrm{C} circuit. It is shown the current decaying time is governed by the quantum dot’s intrinsic state lifetime \tau that is derived from energy level broadening \mathrm{\Gamma } through \tau =\hslash /\mathrm{\Gamma }. With \mathrm{\Gamma }=0.2 in our calculations, we obtain \tau =5, matching the saturated \alpha value within numerical precision. The equivalence \alpha \approx \tau demonstrates that the quantum \mathrm{R}\mathrm{C} circuit’s relaxation timescale is fundamentally dictated by the lifetime of the particles in quantum dot. It implies that the current response J(t) encapsulates crucial quantum information of quantum \mathrm{R}\mathrm{C} circuit.

Upon comparing the \mathrm{R}\mathrm{C} current profile shown in Fig.2, we observe the following fundamental departure from the basic exponential decay predicted by Eq. (9): The current displays persistent decaying oscillations following its maximum peak, with the oscillation frequency showing direct proportionality to the external excitation amplitude U. To systematically investigate these coherent quantum phenomena, we perform spectral decomposition of the alternating current component through Fourier analysis with the following spectral components

(10)J(\mathrm{\Omega })={\displaystyle {\int }_0^T\mathrm{d}t{\mathrm{e}}^{\mathrm{i}\mathrm{\Omega }t}J(t).}

The quasi-continuous frequency spectrum \mathrm{\Omega }=m\omega with finite small radio-frequency (\omega \ll \mathrm{\Gamma }). The distinct traces correspond to different combinations of excitation strength U and quantum dot energy level {ϵ}_d. Our spectral analysis reveals two distinct regimes. (i) Low-frequency regime (): the Fourier components exhibit universal behavior independent of both intrinsic ({ϵ}_d) and extrinsic (U) parameters, with all curves collapsing into a single rapidly decaying profile. (ii) High-frequency regime (\mathrm{\Omega }>U): The spectral response shows {\mathrm{\Omega }}^{-1} scaling with superimposed parameter-dependent features. In this regime, both quantum dot energy {ϵ}_d and driving amplitude U significantly influence the spectral characteristics. Detailed examination of the high-frequency regime (Fig.3) reveals following striking signatures. (i) Resonant case ({ϵ}_d=0, left panels): Spectral discontinuous higher-order derivatives of J(\mathrm{\Omega }) occurs precisely at the fundamental quantum resonance condition \mathrm{\Omega }=U, manifested as a predictable dip in |{\mathrm{\partial }}_{\mathrm{\Omega }}J(\mathrm{\Omega })|. (ii) Detuned case ({ϵ}_d\ne 0, right panels): The single derivative discontinuity splits symmetrically into dual resonances at \mathrm{\Omega }=U\pm {ϵ}_d, reflecting modified quantum interference conditions under energy detuning. These discontinuous higher-order derivatives represent fundamental quantum phenomena with no classical counterparts in \mathrm{R}\mathrm{C} circuit analogs, serving as unambiguous fingerprints of coherent charge transfer dynamics. This dual-regime behavior offers powerful diagnostic tools for characterizing both intrinsic quantum dot properties ({ϵ}_d) and external driving parameters (U) through spectral measurements, establishing Fourier analysis as an essential technique for probing quantum coherence in masoscopic charge transport.

After analyzing the Fourier spectrum in the high-frequency range, let us now turn our attention to the low-frequency range. Given the discrete nature of the frequency spectrum \mathrm{\Omega }=m\omega, we adopt the notation J(\mathrm{\Omega })\equiv J_m for simplicity. Numerical simulations reveal that the current magnitudes during successive half-cycles are nearly identical in amplitude but opposite in polarity. This symmetry suppresses even-order Fourier components, rendering them negligible. Consequently, our analysis focuses exclusively on the odd-order Fourier components (m=1,3,5,\cdots), which dominates the spectral response. Fig.5 highlights the maxima of J_m for the first four odd Fourier components (J_{1,3,5,7}). Fig.5(a) plots modulus |J_m| against the real components \mathrm{R}\mathrm{e}(J_m), while Fig.5(b) presents a Nyquist diagram, where \mathrm{I}\mathrm{m}(J_m) is plotted against \mathrm{R}\mathrm{e}(J_m). In contrast to DC transport, the AC impedance Z=R+\mathrm{i}X inherently possesses a complex character. Here, the real part R quantifies energy dissipation, whereas the imaginary part X reflects energy storage during periodic charging and discharging processes. The Fourier components J_m display a clear correlation with the AC impedance properties: The real part \mathrm{R}\mathrm{e}(J_m) corresponds to the dissipative resistance, while the imaginary part \mathrm{I}\mathrm{m}(J_m) reflects the non-dissipative storage. Fig.5 reveals a significant trend: Higher-order Fourier components show a pronounced dominance of imaginary components, with reactance increasing monotonically as m increases. This hierarchy implies the dissipative effects predominantly reside in low-order harmonics, while reactive behavior, governed by periodic energy redistribution, emerges strongly in higher-order modes. Such presentation of the Fourier spectrum demonstrates the dissipative-storage nature of AC quantum transport, inaccessible to conventional DC measurements.

