Phase transition between two-component and three-component ground states of spin-1 Bose-Einstein condensates

Compared with the scalar Bose-Einstein condensate, the spinor Bose-Einstein condensate, in which internal degrees of freedom are essentially free, has aroused the great interest in the study of topological excitations. In particular, the spinor Bose-Einstein condensate with rotation provides a new opportunity for studying novel quantum states including a coreless vortex and vortex lattice. To date, in the presence of rotation, a great many of studies on the topological excitations have focused on the Bose-Einstein condensate system with the uniform Zeeman field or without external magnetic field. However, the ground state structure of a rotating Bose-Einstein condensate in the presence of in-plane gradient-magnetic-field remains an open question. In this work, by using the imaginary-time propagation method, we study the ground state structure of a rotating Bose-Einstein condensate with in-plane quadrupole field. We first examine the effect of in-plane quadrupole field on trapped spinor Bose-Einstein condensate. The numerical results show that Mermin-Ho vortex can be induced only by the cooperation between quadrupole field and rotation. When magnetic field gradient is increased, the vortices around Mermin-Ho vortex display the symmetrical arrangement. For an even larger magnetic field gradient strength, the system only presents the Mermin-Ho vortex because the in-plane quadrupole field can prevent the vortices around Mermin-Ho vortex from occurring. Next, we examine the effect of the rotation on trapped spinor Bose-Einstein condensate. A phase transition from a polar-core vortex to a Mermin-Ho vortex is found through applying a rotational potential, which is caused by the cooperation between the in-plane quadrupole field and the rotation. We further study the combined effects of spin exchange interaction and density-density interaction. The results confirm that in the presence of the quadrupole field both spin exchange interaction and density-density interaction, acting as controllable parameters, can control the number of the vortices around Mermin-Ho vortex. The corresponding number of the vortices shows step behavior with increasing the ratio between spin exchange interaction and density-density interaction, which behaves as hexagon, pentagon, square and triangle. It is found that two types of topology structures, i.e., the hyperbolic meron and half-skyrmion, can occur in the present system. These vortex structures can be realized via time-of-flight absorption imaging technique. Our results not only provide an opportunity to investigate the exotic vortex structures and the corresponding phase transitions in a controlled platform, but also lay the foundation for the study of topological defect subjected to gauge field and dipolar interaction in future.

Bose–Einstein condensation in a vapor of sodium atoms in an electric field

The first BEC conference in Levico in 1993

To engineer the wavefunction of a Bose–Einstein condensate is to exert control over both the density and phase of the Bose–Einstein condensate order parameter. Having the ability to engineer the condensate order parameter down to the smallest length scale relevant to condensate dynamics—the healing length scale—would enable the study of new combinations of topological defects and may pave the way to using Bose–Einstein condensates as versatile, precise quantum simulators. In this thesis, we present a new wavefunction engineering technique which reaches the sub-optical healing length scale. Influenced by magnetic resonance imaging, we name this technique magnetic resonance control. This technique uses time-varying coupling between internal spin states of a spinor Bose–Einstein condensate within a magnetic field gradient to address spatial regions of the condensate, enabling control over both the phase and the density of the condensate order parameter down to the healing length scale. Techniques already exist to engineer condensate wavefunctions, but not with such a fine degree of control. These techniques primarily rely on either the intensity variation of a laser beam, which limits the resolution to the diffraction limit (larger than the typically sub-optical healing length scale), or the adiabatic inversion of a magnetic trapping potential, which can not be easily changed to produce a variety of structures in the condensate wavefunction. To develop our magnetic resonance control technique, we simulate a spinor condensate in one dimension with time-dependent coupling between spin states and time-dependent external magnetic field gradients using the Gross–Pitaevskii equation. We show that magnetic resonance control can engineer a single black soliton using experimentally feasible parameters. A black soliton is an ideal target state to select for this demonstration because engineering such a state requires control over both the phase and density of the condensate with healing-length resolution. We demonstrate that magnetic resonance control can be extended to engineer more complicated target states by simulating the creation of multiple solitons in a condensate, with control over the initial positions and trajectories of the solitons. When magnetic resonance control is applied to Bose–Einstein condensates in the laboratory, it will be necessary to have an imaging system capable of resolving the fine structures created. As an alternative to high-cost, custom-manufactured lenses, and in-vacuum optical systems, I have designed and bench-tested an objective lens with a high numerical aperture (0.36) and a long working distance (35 mm) consisting of standard catalogue lenses. Using 780 nm light, suitable for imaging rubidium condensates, this objective can achieve a resolution of 1.3 μm across a diffraction-limited field of view of 360 μm through a 5 mm thick glass window of a science cell. By changing the spacing between the lens elements, this objective lens can compensate for the aberrations produced by a glass window up to 15 mm thick, and by changing the aperture size the objective becomes suitable for diffraction-limited monochromatic imaging on the D2 line of all the alkalis. Before performing proof-of-principle magnetic resonance control experiments on real Bose–Einstein condensates, we needed to construct an experimental apparatus capable of producing spinor Bose–Einstein condensates. In this thesis I summarise my main contributions to this group endeavour, including: constructing the ultra-high vacuum system; supervising the bakeout of our vacuum system; designing and aligning the optical systems to produce laser beams of different, tunable frequencies to trap, cool, and image rubidium gas; trapping a cloud of rubidium atoms in a magneto-optical trap; constructing laser beam shutters, photodetectors, and photodetector signal filters; and designing and constructing our side imaging and top imaging systems. Using our spinor Bose–Einstein condensate apparatus, I performed the first proof-of-principle magnetic resonance control experiments. With a pair of condensates side-by-side, separated by 200 μm, we can use magnetic resonance control to invert the spin of one condensate only, while leaving the other condensate unaffected. We anticipate magnetic resonance control being used in the laboratory to engineer the first black soliton in a Bose–Einstein condensate. Looking beyond solitons, magnetic resonance control could have applications to the field of magnon spintronics, and extending the technique to higher dimensions could enable the study of exotic topological defects such as spin knots in a quantum fluid.

