Observation of Shapiro steps in an ultracold atomic Josephson junction
Erik Bernhart Marvin Röhrle Vijay Pal Singh Ludwig Mathey Luigi Amico Herwig Ott


APS/123-QED




Department of Physics and Research Center OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany


Department of Physics and Research Center OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany


Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE


Zentrum für Optische Quantentechnologien and Institut für  Quantenphysik, Universität Hamburg, 22761 Hamburg, Germany
The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, Hamburg 22761, Germany

Luigi Amicoluigi.amico@tii.ae
Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE
Dipartimento di Fisica e Astronomia, Università di Catania, Via S. Sofia 64, 95123 Catania, Italy
INFN-Sezione di Catania, Via S. Sofia 64, 95127 Catania, Italy


Herwig Ott0000-0002-3155-2719ott@physik.uni-kl.de
Department of Physics and Research Center OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany




             

§ ABSTRACT 
  
The current-voltage characteristic of a driven superconducting Josephson junction displays discrete steps. This phenomenon, discovered by Sydney Shapiro, forms today's voltage standard. 
Here, we report the observation of Shapiro steps in a driven Josephson junction in a gas of ultracold atoms. 
We demonstrate that the steps exhibit universal features, and provide key insight into the microscopic dissipative dynamics that we directly observe in the experiment. 
Most importantly, the steps are directly connected to phonon emission and soliton nucleation. 
The experimental results are underpinned by extensive numerical simulations based on classical-field dynamics and represent the transfer of the voltage standard to the realm of ultracold quantum gases.  




Observation of Shapiro steps in an ultracold atomic Josephson junction
    Ludwig Mathey
    September 9, 2024
======================================================================






















§ INTRODUCTION



The Josephson effect is one of the most fundamental phenomena in quantum science and technology: 
originally discovered for superconductors, the  effect features a dissipation-less  electrical current through a tunnel barrier by virtue of the phase difference of the traversing Cooper pairs' wave function.  Above a critical value of the current,  a finite voltage  occurs across the junction as a result of the formation of superconducting quasi-particles <cit.>.  The Josephson effect and Josephson junctions have played a prominent role for understanding the notion of macroscopic quantum coherence, which led to important  technological applications such as SQUIDs <cit.> and are the core units of superconducting qubits <cit.>. If an additional microwave field is applied across the Josephson junction, a staircase current-voltage characteristic emerges, referred to as  Shapiro steps <cit.>.
The origin of this effect is  photon-assisted tunneling of Cooper pairs through the junction <cit.>. 
Since the height of the steps depends only on the frequency of the microwave radiation, Planck's constant, and the electric charge,  Shapiro steps are nowadays used as the voltage standard - see e.g. <cit.>. Shapiro steps in superfluid ^3He have been investigated in Ref. <cit.>.


The Josephson effect can also be observed in ultracold quantum gases <cit.>.   
On the one hand, this enables the fabrication of atomtronic circuits, which have developed into their own field of research in recent years <cit.>. On the other hand, the high degree of control and flexibility over ultracold quantum gases provides unprecedented access to the microscopic physics underlying the system. In particular, Josephson physics has been recently implemented and extensively studied in ultracold atomic systems, both theoretically <cit.> as well as experimentally in bosonic <cit.> and fermionic <cit.> systems. 
In implementations with ultracold atoms, the Josephson current-phase relation is realized by moving the barrier through an ultracold gas which is otherwise at rest <cit.>.
Shapiro steps in ultracold gases are predicted to appear if the linear translation of the barrier (corresponding to a dc particle current) is combined with a periodic modulation of the position of the barrier (corresponding to an ac particle current), thus emulating the external microwave radiation of the superconducting Josephson junction <cit.>.



Here, we demonstrate the emergence of Shapiro steps at an atomic Josephson junction and quantify their global and local properties. The steps occur in the chemical potential difference, in conjunction with a density imbalance across the junction. We confirm that the step heights in the chemical potential difference Δμ are quantized by the external driving frequency f_m, i.e., Δμ = nh f_m, where n is the step index and h is Planck's constant.
By spatially resolving the atomic density, we study the microscopic  dynamics and observe the propagation of phonons and quasi-particles emerging from the barrier, which we identify as solitons.
This paves the way for studying the microscopic dissipative dynamics of Shapiro steps and other related Josephson transport effects.
 Furthermore, since the magnitude of the step height relates the chemical potential difference to a tunable frequency, and since the density imbalance across the barrier is directly measurable, our results suggest an additional insight into the equation of state of strongly correlated superfluids, thus advancing the study of superfluids with a novel approach.



§ EXPERIMENTAL SYSTEM





The experimental setup and the Shapiro protocol are sketched in Fig. <ref>. 
We prepare a ^87Rb Bose-Einstein condensate in an elongated optical dipole trap. The condensate has a tube-like cylindrical geometry with harmonic transverse confinement, see Fig. <ref> A. 
The chemical potential is μ = h× 1900 s^-1.  
A movable repulsive barrier with height V_B = 0.45 ×μ separates the condensate into two parts and thus realizes a weak link <cit.>.
We probe the system by absorption imaging. To achieve a high accuracy, we perform a matter wave imaging scheme <cit.> in order to reduce the optical density <cit.>.

We implement the dc and ac driving protocol following the proposal in Ref. <cit.>. To this end, we move the barrier with constant velocity v and an additional periodic modulation, 

    x(t) = v t + x_msin(2 π f_m t)  ,

where x_m is the modulation amplitude, and f_m the modulation frequency.
We fix the driving time to 33ms, which means that the final position of the barrier depends on v.
At the end, we measure the atom number imbalance z = (N_R - N_L)/N, 
where N is the total atom number and N_L (N_R) is the atom number on the left (right) hand side of the barrier. 
As a reference z_ref , we also measure the corresponding imbalance without the barrier. 
Using a simulation of the Gross-Pitaevskii equation we map Δ z = z -z_ref to the chemical potential difference Δμ <cit.>.
The barrier motion induces an atomic current relative to the barrier of value I, with I = v I_c/v_c, where I_c is the critical current and v_c is the critical velocity <cit.>.
The external current then reads I_ext = I + I_ac = I +  I_msin(2 π f_m t), with the bias current I and the modulation current I_m.
The parameters which we modify in our protocol are I, I_ and f_.
In Fig. <ref> C we show the measurements of the current-chemical potential characteristics 
that are obtained without and with ac driving.  
From the undriven response we characterize the critical velocity and the critical current of the Josephson junction <cit.>. 
For the periodically driven case, the system features plateaus of constant chemical potential difference occurring at integer multiple of the driving frequency. 
Thereby, the steps occur at lower I for stronger I_. We model our system with extensive classical-field simulations <cit.>, 
which capture the feature of our driven Josephson junction well, see Fig. <ref> D, E.



In a superconducting Josephson junction where the Shapiro steps arise from photon-assisted tunneling of the Cooper pairs <cit.>, the voltage V_n of the n-th Shapiro step is directly given by the energy quantization, i.e.,  
V_n = n Φ_0 f_m = n f_m h/(2e), 
where Φ_0 is the flux quantum and e the elementary charge <cit.>.  
For our neutral atom implementation this results in the quantization condition


    Δμ = nh f_m  .


