Assume now that $G=(G_{0},A)$ is a Lie supergroup and $M=(Y,B)$ another
supermanifold. We will say that $G$ acts on $M$ if there is a map
$A(G_{0})^{\ast}\otimes B(Y)^{\ast}\xrightarrow{}B(Y)^{\ast}$, $u\otimes
w\mapsto u\cdot w$, which satisfies

$\Delta u=\sum_{i}u_{i}^{\prime}\otimes u_{i}^{\prime\prime},\qquad\Delta
w=\sum_{j}w_{j}^{\prime}\otimes w_{j}^{\prime\prime}\implies\Delta(u\cdot
w)=\sum_{i,j}(-1)^{|u_{i}^{\prime\prime}||w_{j}^{\prime}|}u_{i}^{\prime}\cdot
w_{j}^{\prime}\otimes u_{i}^{\prime\prime}\cdot w_{j}^{\prime\prime}.$

In this case, the structure algebra $B(Y)$ is a $A(G_{0})^{\ast}$-module
through

$\pi\colon\ A(G_{0})^{\ast}\xrightarrow{}\text{End}(B(Y)),\qquad\langle
w,\pi(u)f\rangle=(-1)^{|u||w|}\langle\sigma(u)\cdot w,f\rangle.$

The later is called the coaction representation of $G$. The action of $G$ is
fully determined by the corresponding coaction representation. Bearing in mind
that $A(G_{0})^{\ast}=E(\mathfrak{g})$, we see that a Lie supergroup action
can be though of as a pair of representations of the underlying group $G_{0}$
and of Lie superalgebra $\mathfrak{g}$ on the vector space $B(Y)$, which
satisfy a compatibility condition.

Dually, there is a map $\varphi\colon B(Y)\xrightarrow{}B(Y)\otimes A(G_{0})$
that makes $B(Y)$ into a comodule-algebra of $A(G_{0})$. This means that
$\varphi$ is a morphism of algebras which is compatible with the Hopf algebra
structure of $A(G_{0})$. For example, $\varphi$ satisfies

$(1\otimes\Delta)\circ\varphi=(\varphi\otimes 1)\circ\varphi\colon\
B(Y)\xrightarrow{}B(Y)\otimes A(G_{0})\otimes A(G_{0}),$

along with a number of other compatibility conditions, see, e.g., [56]. Let
$p$ be a point in $G_{0}$, considered as a morphism $p\colon
A(G_{0})\xrightarrow{}\mathbb{R}$. Then one can form the map $(1\otimes
p)\circ\varphi\colon B(Y)\xrightarrow{}B(Y)$. For obvious reasons, we refer to
such compositions with $p$ as evaluations. Running over all points $p$, we get
a representation of the $G_{0}$ on $B(Y)$. This agrees with the coaction
representation $\pi$ from above. The evaluated action of the bosonic group
$G_{0}$ and the infinitesimal action of the conformal Lie superalgebra fit
together to form a representation of the Lie–Hopf algebra $A(G_{0})^{\ast}$ on
$B(Y)=\mathcal{M}$.

### Acknowledgements

This work reviewed results obtained in collaboration with Volker Schomerus and
Zhenya Sobko, some of which originated in the work with Misha Isachenkov. I
warmly thank all my collaborators. I am also indebted to Maja Burić and Aleix
Gimenez-Grau for a number of illuminating discussions and to anonymous
referees for suggesting a number of improvements of the text. Finally, I
acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306.

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# Pulse driven quantized current steps in ultrasmall Josephson junctions

Fabian Kaap<EMAIL_ADDRESS>Physikalisch-Technische Bundesanstalt,
Bundesallee 100, 38116 Braunschweig, Germany  Christoph Kissling  Victor
Gaydamachenko  Lukas Grünhaupt  Physikalisch-Technische Bundesanstalt,
Bundesallee 100, 38116 Braunschweig, Germany  Sergey Lotkhov  Physikalisch-
Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

###### Abstract

Recent experimental results have validated a prediction made almost four
decades ago, affirming the existence of quantized current steps in ultrasmall
Josephson junctions and superconducting nanowires. These so-called dual
Shapiro steps hold promise for a new current standard and thus could close the
quantum metrological triangle. This is because the steps mark quantized
current levels $I=\pm n\times 2ef$, where the elementary charge $e$ is a fixed
constant in the new SI and the frequency $f$ is the physical quantity measured
with the highest precision. To realize dual Shapiro steps, we embed an
Al/AlOx/Al dc-SQUID in a high impedance environment made from granular
aluminium and oxidized titanium. We successfully demonstrate quantized current
steps in the IV-curves by applying sinusoidal driving signals of frequencies
up to $6\text{\,}\mathrm{GHz}$ resulting in quantized current levels up to
$I\approx\pm$2\text{\,}\mathrm{nA}$$. Remarkably, if changing to a pulsed
drive, the first dual Shapiro step has a larger voltage amplitude, while the
opposite step vanishes, depending on the sign of the pulse. By using the peak
values of the differential resistance as a measure of flatness, we demonstrate
improvement due to the pulsed driving signals by a factor of $\sim 2$ compared
to sinusoidal driving.