Quantum phase transitions universally exist in the ground and excited states of quantum many-body systems, and they have a close relationship with the nonequilibrium dynamical phase transitions, which however are challenging to identify. In the system of spin-1 Bose-Einstein condensates, though dynamical phase transitions with correspondence to equilibrium phase transitions in the ground state and uppermost excited state have been probed, those taking place in intermediate excited states remain untouched in experiments thus far. Here we unravel that both the ground- and excited-state quantum phase transitions in spinor condensates can be diagnosed with dynamical phase transitions. A connection between equilibrium phase transitions and nonequilibrium behaviors of the system is disclosed in terms of the quantum Fisher information. We also demonstrate that near the critical points parameter estimation beyond the standard quantum limit can be implemented. This work not only advances the exploration of excited-state quantum phase transitions via a scheme that can immediately be applied to a broad class of few-mode quantum systems, but also provides a new perspective on the relationship between quantum criticality and quantum enhanced sensing.

Spherical collapse of the Bose-Einstein Condensate (BEC) dark matter model is studied in the Thomas Fermi approximation. The evolution of the overdensity of the collapsed region and its expansion rate are calculated for two scenarios. We consider the case of a sharp phase transition (which happens when the critical temperature is reached) from the normal dark matter state to the condensate one and the case of a smooth first order phase transition where there is a continuous conversion of "normal" dark matter to the BEC phase. We present numerical results for the physics of the collapse for a wide range of the model's space parameter, i.e. the mass of the scalar particle $m_{\chi}$ and the scattering length $l_s$. We show the dependence of the transition redshift on $m_{\chi}$ and $l_s$. Since small scales collapse earlier and eventually before the BEC phase transition the evolution of collapsing halos in this limit is indeed the same in both the CDM and the BEC models. Differences are expected to appear only on the largest astrophysical scales. However, we argue that the BEC model is almost indistinguishable from the usual dark matter scenario concerning the evolution of nonlinear perturbations above typical clusters scales, i.e., $\gtrsim 10^{14}M_{\odot}$. This provides an analytical confirmation for recent results from cosmological numerical simulations [H.-Y. Schive {\it et al.}, Nature Physics, {\bf10}, 496 (2014)].

OpenMP solver for rotating spin-1 spin–orbit- and Rabi-coupled Bose–Einstein condensates