Fig.5 systematically maps the data of Fourier harmonics J_m when quantum dot energy levels {ϵ}_d sweeps from -2 to 2. Distinct symbols mark three excitation strengths: U=0.5 (black squares), 1.0 (red circles), and 3.0 (blue rhombus). There are two distinct spectral hierarchy: (i) Strong driving regime (U=3.0): When U>(|{ϵ}_d|+\mathrm{\Gamma }), the broadened quantum dot level (|{ϵ}_d|\pm \mathrm{\Gamma }) lies entirely within the reservoir’s Fermi window [-U,U]. This full spectral overlap enables maximum electron tunneling efficiency, causing all J_m values to converge to a universal amplitude independent of {ϵ}_d (overlapping blue symbols). Notably, |J_m| decays linearly with harmonic order m [Fig.5(a), blue]. (ii) Moderate driving regime (U=0.5,1.0): For |{ϵ}_d|>U, the quantum dot’s spectral broadening (|{ϵ}_d|\pm \mathrm{\Gamma }) exceeds the Fermi window, resulting in partial tunneling and sub-maximal currents. The |J_m| becomes {ϵ}_d-dependent, approaching its peak only when {ϵ}_d\to 0 (black/red curves). In this regime, the relative contribution of the resistive (\mathrm{R}\mathrm{e}(J_m)) and reactive (\mathrm{I}\mathrm{m}(J_m)) also exhibit two distinct trends: (i) for fundamental mode (m=1), resistive and reactive components vary synchronously with {ϵ}_d, producing a linear correlation between |J_1| and \mathrm{R}\mathrm{e}(J_1) [Fig.5(a), black/red]; (ii) for higher harmonics (m>1), reactance progressively dominates, culminating in \mathrm{R}\mathrm{e}(J_1)\approx 0 with near-pure reactive response at m=7 [Fig.5(b)]. The distinct Fourier spectrum J_m reveals the dual role of AC impedance dynamics: Lower harmonics mediate irreversible dissipation while higher harmonics govern reversible quantum capacitance effects, a character absent in DC regimes.

We now analyze the dominant fundamental harmonic component J_1, where resistive behavior prevails. Fig.6 contrasts two configurations: single-level quantum dot ({ϵ}_d=0) in panels (a) and (b) and multi-level quantum dot ({ϵ}_d=\pm 1) in panels (c) and (d). For both cases, \mathrm{I}\mathrm{m}(J_1) and |(J_1)| are plotted against \mathrm{R}\mathrm{e}(J_1) across excitation strengths U=0.5,1.0,3.0. Data points trace trajectories parameterized by coupling strength \mathrm{\Gamma }\in [0.05,1], with the starting point of \mathrm{\Gamma }=0.05 indicated by cyan triangle. For single-level dynamics [Fig.6(a) and (b)], with the increasing of \mathrm{\Gamma }, resistance is continuously suppressed. Specifically, \mathrm{I}\mathrm{m}(J_1) decreases monotonically, reflecting diminished quantum phase storage. While \mathrm{R}\mathrm{e}(J_1) initially rises with \mathrm{\Gamma } and peaks near \mathrm{\Gamma }=0.7. Beyond this critical coupling, \mathrm{R}\mathrm{e}(J_1) declines, signaling a crossover from quantum-coherent transport to classical resistive dominance. When \mathrm{\Gamma }\gg U, strong hybridization suppresses quantum interference effects, leaving only Ohmic dissipation. In the multi-level system with {ϵ}_d=1 and -1, three distinct curves emerge for different U. For U=0.5 (black symbols), the energy levels {ϵ}_d=\pm 1 lie outside the effective bias window [-U,U]. As a result, the cyclic energy storage and release processes is inactive. Then, the imaginary component \mathrm{I}\mathrm{m}(J_1), representing reactive current storage, approaches zero, while the resistive component \mathrm{R}\mathrm{e}(J_1) exhibits a linear dependence on tunneling strength \mathrm{\Gamma }. When U exceeds 1.0, the {ϵ}_d=\pm 1 states begin entering the bias window, establishing a complete AC response mechanism. Both reactive [\mathrm{I}\mathrm{m}(J_1)] and resistive [\mathrm{R}\mathrm{e}(J_1)] components demonstrate \mathrm{\Gamma }-dependence in this regime. At U>2.0, the {ϵ}_d=\pm 1 levels fully reside within the bias window U(t), enabling periodic storage and release of two quantum charges. This configuration produces nearly doubled current magnitudes compared to lower U values, as evidenced by the U=3 case in Fig.6(d) where |J_1| reaches maximum amplitudes of 4. This contrasts markedly with single-level systems [Fig.6(b)], where maximum current amplitudes remain below 2 due to single-charge cycling limitations. The enhanced current capacity in the two-level system directly reflects the doubled quantum charge participation in the storage-release cycle.

Under the condition \alpha \ll T, electrons achieve complete occupation/depletion in the quantum dot during each half-period, manifesting quantized current transport. As derived from Eq. (9), the m-th Fourier component is given by

J_{RC}(\mathrm{\Omega })={\displaystyle {\int }_0^{\mathrm{\infty }}\mathrm{d}t{\mathrm{e}}^{\mathrm{i}\mathrm{\Omega }t}{J}_{RC}(t)={\displaystyle \frac{2}{1-\mathrm{i}\alpha \mathrm{\Omega }}},}

where \mathrm{\Omega }=m\omega. For fundamental frequency, \mathrm{\Omega }=\omega, and (\alpha \omega \approx \tau \omega )\ll 1, the quantized current approaches |J_1|\approx 2(ef), reflecting two mutually inverse processes across half-periods. Furthermore, for the single-lead RC circuit, the electrochemical capacitance is [11]

C_{\mu }(\omega )={\displaystyle \frac{C_{\mu }}{1-\mathrm{i}\omega C_{\mu }{R}_q}},

where C_{\mu }={\left(\frac{1}{C_0}+\frac{\pi \mathrm{\Gamma }}{e^2}\right)}^{-1} is static electrochemical capacitance [75]. If the geometric capacitance C_0 is not taken into account, C_{\mu }=e^2/(\pi \mathrm{\Gamma }), then with the aid of half-quantized relaxation resistance R_q=h/(2e^2), we obtain

C_{\mu }(\omega )={\displaystyle \frac{e^2}{\pi \mathrm{\Gamma }}}{\displaystyle \frac{1}{1-\mathrm{i}\omega {\displaystyle \frac{\hslash }{\mathrm{\Gamma }}}}}.

