Abstract
On the basis of a recently introduced model for the Bose–Einstein condensate (BEC) trapped in the 2D “funnel” potential, \(\sim \) \(-r^{-1}\), we develop analysis for vortex modes, which are confined in the transverse direction by the self-attraction, or by the trapping potential, in the case of self-repulsion. Linear 3D wave functions are found exactly for eigenstates with an orbital momentum. In the case of self-repulsion, 3D wave functions are obtained by means of the Thomas–Fermi approximation. Then, with the help of the variational method, the underlying Gross–Pitaevskii equation is reduced to a 1D nonpolynomial Schrödinger equation (NPSE) for modes with zero or nonzero embedded vorticity, which are tightly confined by the funnel potential in the transverse plane. Numerical results demonstrate high accuracy of the NPSE reduction for both signs of the nonlinearity. The analysis is performed for stationary modes and for traveling ones colliding with a potential barrier. By means of simulations of NPSE with the self-attraction, collisions between solitons are studied too, demonstrating elastic and inelastic outcomes, depending on the impact velocity and underlying vorticity. A boundary of the stability of 3D vortices with winding number \(S=1\) against spontaneous splitting in two fragments is identified in the case of the self-attraction, all vortices with \(S\ge 2\) being unstable.
Similar content being viewed by others
Data availability statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: No associate data is deposited, as it is not necessary for the presentation of the results in the paper.].
References
F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71, 463 (1999)
L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics (Clarendon Press, 2003)
C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2008)
M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269, 198 (1995)
K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)
C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995)
S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, M. Lewenstein, Phys. Rev. Lett. 83, 5198 (1999)
L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, C. Salomon, Science 296, 1290 (2002)
K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Nature 417, 150 (2002)
S.L. Cornish, S.T. Thompson, C.E. Wieman, Phys. Rev. Lett. 96, 170401 (2006)
J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmer- son, W. P. Reinhardt, S. L. Rolston, B. I. Schneider, and W. D. Phillips, Science 287, 97 (2000)
P. Meystre, Atom Optics (Springer, 2001)
A.D. Martin, J. Ruostekoski, New J. Phys. 14, 043040 (2012)
G.D. McDonald, C.C.N. Kuhn, K.S. Hardman, S. Bennetts, P.J. Everitt, P.A. Altin, J.E. Debs, J.D. Close, N.P. Robins, Phys. Rev. Lett. 113, 013002 (2014)
J.L. Helm, S.L. Cornish, S.A. Gardiner, Phys. Rev. Lett. 114, 134101 (2015)
A. Di Carli, C.D. Colquhoun, G. Henderson, S. Flannigan, G.L. Oppo, A.J. Daley, S. Kuhr, E. Haller, Phys. Rev. Lett. 123, 123602 (2019)
D. Luo, Y. Jin, J.H.V. Nguyen, B.A. Malomed, O.V. Marchukov, V.A. Yurovsky, V. Dunjko, M. Olshanii, R.G. Hulet, Phys. Rev. Lett. 125, 183902 (2020)
D.S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)
D.S. Petrov, G.E. Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016)
C. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, L. Tarruell, Science 359, 301 (2018)
P. Cheiney, C.R. Cabrera, J. Sanz, B. Naylor, L. Tanzi, L. Tarruell, Phys. Rev. Lett. 120, 135301 (2018)
G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wolswijk, F. Minardi, M. Modugno, G. Modugno, M. Inguscio, M. Fattori, Phys. Rev. Lett. 120, 235301 (2018)
C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, C. Fort, Phys. Rev. Res. 1, 033155 (2019)
Z.-H. Luo, W. Pang, B. Liu, Y.-Y. Li, B.A. Malomed, Front. Phys. 16, 1 (2021)
F. Büttcher, J.-N. Schmidt, J. Hertkorn, K.S.H. Ng, S.D. Graham, M. Guo, T. Langen, T. Pfau, Rep. Prog. Phys. 84, 012403 (2021)
A.D. Jackson, G.M. Kavoulakis, C.J. Pethick, Phys. Rev. A 58, 2417 (1998)
M. Chiofalo, M. Tosi, Phys. Lett. A 268, 406 (2000)
L. Salasnich, A. Parola, L. Reatto, Phys. Rev. A 65, 043614 (2002)
P. Massignan, M. Modugno, Phys. Rev. A 67, 023614 (2003)