To confirm this fundamental relation we vary f_ and determine the height Δμ of the first step, see Fig. <ref> A. For the modulation frequencies explored here, we indeed find the behavior Δμ= h f_, see Fig. <ref> B.
This result establishes the transfer of the voltage standard to the realm of ultracold quantum gases <cit.>. 
This is particularly useful in situations, where the relation between the atomic density and the chemical potential is not known, as the protocol allows for the generation of a predetermined chemical potential difference.
 
Next, we analyze the width of the Shapiro steps by varying the modulation amplitude I_m/I_c at a fixed modulation frequency of f_m = 90Hz <cit.>. 
In Fig. <ref> C, D, the widths of the zeroth and first steps display a Bessel-function behavior as a function of I_m/I_c, which is in agreement with the results of numerical simulations <cit.>. 
To support this observation we use the analytical prediction of an ac voltage driven Josephson junction, 
which yields the step width I_n = I_c |J_n(V_/V_n)|, 
where J_n is the Bessel function of the n-th order and V_ the modulation voltage <cit.>.
For an ac current driven junction this can be mathematically transformed into


    I_n =  I_c | J_n (α_th I_m/I_c) |   ,


with α_th=R I_c/f_m.
The resistance R and I_c are determined independently, such that there is no free fitting parameter <cit.>.
In Fig. <ref> C, D the results of Eq. <ref> support the overall trend of our measurements and simulations. 





§ MICROSCOPIC DYNAMICS


In contrast to solid-state physics the time and length scales in quantum gas experiments are much larger, such that the in situ study of excitations, quasi-particles and related microscopic phenomena becomes possible.
To do so, we follow the time evolution of the real space atomic density.
Using only a constant barrier velocity, i.e., no ac particle current, we measure the dc and ac Josephson regime, see Fig. <ref> A.
Below the critical current no imbalance is created and only a single phononic density wave is emitted from the initial acceleration of the barrier (I and II).
The group velocity of the emitted phonons coincides with a reference measurement, where a single perturbation creates a phonon wave packet <cit.>.
In the ac Josephson regime above the critical current, atoms are pushed away by the barrier and a density depletion remains in its wake (III).


In the Shapiro regime instead, we see that the phase dynamics is characterized by two distinct collective excitations: phonons and localized density depletions (see Fig. <ref> B and C). 
Phonons are emitted at distinct times in both directions due to the oscillatory motion of the barrier. This is particularly visible on the zeroth plateau (I).
Increasing the constant current I beyond the zeroth plateau yields a chemical potential difference and atom number imbalance; this phenomenon is characterized by the formation of 'depletion waves' that move backwards compared to the barrier motion (blue lines propagating to the left in II and III). 
These excitations are observed to have a smaller group velocity, i.e., a steeper slope and, by using the phase information of the simulation in Fig. <ref> A, we can conclude that such defects are accompanied by well resolved phase slippages with a phase jump of less than π at the density minimum. Therefore we identify these excitations as grey solitons. 
These solitons are responsible for most of the reduction in density, ultimately  causing the particle imbalance across the junction- see Fig. <ref> B and C. 
Specifically, the simulation suggests that for the n-th plateau n solitons are on-average emitted backwards during one driving cycle, see panels II and III of Fig. <ref> C and <cit.>. This creates the reduced density in the wake of the moving barrier, and therefore gives rise to the resistive regime.

On the other hand, being also defects of the phase field, they play a similar role as  quasi-particles of the driven superconducting Josephson junctions: as single-particle excitations are responsible for the resistive transport and for the actual phase dynamics across the superconducting junction, soliton emission provides the specific mechanism for the evolution of the phase coherence in our system, thus allowing for phase coherent dynamics even in presence of dissipation. 

In this sense, soliton emission in our system plays a similar role to that of single-particle excitations in superconducting Josephson junctions.  
 This statement is corroborated by the fact that the rate of emitted solitons indeed displays a step-like behavior in close parallel with  the staircase displayed by the chemical potential, see Fig. S11 of <cit.>.  
 We note that such a phenomenon resembles the interplay between single-particle excitations and photon-assisted tunneling occurring in superconducting junctions. 
Depending on the geometry the collective excitations can be different. For instance, in the theoretical proposal <cit.>, a larger 2D system was used and vortex-antivortex pairs are generated instead of solitons. 
In earlier studies of the dissipation at an atomic Josephson junction, the creation of vortex ring excitations at the barrier was observed <cit.>.




§ DISCUSSION AND OUTLOOK


We have experimentally demonstrated the emergence of Shapiro steps in an ultracold quantum gas by emulating the external microwave field and an alternating particle current across the barrier with a periodic modulation of the barrier position. Exploiting the know-how of the cold atoms field, we could monitor the coherent dynamics of the system with unprecedented accuracy. 
This way, our work  opens up  new directions both in basic and applied physical science. 
On the fundamental side, different dimensions, geometries, and particle statistics are direct extensions, as well as superfluid mixtures, dipolar superfluids, and superfluids in optical lattices, which can give rise to new physical processes, e.g. by generating other types of solitons, such as bright solitons, or vortices, or defects related to an underlying lattice or competing orders.   
Indeed,  Eq. <ref> provides the possibility to measure the equation of state of strongly correlated many-body systems. 
In the context of  superconducting Josephson junctions, our work  motivates to expand the understanding of the microscopic dynamics of the Shapiro effect as well, including the role of superconducting quasi-particles, solitons and vortices.

On the application side, we remark that our system transfers the voltage standard to the field of ultracold quantum gases.
Towards the development of atomtronic technology, Shapiro steps can be used to fine tune quantum transport,  as they combine both superfluid and resistive transport.
Stacks of Shapiro steps can be used to create even larger differences in chemical potential over a whole system, or as a source for predetermined chemical potential differences.  
We note that Shapiro steps have also been observed recently in an experiment with strongly correlated ultracold fermions in the unitary regime <cit.>.



§ ACKNOWLEDGMENTS


We thank Giacomo Roati and Giulia Del Pace for useful discussions.
Funding:
We gratefully acknowledge financial support by the DFG within the SFB OSCAR (project number 277625399).
L.M. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), namely the Cluster of Excellence ‘Advanced Imaging of Matter’ (EXC 2056), Project No. 390715994. 
The project is co-financed by ERDF of the European Union and by ’Fonds of the Hamburg Ministry of Science, Research, Equalities and Districts (BWFGB)’.

Author contributions:
E.B., M.R. and H.O. conceived the study. E.B. and M.R. performed the experiment and analyzed the data. 
E.B. performed the GPE simulations and V.P.S. modeled classical-field simulations. 
H.O. supervised the experiment. L.M. and L.A. supervised the theoretical part of the project.
E.B. prepared the initial version of the manuscript. 
All authors contributed to the data interpretation and the writing of the manuscript.

Competing interests:
The authors declare no competing interests. 

Data and materials availability:
All data presented in this paper will be deposited at Zenodo.












