# Bose-Einstein condensates

**Introduction**

Bose-Einstein condensation is a manifestation of macroscopic occupation of a single quantum state. The idea of such phenomenon dates back to 1924-1925, when Albert Einstein extended the statistical arguments presented by Satyendra Nath Bose to systems consisting of a conserved number of bosonic particles. Einstein recognised that at sufficiently low temperatures the quantum statistical distribution of an ideal gas of bosons shows condensation of a macroscopic fraction of the material into the ground state of the system. This phenomenon, subsequently termed Bose-Einstein condensation (BEC), is a unique, purely quantum mechanical phase transition in the sense that it occurs in principle even in noninteracting bosonic systems.

A breakthrough in realizing the original idea of BEC occurred in 1995, when research groups at University of Colorado, MIT, and Rice University observed convincing evidence of BEC in dilute alkali atom clouds. These pioneering experiments, from which the Nobel prize was awarded in 2001, launched an avalanche of theoretical and experimental research on the physics of weakly interacting atomic condensates. Due to weak interactions, these systems are rare examples of interacting quantum gases amenable to detailed quantitative analysis, and thus provide unique possibilities for testing the fundamental principles and theories of many-particle quantum physics and even simulating physical phenomena and devices.

**Research at QCD**

One of the most fundamental branches of research in QCD group is the studies on coherent matter fields, that is Bose-Einstein condensates (BECs) and Fermi condensates of dilute atomic gases. See the bottom of this page for a general introduction. In BEC research community, we are well-known from the development of methods to create multi-quantum vortices into the condensate by adiabatic control of the external magnetic fields. Furthermore, we have introduced cyclic vortex creation procedure, which resulted in a so-called vortex pump, that is, a method to cyclically increase the vorticity of the system rendering it possible to create giant vortices [Phys. Rev. Lett. 99, 250406 (2007); J. Low Temp. Phys. 161, pp. 561 (2010)]. Also the dynamics of these giant vortices is of our interest [Phys. Rev. Lett. 97, 110406 (2006); Phys. Rev. Lett. 99, 200403 (2007); Phys. Rev. A 81, 033627 (2010)]. We have also studied so-called coreless vortices in spinor BECs [Phys. Rev. A 79, 023618 (2009)].

In addition to vortices, which are line like topological defects, we are studying point-like defects in BECs. We have investigated the creation of non-Abelian monopoles in these systems [Phys. Rev. Lett. 102, 080403 (2009)]. We have also introduced a robust method to create Dirac monopoles into the condensate utilizing the external magnetic fields [Phys. Rev. Lett. 103, 030401 (2009) PRESS RELEASE]. Our recent research highlight is the experimental observation of Dirac monopoles in the synthetic magnetic field [Nature (2014) PRESS RELEASE], achieved in collaboration with Prof. David S. Hall’s group (Amherst College, USA).

Our research interests also span multi-component and rotating BECs [Phys. Rev. A 85, 043613 (2012)] as well as elementary excitations, both dipolar and spin-orbit coupled condensates [Phys. Rev. A 82, 053616 (2010);Phys. Rev. A 84, 043638 (2011); Phys. Rev. A, 86, 051607(R) (2012)] and vortex dipoles [Phys. Rev. A 83, 011603(R) (2011)].

**Monopoles in BECs**

The question of the existence of a magnetic monopole dates thousands of years back in history when the ancient Greek philosophers found that when breaking a magnetic rock, a new south and north pole is created to the break points. In 1931, Paul Dirac made his initial studies on monopoles in quantum fields. He combined arguments from quantum mechanics and classical electrodynamics and identified electromagnetic potentials consistent with the existence of magnetic monopoles. Dirac found that the vector potential exhibits a nonphysical line singularity, or Dirac string, terminating at the monopole. Because of far-reaching implications to classical, quantum, and particle physics, the experimental observation of monopoles in quantum fields is of great interest. Due to failure of all experiments trying to find monopoles in real electromagnetic fields, there have been many suggestions to experimentally observe monopoles in other analogous quantum fields.

In Ref. [Phys. Rev. Lett. 102, 080403 (2009)], we studied so-called non-Abelian monopoles induced by an artificial magnetic field due to a laser driving transitions between the different hyperfine spin states of the atoms. Although the non-Abelian monopole is conceptually interesting and desirable to realize in the experiments, it has the drawback that the monopole forms to the pseudo spin and not the true hyperfine spin of the atoms. Furthermore, the non-Abelian nature may be challenging to measure in the experiments. Thus we began to study how to create Abelian Dirac monopoles in the condensates by using the phase imprinting method. It turned out that by using the combination of a homogeneous bias field and a three-dimensional quadrupole field, it is possible to create a Dirac monopole [Phys. Rev. Lett. 103, 030401 (2009)] into the vorticity field of the condensate. The figure below shows how the creation of the Dirac monopole by adiabatically moving the zero-point of the magnetic field into the optically trapped BEC that is initially in its defect-free ground state.

**Fig.** (a) Particle density isosurfaces (cut for clarity) and (b) vorticity of the condensate during the creation process of a Dirac monopole. In panel (b), the monopole is manifested in the hedgehog structure of the vorticity and the Dirac string is clearly visible as the red line terminating at the monopole. In panel (a), the Dirac string shows as a depletion of the particle density along it.