Comparing with fundamental Fourier dynamic current J(\omega )=2ef/(1-\mathrm{i}\alpha \omega ) where \alpha =\hslash /\mathrm{\Gamma }, we can find C_{\mu }(\omega ) and J(\omega ) share the same frequency dependence. It means the even-quantized current |J_1|\approx 2ef in low frequency is precisely originated from the half-quantized relaxation resistance R_q=h/(2e^2) [10], they are both the universal features of quantum AC circuits, and together form the two sides of the dual nature of the AC transport. The quantum RC circuit driven by a periodic square wave perfectly illustrates this correspondence between half-integer quantum relaxation resistance and even-integer quantum current, providing new insights and perspectives for understanding AC transport.

Numerical verification of this quantization is performed using Eqs. (5) and (10). Fig.7 displays the fundamental Fourier component |J_1| versus excitation strength U for linearly spaced quantum energy levels {ϵ}_{d,n}=V_g+[-3.0,-1.0,1.0,3.0] [Fig.7(a)] and non-linearly spaced levels {ϵ}_{d,n}=V_g+[-3.0,-1.5,1.2,3.0] [Fig.7(b)] with parameters \omega =0.01 and \mathrm{\Gamma }=0.1 which satisfies \tau \ll T. Energy level positions are globally tuned via gate voltage V_g. The following conclusions hold universally, regardless of the distribution of the energy levels in quantum dots. (i) Plateau formation: The current |J_1| displays distinct step-like plateaus at values of 2, 4, and 6 as U varies. The step-like progression reflects sequential activation of quantum dot levels, where n active levels produce |J_1|\approx 2nef. This contrasts with single-level systems limited to |J_1|\le 2, demonstrating how multi-level configurations amplify quantized current through additive state participation. (ii) Multi-level resolution: When level broadening \mathrm{\Gamma } remains smaller than inter level spacing \mathrm{\Delta }{ϵ}_d, discrete quantization emerges. Each energy level entering the excitation window U(t) contributes a 2-unit current increment. (iii) Transition points: Multi-levels result in step-like quantized current plateaus. At transition points, the quantized current plateau transitions from n-1 to n+1. The invariance of quantization at these points represents their robustness to gate-induced energy shifts.

By comparing Fig.7(a) and (b), it is also evident that the evolution of quantized plateaus depend critically on energy level alignment. As shown in Fig.7(a), the linearly spaced {ϵ}_{d,n} enter the excitation window U(t)\in [-U,U] sequentially with equal intervals. This results in linearly spaced transition points (U=1,2,\cdots). If {ϵ}_{d,n} is randomly arranged, with the increasing of U, {ϵ}_{d,n} enter the excitation window U(t) randomly. It means the arrangement of the quantized current directly reflects the distribution of the quantum energy levels, as shown in Fig.7(b). For the convenience of physical analysis, we have inset the schematic diagram illustrating the positional relationship between the excitation window U(t) (depicted as gray region with U=1,2,3,4 in insets) and the quantum dot energy level {ϵ}_{d,n} (color lines in insets). Different colors and line types in insets correspond to the parameters of the curves with matching colors and line types.

For linearly spaced {ϵ}_{d,n} [Fig.7(a)], V_g=1 (red solid lines) corresponds to {ϵ}_d=-2,0,2,\cdots. In this case, the excitation window U(t) fully encompasses the {ϵ}_d=0 level as long as U>\mathrm{\Gamma }, producing a distinct 2-unit plateau at interval of U\in [\mathrm{\Gamma },2-\mathrm{\Gamma }]. As U increases, higher-energy states, i.e., states with energies \pm 2,\pm 4,\cdots, are sequentially activated. This process leads to the formation of plateaus of 6,10,\cdots at U\in [2+\mathrm{\Gamma },4-\mathrm{\Gamma }], [4+\mathrm{\Gamma },6-\mathrm{\Gamma }], \cdots, respectively. On the other hand, for V_g=0 (black lines), with {ϵ}_{d,n}=[-3,-1,1,3], pairs of symmetric levels (\pm 1,\pm 3,\cdots ) enter the U(t) window as soon as U exceeds \mathrm{\Gamma }+1,3,\cdots, yielding quantized plateaus of 4,8,\cdots surrounding U=2,4,\cdots. Intermediate gate voltages (e.g.,V_g=0.5, black red lines) shift the energy levels as {ϵ}_{d,n}=[-2.5,-0.5,1.5,\cdots ], causing sequential stepwise plateaus of 2,4,6,\cdots at U=1,2,3,\cdots. At the transition points, i.e., U=1,2,3,\cdots, quantization is maintained regardless of any energy shift of {ϵ}_{d,n}, representing the robustness against the gate-induced energy shifts. These transition points satisfy the condition U=n\mathrm{\Delta }/2, where \mathrm{\Delta }=|{ϵ}_{d,i+1}-{ϵ}_{d,i}|=2 defines the fixed level spacing.

As a contrast, the randomly spaced {ϵ}_{d,n} lead random arrangement of the quantized current. Taking V_g=1.0 as an example [the first inset of Fig.7(b)], {ϵ}_{d,n}=[-2,-0.5,2.2,\cdots ]. When , the excitation window U(t) can only encompasses the single level {ϵ}_d=-0.5. As U passes the value of 2, the two quantum dot levels {ϵ}_{d,n}=-2 and 2.2 successively enter the excitation window U(t). Consequently, the current J_1 briefly stabilizes at the plateau of |J_1|=4, then rapidly transitions to the plateau of |J_1|=6. For V_g=0 and 0.3 [the third and second inset of Fig.7(b)], the energy levels are {ϵ}_{d,n}=[-3.0,-1.5,1.2,3.0] and {ϵ}_{d,n}=[-2.7,-1.2,1.5,3.3], respectively. In this case, the corresponding two current curves merge before the third energy level enters the excitation window, i.e., when . This occurs because the first lowest energy level |{ϵ}_{d,1}|=1.2 and the second lowest energy level |{ϵ}_{d,2}|=1.5 enter excitation window U(t) simultaneously. Similar to the case of linearly spaced quantum dot energy levels, discrete multi-level causes step-like quantized current plateaus J_1=2,4,6,\cdots. Moreover, the transition points of the quantum plateaus remain at U=\mathrm{\Delta }/2. Here \mathrm{\Delta }=|{ϵ}_i-{ϵ}_0| defines the distance from any energy level {ϵ}_i within [-U,U] to energy level {ϵ}_0 that is closest to the Fermi level. For the first and second transition points, \mathrm{\Delta }=|-1.5-1.2| and |-3.0-1.2|, corresponding to U=1.35 and 2.1, respectively. Therefore, the distribution of the quantum dot energy levels can also be determined by examining the distribution of the step-like currents as a function of U.

