Dipolar interactions attract a lot of attention because of their strong anisotropic character.
In the field of ultracold dilute atoms there is an additional interest due to the long range of dipolar interactions, compared to short-ranged

-wave scattering.
The properties of dipolar systems were addressed in zero-temperature quantum regime, and at a finite temperature.

An ultracold dilute Bose gas forms a condensate, described by a macroscopic wave function with a certain phase. If we consider two layers with dipoles, separated by a large distance, there will be two condensates with independent phases. For very small separation between layers, dipoles from different layers form dimers (molecules). Being again bosons, dimers form a Bose-Einstein condensate with a single phase, effectively locking the phase between different layers. This transition between two Bose condensates into a single one is an unusual phenomena for which we provide a detailed description.

[1] Tommaso Comparin, Raul Bombin, Markus Holzmann, Ferran Mazzanti, Jordi Boronat, Stefano Giorgini,

*Two-dimensional Mixture of Dipolar Fermions: Equation of State and Magnetic Phases,*

, (2019)

[2] J. Sánchez-Baena, F. Mazzanti, J. Boronat,

*Few trapped quantum dipoles: quantum versus classical structures,*

New Journal of Physics **20**, 013038 (2018)

[3] R. Bombin, J. Boronat, and F. Mazzanti,

*Dipolar Bose Supersolid Stripes,*

Phys. Rev. Lett. **119**, 250402 (2017)

[4] A. Macia, J. Sánchez-Baena, J. Boronat, F. Mazzanti,

*Droplets of trapped quantum dipolar bosons,*

Phys. Rev. Lett. **117**, 205301 (2016)

[5] A. Macia, G. E. Astrakharchik, F. Mazzanti, S. Giorgini, J. Boronat,

*Single-particle versus pair superfluidity in a bilayer system of dipolar bosons,*

Phys. Rev. A **90**, 043623 (2014)

[6] A. Macia, J. Boronat, F. Mazzanti,

*Phase diagram of dipolar bosons in 2D with tilted polarization,*

Phys. Rev. A **90**, 061601(R)/ (2014)

[7] A. Macia, D. Hufnagl, F. Mazzanti, J. Boronat, R. Zillich,

*Excitations and stripe phase formation in a two-dimensional dipolar Bose
gas with tilted polarization,*

Phys. Rev. Lett. **109**, 235307 (2012)

[8] A. Macia, F. Mazzanti, J. Boronat,

*Ground state properties and excitation spectrum of a two dimensional gas of bosonic
dipoles,*

European Physical Journal D **66**, 301 (2012)

[9] M. D. Girardeau and G. E. Astrakharchik,

*Super-Tonks-Girardeau State in an Attractive One-Dimensional Dipolar Gas,*

Phys. Rev. Lett. **109**, 235305 (2012)

[10] A. E. Golomedov, G. E. Astrakharchik, and Yu. E. Lozovik,

*Mesoscopic supersolid of dipoles in a trap,*

Phys. Rev. A **84**, 033615 (2011)

[11] A. Macia, F. Mazzanti, J. Boronat, R. Zillich,

*Microscopic description of anisotropic low-density dipolar Bose gases in two
dimensions,*

Phys. Rev. A **84**, 033625 (2011)

[12] I. L. Kurbakov, Yu. E. Lozovik, G. E. Astrakharchik, and J. Boronat,

*Quasiequilibrium supersolid phase of a two-dimensional dipolar crystal,*

Phys. Rev. B **82**, 014508 (2010)

[13] G. E. Astrakharchik, G. De Chiara, Giovanna Morigi, and J. Boronat,

*Thermal and quantum fluctuations in low dimensional dipolar chains,*

J. Phys. B: At. Mol. Opt. Phys. **42**, 154026 (2009)

[14] G. E. Astrakharchik, Giovanna Morigi, Gabriele De Chiara, and J.
Boronat,

*Ground state of low dimensional dipolar gases: linear, zigzag
and multiple chains,*