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@bib@innerbibempty

[Josephson(1962)]JOSEPHSON1962251
  author author B. Josephson, title title Possible new effects in superconductive tunnelling, https://doi.org/https://doi.org/10.1016/0031-9163(62)91369-0 journal journal Physics Letters volume 1, pages 251 (year 1962)NoStop
[Tinkham(2004)]tinkham2004introduction
  author author M. Tinkham, @noop  title Introduction to superconductivity, Vol. volume 1 (publisher Courier Corporation, year 2004)NoStop
[Fagaly(2006)]10.1063/1.2354545
  author author R. L. Fagaly, title title Superconducting quantum interference device instruments and applications, https://doi.org/10.1063/1.2354545 journal journal Review of Scientific Instruments volume 77, pages 101101 (year 2006), https://arxiv.org/abs/https://pubs.aip.org/aip/rsi/article-pdf/doi/10.1063/1.2354545/11171638/101101_1_online.pdf https://pubs.aip.org/aip/rsi/article-pdf/doi/10.1063/1.2354545/11171638/101101_1_online.pdf NoStop
[Koelle et al.(1999)Koelle, Kleiner, Ludwig, Dantsker, and Clarke]RevModPhys.71.631
  author author D. Koelle, author R. Kleiner, author F. Ludwig, author E. Dantsker, and author J. Clarke, title title High-transition-temperature superconducting quantum interference devices, https://doi.org/10.1103/RevModPhys.71.631 journal journal Rev. Mod. Phys. volume 71, pages 631 (year 1999)NoStop
[Barone and Paterno(1982)]Barone
  author author A. Barone and author G. Paterno, @noop  title Physics and applications of the josephson effect (year 1982)NoStop
[Leggett(1987)]leggett1987macroscopic
  author author A. J. Leggett, title title Macroscopic quantum tunnelling and related matters, @noop  journal journal Japanese Journal of Applied Physics volume 26, pages 1986 (year 1987)NoStop
[Clarke and Wilhelm(2008)]Clarke2008
  author author J. Clarke and author F. K. Wilhelm, title title Superconducting quantum bits, https://doi.org/10.1038/nature07128 journal journal Nature volume 453, pages 1031 (year 2008)NoStop
[Koch et al.(2007)Koch, Yu, Gambetta, Houck, Schuster, Majer, Blais, Devoret, Girvin, and Schoelkopf]PhysRevA.76.042319
  author author J. Koch, author T. M. Yu, author J. Gambetta, author A. A. Houck, author D. I. Schuster, author J. Majer, author A. Blais, author M. H. Devoret, author S. M. Girvin, and author R. J. Schoelkopf, title title Charge-insensitive qubit design derived from the cooper pair box, https://doi.org/10.1103/PhysRevA.76.042319 journal journal Phys. Rev. A volume 76, pages 042319 (year 2007)NoStop
[Johnson et al.(2011)Johnson, Amin, Gildert, Lanting, Hamze, Dickson, Harris, Berkley, Johansson, Bunyk, Chapple, Enderud, Hilton, Karimi, Ladizinsky, Ladizinsky, Oh, Perminov, Rich, Thom, Tolkacheva, Truncik, Uchaikin, Wang, Wilson, and Rose]Johnson2011
  author author M. W. Johnson, author M. H. S. Amin, author S. Gildert, author T. Lanting, author F. Hamze, author N. Dickson, author R. Harris, author A. J. Berkley, author J. Johansson, author P. Bunyk, author E. M. Chapple, author C. Enderud, author J. P. Hilton, author K. Karimi, author E. Ladizinsky, author N. Ladizinsky, author T. Oh, author I. Perminov, author C. Rich, author M. C. Thom, author E. Tolkacheva, author C. J. S. Truncik, author S. Uchaikin, author J. Wang, author B. Wilson, and author G. Rose, title title Quantum annealing with manufactured spins, https://doi.org/10.1038/nature10012 journal journal Nature volume 473, pages 194 (year 2011)NoStop
[Arute et al.(2019)Arute, Arya, Babbush, Bacon, Bardin, Barends, Biswas, Boixo, Brandao, Buell, Burkett, Chen, Chen, Chiaro, Collins, Courtney, Dunsworth, Farhi, Foxen, Fowler, Gidney, Giustina, Graff, Guerin, Habegger, Harrigan, Hartmann, Ho, Hoffmann, Huang, Humble, Isakov, Jeffrey, Jiang, Kafri, Kechedzhi, Kelly, Klimov, Knysh, Korotkov,
  Kostritsa, Landhuis, Lindmark, Lucero, Lyakh, Mandrà, McClean, McEwen, Megrant, Mi, Michielsen, Mohseni, Mutus, Naaman, Neeley, Neill, Niu, Ostby, Petukhov, Platt, Quintana, Rieffel, Roushan, Rubin, Sank, Satzinger, Smelyanskiy, Sung, Trevithick, Vainsencher, Villalonga, White, Yao, Yeh, Zalcman, Neven, and Martinis]Arute2019
  author author F. Arute, author K. Arya, author R. Babbush, author D. Bacon, author J. C. Bardin, author R. Barends, author R. Biswas, author S. Boixo, author F. G. S. L. Brandao, author D. A. Buell, author B. Burkett, author Y. Chen, author Z. Chen, author B. Chiaro, author R. Collins, author W. Courtney, author A. Dunsworth, author E. Farhi, author B. Foxen, author A. Fowler, author C. Gidney, author M. Giustina, author R. Graff, author K. Guerin, author S. Habegger, author M. P. Harrigan, author M. J. Hartmann, author A. Ho, author M. Hoffmann, author T. Huang, author T. S. Humble, author S. V. Isakov, author E. Jeffrey, author Z. Jiang, author D. Kafri, author K. Kechedzhi, author J. Kelly, author P. V. Klimov, author S. Knysh, author A. Korotkov, author F. Kostritsa, author D. Landhuis, author M. Lindmark, author E. Lucero, author D. Lyakh, author S. Mandrà, author J. R. McClean, author M. McEwen, author A. Megrant, author X. Mi, author K. Michielsen, author M. Mohseni, author J. Mutus, author O. Naaman, author M. Neeley, author C. Neill, author M. Y. Niu, author E. Ostby, author A. Petukhov, author J. C. Platt, author C. Quintana, author E. G. Rieffel, author P. Roushan, author N. C. Rubin,
  author D. Sank, author K. J. Satzinger, author V. Smelyanskiy, author K. J. Sung, author M. D. Trevithick, author A. Vainsencher, author B. Villalonga, author T. White, author Z. J. Yao, author P. Yeh, author A. Zalcman, author H. Neven, and author J. M. Martinis, title title Quantum supremacy using a programmable superconducting processor, https://doi.org/10.1038/s41586-019-1666-5 journal journal Nature volume 574, pages 505 (year 2019)NoStop
[Shapiro(1963)]ShapiroOriginal
  author author S. Shapiro, title title Josephson currents in superconducting tunneling: The effect of microwaves and other observations, https://doi.org/10.1103/PhysRevLett.11.80 journal journal Phys. Rev. Lett. volume 11, pages 80 (year 1963)NoStop
[Pöpel(1992)]popel1992josephson
  author author R. Pöpel, title title The josephson effect and voltage standards, https://doi.org/10.1088/0026-1394/29/2/005 journal journal Metrologia volume 29, pages 153 (year 1992)NoStop
[Simmonds et al.(2001)Simmonds, Marchenkov, Davis, and Packard]Shapiro_superfluid