In Fig.8, we present the fundamental Fourier component |J_1| as a function of gate voltage V_g for varying excitation strengths U. For the convenience of physical analysis, we only consider the linear spaced energy levels. From Fig.8, it is evident that the remarkable stability of quantized J_1 persists across wide V_g ranges at the transition points U=1,2,3 (solid curves). This highlights the robustness of quantum current against V_g-induced shifts in energy levels. The weaker the external field U is, the fewer the quantum plateaus are, but the stronger their robustness is. At transition points U=n\mathrm{\Delta }/2 (solid lines with n=1,2,3), n energy levels remain continuously within the bias window regardless of V_g, contributing stable quantized plateaus with maximum quantized value 2n. The quantized value is determined by the number of broadened discrete levels fully contained within the excitation window [-U,U]. As V_g sweeps, quantum dot levels sequentially pass through the transport window. The number of broadened energy levels within [-U,U] increases successively from 1 to 3, and then decreases back to 1. As a result, quantized value of J_1 exhibits a step-like pattern in the shape of 2, 4, 6, 4, 2 for U=3 (thick red solid curve). Deviating from transition point, i.e., U\ne n\mathrm{\Delta }/2 (dashed and dotted lines), the broadened quantum energy levels cannot be fully included in the excitation window, and the current cannot be quantized. This behavior highlights the delicate balance required between excitation strength U and particle level spacing \mathrm{\Delta } for achieving robust quantized current.

Fig.9 presents the modulus of the m-th Fourier component, |J_m|, plotted as a function of gate voltage V_g for odd indices m=1,3,5,\cdots ,119, in a multi-level quantum dot system. Each panel corresponds to a distinct excitation strength U, with all Fourier components J_1 to J_{119} systematically ordered by descending amplitude. Discrete current peaks and plateaus emerge as V_g varies, signifying the sequential entry of quantum energy levels into the excitation window [-U,U]. For U values ranging from 0.1 to 1.0 [Fig.9(a)−(d)], the initial sharp peaks progressively broaden into quantized plateaus. These plateaus exhibit continuous expansion until they coalesce into a unified platform at U=1.0. This behavior confirms U=\mathrm{\Delta }/2 as a stability threshold, where current quantization persists robustly against V_g-induced shifts in energy levels. When U exceeds 1.0 [Fig.9(e)−(h)], a second quantum level enters the [-U,U] energy window. Consequently, J_m transitions from discontinuous spikes to discrete plateaus, ultimately forming a dominant quantized plateau with magnitude 4 at the stable point U=2.0. Prior to the inclusion of the second energy level, a distinct first plateau with magnitude 2 is observed, consistent with the analysis in Fig.8. The red dashed lines in Fig.9(e)−(h) demarcate the position of this initial plateau with magnitude 2.

4 Beyond the wide-band limit

The preceding analysis within WBL provides an intuitive elucidation of quantum current generation mechanisms under periodic pulse excitation. Building upon this foundation, we now proceed to systematically investigate the generalized self-energy effects that transcend WBL. The linewidth function, defined as \mathrm{\Gamma }(ϵ)\equiv 2\pi \rho (ϵ)|t(ϵ){|}^2, adopts a Lorentzian parametrization [76]

\mathrm{\Gamma }(ϵ)={\displaystyle \frac{W^2}{{ϵ}^2+W^2}}\mathrm{\Gamma },

where \mathrm{\Gamma } represents the linewidth magnitude and W characterizes the bandwidth. In the asymptotic limit W\to \mathrm{\infty }, the energy dependence vanishes and \mathrm{\Gamma }(ϵ)\to \mathrm{\Gamma }, recovering the conventional WBL. The analytic structure of this function reveals two first-order poles in the complex plane at ϵ=\pm \mathrm{i}W, with corresponding residues

{\mathrm{R}\mathrm{e}\mathrm{s}}_{\pm }\left[\mathrm{\Gamma }(E)\right]=\pm {\displaystyle \frac{\mathrm{\Gamma }}{2\mathrm{i}}}W.

The spectral representation of the self-energies {\mathrm{\Sigma }}^{r/a}(E) is fundamentally embodied by the linewidth function. This relationship enables the derivation of the retarded/advanced self-energy components through the integral representation

{\mathrm{\Sigma }}^{r/a}(E)={\displaystyle \int {\displaystyle \frac{\mathrm{d}ϵ}{2\pi }}{\displaystyle \frac{\mathrm{\Gamma }(ϵ)}{E-ϵ\pm \mathrm{i}{0}^{+}}}={\displaystyle \frac{\mathrm{\Gamma }}{2}}{\displaystyle \frac{W}{E\pm \mathrm{i}W}}.}

The energy-dependent linewidth function \mathrm{\Gamma }(ϵ) exhibits smooth spectral variations, ensuring that {\mathrm{\Sigma }}^{r/a,0}(t_1-t) maintains dominant contributions from the time-localized region t_1\approx t. This temporal localization justifies the approximate relationship: . Considering the energy dependence of line width function \mathrm{\Gamma }(ϵ), the current pulse formulation becomes

