Gross, E. P. Unified theory of interacting bosons. Phys. Rev. 106, 161 (1957).
Gross, E. P. Classical theory of boson wave fields. Ann. Phys. 4, 57â74 (1958).
Andreev, A. F. & Lifshitz, I. M. Quantum theory of defects in crystals. J. Exp. Theor. Phys. 56, 2057â2068 (1969).
Chester, G. V. Speculations on Bose-Einstein condensation and quantum crystals. Phys. Rev. A 2, 256â258 (1970).
Leggett, A. J. Can a solid be âsuperfluidâ? Phys. Rev. Lett. 25, 1543 (1970).
Li, J.-R. et al. A stripe phase with supersolid properties in spinâorbit-coupled BoseâEinstein condensates. Nature 543, 91â94 (2017).
Léonard, J., Morales, A., Zupancic, P., Esslinger, T. & Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature 543, 87â90 (2017).
Böttcher, F. et al. Transient supersolid properties in an array of dipolar quantum droplets. Phys. Rev. X 9, 011051 (2019).
Tanzi, L. et al. Observation of a dipolar quantum gas with metastable supersolid properties. Phys. Rev. Lett. 122, 130405 (2019).
Chomaz, L. et al. Long-lived and transient supersolid behaviors in dipolar quantum gases. Phys. Rev. X 9, 021012 (2019).
Norcia, M. A. et al. Two-dimensional supersolidity in a dipolar quantum gas. Nature 596, 357â361 (2021).
Chomaz, L. et al. Dipolar physics: a review of experiments with magnetic quantum gases. Rep. Prog. Phys 86, 026401 (2022).
Guo, M. et al. The low-energy Goldstone mode in a trapped dipolar supersolid. Nature 564, 386â389 (2019).
Klaus, L. et al. Observation of vortices and vortex stripes in a dipolar condensate. Nat. Phys. 18, 1453â1458 (2022).
Bland, T., Lamporesi, G., Mark, M. J. & Ferlaino, F. Vortices in dipolar BoseâEinstein condensates. C. R. Phys. 24, 133â152 (2023).
Norcia, M. A. et al. Can angular oscillations probe superfluidity in dipolar supersolids? Phys. Rev. Lett. 129, 040403 (2022).
Bland, T. et al. Two-dimensional supersolid formation in dipolar condensates. Phys. Rev. Lett. 128, 195302 (2022).
Poli, E. et al. Glitches in rotating supersolids. Phys. Rev. Lett. 131, 223401 (2023).
Onsager, L. Discussion on a paper by C. J. Gorter. Nuovo Cimento Suppl. 6, 249â250 (1949).
Feynman, R. P. in Progress in Low Temperature Physics (ed. Gorter, C. J.) 17â53 (Elsevier, 1955).
Yarmchuk, E., Gordon, M. & Packard, R. Observation of stationary vortex arrays in rotating superfluid helium. Phys. Rev. Lett. 43, 214 (1979).
Bewley, G. P., Lathrop, D. P. & Sreenivasan, K. R. Visualization of quantized vortices. Nature 441, 588 (2006).
Abo-Shaeer, J. R., Raman, C., Vogels, J. M. & Ketterle, W. Observation of vortex lattices in Bose-Einstein condensates. Science 292, 476â479 (2001).
Zwierlein, M. W., Abo-Shaeer, J. R., Schirotzek, A., Schunck, C. H. & Ketterle, W. Vortices and superfluidity in a strongly interacting Fermi gas. Nature 435, 1047â1051 (2005).
Lagoudakis, K. G. et al. Quantized vortices in an excitonâpolariton condensate. Nat. Phys. 4, 706â710 (2008).
Wells, F. S., Pan, A. V., Wang, X. R., Fedoseev, S. A. & Hilgenkamp, H. Analysis of low-field isotropic vortex glass containing vortex groups in YBa2Cu3O7âx thin films visualized by scanning SQUID microscopy. Sci. Rep. 5, 8677 (2015).