A.M. Kamchatnov, V.S. Shchesnovich, Phys. Rev. A 70, 023604 (2004)
W. Zhang, L. You, Phys. Rev. A 71, 025603 (2005)
A. Maluckov, L. Hadzievski, B.A. Malomed, L. Salasnich, Phys. Rev. A 78, 013616 (2008)
S.K. Adhikari, L. Salasnich, Phys. Rev. A 77, 033618 (2008)
L. Salasnich, F. Toigo, Phys. Rev. A 78, 053626 (2008)
S. Adhikari, Laser Phys. Lett. 6, 901 (2009)
S.K. Adhikari, B.A. Malomed, Physica D 238, 1402 (2009)
S.K. Adhikari, J. Phys. B At. Mol. Opt. Phys. 43, 085304 (2010)
M. C. P. dos Santos and W, B. Cardoso, Phys. Rev. E 103, 052210 (2021)
A. Muñoz Mateo and V. Delgado, Phys. Rev. A 77, 013617 (2008)
G. Gligorić, A. Maluckov, L. Hadzievski, B.A. Malomed, J. Phys. B At. Mol. Opt. Phys. 42, 145302 (2009)
G. Gligorić, A. Maluckov, L. Hadzievski, B.A. Malomed, Phys. Rev. A 79, 053609 (2009)
L. Salasnich, B.A. Malomed, Phys. Rev. A 79, 053620 (2009)
L. Salasnich, J. Phys. A Math. Theor. 42, 335205 (2009)
Luis E. Young-S., L. Salasnich, and S. K. Adhikari, Phys. Rev. A 82, 053601 (2010)
W.B. Cardoso, A.T. Avelar, D. Bazeia, Phys. Rev. E 83, 036604 (2011)
L. Salasnich, B.A. Malomed, J. Phys. B At. Mol. Opt. Phys. 45, 055302 (2012)
L. Salasnich and B. A. Malomed, Phys. Rev. A - At. Mol. Opt. Phys. 87, 063625 (2013)
L. Salasnich, W.B. Cardoso, B.A. Malomed, Phys. Rev. A 90, 033629 (2014)
M.C. dos Santos, W.B. Cardoso, Phys. Lett. A 383, 1435 (2019)
C. Wang, P. Kevrekidis, T. Horikis, D. Frantzeskakis, Phys. Lett. A 374, 3863 (2010)
A.I. Nicolin, M.C. Raportaru, Phys. A Stat. Mech. its Appl. 389, 4663 (2010)
C.A.G. Buitrago, S.K. Adhikari, J. Phys. B At. Mol. Opt. Phys. 42, 215306 (2009)
H.L.C. Couto, W.B. Cardoso, J. Phys. B At. Mol. Opt. Phys. 48, 025301 (2015)
S. Middelkamp, G. Theocharis, P.G. Kevrekidis, D.J. Frantzeskakis, P. Schmelcher, Phys. Rev. A 81, 053618 (2010)
G. Theocharis, A. Weller, J.P. Ronzheimer, C. Gross, M.K. Oberthaler, P.G. Kevrekidis, D.J. Frantzeskakis, Phys. Rev. A 81, 063604 (2010)
A.M. Mateo, V. Delgado, B.A. Malomed, Phys. Rev. A 83, 053610 (2011)
W.B. Cardoso, J. Zeng, A.T. Avelar, D. Bazeia, B.A. Malomed, Phys. Rev. E 88, 025201 (2013)
A.M. Mateo, V. Delgado, Phys. Rev. E 88, 042916 (2013)
T. Yang, A.J. Henning, K.A. Benedict, J. Phys. B At. Mol. Opt. Phys. 47, 035302 (2014)
H.L.C. Couto, A.T. Avelar, W.B. Cardoso, Ann. Phys. 530, 1700352 (2018)
M.C.P. dos Santos, B.A. Malomed, W.B. Cardoso, J. Phys. B At. Mol. Opt. Phys. 52, 245301 (2019)
B. Mottelson, Phys. Rev. Lett. 83, 2695 (1999)
G.F. Bertsch, T. Papenbrock, Phys. Rev. Lett. 83, 5412 (1999)
A.L. Fetter, Rev. Mod. Phys. 81, 647 (2009)
B.A. Malomed, Physica D 399, 108 (2019)
P. Muruganandam, S.K. Adhikari, Comput. Phys. Commun. 180, 1888 (2009)
J.H. Nguyen, P. Dyke, D. Luo, B.A. Malomed, R.G. Hulet, Nature Physics 10, 918 (2014)
Y.S. Kivshar, B.A. Malomed, Rev. Mod. Phys. 61, 763 (1989)
S. Gatz, J. Herrmann, J. Opt. Soc. Am. B 8, 2296 (1991)
L. Khaykovich, B.A. Malomed, Phys. Rev. A 74, 023607 (2006)
M.C.P. dos Santos, B.A. Malomed, W.B. Cardoso, Phys. Rev. E 102, 042209 (2020)
D. Mihalache, D. Mazilu, B.A. Malomed, F. Lederer, Phys. Rev. A 73, 043615 (2006)
T.J. Alexander, L. Bergé, Phys. Rev. E 65, 026611 (2002)
H. Saito, M. Ueda, Phys. Rev. Lett. 89, 190402 (2002)
E. Shamriz, Z. Chen, B. A. Malomed, and H. Sakaguchi, Singular mean-field states: a brief review of recent results. Condensed Matter 5, 20 (2020)
Acknowledgements
The authors acknowledge financial support from the Brazilian agencies CNPq (#304073/2016-4 & #425718/2018-2), CAPES, and FAPEG (PRONEM #201710267000540 & PRONEX #201710267000503). This work was performed as part of the Brazilian National Institute of Science and Technology (INCT) for Quantum Information (#465469/2014-0). The work B.A.M. is supported, in part, by the Israel Science Foundation through Grant No. 1286/17.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Santos, M.C.P.d., Cardoso, W.B. & Malomed, B.A. An effective equation for quasi-one-dimensional funnel-shaped Bose–Einstein condensates with embedded vorticity. Eur. Phys. J. Spec. Top. 231, 283–295 (2022). https://doi.org/10.1140/epjs/s11734-021-00351-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjs/s11734-021-00351-2