††thanks: the authors contributed equally to this work.††thanks: the authors
contributed equally to this work.

# Evidence of dual Shapiro steps in a Josephson junctions array

Nicolò Crescini<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France  Samuel Cailleaux
<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS, Grenoble INP,
Institut Néel, 38000 Grenoble, France  Wiebke Guichard  Univ. Grenoble Alpes,
CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France  Cécile Naud  Univ.
Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Olivier Buisson  Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel,
38000 Grenoble, France  Kater Murch  Department of Physics, Washington
University, St. Louis, Missouri 63130, USA  Nicolas Roch
<EMAIL_ADDRESS>Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut
Néel, 38000 Grenoble, France

###### Abstract

The modern primary voltage standard is based on the AC Josephson effect and
the ensuing Shapiro steps, where a microwave tone applied to a Josephson
junction yields a constant voltage $hf/2e$ ($h$ is Planck’s constant and $e$
the electron charge) determined by only the microwave frequency $f$ and
fundamental constants 1, 2. Duality arguments for current and voltage 3, 4, 5,
6, 7, 8, 9 have long suggested the possibility of dual Shapiro steps—that a
Josephson junction device could produce current steps with heights determined
only on the applied frequency 10, 11, 12, 13. In this report, we embed an
ultrasmall Josephson junction in a high impedance array of larger junctions to
reveal dual Shapiro steps 8. For multiple frequencies, we detect that the AC
response of the circuit is synchronised with the microwave tone at frequency
$f$, and the corresponding emergence of flat steps in the DC response with
current $2ef$, equal to the tunnelling of a Cooper pair per tone period. This
work sheds new light on phase-charge duality, omnipresent in condensed matter
physics14, 15, and extends it to Josephson circuits. Looking forward, it opens
a broad range of possibilities for new experiments in the field of circuit
quantum electrodynamics 16, 17 and is an important step towards the long-
sought closure of the quantum metrology electrical triangle 18, 19, 20.

Figure 1: The Shapiro steps and their dual. Usual Shapiro steps are realised
by periodically tilting the washboard potential of a Josephson junction with
$E_{J}\gg E_{C}$. Since the phase is a well defined variable, the driving of
the potential yields steps of constant voltage $\Phi_{0}f$ ($\Phi_{0}=h/2e$ is
the flux quantum) that are precisely determined by the metered transport of
fluxons. In contrast, the dual phenomenon (right side) uses the lowest Bloch
band of a junction with $E_{J}\leq E_{C}$ (higher bands in the picture are
shown only for an illustrative purpose). With charge as the well defined
variable, current steps of value $2ef$ determined by the cyclic transport of
Cooper pairs appear: the dual Shapiro steps.

In quantum mechanics, uncertainty relations quantify the incompatibility of
quantum operators. For a Josephson junction these operators are the charge $Q$
and the Josephson phase $\phi$, which respect $[\phi,Q]=2ie$, where $e$ is the
electron charge 21. Because of this incompatibility, the suppression of the
variance of one of these operators causes the conjugate operator’s variance to
increase. The balance between phase and charge fluctuations in a junction is
determined by the ratio of two energy scales, the Josephson energy
$E_{J}=\hbar I_{c}/2e$ and the charging energy $E_{C}=e^{2}/2C$, where $I_{c}$
is the critical current of the junction, $C$ is its capacitance and $\hbar$ is
the reduced Plank constant21. In one limit, $E_{J}/E_{C}\gg 1$, the variance
of $\phi$ is small, allowing it to be treated as a classical variable, and the
dynamics of the system is described by a tilted washboard potential $U(\phi)$
shown in Fig. 1. By applying a microwave tone, the potential is periodically
tilted; one fluxon ($\Phi_{0}$) is transported across the junction at each
period resulting in discrete voltage steps—Shapiro steps—in the IV curve. Au
contraire, if $E_{J}/E_{C}\lesssim 1$, fluctuations in the phase are enhanced
allowing the charge variance to be reduced. In this limit, one expects a
microwave drive to produce _dual_ Shapiro steps; the cyclical tilting of a
potential periodic in charge can meter the transport of Cooper pairs across
the junction, resulting in current steps proportional to the applied
frequency.

Dual transformations
$E_{J}/E_{C}\gg 1$ $E_{J}/E_{C}\leq 1$ Classical (localised) $\phi$ Classical
(localised) $q$ Tilted washboard $U(\phi)$ First Bloch band $U(Q)$ Nonlinear
inductance Nonlinear capacitance Critical current $I_{c}$ Critical voltage
$V_{c}$ AC Josephson effect Bloch oscillations Fluxons transport Cooper pairs
transport Shapiro steps Dual Shapiro steps

Table 1: Summary of dual transformations in a Josephson circuit.

An ultrasmall Josephson junctions (UJJ) has an $E_{J}/E_{C}$ ratio smaller
than one. The junction Hamiltonian is

$H=4E_{C}\left(Q/2e\right)^{2}-E_{J}\cos(\phi),$ (1)

which, using Bloch’s theorem22, can be recast to

$H=\sum_{s}U^{(s)}(q),$ (2)

where we have introduced the quasicharge $q$ and the Bloch bands
$U^{(s)}(q)$10, 10. When this junction is embedded in an environment of
impedance $R$, thermal and quantum fluctuations can further increase the
charge variance. Here, charge fluctuations are bounded by the energy-time
uncertainty relation $\Delta E\Delta\tau>\hbar$, where the charging time of
the junction capacitance is $\Delta\tau=RC$. Hence, large impedance is
necessary to reduce the charge fluctuations, in particular to have $\Delta
E<E_{C}$ requires, $R>h/4e^{2}\simeq 6.45\,\mathrm{k\Omega}$, the quantum of
resistance. Additionally, to suppress thermal fluctuations one has to cool the
environment to ultracryogenic temperatures $T<E_{C}/k_{\mathrm{B}}$. Under
these conditions the charge is localised and thus can be treated as a
classical variable. $Q$ is well approximated by $q$ as its dynamics is
restricted to the first Bloch band ($s=0$), thereby realising the periodic
potential $U(Q)\simeq U^{(0)}(q)=\cos(\pi Q/e)$—dual to $U(\phi)$—shown in
Fig. 1. The periodic potential corresponds to a nonlinear capacitance—dual to
the nonlinear inductance typically associated with a Josephson junction. Table
1 summarises the dual transformations in Josephson circuits. Along these
lines, the amplitude of the first Bloch band is related to the critical
voltage $V_{c}=\pi E_{Q}/e$, the minimum voltage required for a flow of DC
current across the junction. $V_{c}$ is dual to the critical current, the
current below which no voltage is generated6. Upon the application of a DC
current, the system exhibits Bloch oscillations10, 23, i.e. voltage
oscillations arising in the presence of a constant force, a phenomenon dual to
the AC Josephson effect. If these oscillations are synchronised with an
external tone, they yield dual Shapiro steps10, 8, 24.

Prior work has demonstrated some of the essential components for the
exploration of Josephson circuits’ duality. A charge periodic potential has
been obtained in UJJs23, 25, 26, 27 or nanowires 28, 29, 30, 31. Large
impedance environments have been realised by using resistors 23, suspended 32
or low dielectric constant 33 substrates, high kinetic inductance materials
34, 28, 31, 35, or metamaterials 36, 37, 16, 38, 17. Despite this progress a
clear demonstration of dual Shapiro steps is still missing. In this work, we
harness techniques from circuit QED39 to study the interplay of microwave
drive and DC bias of an UJJ embedded in a superconducting high-impedance
environment. Utilising low noise microwave spectroscopy, we observe the onset
of a microwave mode synchronised with an external tone. In turn, we observe
current plateaux proportional to the frequency of the microwaves. These two
experimental observations are consistent with synchronised Bloch oscillations
and dual Shapiro steps.

Figure 2: Overview of device and measurement setup, where the UJJ is shown in
orange and the JJ array in blue. (a) Circuit schematic of the Bloch array. (b)
SEM image of the UJJ and of the beginning of JJ arrays nominally identical to
the one under test. (c) Simplified schematic of the measurement setup
including the device, bias tees, damping resistors, DC and RF electronics, and
a vector-network analyser (VNA).