Embon, L. et al. Imaging of super-fast dynamics and flow instabilities of superconducting vortices. Nat. Commun. 8, 85 (2017).
Hamidian, M. et al. Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x. Nature 532, 343â347 (2016).
Nyéki, J. et al. Intertwined superfluid and density wave order in two-dimensional 4He. Nat. Phys. 13, 455â459 (2017).
Levitin, L. V. et al. Evidence for a spatially modulated superfluid phase of 3He under confinement. Phys. Rev. Lett. 122, 085301 (2019).
Agterberg, D. F. et al. The physics of pair-density waves: cuprate superconductors and beyond. Annu. Rev. Condens. Matter Phys. 11, 231â270 (2020).
Liu, Y. et al. Pair density wave state in a monolayer high-Tc iron-based superconductor. Nature 618, 934â939 (2023).
Maragò, O. M. et al. Observation of the scissors mode and evidence for superfluidity of a trapped Bose-Einstein condensed gas. Phys. Rev. Lett. 84, 2056 (2000).
Tanzi, L. et al. Evidence of superfluidity in a dipolar supersolid from nonclassical rotational inertia. Science 371, 1162â1165 (2021).
Roccuzzo, S., Recati, A. & Stringari, S. Moment of inertia and dynamical rotational response of a supersolid dipolar gas. Phys. Rev. A 105, 023316 (2022).
GallemÃ, A., Roccuzzo, S., Stringari, S. & Recati, A. Quantized vortices in dipolar supersolid Bose-Einstein-condensed gases. Phys. Rev. A 102, 023322 (2020).
Roccuzzo, S., GallemÃ, A., Recati, A. & Stringari, S. Rotating a supersolid dipolar gas. Phys. Rev. Lett. 124, 045702 (2020).
Henkel, N., Cinti, F., Jain, P., Pupillo, G. & Pohl, T. Supersolid vortex crystals in Rydberg-dressed Bose-Einstein condensates. Phys. Rev. Lett. 108, 265301 (2012).
Ancilotto, F., Barranco, M., Pi, M. & Reatto, L. Vortex properties in the extended supersolid phase of dipolar Bose-Einstein condensates. Phys. Rev. A 103, 033314 (2021).
Matsushita, T. Flux Pinning in Superconductors (Springer, 2014).
Prasad, S. B., Bland, T., Mulkerin, B. C., Parker, N. G. & Martin, A. M. Vortex lattice formation in dipolar Bose-Einstein condensates via rotation of the polarization. Phys. Rev. A 100, 023625 (2019).
Stuhler, J. et al. Observation of dipole-dipole interaction in a degenerate quantum gas. Phys. Rev. Lett. 95, 150406 (2005).
Wächtler, F. & Santos, L. Quantum filaments in dipolar Bose-Einstein condensates. Phys. Rev. A 93, 061603 (2016).
Ferrier-Barbut, I., Kadau, H., Schmitt, M., Wenzel, M. & Pfau, T. Observation of quantum droplets in a strongly dipolar Bose gas. Phys. Rev. Lett. 116, 215301 (2016).
Chomaz, L. et al. Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid. Phys. Rev. X 6, 041039 (2016).
Bisset, R. N., Wilson, R. M., Baillie, D. & Blakie, P. B. Ground-state phase diagram of a dipolar condensate with quantum fluctuations. Phys. Rev. A 94, 033619 (2016).
Recati, A., Zambelli, F. & Stringari, S. Overcritical rotation of a trapped Bose-Einstein condensate. Phys. Rev. Lett. 86, 377 (2001).
Sinha, S. & Castin, Y. Dynamic instability of a rotating Bose-Einstein condensate. Phys. Rev. Lett. 87, 190402 (2001).