(11)

where

Next, we solve the spectral function A_n(t_{ϵ}) through Eq. (7). The energy-dependent nature of the self-energy {\mathrm{\Sigma }}^r(E) manifests in modifications to the retarded Green’s function G_n^r(E) under external perturbations. For the single-terminal RC circuit configuration, the application of driving potential U at the lead electrode induces an equivalent energy shift -U at the quantum dot through electrostatic coupling. This leads to time-periodic shift of the quantum level {\overline{ϵ}}_n={ϵ}_{d,n}\mp U, where the upper (lower) sign corresponds to the first (second) half-period. Consequently, the retarded Green’s function acquires the modified form

G_n^r(E)={\displaystyle \frac{1}{E-{\overline{ϵ}}_n-{\mathrm{\Sigma }}^r(E)}}={\displaystyle \frac{E+\mathrm{i}W}{(E-{\widetilde{ϵ}}_d^{+})(E-{\widetilde{ϵ}}_d^{-})}}

with two poles

{\widetilde{ϵ}}_d^{\pm }={\displaystyle \frac{{\overline{ϵ}}_n-\mathrm{i}W\pm \sqrt{({\overline{ϵ}}_n+\mathrm{i}W{)}^2+2\mathrm{\Gamma }W}}{2}}.

The corresponding residues of G_n^r(E) are given by

{\mathrm{R}\mathrm{e}\mathrm{s}}_{\pm }\left[G_n^r(E)\right]=\pm {\displaystyle \frac{{\widetilde{ϵ}}_d^{\pm }+\mathrm{i}W}{{\widetilde{ϵ}}_d^{+}-{\widetilde{ϵ}}_d^{-}}}.

The contour integration in Eq. (7) can be analytically evaluated through residue theorem by enclosing the two poles residing in the lower half complex plane. Importantly, while the Fermi−Dirac distribution in Eq. (11) corresponds to equilibrium conditions (U=0), the spectral function requires energy-axis adjustment to account for nonequilibrium driving. This is implemented through the variable substitution E\to E\mp U applied respectively to the first and second half-period components. Consequently, the generalized spectral function beyond WBL takes the form

(12)A_n(t,ϵ)={\displaystyle \sum _{\pm }{\mathrm{R}\mathrm{e}\mathrm{s}}_{\pm }\left[G_n^r(E)\right]{\mathcal{B}}_n^{\pm }(t,ϵ)}

with

{\mathcal{B}}_n^{\pm }(t,ϵ)={\displaystyle \frac{1}{ϵ-{\widetilde{ϵ}}_d^{\pm }}}+{\displaystyle \frac{2U{\mathrm{e}}^{\mathrm{i}(ϵ-{\widetilde{ϵ}}_d^{\pm })\overline{t}}}{(ϵ-{\widetilde{ϵ}}_d^{\pm })(ϵ-{\widetilde{ϵ}}_d^{\pm }-2U)}}

for the first half cycle and

{\mathcal{B}}_n^{\pm }(t,ϵ)={\displaystyle \frac{1}{ϵ-{\widetilde{ϵ}}_d^{\pm }}}-{\displaystyle \frac{2U{\mathrm{e}}^{\mathrm{i}(ϵ-{\widetilde{ϵ}}_d^{\pm })\overline{t}}}{(ϵ-{\widetilde{ϵ}}_d^{\pm })(ϵ-{\widetilde{ϵ}}_d^{\pm }+2U)}}

for the second half cycle.

By implementing the modified spectral function from Eq. (12) within the current formulation of Eq. (5), we numerically evaluate the pulse current characteristics beyond the WBL. Fig.10(a) and (b) present the time-resolved current profiles J(t) during the first and second half-periods, respectively. Various curves correspond to distinct bandwidth parameters W. The cyan dashed curves demonstrate perfect current antisymmetry between half-periods under WBL (W\to \mathrm{\infty }). However, finite bandwidth effects induce significant current magnitude disparities between the first and the second half-periods, particularly pronounced when W is small. This departure from perfect antisymmetry manifests as deviations in quantization precision of |J_1|, as quantified in the inset displaying |J_{1(U)}| dependencies for different W. The parametric study reveals a systematic improvement in quantization fidelity with the increasing of bandwidth, culminating in complete WBL at W=5.0 (red-solid vs cyan-dashed curve coincidence). Notably, despite inducing subtle quantitative deviations, finite-bandwidth corrections maintain the integrity of core qualitative features: decaying time \alpha \approx \tau, oscillating frequency \propto U, and quantized current |J_1|\approx 2ef.

5 Conclusions

In summary, based on a quantum \mathrm{R}\mathrm{C} circuit and employing the NEGF formalism, we theoretically investigate the dynamic response of single-electron emitters under square-wave driving potentials. The analytic solution reproduces the coherent emission-absorption cycles with J(t)\propto \pm {\mathrm{e}}^{-\overline{t}/\alpha }. Remarkably, our theoretical results reveal the dynamic properties of the single-electron emitter, which include: (i) the characteristic time \alpha of the decaying current J(t)(\propto {\mathrm{e}}^{-t/\alpha }) matches the quantum dot’s intrinsic coherence time \tau; (ii) the decaying current oscillates with a frequency proportional to the external driving amplitude U; (iii) the quantized Fourier component |J_1|=2ef is directly related to the half-quantized relaxation resistance R_q=h/(2e^2), where both of them label the quantized dynamics of the AC transport through single-electron emitters. These findings provide new insights for understanding quantized AC transport phenomena.