  author author R. W. Simmonds, author A. Marchenkov, author J. C. Davis, and author R. E. Packard, title title Observation of the superfluid shapiro effect in a ^3he weak link, https://doi.org/10.1103/PhysRevLett.87.035301 journal journal Phys. Rev. Lett. volume 87, pages 035301 (year 2001)NoStop
[Albiez et al.(2005)Albiez, Gati, Fölling, Hunsmann, Cristiani, and Oberthaler]Oberthaler_2005
  author author M. Albiez, author R. Gati, author J. Fölling, author S. Hunsmann, author M. Cristiani, and author M. K. Oberthaler, title title Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction, https://doi.org/10.1103/PhysRevLett.95.010402 journal journal Phys. Rev. Lett. volume 95, pages 010402 (year 2005)NoStop
[Ryu et al.(2013)Ryu, Blackburn, Blinova, and Boshier]Atom_squid
  author author C. Ryu, author P. W. Blackburn, author A. A. Blinova, and author M. G. Boshier, title title Experimental realization of josephson junctions for an atom squid, https://doi.org/10.1103/PhysRevLett.111.205301 journal journal Phys. Rev. Lett. volume 111, pages 205301 (year 2013)NoStop
[Amico et al.(2021)Amico, Boshier, Birkl, Minguzzi, Miniatura, Kwek, Aghamalyan, Ahufinger, Anderson, Andrei, Arnold, Baker, Bell, Bland, Brantut, Cassettari, Chetcuti, Chevy, Citro, De Palo, Dumke, Edwards, Folman, Fortagh, Gardiner, Garraway, Gauthier, Günther, Haug, Hufnagel, Keil, Ireland, Lebrat, Li, Longchambon, Mompart, Morsch, Naldesi, Neely,
  Olshanii, Orignac, Pandey, Pérez-Obiol, Perrin, Piroli, Polo, Pritchard, Proukakis, Rylands, Rubinsztein-Dunlop, Scazza, Stringari, Tosto, Trombettoni, Victorin, Klitzing, Wilkowski, Xhani, and Yakimenko]roadmap_atomtronic
  author author L. Amico, author M. Boshier, author G. Birkl, author A. Minguzzi, author C. Miniatura, author L.-C. Kwek, author D. Aghamalyan, author V. Ahufinger, author D. Anderson, author N. Andrei, author A. S. Arnold, author M. Baker, author T. A. Bell, author T. Bland, author J. P. Brantut, author D. Cassettari, author W. J. Chetcuti, author F. Chevy, author R. Citro, author S. De Palo, author R. Dumke, author M. Edwards, author R. Folman, author J. Fortagh, author S. A. Gardiner, author B. M. Garraway, author G. Gauthier, author A. Günther, author T. Haug, author C. Hufnagel, author M. Keil, author P. Ireland, author M. Lebrat, author W. Li, author L. Longchambon, author J. Mompart, author O. Morsch, author P. Naldesi, author T. W. Neely, author M. Olshanii, author E. Orignac, author S. Pandey, author A. Pérez-Obiol, author H. Perrin, author L. Piroli, author J. Polo, author A. L. Pritchard, author N. P.  Proukakis, author C. Rylands, author H. Rubinsztein-Dunlop, author F. Scazza, author S. Stringari, author F. Tosto, author A. Trombettoni, author N. Victorin, author W. v. Klitzing, author D. Wilkowski, author K. Xhani, and author A. Yakimenko, title title Roadmap on Atomtronics: State of the art and perspective, https://doi.org/10.1116/5.0026178 journal journal AVS Quantum Science volume 3, pages 039201
  (year 2021), https://arxiv.org/abs/https://pubs.aip.org/avs/aqs/article-pdf/doi/10.1116/5.0026178/19676772/039201_1_5.0026178.pdf https://pubs.aip.org/avs/aqs/article-pdf/doi/10.1116/5.0026178/19676772/039201_1_5.0026178.pdf NoStop
[Amico et al.(2022)Amico, Anderson, Boshier, Brantut, Kwek, Minguzzi, and von Klitzing]Colloquium_Atomtronics
  author author L. Amico, author D. Anderson, author M. Boshier, author J.-P. Brantut, author L.-C. Kwek, author A. Minguzzi, and author W. von Klitzing, title title Colloquium: Atomtronic circuits: From many-body physics to quantum technologies, https://doi.org/10.1103/RevModPhys.94.041001 journal journal Rev. Mod. Phys. volume 94, pages 041001 (year 2022)NoStop
[Giovanazzi et al.(2000)Giovanazzi, Smerzi, and Fantoni]Giovanazzi_2000
  author author S. Giovanazzi, author A. Smerzi, and author S. Fantoni, title title Josephson effects in dilute bose-einstein condensates, https://doi.org/10.1103/PhysRevLett.84.4521 journal journal Phys. Rev. Lett. volume 84, pages 4521 (year 2000)NoStop
[Meier and Zwerger(2001)]Meier_Zwerger_2001
  author author F. Meier and author W. Zwerger, title title Josephson tunneling between weakly interacting bose-einstein condensates, https://doi.org/10.1103/PhysRevA.64.033610 journal journal Phys. Rev. A volume 64, pages 033610 (year 2001)NoStop
[Singh et al.(2020)Singh, Luick, Sobirey, and Mathey]Singh_2020
  author author V. P. Singh, author N. Luick, author L. Sobirey, and author L. Mathey, title title Josephson junction dynamics in a two-dimensional ultracold bose gas, https://doi.org/10.1103/PhysRevResearch.2.033298 journal journal Phys. Rev. Res. volume 2, pages 033298 (year 2020)NoStop
[Saha and Dubessy(2021)]PhysRevA.104.023316
  author author A. K. Saha and author R. Dubessy, title title Dynamical phase diagram of a one-dimensional bose gas in a box with a tunable weak link: From bose-josephson oscillations to shock waves, https://doi.org/10.1103/PhysRevA.104.023316 journal journal Phys. Rev. A volume 104, pages 023316 (year 2021)NoStop
[Grond et al.(2011)Grond, Betz, Hohenester, Mauser, Schmiedmayer, and Schumm]Grond_2011
  author author J. Grond, author T. Betz, author U. Hohenester, author N. J. Mauser, author J. Schmiedmayer, and author T. Schumm, title title The shapiro effect in atomchip-based bosonic josephson junctions, https://doi.org/10.1088/1367-2630/13/6/065026 journal journal New Journal of Physics volume 13, pages 065026 (year 2011)NoStop
[LeBlanc et al.(2011)LeBlanc, Bardon, McKeever, Extavour, Jervis, Thywissen, Piazza, and Smerzi]LeBlanc_2011
  author author L. J. LeBlanc, author A. B. Bardon, author J. McKeever, author M. H. T. Extavour, author D. Jervis, author J. H. Thywissen, author F. Piazza, and author A. Smerzi, title title Dynamics of a tunable superfluid junction, https://doi.org/10.1103/PhysRevLett.106.025302 journal journal Phys. Rev. Lett. volume 106, pages 025302 (year 2011)NoStop
[Eckel et al.(2016)Eckel, Lee, Jendrzejewski, Lobb, Campbell, and Hill]Campbell_2016
  author author S. Eckel, author J. G. Lee, author F. Jendrzejewski, author C. J. Lobb, author G. K. Campbell, and author W. T. Hill, title title Contact resistance and phase slips in mesoscopic superfluid-atom transport, https://doi.org/10.1103/PhysRevA.93.063619 journal journal Phys. Rev. A volume 93, pages 063619 (year 2016)NoStop