Fig. 2 shows the device used in this work, which we refer to as a Bloch array,
and consists in a superconducting quantum interference device (SQUID) formed
by two ultrasmall junctions surrounded by two linear JJs superinductances
which provide the high impedance environment. The array consists of Al/AlOx/Al
junctions in stripline geometry on a fused silica wafer, which reduces the
capacitance to ground 40. The SQUID is formed by junctions of area $0.2\times
0.2\,\mathrm{\mu m^{2}}$ ($E_{J}/E_{C}\simeq 0.4$) each, and the
superinductance comprises an array of $N_{\mathrm{a}}/2=1750$ junctions with
an area of $1\,\mathrm{\mu m^{2}}$ ($E_{J}/E_{C}\simeq 250$) 41. From these
parameters we estimate a first Bloch band width of $E_{Q}/h=2.5\,\mathrm{GHz}$
and a minimum gap of $7.9\,\mathrm{GHz}$ between the first and the second
Bloch band. Each superinductance’s characteristic impedance and inductance are
$Z=8.0\,\mathrm{k\Omega}$ and $L/2=3.3\,\mathrm{\mu H}$, respectively. A
scanning electron microscope (SEM) image is shown in Fig. 2b. Further details
are given in the Methods section. The Bloch array is studied with an
experimental setup allowing for simultaneous DC and RF measurements, as is
shown in Fig. 2c. Measurements are performed at the base temperature of a
dilution refrigerator with $T\simeq 23\,\mathrm{mK}$.

Figure 3: DC and RF characterisation of the Bloch array. Plot (a) is the low
voltage IV curve, which is detailed in the text. The inset schematically shows
the switching mechanism of the junctions in the array with a band model, where
the dashed line is a guide to the eye. The microwave transmission of the Bloch
array’s fourth mode doublet is displayed in (b), where the voltage profile of
odd and even modes is shown below.

We first examine the DC and RF response of the Bloch array separately. The IV
curve is obtained by biasing the device with the voltage $V_{0}$ and reading
out the corresponding current $I_{0}$. Here we consider $|V_{0}|\leq
10\,\mathrm{mV}$ while the whole IV curve is described in the Methods section.
As is shown in the plot, the current-voltage characteristics exhibits several
current peaks42, 43 evenly spaced by $2\Delta$, the superconducting gap of
aluminium. These peaks can be understood as follows. When a voltage is applied
to the Bloch array one of the junctions ($\mathrm{J_{1}}$) is biased to the
sub-gap state, behaving as a large series resistance, and the flow of current
is suppressed. Consequently, $V_{0}$ drops on $\mathrm{J_{1}}$ until it
reaches the gap. When $V_{0}=2\Delta$, $\mathrm{J_{1}}$ turns normal
conducting, and $I_{0}$ sharply increases until a second junction
$\mathrm{J_{2}}$ is biased into the sub-gap state, introducing again a large
series resistance and suppressing the current. To turn both the junctions
resistive the voltage needs to be $4\Delta$, and the process repeats. A
schematic of this mechanism is reported in the inset of Fig. 3a. This
successive switching resets the voltage drop across the junctions every
$2\Delta$. In this way the same physics, and in particular the low voltage
physics, repeats at voltages modulo $2\Delta$ as one more junction switches to
the normal state per step. RCSJ simulations of a voltage biased JJ array,
reported in the Methods, are in agreement with our observations.

Figure 4: Emergence of Bloch oscillations and dual Shapiro steps. In (a) the
frequency of the external tone $f$ is changed to observe the effect it has on
the transmission spectrum of the Bloch array. A similar measurement is
reported in (b) where we vary the power $p$ of the tone at fixed frequency
$f=4.04\,\mathrm{GHz}$. The dashed rectangle evidences the plot region
reported in (c), where the transition between tone interference and microwave
mode is clearly visible. Plot (d) shows the IV characteristics measured for
$f=4.04\,\mathrm{GHz}$ and $p=-5.0\,\mathrm{dBm}$. Between two current peaks
the plateaux current is $\pm 2ef$, as is evidenced by the grey dashed lines.

The microwave transmission of a Bloch array exhibits mode doublets, one with
odd and one with even symmetry 38, as is shown in Fig. 3b. The odd mode is
coupled to the UJJ at the centre of the array, while the even one is not, as
is comprehensively discussed by Leger et al.38. The frequency of the odd mode
is lower than the even mode, which is consistent with the presence of a
capacitive element between the left and right superinductances. This agrees
with duality considerations, according to which the UJJ should behave as a
nonlinear capacitance. Leveraging the DC and microwave capabilities of our
setup, we investigate the effect of an applied voltage on the modes of the
Bloch array. By measuring the $S_{21}$ as a function of $V_{0}$, we observe
that between two current peaks the doublet is visible yet starts to degrade
when current flows through the array. The mode doublet is replicated after
every $2\Delta$-switch, compatible with the mechanism depicted by the IV
curve. The full measurement is detailed in the Methods and in Fig 10.

Figure 5: Current plateaux at different microwave tone frequency and power.
(a) Average plateaux current at different microwave powers $p$ for three
separate voltages. (b) Lock-in measurement collected by modulating the
microwave power $p$ and detecting the corresponding current variation at
$V_{0}=2.4\,\mathrm{mV}$. (c) For three different values of $f$ we follow the
procedure described in the text to observe current plateaux whose amplitude
correspond to the one of the first dual Shapiro step, which is shown as a grey
dashed line. The inset shows the relation between the step current $I_{s}$ and
tone frequency, and the fitted slope is consisted with $2e$. The red point’s
IV curve was omitted for clarity (see the main text and the Methods for
further details).

We now study the behaviour of the Bloch array in the presence of an additional
applied microwave tone. In Fig. 4a we display the microwave transmission of
the Bloch array near the mode doublet at 4.1 GHz for different frequencies,
$f$, of the microwave tone with power $p=-5.0\,\mathrm{dBm}$. We observe that
the array odd mode is influenced by the microwave tone, and drifts away from
$f$, in a mode repulsion fashion. A mode synchronous with the pump appears
when the frequency exceeds $\simeq 4.02\,\mathrm{GHz}$, and fades around
$f\simeq 4.11\,\mathrm{GHz}$. This behaviour is further studied by varying $p$
at $f=4.04\,\mathrm{GHz}$, as reported in Fig. 4b. At low power the array is
essentially unaltered by the tone, until $p$ reaches -20 dBm and the spectrum
shows a profile corresponding to a Fano resonance, whose maximum-to-minimum
separation increases with $p$. The most noteworthy feature is at
$p\simeq-5\,\mathrm{dBm}$, which is presented in Fig. 4c. We clearly observe a
microwave mode with a Lorentzian shape and linewidth of about
$1\,\mathrm{MHz}$, centered at $f$, which disappears at higher $p$. We refer
to this feature as a Bloch mode, as it is consistent with Bloch oscillations
synchronised with the external tone44.

Focusing on this microwave synchronised Bloch mode ($f=4.04\,\mathrm{GHz}$,
$p=-5.0\,\mathrm{dBm}$), we now examine the IV characteristic of the Bloch
array under microwave irradiation as displayed in Fig. 4d. The IV presents a
wide central region with no current or supercurrent peaks. Outside this
central part, we observe that for both positive and negative voltage, current
plateaux form before the $2\Delta$ current peaks. The current corresponding to
their flat part is $2ef$ within a 0.1% accuracy. These observations are
supported by simulations (see Methods) showing that the steps should emerge as
a flat current region right before the upward slope of the $2\Delta$ peaks.

Figure 5 displays a more detailed investigation of these current steps. In
Fig. 5a we display the heights of three different current plateaux versus $p$,
which have a saturation point at $p=-5\,\mathrm{dBm}$, and $I_{0}=2ef$. The
tone power $p$ influences the height of the step and quantized flat steps are
obtained only for a given power range, in a fashion very similar to what is
observed for standard Shapiro steps 45. To further assess the stability of the
step, we modulate the microwave power $p$ and measure the corresponding
$I_{0}$ variation with a lock-in amplifier. The result is displayed in Fig. 5b
and shows that $dI_{0}/dp$ falls to the noise floor at
$p\simeq-5\,\mathrm{dBm}$, suggesting a current that is independent of the
microwave power.