Madison, K. W., Chevy, F., Bretin, V. & Dalibard, J. Stationary states of a rotating Bose-Einstein condensate: routes to vortex nucleation. Phys. Rev. Lett. 86, 4443 (2001).
van Bijnen, R. M., OâDell, D. H., Parker, N. G. & Martin, A. Dynamical instability of a rotating dipolar Bose-Einstein condensate. Phys. Rev. Lett. 98, 150401 (2007).
Sohmen, M. et al. Birth, life, and death of a dipolar supersolid. Phys. Rev. Lett. 126, 233401 (2021).
Matthews, M. R. et al. Vortices in a Bose-Einstein condensate. Phys. Rev. Lett. 83, 2498 (1999).
Madison, K. W., Chevy, F., Wohlleben, W. & Dalibard, J. Vortex formation in a stirred Bose-Einstein condensate. Phys. Rev. Lett. 84, 806 (2000).
Å indik, M., Recati, A., Roccuzzo, S. M., Santos, L. & Stringari, S. Creation and robustness of quantized vortices in a dipolar supersolid when crossing the superfluid-to-supersolid transition. Phys. Rev. A 106, L061303 (2022).
Poli, E. et al. Maintaining supersolidity in one and two dimensions. Phys. Rev. A 104, 063307 (2021).
Pomeau, Y. & Rica, S. Dynamics of a model of supersolid. Phys. Rev. Lett. 72, 2426 (1994).
Blatter, G., Feigelâman, M. V., Geshkenbein, V. B., Larkin, A. I. & Vinokur, V. M. Vortices in high-temperature superconductors. Rev. Mod. Phys. 66, 1125 (1994).
Kwok, W.-K. et al. Vortices in high-performance high-temperature superconductors. Rep. Prog. Phys. 79, 116501 (2016).
Chamel, N. Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids. Phys. Rev. C 85, 035801 (2012).
Tang, Y., Sykes, A., Burdick, N. Q., Bohn, J. L. & Lev, B. L. s-wave scattering lengths of the strongly dipolar bosons 162Dy and 164Dy. Phys. Rev. A 92, 022703 (2015).
Maier, T. et al. Broad universal Feshbach resonances in the chaotic spectrum of dysprosium atoms. Phys. Rev. A 92, 060702 (2015).
Tang, Y. et al. Anisotropic expansion of a thermal dipolar Bose gas. Phys. Rev. Lett. 117, 155301 (2016).
Lima, A. R. P. & Pelster, A. Quantum fluctuations in dipolar Bose gases. Phys. Rev. A 84, 041604 (2011).
Blakie, P., Bradley, A., Davis, M., Ballagh, R. & Gardiner, C. Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques. Adv. Phys. 57, 363â455 (2008).
Cidrim, A., dos Santos, F. E., Henn, E. A. & Macrì, T. Vortices in self-bound dipolar droplets. Phys. Rev. A 98, 023618 (2018).
Lee, A.-C., Baillie, D., Bisset, R. N. & Blakie, P. B. Excitations of a vortex line in an elongated dipolar condensate. Phys. Rev. A 98, 063620 (2018).
Lee, A.-C., Baillie, D. & Blakie, P. B. Numerical calculation of dipolar-quantum-droplet stationary states. Phys. Rev. Res. 3, 013283 (2021).
Li, G. et al. Strongly anisotropic vortices in dipolar quantum droplets. Phys. Rev. Lett. 133, 053804 (2024).
Scherer, D. R., Weiler, C. N., Neely, T. W. & Anderson, B. P. Vortex formation by merging of multiple trapped Bose-Einstein condensates. Phys. Rev. Lett. 98, 110402 (2007).
Stringari, S. & Pitaevskii, L. Bose-Einstein Condensation and Superfluidity (Oxford Univ. Press, 2016).
Powell, M. J. D. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J. 7, 155â162 (1964).
Casotti, E. et al. Data from âObservation of vortices in a dipolar supersolidâ. Zenodo https://doi.org/10.5281/zenodo.10695943 (2024).