Four years later, the long-awaited first experimental realization and observation of the Dirac monopole in a quantum field was achieved using this method. This achievement was a fruit of our collaboration with the group of Prof. David S. Hall (Amherst College, USA) and our results were published in Nature [Nature (2014)]. The figure below shows the comparison between the experimental and the simulated data when the Dirac monopole is near the center of the atomic cloud.

The experimental discovery of the monopole opens up new research avenues. The dynamics and the decay of the monopole are of particular interest in our future studies as the monopole is not created in the ground state [Phys. Rev. A 84, 063627 (2011)]. Other possible future aspects include the study of the interaction of the monopole with other topological defects as well as the phase tomography of the spinor components in the monopole state with spin rotations.

**Fig.** Experimental (a),(c) and simulated (b),(d) particle densities in the spinor components when the monopole is near the center of the condensate. The Dirac string is visible as depleted density in spinor components l-1> and l0>.

**Vortices in BECs**

In a contrast to superfluid 4He, BEC of alkali-metal atoms has the hyperfine spin degree of freedom which can be utilized to create vortices. We have been studying a method which is referred to as the topological phase imprinting. We have shown that with suitable choice of external magnetic fields, it is possible to create two- and four-quantum vortices into the condensate [Phys. Rev. A 66, 013617 (2002); J. Phys.: Condens. Matter 14, 13481 (2002)]. When the fields are adjusted such that the condensate dynamics is adiabatic the result can be interpreted as the accumulation of geometric phase, the Berry’s phase, for individual spins of the condensate atoms. This method has been experimentally verified [Phys. Rev. Lett. 89, 190403 (2002)] together with observations on the splitting of the vortex [Phys. Rev. Lett. 93, 160406 (2004)] which we have also managed to simulate successfully [Phys. Rev. A 68, 023611 (2003); Phys. Rev. Lett. 97, 110406 (2006)]. Since our method relies only on the topology of the external magnetic fields, it is particularly robust for the possible noise in the external fields. We extended this idea of topological vortex formation into a so-called vortex pump, in which the vorticity of the condensate is increasing by a constant amount during each pumping cycle [Phys. Rev. Lett. 99, 250406 (2007)]. A modified pumping cycle utilizing time-averaged orbiting potential trap was recently investigated and the results were published in Ref. [Phys. Rev. A 87, 033623 (2013)]. See also closely related Refs. [J. Low Temp. Phys. 161, 561 (2010); Phys. Rev. A 81, 023603 (2010)]. For more information on the topological vortex formation see our article [Topological vortex creation in spinor Bose-Einstein condensates, in a book Electromagnetic, magnetostatic, and exchange-interaction vortices in confined magnetic structures, E. O. Kamenetskii (Eds.), (Research signpost, Kerala), ISBN 978-81-7895-373-1, 2008].

**Fig.** Pumping vortices in BECs. Particle density (left columns) and complex Phase (right columns) of the condensate.

One of our research interests is the structure and stability of multiply quantized vortices in dilute atomic Bose-Einstein condensates. Doubly and quadruply quantized vortices can be created with the topological phase engineering technique which is also utilized in the vortex pump. Multiply quantized vortices are in general energetically unstable, and they tend to split into singly quantized vortices. The vortices usually intertwine strongly as they split, as shown in the isosurface plot of the condensate density below. Also, dissipation is negligible in the zero temperature limit, and the excess energy is pushed into excitation of surface modes during the splitting process.

We have studied the stability properties of multiquantum vortices in different trap geometries by solving the Bogoliubov excitation spectra in Ref. [Phys. Rev. A 74, 063619 (2006)]. We have also calculated the splitting times of doubly quantized vortices as a function of particle number in elongated condensates in Ref. [Phys. Rev. Lett. 97, 110406 (2006)] and investigated the dynamics of the splitting of quadruply quantized vortices in Ref. [Phys. Rev. Lett. 99, 200403 (2007)]. More recently, we studied how giant vortices with very many circulation quanta split in pancake-shaped condensates and observed that they divide the condensate into a few vortex free regions separated by vortex sheets [Phys. Rev. A 81, 023603 (2010)].

**Fig.** Splitting of a doubly quantized vortex. Cropped particle density isosurface.

**Phase transitions**

We have investigated various phase transitions in the context of spinor BECs. Low-dimensional systems cannot support true long-range order, but the superfluid phase transition is still possible via the Berezinskii-Kosterlitz-Thouless (BKT) mechanism. In the past, we have studied the BKT transition in antiferromagnetic spinor BECs [Phys. Rev. A 81, 033616 (2010)]. In this case, the BKT transition is mediated by half-quantum vortices (HQVs) and we analysed the thermal activation of HQVs in the experimentally relevant trapped quasi-2D systems. We found that the crossover temperature is shifted upwards if skyrmions are allowed, and above the defect binding temperatures we observe transitions corresponding to the onset of a coherent condensate and a quasi-condensate. We have also investigated the instabilities induced by the long-range magnetic dipole-dipole interaction using renormalization group methods [Phys. Rev. A 84, 013605 (2011)]. In the zero-temperature limit where quantum fluctuations prevail, we found the phase diagram to be unaffected by the dipole-dipole interaction. When the thermal fluctuations dominate, polar and ferromagnetic condensates with dipole-dipole interaction become unstable. On the other hand, a so-called spin-singlet condensate remains stable even in the presence of dipole-dipole interactions.

**Fig.** Renormalization group flow for a three dimensional spin-1 Bose gas at fixed temperature.