Today’s voltage standard provided by national metrology institutes [1] is
based on the synchronization of Josephson oscillations [2] with an external
rf-drive. This leads to equidistant voltage steps, the so-called Shapiro steps
[3]. The dual effect of quantized current steps was predicted theoretically
almost 40 years ago [4, 5], originating from the quantum mechanical
description of small Josephson junctions, with charge and phase being
conjugate variables. Quantized currents based on this duality could
potentially close the metrological triangle, uniting the ampere, the volt and
the ohm through fundamental constants and frequency. Significant advantages of
using Josephson junctions for this purpose, as compared to e.g. semiconductor
pumps [6, 7], are higher frequencies in the GHz-range and technological
compatibility with Josephson voltage standards.

To observe dual Shapiro steps, the Josephson junction has to be embedded in an
high impedance environment
$Z_{\mathrm{env}}>R_{\mathrm{Q}}=h/4e^{2}\approx$6.4\text{\,}\mathrm{k\SIUnitSymbolOhm}$$
in order to suppress quantum fluctuations of charge and enable charging
effects to be observed. First indications of dual Shapiro steps were observed
a few years following the original prediction [8] and relied on a purely
resistive environment. However, strong heating effects were found to smear the
quantized current steps [9], such that no detailed study of the
synchronization mechanism could be carried out. To solve this issue on the way
to an unambiguous confirmation of dual Shapiro steps, it was proposed to
implement a biasing circuit combining high-ohmic off-chip resistors and
compact on-chip inductances [10], whose specific impedance exceeds the quantum
resistance. Such so-called superinductances can in principle be realized via
Josephson junction chains [11, 12, 13, 14, 15], nanowires on membranes [16,
17] or high kinetic inductance materials [18, 19, 20, 21, 22, 23, 24, 25].

These technological advancements recently resulted in convincing
demonstrations of dual Shapiro steps [23, 15]. Here, we demonstrate pronounced
quantized current steps in ultrasmall Josephson junctions by synchronizing
their intrinsic voltage oscillations, so called Bloch oscillations (BO), to an
external signal with frequencies up to $6\text{\,}\mathrm{GHz}$. Going beyond
the sinusoidal drive in previously mentioned experiments or synchronizing two
individual Bloch oscillations [26], we show how tailoring the shape of the
external waveform increases the slope of the dual Shapiro steps by a factor of
$\sim 2$. In particular, it has been predicted theoretically [27] and
previously reported for Shapiro steps [28, 29] that the width of the voltage
steps can be enhanced by using pulsed driving signals. Our present work
demonstrates such an effect for dual Shapiro steps.

The behaviour of a Josephson junction is determined by its charging energy
$E_{C}=\frac{e^{2}}{2C}$ and its Josephson energy
$E_{\mathrm{J}}=\frac{\Phi_{0}I_{\mathrm{c}}}{2\pi}$, where $e,\Phi_{0}$ are
the electron charge and the flux quantum and $C$, $I_{\mathrm{c}}$ are the
capacitance and critical current of the Josephson junction. Taking into
account that the charge and phase operators are governed by the canonical
commutation relation $[\hat{Q},\hat{\varphi}]=2ei$, the Hamiltonian is given
by

$H=\frac{\hat{Q}^{2}}{2C}-E_{\mathrm{J}}\cos(\hat{\varphi}).$ (1)

This Hamiltonian can be solved using Blochs theorem [30, 31] and yields
periodic energy bands $E_{s}(q)=E_{s}(q+2e)$, where $s\in\mathbb{N}_{0}$ is
the level number and $q$ is the so-called quasicharge, which represents the
charge injected to the junction [5, 4].

Figure 1: Bloch oscillations: energy diagram and experimental realization. (a)
Energy $E_{s}(q)$ of the two lowest Bloch bands for different ratios
$E_{C}/E_{\mathrm{J}}$. (b) Ground state oscillations of voltage calculated by
taking the derivative $\frac{\partial E_{0}}{\partial q}=V$. These so-called
Bloch oscillations have a frequency $f_{\mathrm{B}}\approx
I_{\mathrm{bias}}/2e$ (see main text). (c-e) Optical and scanning electron
microscope images of a representative device consisting of a dc-SQUID [see
panel(d)] with two Josephson junctions of area
$A_{\mathrm{JJ}}\approx\qtyproduct[product-units=bracket-power]{20x30}{\nano}$
[see panel(e)] embedded in a bias circuitry. The slot line from the top
delivers the rf-driving signal via a capacitive coupling with
$C_{\mathrm{c}}\approx$0.3\text{\,}\mathrm{fF}$$ while the horizontal leads
are used for four-probe dc-measurements. The bias circuitry includes the
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$-long granular aluminium inductor
[20] with kinetic inductance
$L_{\mathrm{grAl}}\approx$50\text{\,}\mathrm{nH}$$ and characteristic
impedance $Z_{\mathrm{grAl}}\approx
10.5\mathrm{k\Omega}>R_{\mathrm{Q}}=h/4e^{2}\approx$6.4\text{\,}\mathrm{k}\Omega$$
and the $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$-long $\mathrm{TiO_{x}}$
resistors [32, 26] of
$R_{\mathrm{TiO_{x}}}\approx$275\text{\,}\mathrm{k\SIUnitSymbolOhm}$$. This
provides a high impedance environment $Z_{\mathrm{env}}\geq R_{\mathrm{Q}}$,
suppresses quantum fluctuations of charge and enables Bloch oscillations [4].