[Ji et al.(2022)Ji, Schweigler, Tajik, Cataldini, Sabino, Møller, Erne, and Schmiedmayer]Ji_2022
  author author S.-C. Ji, author T. Schweigler, author M. Tajik, author F. Cataldini, author J. a. Sabino, author F. S. Møller, author S. Erne, and author J. Schmiedmayer, title title Floquet engineering a bosonic josephson junction, https://doi.org/10.1103/PhysRevLett.129.080402 journal journal Phys. Rev. Lett. volume 129, pages 080402 (year 2022)NoStop
[Valtolina et al.(2015)Valtolina, Burchianti, Amico, Neri, Xhani, Seman, Trombettoni, Smerzi, Zaccanti, Inguscio, and Roati]Roati_dc_jj_scince_2015
  author author G. Valtolina, author A. Burchianti, author A. Amico, author E. Neri, author K. Xhani, author J. A. Seman, author A. Trombettoni, author A. Smerzi, author M. Zaccanti, author M. Inguscio, and author G. Roati, title title Josephson effect in fermionic superfluids across the bec-bcs crossover, https://doi.org/10.1126/science.aac9725 journal journal Science volume 350, pages 1505 (year
  2015), https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.aac9725 https://www.science.org/doi/pdf/10.1126/science.aac9725 NoStop
[Luick et al.(2020)Luick, Sobirey, Bohlen, Singh, Mathey, Lompe, and Moritz]Moritz_jj_science
  author author N. Luick, author L. Sobirey, author M. Bohlen, author V. P. Singh, author L. Mathey, author T. Lompe, and author H. Moritz, title title An ideal josephson junction in an ultracold two-dimensional fermi gas, https://doi.org/10.1126/science.aaz2342 journal journal Science volume 369, pages 89 (year 2020), https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.aaz2342 https://www.science.org/doi/pdf/10.1126/science.aaz2342 NoStop
[Kwon et al.(2020)Kwon, Pace, Panza, Inguscio, Zwerger, Zaccanti, Scazza, and Roati]Roati_acdc_jj_2020
  author author W. J. Kwon, author G. D. Pace, author R. Panza, author M. Inguscio, author W. Zwerger, author M. Zaccanti, author F. Scazza, and author G. Roati, title title Strongly correlated superfluid order parameters from dc josephson supercurrents, https://doi.org/10.1126/science.aaz2463 journal journal Science volume 369, pages 84 (year 2020), https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.aaz2463 https://www.science.org/doi/pdf/10.1126/science.aaz2463 NoStop
[Del Pace et al.(2021)Del Pace, Kwon, Zaccanti, Roati, and Scazza]Roati_acdc_jj_2021
  author author G. Del Pace, author W. J. Kwon, author M. Zaccanti, author G. Roati, and author F. Scazza, title title Tunneling transport of unitary fermions across the superfluid transition, https://doi.org/10.1103/PhysRevLett.126.055301 journal journal Phys. Rev. Lett. volume 126, pages 055301 (year 2021)NoStop
[Singh et al.(2024)Singh, Polo, Mathey, and Amico]singh2023shapiro
  author author V. P. Singh, author J. Polo, author L. Mathey, and author L. Amico, title title Shapiro Steps in Driven Atomic Josephson Junctions, https://doi.org/10.1103/PhysRevLett.133.093401 journal journal Phys. Rev. Lett. volume 133, pages 093401 (year 2024)NoStop
[Asteria et al.(2021)Asteria, Zahn, Kosch, Sengstock, and Weitenberg]Asteria_2021
  author author L. Asteria, author H. P. Zahn, author M. N. Kosch, author K. Sengstock, and author C. Weitenberg, title title Quantum gas magnifier for sub-lattice-resolved imaging of 3d quantum systems, https://doi.org/10.1038/s41586-021-04011-2 journal journal Nature volume 599, pages 571 (year 2021)NoStop
[sup()]supp
  @noop  note See supplementaryNoStop
[Dayem and Martin(1962)]DayemPhotonAssTunel
  author author A. H. Dayem and author R. J. Martin, title title Quantum interaction of microwave radiation with tunneling between superconductors, https://doi.org/10.1103/PhysRevLett.8.246 journal journal Phys. Rev. Lett. volume 8, pages 246 (year 1962)NoStop
[Hamilton(1972)]PhysRevB.5.912
  author author C. A. Hamilton, title title Frequency dependence of the josephson current, https://doi.org/10.1103/PhysRevB.5.912 journal journal Phys. Rev. B volume 5, pages 912 (year 1972)NoStop
[Hamilton(2000)]HamiltonVoltStandard
  author author C. A. Hamilton, title title Josephson voltage standards, https://doi.org/10.1063/1.1289507 journal journal Review of Scientific Instruments volume 71, pages 3611 (year 2000), https://arxiv.org/abs/https://pubs.aip.org/aip/rsi/article-pdf/71/10/3611/19100354/3611_1_online.pdf https://pubs.aip.org/aip/rsi/article-pdf/71/10/3611/19100354/3611_1_online.pdf NoStop
[Hamilton and Shapiro(1971)]PhysRevLett.26.426
  author author C. A. Hamilton and author S. Shapiro, title title Experimental demonstration of the riedel peak, https://doi.org/10.1103/PhysRevLett.26.426 journal journal Phys. Rev. Lett. volume 26, pages 426 (year 1971)NoStop
[Burchianti et al.(2018)Burchianti, Scazza, Amico, Valtolina, Seman, Fort, Zaccanti, Inguscio, and Roati]Roati_dc_jj_2018_phaseslips
  author author A. Burchianti, author F. Scazza, author A. Amico, author G. Valtolina, author J. A. Seman, author C. Fort, author M. Zaccanti, author M. Inguscio, and author G. Roati, title title Connecting dissipation and phase slips in a josephson junction between fermionic superfluids, https://doi.org/10.1103/PhysRevLett.120.025302 journal journal Phys. Rev. Lett. volume 120, pages 025302 (year 2018)NoStop
[Xhani et al.(2020)Xhani, Neri, Galantucci, Scazza, Burchianti, Lee, Barenghi, Trombettoni, Inguscio, Zaccanti, Roati, and Proukakis]Xhani_2020
  author author K. Xhani, author E. Neri, author L. Galantucci, author F. Scazza, author A. Burchianti, author K.-L. Lee, author C. F. Barenghi, author A. Trombettoni, author M. Inguscio, author M. Zaccanti, author G. Roati, and author N. P. Proukakis, title title Critical transport and vortex dynamics in a thin atomic josephson junction, https://doi.org/10.1103/PhysRevLett.124.045301 journal journal Phys.
  Rev. Lett. volume 124, pages 045301 (year 2020)NoStop
[Del Pace et al.(2024)Del Pace, Hernández-Rajkov, Singh, Grani, Frómeta Fernández, Nesti, Seman, Inguscio, Amico, and Roati]DelPace2024
  author author G. Del Pace, author D. Hernández-Rajkov, author V. Singh, author N. Grani, author M. Frómeta Fernández, author G. Nesti, author J. Seman, author M. Inguscio, author L. Amico, and author G. Roati, @noop  title Shapiro steps in strongly-interacting fermi gases (year 2024), note to be publishedNoStop