References

[1] B. J. van Wees , H. van Houten , C. W. J. Beenakker , J. G. Williamson , L. P. Kouwenhoven , D. van der Marel , and C. T. Foxon , Quantized conductance of point contacts in a two-dimensional electron gas, Phys. Rev. Lett. 60(9), 848 (1988)

[2] D. A. Wharam , T. J. Thornton , R. Newbury , M. Pepper , H. Ahmed , J. E. F. Frost , D. G. Hasko , D. C. Peacock , D. A. Ritchie , and G. A. C. Jones , One-dimensional transport and the quantisation of the ballistic resistance, J. Phys. C 21(8), L209 (1988)

[3] R. A. Webb , S. Washburn , C. P. Umbach , and R. B. Laibowitz , Observation of Aharonov−Bohm oscillations in normal-metal rings, Phys. Rev. Lett. 54(25), 2696 (1985)

[4] K. Klitzing , G. Dorda , and M. Pepper , New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45(6), 494 (1980)

[5] D. C. Tsui , H. L. Stormer , and A. C. Gossard , Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48(22), 1559 (1982)

[6] W. Poirier and F. Schopfer , Resistance metrology based on the quantum Hall effect, Eur. Phys. J. Spec. Top. 172(1), 207 (2009)

[7] W. Poirier , F. Schopfer , J. Guignard , O. Thévenot , and P. Gournay , Application of the quantum Hall effect to resistance metrology, C. R. Phys. 12(4), 347 (2011)

[8] S. Djordjevic , R. Behr , and W. Poirier , A primary quantum current standard based on the Josephson and the quantum Hall effects, Nat. Commun. 16(1), 1447 (2025)

[9] M. Büttiker , H. Thomas , and A. Prêtre , Mesoscopic capacitors, Phys. Lett. A 180(4−5), 364 (1993)

[10] J. Gabelli , G. Fève , J. M. Berroir , B. Plaçais , A. Cavanna , B. Etienne , Y. Jin , and D. C. Glattli , Violation of Kirchhoff’s laws for a coherent RC circuit, Science 313(5786), 499 (2006)

[11] J. Wang , B. Wang , and H. Guo , Quantum inductance and negative electrochemical capacitance at finite frequency in a two-plate quantum capacitor, Phys. Rev. B 75(15), 155336 (2007)

[12] C. Bäuerle , D. Christian Glattli , T. Meunier , F. Portier , P. Roche , P. Roulleau , S. Takada , and X. Waintal , Coherent control of single electrons: A review of current progress, Rep. Prog. Phys. 81(5), 056503 (2018)

[13] S. P. Giblin , M. Kataoka , J. D. Fletcher , P. See , T. J. B. M. Janssen , J. P. Griffiths , G. A. C. Jones , I. Farrer , and D. A. Ritchie , Towards a quantum representation of the ampere using single electron pumps, Nat. Commun. 3(1), 930 (2012)

[14] G. Yamahata , K. Nishiguchi , and A. Fujiwara , Gigahertz single-trap electron pumps in silicon, Nat. Commun. 5(1), 5038 (2014)

[15] H. Howe , M. Blumenthal , H. E. Beere , T. Mitchell , D. A. Ritchie , and M. Pepper , Single-electron pump with highly controllable plateaus, Appl. Phys. Lett. 119(15), 153102 (2021)

[16] G. Yamahata , N. Johnson , and A. Fujiwara , Understanding the mechanism of tunable-barrier single-electron pumping: Mechanism crossover and optimal accuracy, Phys. Rev. B 103(24), 245306 (2021)

[17] K. T. Wang , F. Xu , B. Wang , Y. Yu , and Y. Wei , Transport features of topological corner states in honeycomb lattice with multihollow structure, Front. Phys. (Beijing) 17(4), 43501 (2022)

[18] G. Fève , A. Mahé , J. M. Berroir , T. Kontos , B. Plaçais , D. C. Glattli , A. Cavanna , B. Etienne , and Y. Jin , An on-demand coherent single-electron source, Science 316(5828), 1169 (2007)

[19] M. Albert , C. Flindt , and M. Büttiker , Accuracy of the quantum capacitor as a single-electron source, Phys. Rev. B 82(4), 041407 (2010)

[20] Y. Xing , Z. Yang , Q. F. Sun , and J. Wang , Coherent single-spin source based on topological insulators, Phys. Rev. B 90(7), 075435 (2014)

[21] J. P. Pekola , O. P. Saira , V. F. Maisi , A. Kemppinen , M. Möttönen , Y. A. Pashkin , and D. V. Averin , Single-electron current sources: Toward a refined definition of the ampere, Rev. Mod. Phys. 85(4), 1421 (2013)

[22] B. Kaestner and V. Kashcheyevs , Non-adiabatic quantized charge pumping with tunable-barrier quantum dots: A review of current progress, Rep. Prog. Phys. 78(10), 103901 (2015)

[23] F. Stein , H. Scherer , T. Gerster , R. Behr , M. Götz , E. Pesel , C. Leicht , N. Ubbelohde , T. Weimann , K. Pierz , H. W. Schumacher , and F. Hohls , Robustness of singleelectron pumps at sub-ppm current accuracy level, Metrologia 54(1), S1 (2017)

[24] R. Zhao , A. Rossi , S. P. Giblin , J. D. Fletcher , F. E. Hudson , M. Möttönen , M. Kataoka , and A. S. Dzurak , Thermal-error regime in high-accuracy gigahertz single-electron pumping, Phys. Rev. Appl. 8(4), 044021 (2017)

[25] S. P. Giblin , A. Fujiwara , G. Yamahata , M. H. Bae , N. Kim , A. Rossi , M. Möttönen , and M. Kataoka , Evidence for universality of tunable-barrier electron pumps, Metrologia 56(4), 044004 (2019)

[26] S. P Giblin , E. Mykkänen , A. Kemppinen , P. Immonen , A. Manninen , M. Jenei , M. Möttönen , G. Yamahata , A. Fujiwara , and M. Kataoka , Realisation of a quantum current standard at liquid helium temperature with sub-ppm reproducibility, Metrologia 57(2), 025013 (2020)

