 another Josephson junction. By measuring the switching behavior of this junction from the superconducting state to the normal branch with forward and backward pumping, they were able to extract the pumped current of the sluice. However, this type of current detection did not turn out to be as sensitive as the direct measurement with a transimpedance amplifier used in the case of voltage bias. A potential way to improve the sensitivity is, instead of the switching junction, to use a cryogenic current comparator coupled inductively to the superconducting loop. This type of an experiment has not been carried out to date. Instead, \textcite{Gasparinetti2012} have measured a sluice in the vicinity of vanishing voltage bias, where they demonstrated single-Cooper-pair pumping plateaus in both the bias voltage and dc level of the gate voltage, see Fig.~\ref{fig:3_5_2}. The quasiparticle poisoning was reported to be suppressed compared with the previous experiments, and hence they observed clear $2ef$ spacing of the current plateaus.

In addition to the above-mentioned pumping schemes, \textcite{Nguyen2007} have studied how a superconducting quantum bit referred to as quantronium can be used to detect the gate charge ramp arising from a current bias on the gate electrode of the island of the device. The accuracy of this technique on converting the bias current into frequency remains to be studied in detail. \textcite{Hoehne2012} studied another type of a quantum bit, a charge qubit, for pumping Cooper pairs non-adiabatically. The aim here was to increase the pumping speed compared to adiabatic schemes but due to the accumulation of the errors from a pumping cycle to another, a waiting period between the cycles needed to be added. Furthermore, \textcite{Giazotto2011} have showed experimentally how phase oscillations arising from the ac Josephson effect can drive Cooper pairs in a system with no tunnel junctions. However, this kind of pumping was found to be very inaccurate in this proof-of-the-concept experiment.

\begin{figure}[t]
\includegraphics[width=8.3cm]{figures/fig_3_5_2}
\caption{\label{fig:3_5_2} Pumped average charge by \textcite{Gasparinetti2012} for a single pumping cycle of a sluice pump near vanishing voltage bias as a function of the gate charge offset, $n_g$, and span during the pumping cycle, $\Delta n_g$.}
\end{figure}


\subsection{Other realizations and proposals}
\label{sec:otherpumps}
In this subsection we cover various ideas that have been brought up fr experimental demonstration. Although their metrological relevance is still to be proven, we wish to present them for the sake of their complementarity, potential and for completeness.

\subsubsection{Ac-current sources}
The current pumps described in sections \ref{sec:metalpumps}, \ref{sec:hybridpumps} and \ref{sec:dotpumps} can be
considered as single-electron injectors generating dc current. Coulomb blockade ensures a good control of the
electron number on each island during the charge transfer. However, there is no time control of the
electronic injection in metallic and hybrid single-electron pumps because of the stochastic nature of
tunneling. Time-controlled injection can be realized in quantum dots, however, the energy of emitted
electrons spreads randomly in a wide range exceeding $\hbar\Gamma$, where $\Gamma$ is the electron tunneling
rate.