Microwave power is delivered to the UJJ when the tone is on resonance with the
odd microwave mode of the array. Utilising microwave spectroscopy, we identify
other suitable working points (see Methods), corresponding to different mode
doublets of the Bloch array. In Fig. 5c, we display the lowest bias voltage
current step common to three additional frequencies. The parameters and
additional measurements used to obtain these curves are detailed in the
Methods section. For a total of four mode frequencies, we observe that $I_{0}$
flattens at the corresponding $2ef$ for
$f=3.20\,\mathrm{GHz},\,4.04\,\mathrm{GHz},\,5.45\,\mathrm{GHz}$ and
$6.46\,\mathrm{GHz}$. Finally, the inset of Fig. 5c shows the frequency
dependence of $I_{s}$, the average current on the flat part of each step.
Through the slope of the fitting line we can estimate a Cooper pair’s charge
of $(3.23\pm 0.07)\times 10^{-19}\,\mathrm{C}$. Even though the measurement
was not optimised, the average deviation from the expected current $\delta
I=I_{0}/2ef-1$ on the 4 GHz step is $\langle\delta I^{2}\rangle^{1/2}\leq
0.001$, limited by the setup noise in an averaging time of one minute (see
inset of Fig. 5c).

The environment of the UJJ, formed by the Josephson junction arrays, does more
than provide the required high impedance for charge localisation. As we have
discussed, the switching of the junctions in the array absorbs the majority of
the applied voltage, allowing the UJJ to remain in the superconducting state
even at high $V_{0}$. Furthermore, the microwave response of the device allows
us to selectively apply a microwave drive to the UJJ, causing the formation of
the Bloch mode and the corresponding onset of current steps. In contrast, no
effect is observed when driving the even mode which does not couple to the
UJJ.

Our work highlights how the Bloch array can be harnessed to engineer
dissipation, thereby resolving charge at the expense of phase fluctuations, a
regime dual of what is commonly achieved in Josephson circuits (localised
phase and large charge fluctuations). We take advantage of this new
configuration and observe quantised current steps: dual Shapiro steps. The
quantum metrology electrical triangle links current, voltage, and frequency
via three quantum effects. Voltage and frequency can be related via the AC
Josephson effect, while current and voltage via the quantum Hall effect. Here
we study the third edge of the triangle by relating current and frequency via
the charge $2e$ of a Cooper pair. This first step in this direction indicates
the rich physics of the Bloch array will have to be fully mastered in the
process of developing the dual Josephson effect into a metrological definition
of the Ampère46, 19. This also opens exciting opportunities from the
perspective of fundamental physics via tests of quantum electrodynamics
through the closure of the triangle47 or further exploration of phase charge
duality in Josephson devices.

Acknowledgements – The help of Denis Basko is deeply acknowledged and
appreciated. The support of the superconducting circuits team of Institut Néel
is warmly acknowledged. The authors are also grateful to Joe Aumentado, Michel
Devoret, Tim Duty, Serge Florens, David Haviland, Julien Renard, Benjamin
Sacepe and Izak Snyman for the fruitful discussion and comments on our work.
Furthermore, our gratitude goes to the members of the Triangle consortium,
namely Philippe Joyez, Çağlar Girit, Cristiano Ciuti, Helene Le Sueur and
Alexander Wagner, for the valuable discussion and insights. The samples were
fabricated in the clean room facility of Institute Néel, Grenoble; we would
like to thank all the staff for help with the devices fabrication. We would
like to acknowledge Eric Eyraud for his help in the installation of the
experimental setup. This work was supported by the French National Research
Agency (ANR) in the framework of the TRIANGLE project (ANR-20-CE47-0011). N.C.
is supported by the European Union’s Horizon 2020 research and innovation
program under the Marie Skłodowska-Curie grant agreement QMET No. 101029189.
K.W.M acknowledges support from NSF Grant No. PHY-1752844 (CAREER), AFOSR MURI
Grant No. FA9550-21-1-0202, and ONR Grant No. N00014- 21-1-2630.

During the writing of this work, we became aware that the group of Evgeni
Il’ichev and Oleg Astafiev observed quantised current steps in a
superconducting nanowires system. We would like to thank them for the
discussion and their comments.

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##  I Methods

The Methods sections are organised as follows. In the first two sections we
describe the phase charge duality and a semiclassial model that can give
intuition into the observations, and simulation results from a minimal a RCSJ
model. In the third section we detail the device fabrication, parameters and
experimental measurement setup. The fourth section presents the full IV curve
of the device as well as its flux dependence, while the fifth one describes
the microwave properties of the device. The last two sections display the
characterisation of the microwave modes used for the observation of the
current steps and their power and flux dependence.

###  I.1 Semiclassical model and exact duality

Figure 6: Simulation of the CJLR circuit under microwave irradiation with
experimental noise. This IV curves are used to determine how clear are the
steps for different parameters combinations and do not resemble the actual
steps observed in our device. The inset is the simulated circuit, where the
diamond is the non-linear capacitor ($\cos(\frac{\pi}{e}Q)$ element).

The two energy scales in a Josephson junction are the Josephson energy $E_{J}$
and the charging energy $E_{C}$. A large $E_{J}/E_{C}$ ratio is used, for
example for the Josephson voltage standard 48 or in transmon qubits 49, 50,
where an almost classical behaviour of the superconducting phase is
favourable. When $E_{J}\lesssim E_{C}$, the charge of a single electron is
relevant, and quantum phase fluctuations are not negligible. In this regime,
charge dynamics can be dual to the Josephson phase dynamics, provided that the
environmental impedance and inductance $L$ are large8, 24, 40. Eq. (1)
describes a particle in a one-dimensional lattice, whose spectrum consists of
Bloch bands22, as shown in Fig. 1. If the dynamics is restricted to the first
Bloch band, and the bath response is ohmic with resistance $R$8, the equation
of motion for the charge is given by

$L\ddot{Q}+R\dot{Q}+V_{c}\sin\left(\frac{\pi}{e}Q\right)=V_{0}+V_{1},$ (3)

where $V_{c}$ is the critical voltage of the junction, $V_{0}+V_{1}$ are the
DC and AC applied voltages, respectively, and the shape of the first Bloch
band is approximated as $\cos(\pi Q/e)$. Eq. 3 is the exact dual to the
equation of motion of a large $E_{J}/E_{C}$ ratio JJ. While Eq. 3 does not
accurately model the complex physics of the Bloch array, it does provide
useful intuition and serves as a guide for choosing the system parameters.

Following the work of Guichard and Hekking 8, we calculate the time evolution
of the system for a fixed voltage, allowing us to produce the IV
characteristic of the circuit as reported in Fig. 6. The CJRL system under
consideration is reported in the lumped element schematic in the inset of Fig.
6. The usual phase dynamics associated with the Josephson tunneling is
replaced by its dual counterpart: the quasicharge dynamics. In this way,
instead of the voltage Shapiro steps, one can expect quantised current steps.
As with prior work8, here the signal is perturbed with the voltage noise of
the resistor and of the readout amplifier. The circuit parameters are the same
as the ones in the main text.

###  I.2 RCSJ toy model

Figure 7: Simulated IV curve of a chain of four junctions in series with an
inductance, a resistor, and a non-linear capacitor with and without an AC
drive of frequency $f$. Current plateaux of height $2ef$ are observed before
each $2\Delta$ current peak in qualitative agreement with the experimental
observations.

The IV curve of the array is simulated by numerically solving the equations of
motion of a chain of $N$ junctions in series with a non-linear capacitor, an
inductance and a damping resistor under a voltage drive. Each junction is
modeled using the resistively and capacitively shunted junction model (RCSJ).
The total current $I_{\mathrm{tot}}$ through a junction is thus the sum of a
capacitive current $I_{C}$, a resistive current $I_{R}$ and a Josephson
current $I_{J}$:

$\displaystyle\begin{split}&I_{\mathrm{tot}}=I_{C}+I_{R}+I_{J},\\\
&I_{C}=C\ddot{\phi}_{i},\\\
&I_{R}=s(\dot{\phi}_{i})\frac{1}{R_{N}}\dot{\phi}_{i},\\\
&I_{J}=\left(1-s(\dot{\phi}_{i})\right)I_{c}\sin\left(\frac{\phi_{i}}{\phi_{0}}\right).\end{split}$
(4)

In Eqs. (4), $\phi_{i}$ is the superconducting flux drop across the
$i^{\mathrm{th}}$ junction, $R_{N}$ is its normal state resistance, $C$ is its
capacitance and $I_{c}$ its critical current. We model the gap transition of
the junction through the factor $s(\dot{\phi}_{i})$ which goes smoothly from 0
when $|\dot{\phi}_{i}|<2\Delta$ to 1 when $|\dot{\phi}_{i}|>2\Delta$. We found
that the exact shape of the function $s$ does not impact the resulting IV
characteristics as long as the transition is sharp. All the results presented
used a scaled sigmoid for the transition function. The charge and its
associated current $I_{\mathrm{tot}}=\dot{Q}$ are related to the bias voltage
$V$ by an equation similar to (3)

$V=L\ddot{Q}+R\dot{Q}+V_{C}\sin\left(\frac{\pi}{e}Q\right)+\sum_{i}\dot{\phi}_{i}.$
(5)

We then solve Eq. (4) and (5) using a Runge-Kutta method of order four with a
time-step $dt=0.02\,RC$. We then average the current $\langle
I_{\mathrm{tot}}\rangle$ to obtain its DC component and plot the IV
characteristic.