[27] C. L. Degen , F. Reinhard , and P. Cappellaro , Quantum sensing, Rev. Mod. Phys. 89(3), 035002 (2017)

[28] N. Johnson , J. D. Fletcher , D. A. Humphreys , P. See , J. P. Griffiths , G. A. C. Jones , I. Farrer , D. A. Ritchie , M. Pepper , T. J. B. M. Janssen , and M. Kataoka , Ultrafast voltage sampling using single-electron wavepackets, Appl. Phys. Lett. 110(10), 102105 (2017)

[29] G. Yamahata , S. Ryu , N. Johnson , H. S. Sim , A. Fujiwara , and M. Kataoka , Picosecond coherent electron motion in a silicon single-electron source, Nat. Nanotechnol. 14(11), 1019 (2019)

[30] M. Yamamoto , S. Takada , C. Bäuerle , K. Watanabe , A. D. Wieck , and S. Tarucha , Electrical control of a solid state flying qubit, Nat. Nanotechnol. 7(4), 247 (2012)

[31] N. Ubbelohde , F. Hohls , V. Kashcheyevs , T. Wagner , L. Fricke , B. Kästner , K. Pierz , H. W. Schumacher , and R. J. Haug , Partitioning of on-demand electron pairs, Nat. Nanotechnol. 10(1), 46 (2015)

[32] N. Johnson , C. Emary , S. Ryu , H. S. Sim , P. See , J. D. Fletcher , J. P. Griffiths , G. A. C. Jones , I. Farrer , D. A. Ritchie , M. Pepper , T. J. B. M. Janssen , and M. Kataoka , LO-phonon emission rate of hot electrons from an ondemand single-electron source in a GaAs/AlGaAs heterostructure, Phys. Rev. Lett. 121(13), 137703 (2018)

[33] J. D. Fletcher , N. Johnson , E. Locane , P. See , J. P. Griffiths , I. Farrer , D. A. Ritchie , P. W. Brouwer , V. Kashcheyevs , and M. Kataoka , Continuous-variable tomography of solitary electrons, Nat. Commun. 10(1), 5298 (2019)

[34] L. P. Kouwenhoven , A. T. Johnson , N. C. van der Vaart , C. J. P. M. Harmans , and C. T. Foxon , Quantized current in a quantum-dot turnstile using oscillating tunnel barriers, Phys. Rev. Lett. 67(12), 1626 (1991)

[35] A. Fujiwara , N. M. Zimmerman , Y. Ono , and Y. Takahashi , Current quantization due to single-electron transfer in si-wire charge-coupled devices, Appl. Phys. Lett. 84(8), 1323 (2004)

[36] M. D. Blumenthal , B. Kaestner , L. Li , S. Giblin , T. J. B. M. Janssen , M. Pepper , D. Anderson , G. Jones , and D. A. Ritchie , Gigahertz quantized charge pumping, Nat. Phys. 3(5), 343 (2007)

[37] L. D. Contreras-Pulido , J. Splettstoesser , M. Governale , J. König , and M. Büttiker , Time scales in the dynamics of an interacting quantum dot, Phys. Rev. B 85(7), 075301 (2012)

[38] V. I. Kleshch , V. Porshyn , A. S. Orekhov , A. S. Orekhov , D. Lützenkirchen-Hecht , and A. N. Obraztsov , Carbon single-electron point source controlled by Coulomb blockade, Carbon 171, 154 (2021)

[39] J. M. Shilton , V. I. Talyanskii , M. Pepper , D. A. Ritchie , J. E. F. Frost , C. J. B. Ford , C. G. Smith , and G. A. C. Jones , High-frequency single-electron transport in a quasi-one-dimensional GaAs channel induced by surface acoustic waves, J. Phys. : Condens. Matter 8(38), L531 (1996)

[40] V. I. Talyanskii , J. M. Shilton , M. Pepper , C. G. Smith , C. J. B. Ford , E. H. Linfield , D. A. Ritchie , and G. A. C. Jones , Single-electron transport in a one-dimensional channel by high-frequency surface acoustic waves, Phys. Rev. B 56(23), 15180 (1997)

[41] S. Hermelin , S. Takada , M. Yamamoto , S. Tarucha , A. D. Wieck , L. Saminadayar , C. Bäuerle , and T. Meunier , Electrons surfing on a sound wave as a platform for quantum optics with flying electrons, Nature 477(7365), 435 (2011)

[42] C. J. B. Ford , Transporting and manipulating single electrons in surface-acoustic-wave minima, Phys. Status Solidi B 254(3), 1600658 (2017)

[43] L. S. Levitov , H. Lee , and G. B. Lesovik , Electron counting statistics and coherent states of electric current, J. Math. Phys. 37(10), 4845 (1996)

[44] J. Keeling , I. Klich , and L. S. Levitov , Minimal excitation states of electrons in one-dimensional wires, Phys. Rev. Lett. 97(11), 116403 (2006)

[45] J. Dubois , T. Jullien , C. Grenier , P. Degiovanni , P. Roulleau , and D. C. Glattli , Integer and fractional charge Lorentzian voltage pulses analyzed in the framework of photon-assisted shot noise, Phys. Rev. B 88(8), 085301 (2013)

[46] B. Bertin-Johannet , J. Rech , T. Jonckheere , B. Grémaud , L. Raymond , and T. Martin , Microscopic theory of photo-assisted electronic transport in normal-metal/BCS-superconductor junctions, Phys. Rev. B 105(11), 115112 (2022)

[47] M. R. Connolly , K. L. Chiu , S. P. Giblin , M. Kataoka , J. D. Fletcher , C. Chua , J. P. Griffiths , G. A. C. Jones , V. I. Fal’ko , C. G. Smith , and T. J. B. M. Janssen , Gigahertz quantized charge pumping in graphene quantum dots, Nat. Nanotechnol. 8(6), 417 (2013)