Virtual Journal of Nanoscale Science &
Technology **19**, Issue 2 Jan (2009)

[15] G. E. Astrakharchik and Yu. E. Lozovik,

*Super-Tonks-Girardeau regime in trapped one-dimensional dipolar gases ,*

Phys. Rev. A **77**, 013404 (2008)

[16] G. E. Astrakharchik, Giovanna Morigi, Gabriele De Chiara, and J.
Boronat,

*Ground state of low-dimensional dipolar gases: Linear and zigzag chains,*

Phys. Rev. A **78**, 063622 (2008)

[17] G. E. Astrakharchik, J. Boronat, J. Casulleras, I. L. Kurbakov, and
Yu. E. Lozovik,

*Ground state properties of a homogeneous 2D system of Bosons with dipolar interactions,*

Recent Progress in Many Body Theories, Proceedings of the Fourteenth International Conference, edited J. Boronat, G. E. Astrakharchik and F. Mazzanti, (World Scientific, Singapore, 2008 , 245 (2008)

[18] Yu. E. Lozovik, I. L. Kurbakov, G. E. Astrakharchik, M. Willander,

*Bose condensation of two-dimensional dipolar excitons: Simulation by the quantum Monte Carlo method,*

JETP **106**, 296 (2008)

[19] G.E. Astrakharchik, J. Boronat, I.L. Kurbakov, and Yu.E. Lozovik,

*Quantum phase transition in a two-dimensional system of dipoles,*

Phys. Rev. Lett. **98**, 060405 (2007)

[20] G.E. Astrakharchik, J. Boronat, J. Casulleras, I.L. Kurbakov, and
Yu.E. Lozovik,

*Weakly interacting two-dimensional system of dipoles: limitations of the mean-field theory,*

Phys. Rev. A **75**, 063630 (2007)

[21] Yu. E. Lozovik, I.L. Kurbakov, G.E. Astrakharchik, J. Boronat, and
Magnus Willander,

*Effects of strong correlations for 2D Bose-Einstein condensed dipolar excitons,*

Solid State Communications **144**, 399 (2007)

[22] A. S. Arkhipov, G. E. Astrakharchik, A. V. Belikov, and Yu. E. Lozovik ,

*Ground-state properties of a one-dimensional system of dipoles,*

Pis’ma Zh. Eksp. Teor. Fiz. **82**, 41 (2005)

[1] A. Macia, J. Boronat, F. Mazzanti,

*Phase diagram of dipolar bosons in 2D with tilted polarization,*

Phys. Rev. A **90**, 061601(R)/ (2014)

[2] A. Macia, F. Mazzanti, J. Boronat, R. Zillich,

*Microscopic description of anisotropic low-density dipolar Bose gases in two
dimensions,*

Phys. Rev. A **84**, 033625 (2011)

[3] G. E. Astrakharchik, G. De Chiara, Giovanna Morigi, and J. Boronat,

*Thermal and quantum fluctuations in low dimensional dipolar chains,*

J. Phys. B: At. Mol. Opt. Phys. **42**, 154026 (2009)

[4] G. E. Astrakharchik, Giovanna Morigi, Gabriele De Chiara, and J.
Boronat,

*Ground state of low dimensional dipolar gases: linear, zigzag
and multiple chains,*

Virtual Journal of Nanoscale Science &
Technology **19**, Issue 2 Jan (2009)

[5] G. E. Astrakharchik, Giovanna Morigi, Gabriele De Chiara, and J.
Boronat,

*Ground state of low-dimensional dipolar gases: Linear and zigzag chains,*

Phys. Rev. A **78**, 063622 (2008)

[6] G. E. Astrakharchik and S. Giorgini,

*Quantum Monte Carlo study of the three- to one-dimensional crossover for a trapped Bose gas,*

Phys. Rev. A **66**, 053614 (2002)