Due to the complexity of simulating the response of $N_{\mathrm{a}}=3500$ JJs,
we focus on a system with $N=4$ junctions embedded in a circuit with
inductance $L=5\,\mathrm{\mu H}$ which captures the inductance of the
remaining JJs that form the superinductance.

Fig. 7 displays the IV characteristic for the chosen parameters, with and
without an applied microwave tone. Many features that we observe in the
experimental IV curve are qualitatively reproduced by the simulation. First,
we observe current peaks, spaced by $2\Delta$, corresponding to the successive
switching of junctions in the array. Second, when the microwave tone is
applied we observe the formation of current plateaux just before the voltages
at which the current peaks occur. These current plateaux are repeated for each
current peak.

Other features of this simple simulation do not qualitatively match the
experimental IV curves. Whereas the height of the simulated current peaks is
of order $I_{c}$, and monotonically decreasing with voltage, the experimental
current peaks are all well below the estimated $I_{c}$, with heights
suppressed for small $V_{0}$. This qualitative difference points to aspects of
the UJJ, which is simplified in the model. The width of the simulated plateaux
appear to be of order $V_{c}$, while experimentally we measure different
widths, as is shown in Fig. 5.

###  I.3 Device and experimental setup

The device consists of Al/AlOx/Al junctions in a microstripline geometry,
fabricated with the shadow evaporation technique and electron beam lithography
on a fused silica wafer. The substrate choice is related to the minimisation
of ground capacitances which, being proportional to the dielectric constant,
are reduced of about four-folds with respect to silicon. Reduced ground
capacitance and large inductance produce a high impedance and prevent Landau-
Zener transitions between Bloch bands which might suppress the amplitude of
Bloch oscillations 40. Details on the fabrication procedure can be found in
the data repository of this work51.

The superinductance is formed from JJs with capacitance $C_{\mathrm{a}}\simeq
45\,\mathrm{fF}$ and critical current $I_{c}\simeq 150\,\mathrm{nA}$,
estimated from their area by using the expected capacitance per unit of area
$\sigma_{C}=45\,\mathrm{fF/\mu m^{2}}$ and critical current density of
$\sigma_{I}=15\,\mathrm{A/cm^{2}}$, respectively. Each UJJ has an area of
$0.04\,\mathrm{\mu m^{2}}$, from which we can similarly estimate a capacitance
of $C=1.8\,\mathrm{fF}$ and a critical current of $6\,\mathrm{nA}$.

Figure 8: Experimental setup used in this work. The yellow boxes show the
different stages of the dilution refrigerator with their temperatures on the
right side, while the part of the apparatus at room temperature is in green.
The device is in gray at the bottom of the figure, the reader is referred to
the main text for its description.

A detailed schematic of the setup used in this work is reported in Fig. 8. The
experimental apparatus can simultaneously perform DC and RF measurements, as
is shown in Fig. 2c. This is achieved using two off-chip bias-tees that
decouple kHz frequency measurements from the ones in the GHz range. The two
ports of the Bloch array are connected to the bias-tees via
$50\,\mathrm{\Omega}$-matched lines. At the lowest temperature stage, the DC
lines feature $45\,\mathrm{MHz}$ $\pi$-filters, $50\,\Omega$ damping resistors
in series and Eccosorb filters. Further $RC$ and $\pi$-filters are applied at
room temperature, which are the main source of damping in the circuit. The
current $I_{0}$ is amplified with a low noise transimpedance amplifier, while
the readout and voltage bias are performed with a real-time analog-to-digital
converter (Adwin-Gold II).

Figure 9: Full IV curve of the Bloch array. Plots (a) to (d) show the IV curve
at different scales. The black (grey) line indicates increasing (decreasing)
$|V_{0}|$, as shown by the arrows in (a), while in the same plot the blue
dashed line is a guide for the eye. The voltage scale is gradually reduced
from (a) to (d); discrepancies between the current amplitude in the plots are
related to measurement conditions. Finally, (e) shows the flux dependence of
the Bloch array’s IV characteristic at low voltages.

The IV curves are collected by biasing the device with triangular voltage
ramps with period ranging from 10 ms to 10 s, depending on the measurement
conditions and voltage range. To apply a magnetic field on the sample we
employ a supeconducting coil anchored to the sample holder. The measured
periodicity in the IV response agrees with what is estimated considering the
SQUID area and the magnetic field produced by the coil. Microwaves are applied
to the sample via a a high frequency line with an attenuation of about 60 dB,
which is connected to one of the bias-tees, while spectroscopic measurements
are performed with extra 30 dB attenuation. The second bias-tee’s high
frequency port is connected to an isolator and then to a high electron
mobility transistor (HEMT) amplifier working in the 4-8 GHz frequency range
and with a noise temperature of about $5\,\mathrm{K}$. Further amplification
is performed at room temperature, before reading out the signal with a vector
network analyser. The setup is housed in an inverted dilution refrigerator
with a base temperature of $23\,\mathrm{mK}$.

###  I.4 Full IV curve and flux dependence

The current-voltage characteristics of the Bloch array are reported in Fig. 9.
Figure 9a shows the IV curves over the largest voltage span, with switching to
the resistive branch of the whole array at a voltage of about
$2N_{\mathrm{a}}\Delta\simeq\pm 1.6\,\mathrm{V}$, marked by a sharp peak in
the device conductance. From this, we estimate $\Delta=0.23\,\mathrm{mV}$,
compatible with the superconducting gap of aluminium thin films 52. For
applied voltages $V_{0}\sim 4$ V, we estimate the normal state resistance of
each JJ in the array
$R_{\mathrm{a}}=7.75\,\mathrm{M\Omega}/N_{\mathrm{a}}=2.21\,\mathrm{k\Omega}$.
Since, the resistance of the UJJ is expected to be $\sim
20\,\mathrm{k}\Omega$, its contribution to the total array resistance is
negligible. Using Ambegaokar–Baratoff formula,
$I_{c}=\pi\Delta/2R_{\mathrm{a}}$, we estimate the critical current of the
large junctions $I_{c}\simeq 164\,\mathrm{nA}$, compatible with the estimation
obtained with SEM imaging.

At applied voltages $V_{0}<\pm 1.6$ V the IV has a more resistive branch and a
central plateau of about $\Delta_{\mathrm{s}}\simeq 1\,\mathrm{V}$, where its
resistance is larger than $100\,\mathrm{M\Omega}$, as shown in Fig. 9b. This
is in agreement with prior measurements and theory 53. At a lower voltage
scale, see Figs. 3a and 9c, more features appear 42, 43, whose explanation is
reported in the main text. Current peaks are present when the voltage is an
integer multiple of $2\Delta$, and their spacing is understood in terms of a
sub-gap to resistive switching mechanism as explained in the main text.

Figure 9d displays the IV curve for the finest voltage scale. For
$V_{0}<\Delta$ we detect a low-voltage-current with non-trivial shape which we
associate with a combination of supercurrent and inelastic Cooper pair
tunnelling 5, 54, 55. Small applied magnetic fields are expected to tune the
$E_{J}/E_{C}$ ratio of the UJJ, while the superinductances, being formed of
single junctions are to first order not affected. Figure 9e shows the flux
dependence of the low voltage IV curve of the Bloch array, where at $\pm\pi$
flux we observe suppression of the low-voltage-current54, 55.

###  I.5 Microwave measurements of the Bloch array

Microwave spectroscopy is used to measure the resonances of the Bloch array
16. In particular, we identify several peak doublets, which can be further
studied with two-tone measurements 56. We probe the transmission of the array
near one of the observed resonances at 4.13 GHz and sweep the power and
frequency of an additional applied tone. When the tone is resonant with an
additional mode of the array the transmission at the probe mode is affected.
From these data we observe a free spectral range of about $1\,\mathrm{GHz}$
between two peaks doublets, and a JJ plasma frequency of 16 GHz, beyond which
no modes are observed. From these measurements, we obtain the JJ inductance
$L_{\mathrm{a}}\simeq 1.9\,\mathrm{nH}$, capacitance $C_{\mathrm{a}}\simeq
45\,\mathrm{fF}$, and ground capacitance $C_{\mathrm{g}}\simeq 7\times
10^{-4}C_{\mathrm{a}}=0.3\,\mathrm{aF}$. These results are in agreement with
the ones obtained by measuring the dispersion relation of a JJ array resonator
fabricated in the same batch.

