Quantized Boundary Charge Pumping

• Quantized boundary charge pumping is the controlled, cycle-wise transfer of integer charges along a quantum system's boundaries, driven by time-dependent modulation and protected by topological invariants.
• It utilizes a dynamic quasi-bound state in semiconductor nanowires where a single electrostatic barrier is modulated by an RF signal to load and unload electrons with high precision.
• The non-adiabatic blockade effectively prevents unwanted tunneling, making the method scalable for quantum metrology and single-electron applications in cutting-edge nanoelectronic devices.

Quantized boundary charge pumping refers to the robust, cycle-wise transfer of integer multiples of the elementary charge along the boundaries of a low-dimensional quantum system through the time-dependent modulation of system parameters. The quantization originates from underlying topological invariants, is protected against a broad class of perturbations, and, depending on the physical regime, may be realized even with a minimal set of control parameters and without the need for external voltage bias.

1. Fundamental Mechanism: Dynamic Quasi-bound State and Single-Parameter Modulation

The essential physical configuration is a dynamically generated potential well in a semiconductor nanowire, typically realized using two electrostatic barriers. In the experimental realization with an AlGaAs/GaAs gated nanowire, only one barrier is modulated by an external radio-frequency (RF) signal while the other remains fixed. The time-dependent potential profile is given by

$$U(x, t) = U_1(t) \exp\left[-\frac{(x + x_0)^2}{w^2}\right] + U_2 \exp\left[-\frac{(x - x_0)^2}{w^2}\right]$$

where the left barrier height is modulated,

$$U_1(t) = U_1^{(\mathrm{dc})} - U_1^{(\mathrm{ac})} \cos(2\pi f t)$$

and the right barrier $U_2$ is held constant. The time dependence of $U_1$ modulates both the depth of the well and the energy level $\epsilon_0(t)$ of the quasi-bound state formed in the well.

During a modulation cycle, when $\epsilon_0(t)$ drops well below the chemical potential $\mu$ of the leads, the quasi-bound state loads an electron from the source (left contact). As the cycle progresses and $\epsilon_0(t)$ is raised above $\mu$ (or when tunneling rates alter appropriately), the electron is unloaded to the drain (right contact). With one electron transferred per cycle at frequency $f$, the resulting current is quantized: $I = -n e f \qquad (n = 1, 2, 3, \ldots)$

2. Role of Non-Adiabaticity and the Blockade of Unwanted Tunneling

Contrasting with traditional multi-parameter (often adiabatic) pumps, the single-parameter scheme without non-adiabatic effects would not permit quantized, directional current due to system symmetry. Crucially, non-adiabatic driving with a finite modulation frequency $f$ introduces temporal asymmetry: loading and unloading steps are delayed relative to energy-level crossings. This delay causes the dominant tunneling coupling to switch between the source and drain:

• During loading, the left barrier opens ($\Gamma_l \gg \Gamma_r$), prioritizing electronic transfer from the source.
• During unloading, the right barrier opens ($\Gamma_l \ll \Gamma_r$), favoring discharge to the drain.

This “non-adiabatic blockade” blocks back-tunneling events that would otherwise compromise current quantization—substituting, in effect, for the explicit phase-shifted second parameter used in adiabatic two-parameter pumps.

3. Theoretical Model: Rate Equation Formalism

The occupation probability $P(t)$ of the quasi-bound state is governed by a non-equilibrium, time-dependent rate equation: $$\hbar \frac{dP}{dt} = [\Gamma_l(t) + \Gamma_r(t)][f_F(\epsilon_0(t)) - P(t)]$$ where $f_F(\epsilon_0)$ is the Fermi function, and $\Gamma_{l,r}(t)$ are the instantaneous tunneling rates to the left and right leads, respectively. In the regime where level spacing $\Delta\epsilon$ exceeds broadening and thermal smearing, and $\Gamma \ll k_BT$, this equation robustly predicts near-complete transfer of an integer number of electrons per modulation cycle. The quantized current is given by

$I = -e N f$

where $N$ is the number of electrons transferred per cycle (typically $N=1$ at optimal operating points).

4. Frequency Dependence and Quantization Plateaus

Experimental mapping of the pumped current as a function of control parameters reveals distinct current plateaus, with steps of $\Delta I = e f$, confirming the quantization. The operational regime is constrained by frequency:

• At low frequencies (near adiabatic driving), electrons may escape back to the source during the loading phase, reducing quantization fidelity.
• At properly tuned, intermediate frequencies, non-adiabaticity blocks back-tunneling yet allows efficient unloading—maximizing pumped current quantization (i.e., $n_p \approx 1$ per cycle).
• At excessively high frequencies, unloading to the drain becomes incomplete within a cycle, reducing $n_p$ below unity.

The optimization of parameters for robust quantization is thus governed by this competing interplay, as experimentally probed by tracing current plateaus over gate voltage and modulation amplitude.

5. Robustness Against Parameter Choice and Quantum Metrology Implications

Although previously the use of two synchronized, phase-shifted periodic signals (two-parameter driving) was considered necessary for electron pumps, non-adiabatic single-parameter schemes demonstrate intrinsic temporal asymmetry sufficient for quantitative current control. This simplification is significant for scalable implementations—single-parameter modulation enables the parallel operation of multiple pumps on a single chip, with substantial implications for quantum metrology (e.g., closing the quantum metrological triangle). The accuracy of plateaus, quantized to within $e f$, is established experimentally, providing a platform for improved quantum current standards.

6. Practical Implementation and Device Considerations

Key implementation aspects include:

• Device geometry: AlGaAs/GaAs gated nanowire or quantum dot with two gate-defined barriers.
• Modulation protocol: one barrier modulated via RF drive, the other fixed.
• Key design parameters: position and width of barriers, tunneling rates (tailored via gate voltages), and modulation amplitude/frequency.
• Operating window: frequency and gate voltages must be tuned such that the dynamical phase-shift (delay) induced by non-adiabaticity blocks unwanted tunneling while ensuring efficient loading/unloading.

Scalability and robustness benefit from this minimalist architecture, holding promise for integration into metrological devices or quantum electronic applications requiring precise single-electron manipulation.

7. Summary Table: Key Features

Feature	Description	Reference Quantity/Formula
Principle	Single-parameter, non-adiabatic pump	$U_1(t)$ modulated, $U_2$ fixed
State involved	Dynamic quasi-bound state	Energy $\epsilon_0(t)$ varies with time
Current quantization	Integer charge per cycle	$I = -n e f$
Rate equation	Time-dependent occupation dynamics	$$\hbar \frac{dP}{dt} = (\Gamma_l+\Gamma_r)[f_F(\epsilon_0) - P]$$
Non-adiabatic blockade	Selective loading/unloading, time delay	$\Gamma_l \gg \Gamma_r$ (loading); $\Gamma_l \ll \Gamma_r$ (unloading)
Metrological relevance	Quantum current standard application	$\Delta I = e f$ plateau spacing

Quantized boundary charge pumping in this context represents a paradigm shift: non-adiabatic effects, long considered detrimental for pump accuracy, are revealed to be essential enabling mechanisms for robust and scalable single-parameter pumping protocols. This realization simplifies device architecture, advances the prospect of high-precision current sources for metrology, and provides a foundation for quantum electronics exploiting controlled, deterministic single-electron transfer.