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Supplementary Information for
 Observation of Shapiro steps in an ultracold atomic Josephson junction
    Ludwig Mathey
    September 9, 2024
=====================================================================================================



§ SUPPLEMENTARY MATERIAL



































§ EXPERIMENTAL DETAILS






  §.§.§ Experimental procedure


We start by preparing a Bose-Einstein condensate (BEC) of around 180e3 ^87Rb atoms in a crossed optical dipole trap, consisting of a High Power (HP) beam and a Low Power (LP) beam. 



To reach the required tube like geometry we ramp the HP linearly in 400ms from 20mW up to 150mW and the LP within the same time down to 0mW.
This results in a trapping geometry with trapping frequencies ω = 2 π× [1.6; 252; 250] Hz. 
Because the atoms can expand in the dipole trap during this procedure and to set a defined length of the sample we use two repulsive barriers at positions x = ±37.5μ m from the center of the initial position of the BEC.
The barriers are ramped linearly within 40ms up to a intensity resulting in a potential much higher than the chemical potential of the cloud (≈ 10 μ). 
      
When the ramp of the dipole trap is finished we wait 200ms, letting the system equilibrate, before starting the experiment.
After this wait time we ramp up the center barrier in 45ms to its intended value with an initial position which is 20μ m off center. Subsequently, the Shapiro protocol starts, see main text. 




  §.§.§ Barrier generation


The realization of the optical barriers are sketched in the following.
We use a 532nm laser to create a repulsive potential and an AOM to stabilize its intensity.
To generate the barriers we guide the beam through a two axis acousto-optical deflector (AOD).
By changing the AODs RF driving frequency we control the axial position and the transverse extent of the barriers.
After the AOD we use a f = 100mm scan lens to convert the beams angular displacement in a lateral displacement. 
The light is then collected by a f = 750mm tube lens and guided to the NA=0.3 in-vacuum objective, which focus the beam down to the atoms.     
The tube lens is chosen in a way that we get a diffraction limited spot at the atoms.
We estimate the size of a single spot by a measurement with a test target in the chamber. The Gaussian beam radius is round 1.1μ m.










To realize in the transverse direction a homogeneously extended and in the axial direction movable barrier, we drive the AOD by multiple  RF frequencies <cit.>. 
By applying a multitone RF signal of the form

    S(t) = ∑_i S_i sin(ω_i t + ϕ_i),

to each of the AOD axis, we create three individual barriers, which are independently movable, when using time dependent ϕ_i(t). 


For the experiments described in the main text we use 40mW laser power in total.  









  §.§.§ Calibration of the barrier height




To estimate the height of the barrier used in the experiment we measure the equation of state n(μ). The method is close to the one used in <cit.>. 

We generate a block of 5 × 9 barrier spots, which corresponds to 5 times the experiments barrier size and measure their mean intensity separately on a camera. 
This barrier block is projected onto the center of the BEC consisting of 180e3 atoms. We count the atoms at the position of the block by absorption imaging in the y-direction, applying the matter wave imaging scheme.
The result is shown in fig. S<ref>.
The initial decrease is linearly fitted and the zero crossing is set as the barrier height corresponding to the chemical potential V_0 = μ.




  §.§.§ Imaging system


The atom cloud is probed via absorption imaging, using the 780nm Rubidium D2 transition. 
Our experiment uses three different imaging systems.
First we have a standard time of flight imaging along the y-axis (horizontal direction) with a magnification of M_hor = 3.77, which is used to measure the atom number of the BEC.
Second we use the same imaging setup in a different mode, known as matter wave imaging <cit.>.
This procedure enables us to image the density distribution of the cloud in position space after a certain time of flight with a magnification in the axial direction of M_mwi = M_mwi' · M_hor = 32.8, were M_mwi' = 8.7. 

the benefit of the matter wave imaging system is a strongly reduced optical density, which allows for a precise atom number determination.

Last, we can perform in-situ absorption imaging through the NA = 0.3 objective along the z-axis (vertical direction).
With this imaging scheme we can get a theoretical optical  resolution of d_vert = 1.6μ m and a magnification of M_vert = 19.5.
However, the high optical density of the sample in the trap prevents us from correctly determining the density with in-situ imaging.




§ CHARACTERIZATION OF THE WEAK LINK
 




To effectively describe a Josephson junction in the undriven case, one usually applies the so-called resistively capacitively shunted junction (RCSJ) model. 
There, an additional resistance R and capacity C is assigned to the junction and a simple circuit, see fig. S<ref>, is used to quantitatively model the junctions behaviour.
With Kirchoff's law we get the basic equation of the circuit

    I_ext = I_c sin(ϕ) -1/RΔμ - C Δμ̇.

Here, the first Josephson equation I(ϕ) = I_csin(ϕ) is used to describe the supercurrent across the junction.
In case of ultracold atoms, where the chemical potential difference Δμ plays the role of the voltage, the second Josephson equation reads <cit.>

    Δμ = - ħϕ̇.
 
To be able to fully describe the junction we use these equations to experimentally determine the systems parameters R, C, and I_c.




To this end, we measure Josephson plasma oscillations <cit.>. We first prepare an initial atom number imbalance across the junction and let the system evolve freely in time. 
We then measure both, the atom number difference N = N_L - N_R and the relative phase ϕ = ϕ_L - ϕ_R between the two superfluids.   
Applying a Gaussian filter to the experimental data and numerical differentiation of N gives dN/dt, and by fitting  I_c to  I(t) = dN/dt = I_csin(ϕ(t)) (see fig. S<ref>), we find I_c = 192e31/s.
In the same way, we can determine the capacity C.
Using the same measurement, we numerically differentiate ϕ(t) and together with Eq. <ref> we get - V(t) = - N(t)/C = dϕ/dt. 
We find C = 59.6s/h.  



The resistance R can be determined by measuring the BEC critical velocity v_c. Thereby, we run a dc Josephson protocol, by moving the barrier with constant velocity through the condensate. 
The result is shown in fig. S<ref>. Below the critical velocity v_c, we find Δμ = 0, indicating the absence of any ”voltage” drop. Above v_c, a finite chemical potential difference Δμ builds up, signaling the onset of the ac Josephson branch.
Fitting Δ z = R √(v^2 - v^2_c), gives v_c = 0.42mm/s. 
With I = v · I_c/v_c and Eq. <ref> we can rescale the measurement data and use again the fit function Δμ = R √(I^2 - I^2_c), and obtain R=0.9e-3h.   




 §.§ Current phase relation




The Josephson junction in our experiments has a barrier height which is significantly lower than the chemical potential. It is necessary to independently measure the current phase relation in order to verify, whether higher order terms in the Josephson relation are relevant <cit.>.
We repeat the experiment for the critical velocity and measure the phase between the two condensates at the end of the motion.
The phase is determined via interference of both condensates in time of flight.
We find the best contrast for 5ms time of flight, for which a substantial part of the cloud has already interfered with each other.
After subtracting a reference image without barrier we fit a sinusoidal function to extract the phase.
The resulting current phase relation is shown in fig. S<ref>.
Following Ref. <cit.> one can describe the current phase relation of a Josephson junction with I(φ) = ∑_n> 0 I_n sin(nφ) if the system exhibits time reversal symmetry. 
In the limit of small coupling the above expression reduces to the first Josephson equation, I(φ) = I_csin(φ).