In Fig. 1(a) the two lowest energy bands ($s=0,1$) are shown for different
ratios $E_{C}/E_{\mathrm{J}}$. In the ground-state approximation, the dynamics
of this system is given by the so-called Langevin equation:

$\dot{q}=I_{\mathrm{bias}}-V[q(t)]/R+I_{\rm{rf}}(t)+\tilde{I}(t),$ (2)

with $I_{\mathrm{bias}}$ a dc bias current, $V=\frac{\partial E_{0}}{\partial
q}$, $R$ the real part of the impedance of the environment, which we for
simplicity assume to be purely ohmic, $\tilde{I}(t)$ the fluctuation current
and $I_{\mathrm{rf}}$ an external rf-signal [4, 5]. If $I_{\rm{rf}}(t)=0$,
$\tilde{I}(t)=0$ and $I_{\mathrm{bias}}<\max[\frac{\partial E_{0}}{\partial
q}]/R$, there is a static solution for Eq. (2), a Coulomb blockade. For
currents $I_{\mathrm{bias}}>\max[\frac{\partial E_{0}}{\partial q}]/R$ the
solution is periodic in time and results in BO with frequency $f\approx
I_{\mathrm{bias}}/2e$ [see Fig. 1(b)]. These BO can synchronize to an external
rf-current $I_{\mathrm{rf}}$, generating quantized current steps at $I=\pm
n\times 2ef$, referred to as dual Shapiro steps, where $n$ is a natural
number.

Figure 2: External flux dependent Coulomb blockade of the device. (a)
$IV$-curves of the sample [cf. Fig. 1(c-e)] for external flux values
$0\leq\Phi/\Phi_{0}\leq 0.45$. By approaching $\Phi_{0}/2$ and thus increasing
the ratio $E_{C}/E_{\mathrm{J}}$ we observe wider ranges of the Coulomb
blockade and steeper slopes of the resistive branch caused by Landau-Zener
tunnelling (LZT) [33, 34, 35] (see main text). (b) Zoom-in on the $IV$-curves
for $\Phi/\Phi_{0}=0,0.3$. A back-bending to almost zero voltage can be
observed, which is a hallmark of Bloch oscillations in a dc-measurement. We
identify flux values $0\leq\Phi/\Phi_{0}\leq 0.3\>(1.3\leq
E_{C}/E_{\mathrm{J}}\leq 2.3)$ as a suitable range for observing dual Shapiro
steps. Figure 3: Dual Shapiro steps occurring due to sinusoidal rf-drive. (a)
Measured $IV$-curve with $f_{\mathrm{drive}}=2.8\mskip
3.0mu\unit{}\>(\mathrm{brown}),\>4.6\mskip 3.0mu\unit{}\>$
$(\mathrm{violet}),\>7.8\mskip 3.0mu\unit{}\>(\mathrm{blue})$ for rf-power at
the lowest temperature stage of
$p_{\mathrm{drive}}=$-49\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$. The dashed
lines indicate the due positions of the first dual Shapiro steps according to
$I_{\mathrm{dSs}}=\pm 2ef_{\mathrm{drive}}\approx\pm 0.93\>(1.53)\mskip
3.0mu\unit{\nano}$. (b) Differential resistance
$R_{\mathrm{diff}}=\frac{dU}{dI}$ measured using a lock-in amplifier for
different drive frequencies $f_{\mathrm{drive}}$ at constant drive power
$p_{\mathrm{drive}}=$-49\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$. Dashed
white lines confirm the expected position of dual Shapiro steps according to
$I_{\mathrm{dSs}}=\pm n\>2ef_{\mathrm{drive}}$. (c) $R_{\mathrm{diff}}$ for
different powers $p_{\mathrm{drive}}$ at fixed frequency
$f_{\mathrm{drive}}=$3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$, leading to
steps at $I_{\mathrm{dSs}}\approx\pm n\cdot$0.96\text{\,}\mathrm{nA}$$. The
amplitude of the Coulomb blockade and the dual Shapiro steps vary with
$p_{\mathrm{drive}}$ in qualitative agreement with the results of Ref. [28]
and with the Tien-Gordon formula [36, 23].