We then obtain the two superinductance’s impedance and inductance
$\sqrt{L_{\mathrm{a}}/C_{\mathrm{g}}}=\mathrm{8.0\,k\Omega}$ and
$N_{\mathrm{a}}L_{\mathrm{a}}/2=3.3\,\mathrm{\mu H}$. The critical current of
the large JJs, $I_{c}=\hbar/2eL_{\mathrm{a}}\simeq 173\,\mathrm{nA}$ is in
good agreement with the values estimated in the previous section. The
spectroscopic data also give information on the UJJs. Referring to Fig. 3c,
the observed modes splitting yields a capacitance of about $1\mathrm{fF}\simeq
C/2$, in agreement with the value estimated through the ultrasmall junction’s
area.

Figure 10: Plot (a) is the variation of the transmission with respect to
$V_{0}$, where the IV was superimposed to the plot as a dashed line for
illustrative purposes. In (b) we show slices of (a) at different voltages,
where a 1 dB offset for each curve is added for clarity.

The effect of $V_{0}$ on the $S_{21}$ of this mode doublet is displayed in
Fig. 10. Here, we see how a current flowing in the Bloch array degrades the
microwave response of both modes, reducing the transmission. The microwave
mode doublet is recovered when $I_{0}$ is close to zero, further confirming
the interpretation of the IV curve reported in the main text.

Figure 11: Power sweep on the peak doublet for some of the modes identified in
the Bloch array, ranging from (a) at 3.2 GHz to (f) at 7.5 GHz. These
measurements are used to identify the power to use to obtain the steps.

###  I.6 Microwave modes used for observing current steps

To observe the steps at different frequencies of the microwave tone, we
identified the suitable peak doublets and their response to the microwave tone
using the method displayed in Fig. 4b. The corresponding power dependence for
the other observed steps are displayed in Figure 11. In each case, the
characteristic shape of the power response guides the choice of $p$ to observe
the steps. A summary of the identified frequencies and powers suitable for the
emergence of the steps is reported in Table 2.

$f$ | $p$
---|---
3.20 GHz | -11.1 dBm
4.04 GHz | -5.0 dBm
5.45 GHz | 2.8 dBm
6.11 GHz | Not identified
6.46 GHz | 3.8 dBm
7.48 GHz | Not identified
Table 2: Microwave tone parameters used to observe the steps in Fig. 5,
obtained through the measurements in Fig. 11.

Referring to Fig. 5c, we note that the 5 GHz step is not displayed since the
corresponding IV curve suffers from a much higher noise with respect to the
other three. Repeated measurements show that with this tone frequency, the
Bloch array experiences a significant $1/f$ noise which, shifting the position
of the peaks during the measurement, compromises the averaging reliability.
For this reason the averaging performed on this curve is extremely limited,
leading to a higher current noise, as can be seen from the errorbar in the
inset of Fig. 5c.


2022

[1]Matthias Wellershoff

[1]Department of Mathematics, University of Maryland, 4176 Campus Drive,
College Park, MD, 20742, USA

# Sampling at twice the Nyquist rate in two frequency bins guarantees
uniqueness in Gabor phase retrieval

<EMAIL_ADDRESS>*

###### Abstract

We show that bandlimited signals can be uniquely recovered (up to a constant
global phase factor) from Gabor transform magnitudes sampled at twice the
Nyquist rate in two frequency bins.

###### keywords:

Phase retrieval, Gabor transform, Nyquist–Shannon sampling, Hadamard
factorisation theorem, Müntz–Szász type result

###### pacs:

[

MSC Classification]94A12, 42B10

Acknowledgments

The author would like to thank A. Bandeira for posing the question which is
being answered in the present letter. In addition, the author would like to
thank the reviewers for their comments which allowed for a significant
improvement in the presentation of the paper. Finally, the author acknowledges
funding through the Swiss National Science Foundation grant 200021_184698.

##  1 Introduction

We consider the recovery of square-integrable signals $f\in L^{2}(\mathbb{R})$
from the magnitude of their _Gabor transforms_

$\mathcal{G}f(x,\omega):=\sqrt[4]{2}\int_{\mathbb{R}}f(t)\mathrm{e}^{-\pi(t-x)^{2}}\mathrm{e}^{-2\pi\mathrm{i}t\omega},\qquad(x,\omega)\in\mathbb{R}^{2},$
(1)

also called the _Gabor phase retrieval problem_. Specifically, we are
interested in questions related to the reconstruction of $f$ from
$\left\lvert\mathcal{G}f\right\rvert$ measured on lattices111A _lattice_
$\Lambda\subset\mathbb{R}^{2}$ is a discrete subset of the time-frequency
plane that can be written as $\Lambda=L\mathbb{R}^{k}$, where
$L\in\mathbb{R}^{2\times k}$ is a matrix with linearly independent columns and
$k\in\\{1,2\\}$. $\Lambda\subset\mathbb{R}^{2}$, which we refer to as _sampled
Gabor phase retrieval_.

The first uniqueness result for sampled Gabor phase retrieval was presented in
alaifari2021uniqueness  where it is shown that real-valued, bandlimited,
square-integrable signals $f$ with bandwidth $B>0$ can be recovered (up to a
constant global sign factor) from $\left\lvert\mathcal{G}f\right\rvert$
sampled on $\tfrac{1}{4B}\mathbb{Z}\times\\{0\\}$. Just two months later, the
paper grohs2023injectivity  appeared in which it is revealed that the real-
valuedness assumption from alaifari2021uniqueness  can be dropped:
specifically, the bandlimited square-integrable signals $f$ whose Fourier
transform is in $L^{4}$ can be recovered (up to a constant global phase
factor) from $\left\lvert\mathcal{G}f\right\rvert$ sampled on
$\tfrac{1}{4B}\mathbb{Z}\times\mathbb{Z}$. Another two months later, the paper
alaifari2021phase  appeared, showing that the bandlimitedness assumption
cannot be dropped. This is achieved by constructing two square-integrable
functions that do not agree (up to a constant global phase factor) but whose
Gabor transform magnitudes agree on $\Lambda$, where
$\Lambda\subset\mathbb{R}^{2}$ is a general lattice in the time-frequency
plane.

Finally, we mention the recent preprint wellershoff2021injectivity  in which
results from grohs2023injectivity  are extended to show that all bandlimited
square-integrable signals $f$ can be recovered (up to a constant global phase
factor) from $\left\lvert\mathcal{G}f\right\rvert$ sampled on
$\tfrac{1}{4B}\mathbb{Z}\times\mathbb{N}$. In this paper, we further
strengthen this result and prove that in fact bandlimited square-integrable
signals $f$, with bandwidth $B>0$, can be recovered (up to a constant global
phase factor) from $\left\lvert\mathcal{G}f\right\rvert$ sampled on
$\tfrac{1}{4B}\mathbb{Z}\times\\{\omega_{0},\omega_{1}\\}$, where
$\omega_{0}\neq\omega_{1}$. As $\tfrac{1}{4B}$ is exactly twice the Nyquist
rate, we therefore show that sampling at twice the Nyquist rate in two
frequency bins guarantees uniqueness in Gabor phase retrieval as advertised in
the title. We point out that a similar result was already known for the Cauchy
wavelet transform alaifari2020phase : more precisely, sampling at twice the
Nyquist rate at two scales guarantees uniqueness in Cauchy wavelet phase
retrieval.

######  Remark 1.

The original motivation for this paper stems from a resemblance of the result
in alaifari2021uniqueness  on Gabor sign retrieval with the work in
balan2006signal  on finite-dimensional sign retrieval: in the prior, it is
shown that sampling at twice the Nyquist rate in a single frequency bin
guarantees uniqueness in Gabor sign retrieval while, in the latter, it is
shown that $2n-1$ generic measurement vectors are necessary and sufficient for
uniqueness in finite-dimensional sign retrieval. As it is also known that on
the order of $4n$ generic measurement vectors are necessary and sufficient for
uniqueness in finite-dimensional phase retrieval balan2006signal ;
heinosaari2013quantum , it seems natural to ask whether sampling at four times
the Nyquist rate in one frequency bin or at twice the Nyquist rate in two
frequency bins would guarantee uniqueness in Gabor phase retrieval. The former
is clearly untrue as can be seen from considering $f,g\in\mathrm{PW}_{B}^{2}$
real-valued and

$\left\lvert\mathcal{G}(f+\mathrm{i}g)\right\rvert=\left\lvert\mathcal{G}(f-\mathrm{i}g)\right\rvert\mbox{
on }\mathbb{R}\times\\{0\\}.$

The latter is shown to be true in this paper.