We fit our data up to the second order term and find I_1=0.995I_c and I_2=0.054I_c. The Josephson junction is therefore close to the ideal Josephson junction limit.




 §.§ Speed of sound



We measure the speed of sound following the original protocol from Ref. <cit.>.
We prepare the same sample as in the Shapiro step measurements. We then instantaneously switch on a barrier with height V_B≈μ in the center of the condensate.
We follow the resulting phonon propagation via matter wave imaging, subtracting a reference image without barrier (see fig. S<ref>). We extract from a linear fit the speed of sound. 
The density wave traveling to the right has a speed of c_s = 1.63 ± 0.03 mm/s, and the one traveling to the left has a speed of c_s = -1.38 ± 0.02 mm/s. The small difference probably rises from a residual motion of the condensate in the shallow trap.





 §.§ Shapiro steps for different barrier height


To see the influence of the barrier height on the occurrence of the Shapiro steps, we repeat the same protocol for a large range of different barrier heights as shown in fig. S<ref>.
Shapiro steps are visible over the whole parameter range. However, the steps tend to smear out for very high barriers (corresponding to a smaller critical current), while for lowest barrier height (V_B =0.25 μ), the step height is reduced. We attribute the latter to the fact that the “weak link regime” is no longer valid for such low barriers.









§ MEASUREMENT EVALUATION




 §.§ Chemical potential




To map the measured atom number differences Δ z to a chemical potential difference Δμ, we calculate μ for different total atom numbers N, see fig. S<ref>, using imaginary time evolution of the GPE <cit.>.


We then extract the chemical potential from the calculated ground state wave function, and use a phenomenological fit to get a function μ (N). 
With 

    Δμ (z) = μ(N·(1+z)) - μ(N·(1-z))

we calculate Δμ(z). 






  §.§.§ Step height


To evaluate the step height, we rescale the data to chemical potential differences using Eq. <ref> and fit a logistic function 

    f(x) = ∑_i S/1+exp(-k_i(x-x_i)) + b,

where S is the step height, k_i the steepness of the steps and i-1 the number of visible steps.  





  §.§.§ Step width


We extract the width of the steps as illustrated in fig. S<ref>.  
We first a apply a Gaussian filter to the data to reduce high frequency noise before we numerically calculate the first derivative ∂Δ z/∂ v of the recorded atom number imbalance Δ z. 
For each step n we search the maxima of the derivative in the area where Δμ = n f_m±ϵ, with ϵ = 45Hz.  
We define the width of a step I_n as the distance between the two maxima. 
For the 0-th step we calculate the distance with respect to zero.
I_n is normalized by I_0, which is the width of 0-th step in the dc Josephson protocol. 













§ CLASSICAL-FIELD SIMULATION METHOD


We simulate the dynamics of a driven atomic Josephson junction using a classical-field method within the truncated Wigner approximation <cit.>. 
We consider a three-dimensional (3D) condensate of  ^87Rb atoms confined in a cigar-shaped geometry. 
The system is described by the Hamiltonian


    Ĥ   = ∫d r[  ħ^2/2m∇ψ̂^†( r) ·∇ψ̂( r)  + V( r) ψ̂^†( r)ψ̂( r)   
       + g/2ψ̂^†( r)ψ̂^†( r)ψ̂( r)ψ̂( r)],


where ψ̂( r) and ψ̂^†( r) are the bosonic annihilation and creation field operators, respectively. 
The 3D interaction parameter is given by g=4π a_s ħ^2/m, where a_s is the s-wave scattering length and m is the mass. 
For ^87Rb atoms a_s is 5.3 nm.  
The external potential V( r) represents the harmonic trap V_trap( r)=m(ω_x^2x^2+ ω_y^2 y^2 + ω_z^2 z^2)/2,  where the trap frequencies are chosen according to the experiment, i.e.,  (ω_x, ω_y, ω_z)= 2π×(1.6, 252, 250).  

Within the classical-field approximation we replace the operators ψ̂ in Eq. <ref> and in the equations of motion by complex numbers ψ.  We map real space on a lattice system of  760 × 25 × 25 sites with the lattice discretization length l= 0.1 μ. 
We note that the continuum limit is satisfied by choosing l to be comparable or smaller than the healing length ξ = ħ/√(2mgn) and the de Broglie wavelength, where n is the density <cit.>. 
We generate the initial states ψ(t=0) in a grand canonical ensemble of temperature T and chemical potential μ via a classical Metropolis algorithm. We use T= 35  and adjust μ such that the total atom number N is close to the experimental one. 
We propagate each initial state using the classical equations of motion. 

To create a Josephson junction and excite Shapiro steps we add a perturbation term ℋ_ex = ∫d r  V(x,t) n( r, t), where n( r, t) = |ψ( r , t)|^2 is the local density and V(x, t) is the Gaussian barrier potential of the form 


    V(x,t)  = V_0 (t) exp[-  2( x-x_0- x(t) )^2/w^2].


V_0 is the time-dependent strength and w is the width. x_0 is the initial location of the barrier and x(t) is the dc and ac driving term. 
We choose w=1 and x_0= 19. 
We linearly ramp up V_0(t) to the value V_0/μ=0.45 over 200 and then wait for 50.
This creates a weak link by suppressing the tunneling at location x_0. 
While we move the barrier at a constant velocity v,  we also periodically modulate its position as 


    x(t) = v t + x_sin(2π f_ t),


where x_ is the driving amplitude and f_ is the driving frequency <cit.>.   
This induces an atom current I relative to the barrier motion, i.e., I = v I_c/v_c, 
where I_c is the critical current and v_c is the critical velocity. 
Similarly, the amplitude of the ac current is given by I_= 2π f_ x_ I_c/v_c. 
For the calculation of the atom imbalance we fix the driving time to 33. 
The atom imbalance  z is determined by z=(N_R - N_L)/N, 
where N_L (N_R) is the atom number in the left (right) reservoir, and N= N_L +N_R is the total atom number.  
For various values of I, I_, and f_ we determine Δ z = z - z_ and convert it to the chemical potential difference Δμ using Eq. <ref>. 
z_ is the imbalance determined at the final location without the barrier. 
Without the ac driving the onset of a nonzero Δμ occurs above a certain v_c, which we determine using the prediction of the RCSJ circuit model, see Fig. <ref> A. We obtain v_c=0.42 mm/s in excellent agreement with the measurement of v_c. 
In the presence of ac drive we find the creation of Shapiro steps whose height depends on the driving frequency f_. Using sigmoid fits we determine the height of the first Shapiro step, see Fig. <ref> B. 
We determine the step height for f_ in the range between 50 and 185 by analyzing the driven response after 3 to 6 driving periods, and also average these results over a few values of I_. 
We determine the width of zeroth and first step following the same procedure as in the experiment, 
which involves calculating the differential resistance d Δμ/dI and 
then determining the step width from the maximum of d Δμ /dI, see Fig. S<ref>. 
The simulation results of the step width for f_=90 and varying I_/I_c are 
presented in the main text of the paper. 
For the phase evolution we calculate the local phase δϕ ( r)=  ϕ ( r) -  ϕ( r_) from the time evolution of a single trajectory, 
where ϕ( r_) is the reference phase. 