Fig. 1(c-e) shows optical and scanning electron microscope (SEM) pictures of a
representative device. We fabricate the chip using three lithographic steps on
a thermally oxidized silicon wafer [25]. First, we pattern a wiring structure
from Au (not shown) using optical lift-off lithography. Subsequently, we
fabricate $150\text{\,}\mathrm{nm}$ wide and $15\text{\,}\mathrm{nm}$ thick
TiOx resistors [32]. Finally, we in-situ combine a $20\text{\,}\mathrm{nm}$
thick granular aluminium superinductor [20] with double-angle evaporation
Al/AlOx/Al Josephson junctions [37]. To implement the last lithographic step
we utilize a PMMA/copolymer liftoff mask with a controllable undercut [38].
Each of the two Josephson junctions of the SQUID has an estimated area
$A_{\mathrm{JJ}}\approx\qtyproduct[product-units=bracket-power]{20x30}{\nano}$
[see Fig. 1(e)] leading to an upper bound of the charging energy
$E_{C}=e^{2}/(4A_{\mathrm{JJ}}\times$50\text{\,}\mathrm{f}\mathrm{F}\mathrm{/}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$)=$1.3\text{\,}\mathrm{meV}$$.
The true experimental value of $E_{C}$, however, includes the full 3D junction
topology as well as the stray capacitances of the leads, and is measured to be
$E_{C}\approx$80\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ (see supplementary).
In order to keep the Josephson energy $E_{\mathrm{J}}$ comparable to the
charging energy, we dynamically oxidize at
$P_{\mathrm{O_{2}}}=$0.1\text{\,}\mathrm{Pa}$$ for $300\text{\,}\mathrm{s}$,
resulting in $E_{\mathrm{J}}\approx$60\text{\,}\mathrm{\SIUnitSymbolMicro
eV}$$ (see supplementary). The leads of the SQUID are connected to granular
aluminium strips, which split up in two oxidized titanium resistors [32, 26]
[see Fig. 1(c)]. Aside from their inductive contribution, the grAl strips
might act as quasiparticle filters due to their larger superconducting energy
gap as compared to the Al films [39]. An external rf-signal for
synchronization of the BO is supplied via a
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$ slot line and capacitively couples to
the SQUID [see Fig. 1 (c-d)].

All measurements were performed in a dilution refrigerator at
$\sim$15\text{\,}\mathrm{mK}$$, such that
$k_{\mathrm{B}}T\ll\Delta^{(0)}=\mathrm{min}(E_{1})-\mathrm{max}(E_{0})$,
which confines the dynamics of the system to the lowest energy band. All dc
lines include three meter long ThermocoaxTM cable filters [40] connecting the
room temperature electronics to the mK sample stage. The rf driving signals
applied to the sample are attenuated by $50\text{\,}\mathrm{dB}$
($40\text{\,}\mathrm{dB}$ for results of Fig. 4(e)) at different temperature
stages of the dilution cryostat (see supplementary). The rf-powers quoted in
the following refer to the power level at the mK-stage.

Fig. 2(a) shows the $IV$-curves of the sample without external rf-drive for
values of the external flux between $0\leq\Phi/\Phi_{0}\leq 0.45$. As can be
seen, the effective Josephson energy of the dc-SQUID
$E_{\mathrm{J}}(\Phi)=E_{\mathrm{J,0}}|\cos(\pi\frac{\Phi}{\Phi_{0}})|$,
therefore the shape of the Bloch bands and, consequently, the $IV$-curve are
tuned. Close to $\Phi\approx{\Phi_{0}}/{2}$ the Coulomb blockade increases,
which can be explained by the increasing amplitude of the first Bloch band
[cf. Fig. 1(a)]. Moreover, for large bias currents $I_{\mathrm{bias}}$ the
probability to tunnel from $E_{0}$ to $E_{1}$ increases and the so-called
Landau-Zener-tunneling (LZT) [33, 34, 35] destroys the coherence of the BO.
Since the gap between the lowest two bands is approximately proportional to
$E_{\mathrm{J}}$, the onset current of the resistive LZT branch rapidly
decreases with the external flux and reaches a minimum at $\Phi/\Phi_{0}=0.5$.

Fig. 2(b) shows a zoom-in of the $IV$-curves for $\Phi/\Phi_{0}=0$ and $0.3$.
In the blue curve a clear backbending from the initial Coulomb blockade of
$\sim$4\text{\,}\mathrm{\SIUnitSymbolMicro V}$$ down to $U\approx 0$ can be
observed, indicating negligible LZT up to currents of
$\sim\pm$5\text{\,}\mathrm{nA}$$. In comparison, the orange curve shows an
extended blockade range of $\sim$18\text{\,}\mathrm{\SIUnitSymbolMicro V}$$,
but at the same time a reduced onset current of LZT of
$\sim$2\text{\,}\mathrm{nA}$$. As a compromise between maximizing the Coulomb
blockade width, which sets an upper limit for the dual Shapiro step width, and
the range of quantized currents limited by LZT, all measurements shown in this
work are done at $\Phi/\Phi_{0}=0.3$, resulting in
$E_{C}/E_{\mathrm{J}}\approx 2.3$.