### Notation

We denote the normalised Gaussian by $\phi(t)=\sqrt[4]{2}\exp(-\pi t^{2})$,
where $t\in\mathbb{R}$. For $-\infty<a<b<\infty$ and $f\in L^{p}(\mathbb{R})$,
with $p\in[1,\infty)$, we write $\operatorname{supp}f\subset[a,b]$ if
$f(t)=0$, for a.e. $t\not\in[a,b]$. Moreover, we define the families of
_translation_ and _modulation_ operators
$(\operatorname{T}_{x})_{x\in\mathbb{R}}:L^{p}(\mathbb{R})\to
L^{p}(\mathbb{R})$ and
$(\operatorname{M}_{\omega})_{\omega\in\mathbb{R}}:L^{p}(\mathbb{R})\to
L^{p}(\mathbb{R})$ by

$\operatorname{T}_{x}f(t):=f(t-x),\qquad\operatorname{M}_{\omega}f(t):=f(t)\mathrm{e}^{2\pi\mathrm{i}t\omega}.$

We furthermore use the convention

$\mathcal{F}f(\xi)=\int_{\mathbb{R}}f(t)\mathrm{e}^{-2\pi\mathrm{i}t\xi}\,\mathrm{d}t,\qquad\xi\in\mathbb{R},$

for the Fourier transform of $f\in L^{1}(\mathbb{R})\cup L^{2}(\mathbb{R})$
and note that the Fourier transform of the normalised Gaussian is the
normalised Gaussian itself, i.e. $\mathcal{F}\phi=\phi$. Additionally, as is
usual in the phase retrieval literature, we will introduce an equivalence
relation on the set of complex-valued functions via

$f\sim g:\iff\exists\alpha\in\mathbb{R}:f=\mathrm{e}^{\mathrm{i}\alpha}g$

and say that $f$ and $g$ _agree up to global phase_ if $f\sim g$. The space of
polynomials with complex argument and complex coefficients is denoted by
$\mathbb{C}[z]$. Similarly, the subspace of degree $n\in\mathbb{N}$
polynomials in $\mathbb{C}[z]$ is denoted by $\mathbb{C}_{n}[z]$. Finally, for
an entire function $F:\mathbb{C}\to\mathbb{C}$, we denote its zero set by
$Z(F)\subset\mathbb{C}$ and define a function
$m_{F}:\mathbb{C}\to\mathbb{N}_{0}$ which assigns the multiplicity of $z$ as a
zero of $F$ to every $z\in\mathbb{C}$. Note that we use the convention
$m_{F}(z)=0$, for $z\not\in Z(F)$.

### Definitions and basic notions

We will work with the _Paley–Wiener spaces_ of bandlimited functions

$\mathrm{PW}_{B}^{p}:=\\{f\in L^{p}(\mathbb{R})\,\mathchar
24635\relax\;\,\operatorname{supp}(\mathcal{F}f)\subset[-B,B]\\},\qquad
p\in[1,\infty],$

where $B>0$. With this definition, it is well known that
$\mathrm{PW}_{B}^{1}\subset\mathrm{PW}_{B}^{2}$. It turns out to be useful to
consider the _Bargmann transform_ of square-integrable signals $f\in
L^{2}(\mathbb{R})$ given by

$\mathcal{B}f(z):=\sqrt[4]{2}\int_{\mathbb{R}}f(t)\mathrm{e}^{2\pi tz-\pi
t^{2}-\tfrac{\pi}{2}z^{2}}\,\mathrm{d}t,\qquad z\in\mathbb{C}.$

One of the two main reasons for this is that the Bargmann transform and the
Gabor transform (as defined in equation (1)) are related via the formula
(grochenig2001foundations, , Proposition 3.4.1 on p. 54)

$\mathcal{G}f(x,-\omega)=\mathrm{e}^{\pi\mathrm{i}x\omega}\mathcal{B}f(x+\mathrm{i}\omega)\mathrm{e}^{-\tfrac{\pi}{2}\left(x^{2}+\omega^{2}\right)},\qquad(x,\omega)\in\mathbb{R}^{2}.$

The other main reason is that the Bargmann transform of a square-integrable
signal is an entire function of finite order. More precisely, we define the
_order_ of an entire function $F:\mathbb{C}\to\mathbb{C}$ to be

$\rho:=\limsup_{r\to\infty}\frac{\log(\log(\sup_{\left\lvert
z\right\rvert=r}\left\lvert F(z)\right\rvert))}{\log r}$

and say that $F$ is of _finite order_ if $\rho<\infty$. For $F=\mathcal{B}f$,
with $f\in L^{2}(\mathbb{R})$, one can use (grochenig2001foundations, ,
Theorem 3.4.2 on p. 54) to show that $\rho\leq 2$: we say that $\mathcal{B}f$
is of _second-order_.

##  2 Preliminaries

Our goal is to recover a bandlimited function $f$ with bandwidth $B$ from the
magnitudes of its Gabor transform $\left\lvert\mathcal{G}f\right\rvert$
sampled on the set $\tfrac{1}{4B}\mathbb{Z}\times\\{\omega_{0},\omega_{1}\\}$,
where $\omega_{0}\neq\omega_{1}$. In order to do this, we follow a three-step
procedure.

  1. 1.

We note that $\left\lvert\mathcal{G}f(\cdot,\omega)\right\rvert^{2}$ is
bandlimited, for all $\omega\in\mathbb{R}$, and use the Nyquist–Shannon
sampling theorem to recover $\left\lvert\mathcal{G}f\right\rvert^{2}$ on
$\mathbb{R}\times\\{\omega_{0},\omega_{1}\\}$.

  2. 2.

Relating this to the Bargmann transform of $f$ shows that it suffices to
analyse the recovery of a second-order entire function from magnitude
measurements on two parallel lines. In this direction, we show that a second-
order entire function is either uniquely determined (up to global phase) by
its magnitude on two parallel lines or it has infinitely many evenly spaced
zeroes.

  3. 3.

Finally, we make use of the bandlimitedness of $f$ again and show that the
Bargmann transform of $f$ can only have infinitely many evenly spaced zeroes
if $f=0$.

Let us start with the realisation of item 1. We note that the following lemma
already follows from the considerations in wellershoff2021injectivity . A
considerably simpler proof based on a different convention for the
Paley–Wiener spaces is given here.

######  Lemma 1.

Let $B>0$, $\omega\in\mathbb{R}$ and $f\in\mathrm{PW}_{B}^{2}$. Then,
$x\mapsto\left\lvert\mathcal{G}f(x,\omega)\right\rvert^{2}\in\mathrm{PW}_{2B}^{2}$.

Proof: We have
$x\mapsto\mathrm{e}^{2\pi\mathrm{i}x\omega}\mathcal{G}f(x,\omega)\in\mathrm{PW}_{B}^{2}$
since it is the (inverse) Fourier transform of
$\mathcal{F}f\cdot\operatorname{T}_{\omega}\phi\in L^{2}(\mathbb{R})$ which
satisfies
$\operatorname{supp}(\mathcal{F}f\cdot\operatorname{T}_{\omega}\phi)\subset[-B,B]$
(grochenig2001foundations, , Equation (3.5) on p. 39). Therefore,
$x\mapsto\left\lvert\mathcal{G}f(x,\omega)\right\rvert^{2}\in
L^{1}(\mathbb{R})$ and applying the Fourier convolution theorem to

$x\mapsto\left\lvert\mathcal{G}f(x,\omega)\right\rvert^{2}=\mathcal{G}f(x,\omega)\overline{\mathcal{G}f(x,\omega)}$

shows that
$x\mapsto\left\lvert\mathcal{G}f(x,\omega)\right\rvert^{2}\in\mathrm{PW}_{2B}^{1}\subset\mathrm{PW}_{2B}^{2}$.