To count total number of solitons we analyze the phase evolution δϕ (x, t)=  ϕ (x+l, t) -  ϕ(x, t) of a single line along the x direction of a single trajectory of the driven system, 
which is because the phase profile in the yz plane is almost uniform. 
The oscillating motion of the barrier results in the creation of solitons, which we identify by a phase jump of less than π. 
We count total number of such phase jumps to determine the soliton number N_s. 
In Fig. S<ref>B we show N_s, averaged over few samples and total number of driving cycles, as a function of I/I_c, which features a step-like behavior, coinciding with the one observed in the I- Δμ characteristic in Fig. S<ref>A. There are on-average one, two and three solitons at the first, second and third Shapiro steps, respectively.  

















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@bib@innerbibempty

[Endres et al.(2016)Endres,
  Bernien, Keesling, Levine,
  Anschuetz, Krajenbrink, Senko, Vuletic, Greiner, and  Lukin]EndresTweezer
  author author M. Endres, author H. Bernien,
  author A. Keesling, author H. Levine, author
  E. R. Anschuetz, author
  A. Krajenbrink, author
  C. Senko, author V. Vuletic, author M. Greiner, and author M. D.  Lukin, title title Atom-by-atom
  assembly of defect-free one-dimensional cold atom arrays, https://doi.org/10.1126/science.aah3752 journal journal Science volume 354, pages
  1024 (year 2016), https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.aah3752
  https://www.science.org/doi/pdf/10.1126/science.aah3752 NoStop
[Kwon et al.(2020)Kwon,
  Pace, Panza, Inguscio,
  Zwerger, Zaccanti, Scazza,  and Roati]Roati_acdc_jj_2020
  author author W. J. Kwon, author G. D. Pace,
  author R. Panza, author M. Inguscio, author
  W. Zwerger, author M. Zaccanti, author F. Scazza, and author G. Roati, title title Strongly
  correlated superfluid order parameters from dc josephson supercurrents,  https://doi.org/10.1126/science.aaz2463 journal
  journal Science volume 369, pages 84 (year 2020), https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.aaz2463
  https://www.science.org/doi/pdf/10.1126/science.aaz2463 NoStop
[Asteria et al.(2021)Asteria, Zahn, Kosch, Sengstock, and Weitenberg]Asteria_2021
  author author L. Asteria, author H. P. Zahn,
  author M. N. Kosch, author K. Sengstock, and author C. Weitenberg, title
  title Quantum gas magnifier for sub-lattice-resolved imaging of
  3d quantum systems, https://doi.org/10.1038/s41586-021-04011-2
  journal journal Nature volume 599, pages 571 (year
  2021)NoStop
[Meier and Zwerger(2001)]Meier_Zwerger_2001
  author author F. Meier and author W. Zwerger, title title Josephson tunneling
  between weakly interacting bose-einstein condensates, https://doi.org/10.1103/PhysRevA.64.033610 journal journal Phys. Rev. A volume 64, pages 033610 (year 2001)NoStop
[Valtolina et al.(2015)Valtolina, Burchianti, Amico, Neri, Xhani, Seman, Trombettoni, Smerzi, Zaccanti,
  Inguscio, and Roati]Roati_dc_jj_scince_2015
  author author G. Valtolina, author A. Burchianti, author A. Amico,
  author E. Neri, author
  K. Xhani, author J. A. Seman, author A. Trombettoni, author A. Smerzi, author M. Zaccanti, author M. Inguscio, and author G. Roati, title title Josephson
  effect in fermionic superfluids across the bec-bcs crossover, https://doi.org/10.1126/science.aac9725 journal journal Science volume 350, pages
  1505 (year 2015), https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.aac9725
  https://www.science.org/doi/pdf/10.1126/science.aac9725 NoStop
[Golubov et al.(2004)Golubov, Kupriyanov, and Il'ichev]RevModPhys.76.411
  author author A. A. Golubov, author M. Y. Kupriyanov, and author E. Il'ichev, title title The current-phase
  relation in josephson junctions, https://doi.org/10.1103/RevModPhys.76.411 journal journal Rev. Mod. Phys. volume 76, pages 411 (year 2004)NoStop
[Andrews et al.(1997)Andrews, Kurn, Miesner, Durfee, Townsend, Inouye, and  Ketterle]Andrews_1997
  author author M. R. Andrews, author D. M. Kurn,
  author H.-J. Miesner, author D. S. Durfee, author
  C. G. Townsend, author
  S. Inouye, and author
  W. Ketterle, title title Propagation of sound in a bose-einstein condensate, https://doi.org/10.1103/PhysRevLett.79.553 journal journal Phys. Rev. Lett. volume 79, pages 553 (year 1997)NoStop
[Baals et al.(2021)Baals,
  Moreno, Jiang, Benary, and  Ott]BaalsStability
  author author C. Baals, author A. G. Moreno,
  author J. Jiang, author J. Benary, and author
  H. Ott, title title Stability analysis and attractor dynamics of three-dimensional dark
  solitons with localized dissipation, https://doi.org/10.1103/PhysRevA.103.043304 journal journal Phys. Rev. A volume 103, pages 043304 (year 2021)NoStop
[Singh et al.(2016)Singh,
  Weimer, Morgener, Siegl,
  Hueck, Luick, Moritz, and  Mathey]Singh2016
  author author V. P. Singh, author W. Weimer,
  author K. Morgener, author J. Siegl, author
  K. Hueck, author N. Luick, author H. Moritz, and author L. Mathey, title title Probing
  superfluidity of Bose-Einstein condensates via laser stirring, https://doi.org/10.1103/PhysRevA.93.023634 journal journal Phys. Rev. A volume 93, pages 023634 (year 2016)NoStop
[Kiehn et al.(2022)Kiehn,
  Singh, and Mathey]Kiehn2022
  author author H. Kiehn, author V. P. Singh,  and author L. Mathey,  title title Superfluidity of a laser-stirred
  Bose-Einstein condensate, https://doi.org/10.1103/PhysRevA.105.043317 journal journal Phys. Rev. A volume 105, pages 043317 (year 2022)NoStop
[Mora and Castin(2003)]Mora2003
  author author C. Mora and author Y. Castin,  title title Extension of Bogoliubov theory to
  quasicondensates, https://doi.org/10.1103/PhysRevA.67.053615
  journal journal Phys. Rev. A volume 67, pages 053615 (year
  2003)NoStop
[Singh et al.(2024)Singh,
  Polo, Mathey, and Amico]singh2023shapiro
  author author V. P. Singh, author J. Polo,
  author L. Mathey, and author L. Amico, title
  title Shapiro Steps in Driven Atomic Josephson Junctions,  https://doi.org/10.1103/PhysRevLett.133.093401 journal
  journal Phys. Rev. Lett. volume 133,  pages 093401 (year 2024)NoStop