[48] V. Siegle , C. W. Liang , B. Kaestner , H. W. Schumacher , F. Jessen , D. Koelle , R. Kleiner , and S. Roth , A molecular quantized charge pump, Nano Lett. 10(10), 3841 (2010)

[49] C. Mora and K. Le Hur , Universal resistances of the quantum resistance–capacitance circuit, Nat. Phys. 6(9), 697 (2010)

[50] M. Moskalets and M. Büttiker , Floquet scattering theory of quantum pumps, Phys. Rev. B 66(20), 205320 (2002)

[51] M. Moskalets , P. Samuelsson , and M. Büttiker , Quantized dynamics of a coherent capacitor, Phys. Rev. Lett. 100(8), 086601 (2008)

[52] P. Portugal , F. Brange , and C. Flindt , Heat pulses in electron quantum optics, Phys. Rev. Lett. 132(25), 256301 (2024)

[53] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995

[54] H. Haug,A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer Series in Solid-State Sciences No. 123, Springer Berlin, Heidelberg, 2008

[55] M. I. Alomar , J. S. Lim , and D. Sánchez , Coulomb-blockade effect in nonlinear mesoscopic capacitors, Phys. Rev. B 94(16), 165425 (2016)

[56] C. Zhang , F. Xu , and J. Wang , Full counting statistics of phonon transport in disordered systems, Front. Phys. (Beijing) 16(3), 33502 (2021)

[57] Z. Z. Yu , G. H. Xiong , and L. F. Zhang , A brief review of thermal transport in mesoscopic systems from nonequilibrium Green’s function approach, Front. Phys. (Beijing) 16(4), 43201 (2021)

[58] J. Zeng , R. Xue , T. Hou , Y. Han , and Z. Qiao , Formation of topological domain walls and quantum transport properties of zero-line modes in commensurate bilayer graphene systems, Front. Phys. (Beijing) 17(6), 63503 (2022)

[59] K. Fukuzawa , T. Kato , T. Jonckheere , J. Rech , and T. Martin , Minimal alternating current injection into carbon nanotubes, Phys. Rev. B 108(12), 125307 (2023)

[60] L. Zhang , F. Xu , J. Chen , Y. Xing , and J. Wang , Quantum nonlinear ac transport theory at low frequency, New J. Phys. 25(11), 113006 (2023)

[61] G. Li , F. Xu , and J. Wang , Universal behaviors of magnon-mediated spin transport in disordered nonmagnetic metal-ferromagnetic insulator heterostructures, Front. Phys. (Beijing) 18(3), 33310 (2023)

[62] M. Wei , B. Wang , Y. Yu , F. Xu , and J. Wang , Nonlinear hall effect induced by internal coulomb interaction and phase relaxation process in a four-terminal system with time-reversal symmetry, Phys. Rev. B 105(11), 115411 (2022)

[63] P. W. Brouwer , Scattering approach to parametric pumping, Phys. Rev. B 58(16), R10135 (1998)

[64] Y. Blanter and M. Büttiker , Shot noise in mesoscopic conductors, Phys. Rep. 336(1−2), 1 (2000)

[65] J. Wang and B. Wang , Quantization of adiabatic pumped charge in the presence of superconducting lead, Phys. Rev. B 65(15), 153311 (2002)

[66] J. Gabelli , G. Fève , J. M. Berroir , and B. Plaçais , A coherent RC circuit, Rep. Prog. Phys. 75(12), 126504 (2012)

[67] S. U. Cho , W. Park , B. K. Kim , M. Seo , D. T. Park , H. Choi , N. Kim , H. S. Sim , and M. H. Bae , One-lead single-electron source with charging energy, Nano Lett. 22(23), 9313 (2022)

[68] Y. Xing , B. Wang , and J. Wang , First-principles investigation of dynamical properties of molecular devices under a steplike pulse, Phys. Rev. B 82(20), 205112 (2010)

[69] T. Jonckheere , T. Stoll , J. Rech , and T. Martin , Realtime simulation of finite-frequency noise from a single-electron emitter, Phys. Rev. B 85(4), 045321 (2012)

[70] S. E. Nigg , R. López , and M. Büttiker , Mesoscopic charge relaxation, Phys. Rev. Lett. 97(20), 206804 (2006)

[71] Y. Liu , Y. Sun , X. Yan , B. Li , L. Wang , J. Li , J. Sun , Y. Guo , W. Liu , B. Hu , Q. Lin , F. Fan , and H. Shen , Realizing low voltage-driven bright and stable quantum dot light-emitting diodes through energy landscape flattening, Light Sci. Appl. 14(1), 50 (2025)

[72] B. Harpt , J. Corrigan , N. Holman , P. Marciniec , D. Rosenberg , D. Yost , R. Das , R. Ruskov , C. Tahan , W. D. Oliver , R. McDermott , M. Friesen , and M. A. Eriksson , Ultra-dispersive resonator readout of a quantum-dot qubit using longitudinal coupling, npj Quantum Inf. 11, 5 (2025)

[73] Q. F. Sun , J. Wang , and T. H. Lin , Photon-assisted Andreev tunneling through a mesoscopic hybrid system, Phys. Rev. B 59(20), 13126 (1999)

[74] N. S. Wingreen , A. P. Jauho , and Y. Meir , Time-dependent transport through a mesoscopic structure, Phys. Rev. B 48(11), 8487 (1993)

[75] F. Xu and J. Wang , Statistical properties of electrochemical capacitance in disordered mesoscopic capacitors, Phys. Rev. B 89(24), 245430 (2014)

[76] J. Maciejko , J. Wang , and H. Guo , Time-dependent quantum transport far from equilibrium: An exact non-linear response theory, Phys. Rev. B 74(8), 085324 (2006)

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