Next, we move on to item 2 of our three-step procedure: the analysis of the
recovery of a second-order entire function from magnitude measurements on two
parallel lines. Interestingly, it can be shown that it is impossible to
reconstruct a finite order entire function (up to global phase) from magnitude
information on two parallel lines wellershoff2022phase . It is therefore also
impossible to recover a square-integrable signal from Gabor phase retrieval
measurements on two parallel lines; a fact which has been used to construct
the counterexamples to Gabor phase retrieval in alaifari2021phase . We are not
considering general square-integrable signals here however and the
bandlimitedness assumption turns out to be sufficient to exclude
counterexamples.

Let us start by realising that fixing the magnitude of an entire function on
two parallel lines enforces a periodicity in its zeroes. More precisely, we
show that, if two entire functions $F$ and $G$ have magnitudes that agree on
$\mathbb{R}\cup(\mathbb{R}+\mathrm{i}\tau)$, then $m_{F}-m_{G}$ is
$(2\mathrm{i}\tau)$-periodic.

######  Lemma 2.

Let $\tau>0$ and let $F,G\in\mathbb{C}\to\mathbb{C}$ be two entire functions
such that $\left\lvert F\right\rvert=\left\lvert G\right\rvert$ on
$\mathbb{R}\cup(\mathbb{R}+\mathrm{i}\tau)$. Then,

$m_{F}(z+2\mathrm{i}\tau)-m_{G}(z+2\mathrm{i}\tau)=m_{F}(z)-m_{G}(z),\qquad
z\in\mathbb{C}.$

Proof: Let $z\in\mathbb{C}$ denote an arbitrary complex number. According to
(mcdonald2004phase, , Proposition 1 on p. 261), $\left\lvert
F\right\rvert=\left\lvert G\right\rvert$ on $\mathbb{R}$ implies that

$F(z)\overline{F(\overline{z})}=G(z)\overline{G(\overline{z})}.$

Therefore, after looking at the zeroes of the above equation exclusively, we
have

$m_{F}(z)+m_{F}(\overline{z})=m_{G}(z)+m_{G}(\overline{z}).$

The same argument applied to $F_{\tau}(z):=F(z+\mathrm{i}\tau)$ and
$G_{\tau}(z):=G(z+\mathrm{i}\tau)$ yields

$m_{F}(z+\mathrm{i}\tau)+m_{F}(\overline{z}+\mathrm{i}\tau)=m_{G}(z+\mathrm{i}\tau)+m_{G}(\overline{z}+\mathrm{i}\tau)$

such that we can conclude that

$m_{F}(z+2\mathrm{i}\tau)-m_{G}(z+2\mathrm{i}\tau)=m_{G}(\overline{z})-m_{F}(\overline{z})=m_{F}(z)-m_{G}(z).$

The periodicity in $m_{F}-m_{G}$ directly implies that the zeroes (with
multiplicities) of $F$ and $G$ agree everywhere if they agree on the strip
$\mathbb{R}+\mathrm{i}(-\tau,\tau]$. Combining this insight with the Hadamard
factorisation theorem yields that a second-order entire function is either
uniquely determined (up to global phase) by its magnitude on the two parallel
lines $\mathbb{R}\cup(\mathbb{R}+\mathrm{i}\tau)$ or that it has at least one
zero in the strip $\mathbb{R}+\mathrm{i}(-\tau,\tau]$.

######  Corollary 3.

Let $\tau>0$ and let $F,G\in\mathbb{C}\to\mathbb{C}$ be two entire functions
_of second-order_ such that $\left\lvert F\right\rvert=\left\lvert
G\right\rvert$ on $\mathbb{R}\cup(\mathbb{R}+\mathrm{i}\tau)$. If
$m_{F}-m_{G}=0$ on $\mathbb{R}+\mathrm{i}(-\tau,\tau]$, then $F\sim G$.

Proof: If $m_{F}-m_{G}=0$ on $\mathbb{R}+\mathrm{i}(-\tau,\tau]$, then Lemma 2
implies $m_{F}-m_{G}=0$ such that the zeroes (with multiplicity) of $F$ and
$G$ agree. It therefore follows from Hadamard’s factorisation theorem (cf.
(titchmarsh1939theory, , Subsection 8.24 on p. 250)) that

$F(z)=\mathrm{e}^{Q(z)}G(z),\qquad z\in\mathbb{C},$

where $Q\in\mathbb{C}_{2}[z]$ is a quadratic polynomial. As $\left\lvert
F\right\rvert=\left\lvert G\right\rvert$ on $\mathbb{R}$, we furthermore have

$\left\lvert F(x)\right\rvert=\mathrm{e}^{\operatorname{Re}Q(x)}\left\lvert
G(x)\right\rvert=\mathrm{e}^{\operatorname{Re}Q(x)}\left\lvert
F(x)\right\rvert,\qquad x\in\mathbb{R},$

which implies that $\exp(\operatorname{Re}Q)=1$ holds on $\mathbb{R}\setminus
Z(F)$. As $F$ is entire, $Z(F)$ has no accumulation point in $\mathbb{R}$ and
is thus of measure zero. It follows that $\operatorname{Re}Q=0$ almost
everywhere (and thus everywhere) on $\mathbb{R}$. We can now write

$Q(z)=\mathrm{i}\left(\alpha+\lambda_{1}z+\lambda_{2}z^{2}\right),\qquad
z\in\mathbb{C},$

for some $\alpha,\lambda_{1},\lambda_{2}\in\mathbb{R}$. The same argument as
above shows that $\left\lvert F\right\rvert=\left\lvert G\right\rvert$ on
$\mathbb{R}+\mathrm{i}\tau$ implies $\operatorname{Re}Q=0$ on
$\mathbb{R}+\mathrm{i}\tau$. Therefore, we find

$\operatorname{Re}Q(x+\mathrm{i}\tau)=\lambda_{1}\tau+2\lambda_{2}\tau
x=0,\qquad x\in\mathbb{R},$

which proves that $\lambda_{1}=\lambda_{2}=0$ and thus
$F=\mathrm{e}^{\mathrm{i}\alpha}G$.

######  Remark 2.

Corollary 3 actually holds for general entire functions of finite order
$F,G\in\mathbb{C}\to\mathbb{C}$. The proof remains mostly the same: only the
polynomial $Q\in\mathbb{C}[z]$ is of arbitrary order instead of quadratic. We
therefore have

$Q(z)=\mathrm{i}\sum_{j=0}^{n}\lambda_{j}z^{j},\qquad z\in\mathbb{C},$

for some $n\in\mathbb{N}$ and $(\lambda_{j})_{j=0}^{n}\in\mathbb{R}$, and, as
$\operatorname{Re}Q=0$ on $\mathbb{R}+\mathrm{i}\tau$, we find

$\sum_{j=0}^{n}\lambda_{j}\operatorname{Im}\left((x+\mathrm{i}\tau)^{j}\right)=0,\qquad
x\in\mathbb{R}.$

The binomial theorem, then shows

$0=\sum_{j=0}^{n}\lambda_{j}\sum_{k=0}^{j}\binom{j}{k}x^{k}\tau^{j-k}\operatorname{Im}\left(\mathrm{i}^{j-k}\right)=\sum_{k=0}^{n}x^{k}\sum_{j=k}^{n}\binom{j}{k}\lambda_{j}\tau^{j-k}\operatorname{Im}\left(\mathrm{i}^{j-k}\right),$

for $x\in\mathbb{R}$. Comparing coefficients yields

$\displaystyle 0$
$\displaystyle=\sum_{j=k}^{n}\binom{j}{k}\lambda_{j}\tau^{j-k}\operatorname{Im}\left(\mathrm{i}^{j-k}\right)=\sum_{j=0}^{n-k}\binom{j+k}{k}\lambda_{j+k}\tau^{j}\operatorname{Im}\left(\mathrm{i}^{j}\right)$
$\displaystyle=\sum_{j=0}^{\lfloor(n-k-1)/2\rfloor}\binom{2j+k+1}{k}\lambda_{2j+k+1}\tau^{2j+1}(-1)^{j},$

for $k=0,\dots,n-1$, which can be used inductively to show that
$\lambda_{j}=0$, for $j=1,\dots,n$. Indeed, for $k=n-1$, we find

$0=\binom{n}{n-1}\lambda_{n}\tau\implies\lambda_{n}=0.$

If we assume that $\lambda_{j}=0$, for $j=k+2,\dots,n$, with
$k\in\\{0,\dots,n-2\\}$, then

$0=\sum_{j=0}^{\lfloor(n-k-1)/2\rfloor}\binom{2j+k+1}{k}\lambda_{2j+k+1}\tau^{2j+1}(-1)^{j}=\binom{k+1}{k}\lambda_{k+1}\tau\\\
\implies\lambda_{k+1}=0.$

We can finally turn to item 3 of our three step procedure: using
bandlimitedness to show that $f=0$ if $\mathcal{B}f$ has infinitely many
evenly spaced zeroes. The inspiration for this step actually comes from
grohs2023injectivity ; wellershoff2021injectivity , where a Müntz–Szász type
result from zalik1978approximation  is used in order to recover bandlimited
$f$ from $\left\lvert\mathcal{G}f\right\rvert$ on
$\mathbb{R}\times\mathbb{N}$. We rely on a slight generalisation of that same
Müntz–Szász type result for this paper.