A time-controlled single-electron source generating ac current was reported by \textcite{Feve2007}. The
source was made of a quantum dot, realized in a GaAlAs/GaAs heterojunction of nominal density $n_s = 1.7
\times 10^{15}~\rm{m}^{-2}$ and mobility $\mu = 260~\rm{V}^{-1} \rm{m}^2 s^{-1}$ and tunnel-coupled to a
large conductor through a quantum point contact (Fig.~\ref{fig:ACsource_schematic}). The discrete energy
levels of the quantum dot were controlled by the voltage applied to the capacitively coupled metallic gate
located 100~nm above the 2DEG. When the applied voltage is high enough, the electron occupying the highest
energy level tunnels to the conductor. This process is called electron emission. If the applied voltage
is changed abruptly at the subnanosecond time scale (a sudden jump), the emission process is coherent and the
final state of the electron is a coherent wave packet propagating away in the conductor. The energy width of the wave packet is
given by the inverse tunneling time, as required for an on-demand single-particle source. It is also independent
of temperature. Its mean energy is adjusted by the amplitude of the voltage step, $V_{\rm exc}$. For all
measurements, the electronic temperature was about 200~mK. A magnetic field $B \approx 1.3~$T was applied to
the sample so as to work in the quantum Hall regime with no spin degeneracy. The QPC dc gate voltage
$V_{\mathrm G}$ controlled the transmission $D$ of a single edge state as well as the dc dot potential. As
reported by \textcite{Gabelli2006}, this circuit constitutes the effective quantum-coherent $RC$ circuit,
where coherence strongly affects the charge relaxation dynamics. The effective quantum resistance $R$ and
capacitance $C$ are defined as $R=h/2e^2$ and $C=e^2(dN/d\varepsilon)$, where $dN/d\varepsilon$ is the
local density of states of the mode propagating in the dot, taken at the Fermi energy~\cite{Pretre1996}. Interestingly, the QPC
transmission probability $D$ affects the quantum capacitance but not the quantum resistance. In the
experiment, the addition energy $\Delta + e^2/C \approx 2.5$~K was extracted. The large top-gate capacitance
makes the Coulomb energy $e^2/2C$ unusually small, and the total addition energy reduces to the energy-level
spacing $\Delta$.

\begin{figure}[t]
\includegraphics[width=8cm]{figures/ACsource_schematic}
\caption{\label{fig:ACsource_schematic} Schematic of the single-charge injector and its operation principle.
Starting from step 1 where the Fermi energy level of the conductor lies in between two energy levels of the
dot, the dot potential is increased by $\Delta$ moving one occupied dot level above the Fermi energy (step
2). One electron then escapes from the dot. After that the dot potential is brought back to the initial value
(step 3), where one electron can enter the dot leaving a hole in the conductor. One-edge channel of the
quantum $RC$ circuit is transmitted into the dot, with transmission $D$ tuned by the QPC gate voltage
$V_{\rm G}$. The dot potential is controlled by a radio-frequency excitation $V_{\rm exc}$ applied on a
macroscopic gate located on top of the dot \cite{Feve2007}.}
\end{figure}

The single-charge injection was achieved by the application of a high-amplitude excitation voltage $V_{\mathrm{exc}} \sim
\Delta/e$ to the top gate. When an electron level is brought suddenly above the Fermi energy of the
lead, the electron escapes the dot at a typical tunnel rate $\tau^{-1} = D\Delta/h$, where $\Delta/h$ is
the attempt frequency. Typically, the tunnel rates are in the nanosecond time scales, and this makes the single-shot charge detection a
challenging task. In the experiment, a statistical average over many
individual events was used by repeating cycles of single-electron emission with period $T$ followed by
single-electron absorption (or hole emission) as shown in Fig.~\ref{fig:ACsource_schematic}. This was done by
applying a periodic square wave voltage of amplitude $\approx \Delta/e$ to the top gate. Thus, the signal-to-noise
ratio was increased.

Under certain conditions single-electron repeatable injection leads to a good quantization of the ac current. 
$| I_\omega |$ as a function of $V_{\mathrm{exc}}$ for two
values of the dc dot potential at $D \approx$ 0.2 and $D \approx$ 0.9 is shown in Fig.
\ref{fig:ACsource_pumping}. Apparently, a good current plateau is expected when the charge on the dot is well
defined. The latter depends on the transmission and the working point set by the gate voltage. For example, transmission $D
\approx$ 0.2 is low enough and the electronic states in the dot are well resolved, as shown in the inset of Fig.~\ref{fig:ACsource_pumping}
(left). On the other hand, the transmission is large enough for the escape time to be shorter than $T$/2. When the Fermi energy lies
exactly in the middle of the density-of-states valley (blue vertical line in the left inset)), a well-pronounced $| I_\omega |$ = $2ef$ current
plateau is observed centered at $2eV_{\mathrm{exc}}/\Delta = 1$. At this working point, the plateau is rather flat. It is claimed that
the current uncertainty at the plateau is 5\% due to the systematic calibration error.