Figure 4: Shaping dual Shapiro steps by using a tailored waveform. (a)
Dedicated waveforms $U_{\mathrm{d}}(t)$ used to drive the dual Shapiro steps.
(b) RF-current injection according to $I_{\mathrm{rf}}\propto
C_{\mathrm{c}}\times\frac{dU_{\mathrm{d}}}{dt}$. The symmetric waveform
(brown) leads to a rectangular current profile, while the negative (positive)
sloped sawtooth signal lead to short positive (negative) current pulses (black
and beige, respectively). (c) Differential resistance $R_{\mathrm{diff}}$ for
the three current profiles shown in (b) and a sinusoidal drive (red) with
$f_{\mathrm{drive}}=$2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and
$p_{\mathrm{drive}}=$-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$ (see
supplementary). Depending on the pulse, either the positive or negative dual
Shapiro step is enhanced and reaches almost twice as high peak values
$R_{\mathrm{diff}}$ as compared to the symmetric drives (triangular and
sinusoidal), while the opposite step is suppressed. (d) $R_{\mathrm{diff}}$
measured for linearly sweeping back and forth between a positive (beige) and
negative pulse drives (black). The colorbar on the y-axis indicates the
gradual change of the sawtooth polarity. Colored lines mark the curves shown
in (c). (e) Power dependence of $R_{\mathrm{diff}}$ using a pulsed drive at
$2\text{\,}\mathrm{GHz}$. Increasing the power of the pulse lead to a gradual
activation of the consecutive dual Shapiro steps. (f) Differential resistance
$R_{\mathrm{diff}}$ for different frequencies $f_{\mathrm{drive}}$ of the
pulsed drive. Over the whole frequency range the position of the positive dual
Shapiro step (black dashed line) aligns well with
$I_{\mathrm{dSs}}=2ef_{\mathrm{drive}}$, while the negative step is
suppressed. Inset: Peak values $R_{\mathrm{diff,peak}}$ extracted from the
main plot (beige) compared to similar data measured for a sinusoidal drive
(red). The values were attained by Gaussian fits to the peaks (see
supplementary). The higher $R_{\mathrm{diff}}$ peak values indicate the
improved flatness of the dual Shapiro steps.

In the following we perform measurements with a sinusoidal drive applied to
the SQUID at different GHz-frequencies and rf-power levels. Fig. 3(a) shows
the $IV$-curves measured at a drive power of
$p_{\mathrm{drive}}=$-49\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$ and
frequency $f_{\mathrm{drive}}=2.8\>(4.6)\mskip 3.0mu\unit{GHz}$ featuring two
distinct steps around the quantized values $I=\pm 2ef_{\mathrm{drive}}\approx
0.93\>(1.53)\mskip 3.0mu\mathrm{GHz}$ (brown and violet dotted lines). The
Coulomb blockade as well as the dual Shapiro steps are skewed as compared to
the non-rf case [see orange curve in Fig. 2(b)], which we attribute to
additional heating caused by the external rf-drive. The continuous raise of
voltage outside the range $|I|>$2\text{\,}\mathrm{nA}$$ is caused by the LZT
and resembles the LZT branch in the orange curve in Fig. 2(b).

Fig. 3(b) shows the differential resistance $R_{\mathrm{diff}}=\frac{dU}{dI}$
measured with a lock-in amplifier for different drive frequencies
$f_{\mathrm{drive}}$ at a constant power
$p_{\mathrm{drive}}=$-49\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$. The white
dashed lines indicate the due positions of the first and second dual Shapiro
step, $I_{\mathrm{peak}}=\pm n\times 2ef_{\mathrm{drive}}$. Up to frequencies
of $\sim$6\text{\,}\mathrm{GHz}$$ the first step can clearly be identified,
while towards higher frequencies the steps are vanishing [see Fig. 3(a)] and
the Coulomb blockade is restoring for higher frequencies, which indicates
lower rf-power reaching the SQUID. We attribute this to stronger reflection of
the rf-signal at higher frequencies in the slot line. Additionally, for
synchronization at higher frequencies larger power is needed [cf. Fig. 3(c)].

Fig. 3(c) shows the dependence of $R_{\mathrm{diff}}$ for rf-power in the
range $$-60\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$\leq
p_{\mathrm{drive}}\leq$-40\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$ at a fixed
drive frequency $f_{\mathrm{drive}}=$3\text{\,}\mathrm{GHz}$$. The equidistant
set of peaks in $R_{\mathrm{diff}}(I)$ indicates the dual Shapiro steps
separated by $\Delta I=$0.96\text{\,}\mathrm{n}\mathrm{A}$$. At low power,
$p_{\mathrm{drive}}\approx$-60\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$, the
Coulomb blockade peak dominates, while the first dual Shapiro steps are barely
visible. By increasing $p_{\mathrm{drive}}$ the first current steps grow
larger, as indicated by the higher resistance peaks, and they reach the
maximum peak height at
$p_{\mathrm{drive}}=$-51\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$, while the
Coulomb blockade disappears. Up to the higher power levels around
$p_{\mathrm{drive}}\approx$-47\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$, the
second and around $-43\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$ the third dual
Shapiro steps can be seen. Higher-order dual Shapiro steps are not visible due
to LZT destroying the coherence of the Bloch oscillations. This measured
pattern of dual Shapiro steps can be modelled using Eq. (1) and qualitatively
reproduces the data obtained in Refs. [23, 28, 36] (see supplementary). We
note that for higher rf-power the increased heating leads to smearing of the
features.