######  Theorem 4 (Zalik’s theorem; cf. Theorem 4 in zalik1978approximation ).

Let $p\in[1,\infty)$, let $a,b\in\mathbb{R}$ be such that $a<b$, let $r>0$ and
let $(c_{n})_{n\in\mathbb{N}}\in\mathbb{C}$ be a sequence of distinct complex
numbers such that there exists a $\delta>0$ and an $N_{0}\in\mathbb{N}$ with

$\left\lvert\operatorname{Re}\left[c_{n}-\tfrac{1}{2}\right]\right\rvert\geq\delta\left\lvert
c_{n}-\tfrac{1}{2}\right\rvert,\qquad n\geq N_{0}.$

Then,

$\left\\{t\mapsto\mathrm{e}^{-r^{2}(t-c_{n})^{2}}\,\mathchar
24635\relax\;\,n\in\mathbb{N}\right\\}$

is complete in $L^{p}([a,b])$ if and only if

$\sum_{n\in\mathbb{N},~{}c_{n}\neq 0}\left\lvert c_{n}\right\rvert^{-1}$

diverges.

Proof: The theorem follows from the original proof in zalik1978approximation
with some small modifications.

######  Remark 3.

Zalik’s original result is stated for $p=2$ and _real numbers_
$(c_{n})_{n\in\mathbb{N}}$. We do not make use of the added generality in the
integrability parameter $p$ here but the proof of our main result does require
the sequence $(c_{n})_{n\in\mathbb{N}}$ to be complex-valued.

##  3 The main result

We are now in a position to state and prove our main result: general
bandlimited signals can be recovered from their Gabor transform magnitudes
sampled at twice the Nyquist rate in two frequency bins.

######  Theorem 5 (Main result).

Let $B>0$, let $\omega_{0},\omega_{1}\in\mathbb{R}$ be such that
$\omega_{0}<\omega_{1}$ and let $f,g\in\mathrm{PW}_{B}^{2}$. Then, $f\sim g$
if (and only if)
$\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$ on
$\tfrac{1}{4B}\mathbb{Z}\times\\{\omega_{0},\omega_{1}\\}$.

Proof: Let us suppose that
$\left\lvert\mathcal{G}f\right\rvert=\left\lvert\mathcal{G}g\right\rvert$ on
$\tfrac{1}{4B}\mathbb{Z}\times\\{\omega_{0},\omega_{1}\\}$ and fix
$j\in\\{0,1\\}$. According to Lemma 1, the functions

$x\mapsto\lvert\mathcal{G}f(x,\omega_{j})\rvert^{2},\qquad
x\mapsto\lvert\mathcal{G}g(x,\omega_{j})\rvert^{2}$

are in $\mathrm{PW}_{2B}^{2}$. By the Nyquist–Shannon sampling theorem, we
therefore find that

$\left\lvert\mathcal{G}f(x,\omega_{j})\right\rvert^{2}=\left\lvert\mathcal{G}g(x,\omega_{j})\right\rvert^{2},\qquad
x\in\mathbb{R}.$ (2)

Next, we define the second-order entire functions
$F(z):=\mathcal{B}f(z+\mathrm{i}\omega_{0})$ and
$G(z):=\mathcal{B}g(z+\mathrm{i}\omega_{0})$, for $z\in\mathbb{C}$, which
satisfy

$\left\lvert F(x)\right\rvert=\left\lvert G(x)\right\rvert,\qquad\left\lvert
F(x+\mathrm{i}\tau)\right\rvert=\left\lvert
G(x+\mathrm{i}\tau)\right\rvert,\qquad x\in\mathbb{R},$

for $\tau:=\omega_{1}-\omega_{0}>0$, according to equation (2). In the rest of
this proof, we will distinguish between two cases: $m_{F}-m_{G}=0$ on
$\mathbb{R}\times\mathrm{i}(-\tau,\tau]$ and $m_{F}-m_{G}\neq 0$ on
$\mathbb{R}\times\mathrm{i}(-\tau,\tau]$. In the first case, i.e. when
$m_{F}-m_{G}=0$ on $\mathbb{R}\times\mathrm{i}(-\tau,\tau]$, Corollary 3
implies that $F\sim G$. Since the Bargmann transform is injective222Actually,
the range of the Bargmann transform can be equipped with an inner product and
thereby turned into a Hilbert space which known as the Fock space
$\mathcal{F}^{2}(\mathbb{C})$. The Bargmann transform then turns out to be a
unitary operator mapping $L^{2}(\mathbb{R})$ onto
$\mathcal{F}^{2}(\mathbb{C})$ (grochenig2001foundations, , Theorem 3.4.3 on p.
56)., it follows that $f\sim g$.

In the second case, we may without loss of generality assume that there exists
a complex number $z_{0}\in\mathbb{R}\times(-\tau,\tau]$ such that
$m_{F}(z_{0})-m_{G}(z_{0})>0$: indeed, there exists a complex number at which
$m_{F}-m_{G}$ is non-zero and we can exchange $F$ and $G$ if we have
$m_{F}(z_{0})-m_{G}(z_{0})<0$. By Lemma 2, we therefore find that

$m_{F}(z_{0}+2\mathrm{i}k\tau)-m_{G}(z_{0}+2\mathrm{i}k\tau)=m_{F}(z_{0})-m_{G}(z_{0})>0,$

for $k\in\mathbb{Z}$. It follows that
$(z_{0}+2\mathrm{i}k\tau)_{k\in\mathbb{Z}}\in\mathbb{C}$ forms a sequence of
zeroes of $F$, i.e.

$\mathcal{B}f(z_{0}+\mathrm{i}(\omega_{0}+2k\tau))=0,\qquad k\in\mathbb{Z}.$

The Bargmann transform satisfies the nice symmetry
$\mathcal{B}f(-\mathrm{i}z)=\mathcal{B}\mathcal{F}f(z)$, for $z\in\mathbb{C}$
(c.f. (bargmann1961Hilbert, , Equation (3.10a) on p. 207)). Therefore,

$\displaystyle 0$
$\displaystyle=\mathcal{B}f(z_{0}+\mathrm{i}(\omega_{0}+2k\tau))=\mathcal{B}\mathcal{F}f(\mathrm{i}z_{0}-\omega_{0}-2k\tau)$
$\displaystyle=\mathrm{e}^{\tfrac{\pi}{2}(\mathrm{i}z_{0}-\omega_{0}-2k\tau)^{2}}\int_{-B}^{B}\mathcal{F}f(\xi)\mathrm{e}^{-\pi(\xi-\mathrm{i}z_{0}+\omega_{0}+2k\tau)^{2}}\,\mathrm{d}\xi$

which implies that $\mathcal{F}f$ is orthogonal to
$\xi\mapsto\mathrm{e}^{-\pi(\xi-\mathrm{i}z_{0}+\omega_{0}+2k\tau)^{2}}$ in
$L^{2}([-B,B])$, for all $k\in\mathbb{Z}$. Theorem 4 now implies that
$\mathcal{F}f=0$ and thus $f=0$. Hence, $F=0$ and as $\left\lvert
F\right\rvert=\left\lvert G\right\rvert$ the identity theorem of complex
analysis can be used to show that $G=0$ such that $g=0=f$.

Our main result may alternatively be stated without reference to the Gabor
transform. Indeed, it is equivalent to the following theorem.

######  Theorem 6.

Let $B>0$ and let $\omega_{0},\omega_{1}\in\mathbb{R}$ be such that
$\omega_{0}\neq\omega_{1}$. If $f,g\in\mathrm{PW}_{B}^{2}$ satisfy

$\left\lvert\phi\ast(\operatorname{M}_{-\omega_{j}}f)\right\rvert=\left\lvert\phi\ast(\operatorname{M}_{-\omega_{j}}g)\right\rvert,\qquad
j=0,1,$

then $f\sim g$.

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