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[36] Being the e ciency of the Bragg pulse, the populations of the two interfering condensates at port A are NA1 = N 2 and NA2 = N (1 )2, while at port B are NB1 = NB2 = N (1 ) respectively. As a consequence the contrast of the port A interferogram is CA = 2 (1 )=[ 2 + (1 )2] and varies with , while CB = 1 independently from .
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[38] We rst solve the stationary GP equation in the presence of a harmonic trap (as in the experiment), by means of a conjugate gradient algorithm [44, 45]. The timedependent GP equation is solved by using the FFT splitstep method discussed in [46]. The typical size of the numerical grid is 96 m 96 m 48 m, with 1024 128 64 points. The simulations are performed in the reference frame of the initial condensate (that is, by considering a moving optical lattice). In order to simplify the calculations, at tT OF = 6 ms we manually remove the momentum components that do not contribute to the interferograms in ports A and B, namely those with momentum components with jkj > 1:33kL. This cut in momentum space permits to reduce the size of the numerical box. Then, thanks to the fact that the system becomes almost noninteracting already at tT OF = 8 ms, the subsequent expansion dynamics up to tT OF = 33 ms is via a free expansion. The latter amounts to a single multiplication in Fourier space.
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[40] Following the discussion in Ref. [30], here we include an extra factor of 2 in the force owing to the fact that the two condensates are in the same internal state. The force is then normalized by a factor 1=2 each time a =2 pulse is applied, in order to account for the halving of the population of each component, see again Ref. [30].
[41] Notice that in both Figs. 4 and 6 we have considered only the mutual e ect of the two wavepackets belonging to the same port of the interferometer. We have veri ed that by including all the four wavepackes (produced after the second Bragg pulse) in the description, this results in additional forces that make the wavevector Kf decrease even further. Similarly, if one corrects the Castin-Dum equations [29] for the scaling parameters (t) in order to account for the splitting of the total number of atoms across the two interferometer ports, this also lowers the dotted line.
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[43] Here we employ a direct t in coordinate space instead of using the FT approach because we are interested in measuring the local e ect of the phase.
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[9] S. Abend, M. Gebbe, M. Gersemann, H. Ahlers, H. Müntinga, E. Giese, N. Gaaloul, C. Schubert, C. Lämmerzahl, W. Ertmer, W. P. Schleich, and E. M. Rasel, Phys. Rev. Lett. 117, 203003 (2016).
[10] G. E. Marti, R. Olf, and D. M. Stamper-Kurn, Phys. Rev. A 91, 013602 (2015).
[11] P. Navez, S. Pandey, H. Mas, K. Poulios, T. Fernholz, and W. v. Klitzing, New J. Phys. 18, 075014 (2016).
[12] S. Pandey, H. Mas, G. Drougakis, P. Thekkeppatt, V. Bolpasi, G. Vasilakis, K. Poulios, and W. v. Klitzing, Nature (London) 570, 205 (2019).
[13] S. Gupta, K. Dieckmann, Z. Hadzibabic, and D. E. Pritchard, Phys. Rev. Lett. 89, 140401 (2002).
[14] A. O. Jamison, B. Plotkin-Swing, and S. Gupta, Phys. Rev. A 90, 063606 (2014).
[15] M. Olshanii and V. Dunjko, arXiv:cond-mat/0505358.
[16] M. Horikoshi and K. Nakagawa, Phys. Rev. A 74, 031602(R) (2006).
[17] R. Jannin, P. Cladé, and S. Guellati-Khélifa, Phys. Rev. A 92, 013616 (2015).
[18] L. Pezzé, L. A. Collins, A. Smerzi, G. P. Berman, and A. R. Bishop, Phys. Rev. A 72, 043612 (2005).
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[26] A. O. Jamison, J. N. Kutz, and S. Gupta, Phys. Rev. A 84, 043643 (2011).
[27] S. Watanabe, S. Aizawa, and T. Yamakoshi, Phys. Rev. A 85, 043621 (2012).
[28] G. Yang, J. Guo, and S. Zhang, Int. J. Mod. Phys. B 33, 1950048 (2019).