For a more specific study, we apply a sawtooth signal $U_{\rm d}(t)$ to the
slot line thus implementing a pulsed drive for the SQUID, $I_{\rm{rf}}\propto
C_{\rm c}\times dU_{\mathrm{d}}/dt$. Fig. 4(a) depicts the signal profile of
three different waveforms, namely a positive sawtooth, a negative sawtooth and
a symmetric triangular waveform. The derived rf-current through the SQUID is
sketched in Fig. 4(b). For the triangle we assume a symmetric rectangular
current profile of low amplitude, while both sawtooth waveforms will generate
a unipolar current pulse of larger amplitude. In Fig. 4(c) the differential
resistance $R_{\mathrm{diff}}$ is shown for the three current profiles
displayed in Fig. 4(b). For comparison, we plot similar data obtained for an
optimized sinusoidal drive with $f_{\mathrm{drive}}=$2\text{\,}\mathrm{GHz}$$
and $p_{\mathrm{drive}}=$-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$ (see
supplementary). For the triangular waveform both positive and negative branch
exhibit a similar peak height as in the case with a sinusoidal drive. In
contrast, if using a sawtooth-like waveform generating current pulses, the
peak height of $R_{\mathrm{diff}}$ of one of the two opposite steps is
approximately doubled in magnitude, while the other step is suppressed (see
supplementary, cf. Ref. [28]). As can be seen in Fig. 4(d), by changing the
drive waveform between a positive and a negative sawtooth (pulses) [beige and
black curves in Fig. 4(a,b)], we observe a modulation of measured curves
between the three representative cases shown in Fig. 4(c), which are indicated
by the three colored lines in panel (d). We note, that the largest
differential resistance $R_{\mathrm{diff,peak}}$ of the measured steps occurs
for maximally tilted sawtooth signals leading to the highest current pulse in
Fig. 4(b).

The power dependence of $R_{\mathrm{diff}}$ measured with pulse driving at
$2\text{\,}\mathrm{GHz}$ is shown in Fig. 4(e). By increasing the power, the
first dual Shapiro step appears at
$-50\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$, where the Coulomb blockade
fades. Further increasing the power above
$-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$ decreases the first dual Shapiro
step and the second step appears at
$-43\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$. This power dependence of the
individual steps is in good qualitative agreement with numerical simulations
(see supplementary, cf. Ref. [28]).

In Fig. 4(f) we plot $R_{\mathrm{diff}}$ for pulse drives at different
frequencies up to $2\text{\,}\mathrm{GHz}$, showing that the asymmetry in the
$IV$-curve persists and the position of the step follows the fundamental
relation $I=2ef_{\mathrm{drive}}$ (black dashed line). To quantify the
enhancement of the quantized current step due to the pulsed drive [see inset
of Fig. 4(f)], we extract the peak values of $R_{\mathrm{diff}}$ and compare
to the case of a sinusoidal drive (see supplementary). The power of the
sinusoidal drive,
$p_{\mathrm{drive}}=$-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$, was set to
maximize the step width and is comparable to the power of the sawtooth drives
with $-47.8\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$. For the sinusoidal drive
at frequencies lower than $f_{\mathrm{drive}}=$1.25\text{\,}\mathrm{GHz}$$,
the fit results are not shown in Fig. 4(d) inset, as long as the first two
dual Shapiro steps are overlapping due to thermal broadening. Note that over
the experimentally accessible range of frequencies we observe an enhancement
by a factor of at least 2.2 by using a sawtooth drive.

In conclusion, we experimentally show the occurrence of Bloch oscillations in
Josephson junctions embedded in a high impedance environment made from
granular aluminium and oxidized titanium. By synchronizing the Bloch
oscillations with an external sinusoidal rf-drive, we demonstrate pronounced
dual Shapiro steps and quantized currents up to $2\text{\,}\mathrm{nA}$. By
changing to the tailored waveform, we are able to increase the differential
resistance of the first dual Shapiro step by a factor of $\sim 2$. This
research serves as a stepping stone for future metrological applications,
namely the development of new quantum standards for currents in the nA-range
and thus, for closing the metrological triangle. For future experiments, the
detailed studies of dual Shapiro steps arising due to a wide variety of
driving waveforms appear to be of particular interest. Combining the
demonstrated technique of drives with further improvements in the circuit
layout like an optimized rf-line geometry and using even smaller, point-like
contacts could lead to even wider current steps.

We thankfully acknowledge useful discussions with A. B. Zorin, D. Scheer, F.
Hassler, and M. Bieler. The authors also acknowledge technical support from H.
Marx and N. Ubbelohde, and A. Fernandez-Scarioni, M. Schröder, J. Blohm, P.
Hinze and T. Weimann. This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) under Grant No. LO 870/2-1.

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##  I Supplementary

###  S0.1 Experimental setup

Figure S1: Schematics of the experimental setup. Different shades of blue
indicate the temperature stages within the cryostat. See the text for detailed
description of the used cabling and measuring schemes.

Fig. S1 depicts the experimental setup. The measurements were performed in an
Oxford Instruments Triton 400 dilution refrigerator. The DUT is mounted in a
rf-tight copper box, which has an integrated coil to apply the magnetic flux
to the SQUID and thus, control $E_{\mathrm{J,eff}}$. The dc-lines from the
sample box are connected to 3 m long ThermocoaxTM cable filters 40, reaching
from the mixing chamber stage ($\sim$15\text{\,}\mathrm{mK}$$) to the breakout
connectors at the top of the cryostat. The $IV$-curves shown in Fig. 2 and
Fig. 3(a) of the main text are measured by using CompactDAQ (cDAQ) converter
from National Instruments. By applying dc-voltage $V_{\mathrm{b}}$ to two
biasing resistors $R_{\mathrm{b}}=$1\text{\,}\mathrm{G\SIUnitSymbolOhm}$$, a
current $I=(V_{\mathrm{b}}-U)/(2R_{\mathrm{b}}+R_{\mathrm{add}})$ can be
biased through the sample. $R_{\mathrm{add}}$ is the additional resistance
from the cabling, the cable filters and the on-chip resistors. The voltage
drop $U$ over the DUT is amplified by the low-noise voltage amplifier Femto
DLPVA-100-F-D by a factor of $10^{4}$ ($80\text{\,}\mathrm{dB}$) and then
digitalized using the cDAQ. For the measurements of differential resistance
$R_{\mathrm{diff}}$ shown in Fig. 3(b-c) and Fig. 4(c-f) we use a Lock-in
amplifier SR860 from Stanford Research Systems, where biasing resistors with
$100\text{\,}\mathrm{M\SIUnitSymbolOhm}$, the modulation frequency of
$13.97\text{\,}\mathrm{Hz}$ and a amplitude of $1\text{\,}\mathrm{mV}$ are
used. The sinusoidal drives are provided by an Anapico APMS40G rf-generator
and the arbitrary waveforms are generated by a Tektronix AWG7102. The rf-
signals are attenuated by $10\text{\,}\mathrm{dB}$ at each temperature stage
of the cryostat, resulting in a total attenuation of $50\text{\,}\mathrm{dB}$,
neglecting additional losses from cables and connectors. The current for the
flux coil is provided by either an Agilent B2901A SMU or by using an
additional cDAQ channel.

###  S0.2 Fabrication techniques

The granular aluminium and the titanium oxide layers were fabricated in the
same electron beam evaporation system as the aluminium layers of the Josephson
junctions. For the grAl films, Al was evaporated with an evaporation rate of
$2\text{\,}\mathrm{\SIUnitSymbolAngstrom}\mathrm{/}\mathrm{s}$, while the
wafer was placed in a chamber with an oxygen pressure of
$p_{\mathrm{O_{2}}}\approx$1.4\text{\times}{10}^{-5}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$$.
The same procedure was used for the TiOx, where titanium was evaporated at a
rate of $2\text{\,}\mathrm{\SIUnitSymbolAngstrom}\mathrm{/}\mathrm{s}$ and the
oxygen pressure was set to
$p_{\mathrm{O_{2}}}\approx$3.0\text{\times}{10}^{-6}\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$$.
To ensure good electrical contact between the TiOx and the following grAl and
Al-layers, we sealed the end of the resistors with AuPd patches, which are
less prone to oxidation in the air in between of the two fabrication steps.

###  S0.3 Characterization of the SQUID

Figure S2: Characterization measurements: (a) Measured $IV$-curve for
$\Phi=n\>\Phi_{0}$. From the critical current
$I_{c}\approx$30\text{\,}\mathrm{nA}$$ a Josephson energy
$E_{\mathrm{J}}\approx$60\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ can be
estimated. (b) $IV$-curve at $\Phi=(n+1/2)\>\Phi_{0}$. We estimate the
charging energy $E_{C}\approx$80\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$. (c)
$IV$-curves for different coil currents $I_{\mathrm{coil}}$. $E_{\mathrm{J}}$
is maximally suppressed at $I_{\mathrm{coil}}\approx$11\text{\,}\mathrm{mA}$$.

Fig. S2(a) shows the $IV$-curves for $\Phi=n\>\Phi_{0}$, where the critical
current $I_{\mathrm{c}}\approx$30\text{\,}\mathrm{nA}$$, defined as the onset
of the resistive branch, is maximized. The unsuppressed Josephson energy is
thus
$E_{\mathrm{J}}=\frac{\Phi_{0}I_{\mathrm{c}}}{2\pi}\approx$60\text{\,}\mathrm{\SIUnitSymbolMicro
eV}$$. From the $IV$-curve in Fig. S2(b) for $\Phi\to(n+1/2)\>\Phi_{0}$, where
the Coulomb blockade is maximized, we can estimate the charging energy
$E_{C}=$80\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$. In Fig. S2(c) the
$IV$-curves as a function of the coil current $I_{\mathrm{coil}}$ are shown.
At $I_{\mathrm{coil}}\approx$11\text{\,}\mathrm{mA}$$ the flux value
$\Phi=\Phi_{0}/2$ is reached. From measurements up to
$I_{\mathrm{coil}}=$35\text{\,}\mathrm{mA}$$ we extract a current-to-flux
coefficient of
$\sim$22.5\text{\,}\mathrm{m}\mathrm{A}\mathrm{/}\Phi_{\mathrm{0}}$$.