[29] P. Tang, P. Peng, Z. Li, X. Chen, X. Li, and X. Zhou, Phys. Rev. A 100, 013618 (2019).
[30] Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996).
[31] Q. Guan, T. M. Bersano, S. Mossman, P. Engels, and D. Blume, Phys. Rev. A 101, 063620 (2020).
[32] P. W. Anderson, A Career in Theoretical Physics (World Scientific Publishing Company, Singapore, 1994).
[33] F. Sols, in Proceedings of the International School of Physics “Enrico Fermi,”edited by M. Inguscio, S. Stringari, and C. E. Wieman (IOS Press, Amsterdam, 1999), p. 453.
[34] M. Naraschewski, H. Wallis, A. Schenzle, J. I. Cirac, and P. Zoller, Phys. Rev. A 54, 2185 (1996).
[35] A three-pulse sequence (π /2 − π − π /2) is often used [2], where the π pulse acts as a mirror in analogy with the MachZehnder optical counterpart.
[36] Y.-J. Lin, A. R. Perry, R. L. Compton, I. B. Spielman, and J. V. Porto, Phys. Rev. A 79, 063631 (2009).
[37] A. Burchianti, C. D'Errico, S. Rosi, A. Simoni, M. Modugno, C. Fort, and F. Minardi, Phys. Rev. A 98, 063616 (2018).
[38] With being the efficiency of the Bragg pulse, the populations of the two interfering condensates at port A are NA1 = N 2 and NA2 = N (1 − )2, while at port B are NB1 = NB2 = N (1 − ), respectively. As a consequence the contrast of the port A interferogram is CA = 2 (1 − )/[ 2 + (1 − )2] and varies with , while CB = 1 independently from .
[39] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[40] We first solve the stationary GP equation in the presence of a harmonic trap (as in the experiment) by means of a conjugate gradient algorithm [47,48]. The time-dependent GP equation is solved by using the FFT split-step method discussed in [49]. The typical size of the numerical grid is 96 μm × 96 μm × 48 μm, with 1024 × 128 × 64 points. The simulations are performed in the reference frame of the initial condensate (that is, by considering a moving optical lattice). In order to simplify the calculations, at tTOF = 6 ms we manually remove the momentum components that do not contribute to the interferograms in ports A and B, namely those with momentum components with |k| > 1.33kL. This cut in momentum space permits one to reduce the size of the numerical box. Then, thanks to the fact that the system becomes almost noninteracting already at tTOF = 8 ms, the subsequent expansion dynamics up to tTOF = 33 ms is via a free expansion. The latter amounts to a single multiplication in Fourier space.
[41] C. Fort, P. Maddaloni, F. Minardi, M. Modugno, and M. Inguscio, Opt. Lett. 26, 1039 (2001).
[42] Following the discussion in Ref. [31], here we include an extra factor of 2 in the force owing to the fact that the two condensates are in the same internal state. The force is then normalized by a factor of 1/2 each time a π /2 pulse is applied, in order to account for the halving of the population of each component; see again Ref. [31].
[43] Notice that in both Figs. 4 and 6 we have considered only the mutual effect of the two wave packets belonging to the same port of the interferometer. We have verified that by including all the four wave packets (produced after the second Bragg pulse) in the description, this results in additional forces that make the wave vector Kf decrease even further. Similarly, if one corrects the Castin-Dum equations [30] for the scaling parameters λν (t ) in order to account for the splitting of the total number of atoms across the two interferometer ports, this also lowers the dotted line.
[44] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).
[45] In order to represent the phase as a smooth curve, we remove artificial jumps produced by the fact that the phase can only be determined modulo 2π . This permits one to “unwrap” the values obtained from the numerics, in order to keep a (piecewise) monotonic behavior.
[46] Here we employ a direct fit in coordinate space instead of using the FT approach because we are interested in measuring the local effect of the phase.
[47] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipies: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, 2007).
[48] M. Modugno, L. Pricoupenko, and Y. Castin, Eur. Phys. J. D 22, 235 (2003).
[49] B. Jackson, J. F. McCann, and C. S. Adams, J. Phys. B: At. Mol. Opt. Phys. 31, 4489 (1998).