Figure S3: Detailed plots for Fig. 4 of the main text. (a) $R_{\mathrm{diff}}$
for different drive powers $p_{\mathrm{drive}}$. To find the best comparison
with the measurement using a pulsed drive, we chose
$p_{\mathrm{drive}}=$-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$ (indicated
with red line), at which the first dual Shapiro step is maximized. (b)
$R_{\mathrm{diff}}$ for different frequencies $f_{\mathrm{drive}}$ of the
sinusoidal drive. (c) Gaussian fits to the data shown in (b). For
visualization purposes an offset of $3\text{\,}\mathrm{k\SIUnitSymbolOhm}$ is
added to each line. (d) Gaussian fits to the data shown in Fig. 4(f) of the
main text. An offset of $5\text{\,}\mathrm{k\SIUnitSymbolOhm}$ is added to
each line for clarity. The data shown in the inset of Fig. 4(f) are the
amplitudes of the gaussian fits.

###  S0.4 Quantifying dual Shapiro steps

The differential resistance $R_{\mathrm{diff}}$ for values between
$$-52.8\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$\leq
p_{\mathrm{drive}}\leq$-44.2\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$ is shown
in FIG. S3(a). The red lines at $-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$
indicates, where the measured peak $R_{\mathrm{diff,peak}}$ of the first dual
Shapiro step is maximized and is used for comparison in Fig. 4(c) in the main
text. To quantify the difference between the sinusoidal and the pulsed drive,
we measured, analogue to Fig. 4(f) of the main text, the differential
resistance for sinusoidal drive frequencies $$1\text{\,}\mathrm{GHz}$\leq
f_{\mathrm{drive}}\leq$2\text{\,}\mathrm{GHz}$$ at
$p_{\mathrm{drive}}=$-48\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$$, as shown in
Fig. S3(b). The peak values $R_{\mathrm{diff,peak}}$ shown in the inset of
Fig. 4(f) of the main text are extracted by fitting Gaussians to the
measurements of Fig. S3(b) and Fig. 4(f) of the main text. In Fig. S3(c) and
Fig. S3(d) the fits for a set of frequencies are shown. Supporting the
assumption that the smearing of the dual Shapiro steps occurs due to thermal
broadening, fitting a Gaussian leads to good agreement with the measurements.

###  S0.5 Numerical analysis of different drives

To evaluate the influence of tailored drives on the dual Shapiro steps, we
make use of the duality principle. For simplicity, we assume a sinusoidal
shape of the lowest Bloch band and neglect all higher energy bands. With this
single band approximation the dynamics of the quasicharge $q$ is described by

$L\ddot{q}+R\dot{q}+V_{\mathrm{c}}\sin\left(\frac{\pi}{e}q\right)=V_{\mathrm{tot}}=V_{\mathrm{dc}}+V_{\mathrm{ac}}(t).$
(S1)

With $\tau=t\frac{\pi V_{\mathrm{c}}}{e}$, $\beta=\frac{\pi
V_{\mathrm{c}}L}{eR}$, $\tilde{q}=\frac{\pi}{e}q$ and
$\alpha_{\mathrm{dc,ac}}=V_{\mathrm{dc,ac}}/V_{\mathrm{c}}$ the equation (S1)
simplifies to

$\beta\frac{d^{2}\tilde{q}}{d\tau^{2}}+\frac{d\tilde{q}}{d\tau}+\sin\tilde{q}=\alpha_{\mathrm{dc}}+\alpha_{\mathrm{ac}}f(\tau)$
(S2)

and can be solved using a fourth-order Runge-Kutta method [28]. Fig. S4 shows
the calculated $IV$-curves for different drive powers for a pulsed (a) and a
sinusoidal (b) driving signal. For pulsed driving, the Coulomb blockade
(colored region indicated by ”0”) gradually vanishes with increasing power and
the first dual Shapiro step (colored region indicated by ”+1”) grows up the
the point $\alpha_{\mathrm{ac}}\approx 10$, where the second dual Shapiro
steps starts appearing. The negative dual Shapiro steps never appear in this
calculation due to the sign of the current pulse. Changing the sign of the
pulse would lead to a similar pattern of steps at negative currents. The
measured power dependence shown in Fig. 4(e) of the main text qualitatively
supports this calculation. For a sinusoidal drive shown in Fig. S4(b),
increasing the power leads to a suppression of the Coulomb blockade and an
appearance of both positive and negative steps symmetrically until at
$\alpha_{\mathrm{ac}}=2.5$ the Coulomb blockade completely vanishes. Further
increasing the power leads to a reappearing of the Coulomb blockade and a
pronounced second dual Shapiro step. This pattern is matching the measurement
shown in Fig. 3(c) in the main text.

Figure S4: Simulated $IV$-curves for different drive powers obtained by
solving Eq. (S2) using the fourth-order Runge-Kutta method. (a) Pulsed drive
with pulse duration $t_{p}=0.05\cdot(1/f_{\mathrm{drive}})$. The region 0
indicates the Coulomb blockade, while the numbers and sign indicate the order
and sign of the dual Shapiro step. For a pulsed drive dual Shapiro steps only
occur for one current direction. (b) Sinusoidal drive leads to a symmetric
power dependence of the $IV$-curves and appearance of positive and negative
dual Shapiro steps as indicated in the plot.


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