In this paper, as a personal review, we suppose a possible extension of Gibbs ensembletheory so that it can provide a reasonable description of phase transitions and spontaneoussymmetry breaking. The extension is founded on three hypotheses, and can beregarded as a microscopic edition of the Landau phenomenological theory of phasetransitions. Within its framework, the stable state of a system is determined by theevolution of order parameter with temperature according to such a principle thatthe entropy of the system will reach its minimum in this state. The evolution oforder parameter can cause a change in representation of the system Hamiltonian;different phases will realize different representations, respectively; a phase transitionamounts to a representation transformation. Physically, it turns out that phasetransitions originate from the automatic interference among matter waves as thetemperature is cooled down. Typical quantum many-body systems are studied with thisextended ensemble theory. We regain the Bardeen–Cooper–Schrieffer solution forthe weak-coupling superconductivity, and prove that it is stable. We find thatnegative-temperature and laser phases arise from the same mechanism as phase transitions,and that they are unstable. For the ideal Bose gas, we demonstrate that it willproduce Bose–Einstein condensation (BEC) in the thermodynamic limit, whichconfirms exactly Einstein’s deep physical insight. In contrast, there is no BEC eitherwithin the phonon gas in a black body or within the ideal photon gas in a solidbody. We prove that it is not admissible to quantize the Dirac field by usingBose–Einstein statistics. We show that a structural phase transition belongs physically tothe BEC happening in configuration space, and that a double-well anharmonicsystem will undergo a structural phase transition at a finite temperature. For theO(N)-symmetric vector model, we demonstrate that it will yield spontaneous symmetry breakingand produce Goldstone bosons; and if it is coupled with a gauge field, the gauge field willobtain a mass (Higgs mechanism). Also, we show that an interacting Bose gas is stable onlyif the interaction is repulsive. For the weak interaction case, we find that the BEC is a‘λ-transition’ and its transition temperature can be lowered by the repulsive interaction. In connection withliquid 4He, it is found that the specific heat at constant pressureCP willshow a T3 law at low temperatures, which is in agreement with the experiment.If the system is further cooled down, the theory predicts thatCP will vanish linearly as , which is anticipating experimental verifications.

The continuous crossover between a Bardeen-Cooper-Schrieffer (BCS)-type superfluid of fermion pairs and a Bose-Einstein condensate (BEC) of tightly bound bosonic molecules can be attributed to the spontaneous breaking of global $U(1)$ gauge symmetry which underlies both quantum condensation phenomena. Recently much attention has been paid to this problem, since Feshbach resonances allow for an experimental implementation of crossover physics in cold fermion gases. The strong interactions close to resonance call for an analysis beyond Mean Field Theory. We develop a systematic functional integral approach for the description of this phenomenon. Starting from a Yukawa-type atom-molecule model, a symmetry analysis allows to both construct the equation of state and to classify the thermodynamic phases in a unified way. The onset of superfluidity is signalled by the emergence of a massless Goldstone mode associated with the broken continuous U(1) symmetry. Beyond Mean Field, we include fluctuations of the molecule field self-consistently via the solution of suitable Schwinger-Dyson equations. The phase diagram is computed, and a variety of universal features are established. A new form of crossover from an exactly solvable narrow resonance limit to broad resonances or pointlike interactions is found. At low temperature our results agree well with quantum Monte Carlo simulations and recent experiments. Our approach is further developed in the frame of functional renormalization group equations. While the effective bosonic theory in the BEC regime shows the characteristics of a Bogoliubov theory for small temperatures, the phase transition is of second order.

We study the kinetic regime of the Bose-condensation of scalar particles with weak $\lambda \phi^4$ self-interaction. The Boltzmann equation is solved numerically. We consider two kinetic stages. At the first stage the condensate is still absent but there is a nonzero inflow of particles towards ${\bf p} = {\bf 0}$ and the distribution function at ${\bf p} ={\bf 0}$ grows from finite values to infinity in a finite time. We observe a profound similarity between Bose-condensation and Kolmogorov turbulence. At the second stage there are two components, the condensate and particles, reaching their equilibrium values. We show that the evolution in both stages proceeds in a self-similar way and find the time needed for condensation. We do not consider a phase transition from the first stage to the second. Condensation of self-interacting bosons is compared to the condensation driven by interaction with a cold gas of fermions; the latter turns out to be self-similar too. Exploiting the self-similarity we obtain a number of analytical results in all cases.

Two-component Bose-Einstein condensate offers an ideal platform for investigating many intriguing topological defects due to the interplay between intraspecies and interspecies interactions. The recent realization of spin-orbit coupling in two-component Bose-Einstein condensate, owing to coupling between the spin and the centre-of-mass motion of the atom, provides possibly new opportunities to search for novel quantum states. In particular, the gradient magnetic field in the Bose-Einstein condensate has brought a new way to create topologically nontrivial structures including Dirac monopoles and quantum knots. Previous studies of the gradient magnetic field effect in the Bose-Einstein condensate mainly focused on the three-component case. However, it remains unclear how the gradient magnetic field affects the ground state configuration in the rotating two-component Bose-Einstein condensate with spin-orbit coupling. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of a rotating two-component Bose-Einstein condensate with spin-orbit coupling and gradient magnetic field. We concentrate on the effects of the spin-orbit coupling and the gradient magnetic field on the ground state. The numerical results show that increasing the strength of the spin-orbit coupling can induce a phase transition from skyrmion lattice to skyrmion chain in the presence of the gradient magnetic field. Unlike the study of skyrmion in rotating two-component Bose-Einstein condensate with only spin-orbit coupling, the skyrmion chain can occur under the isotropic spin-orbit coupling when the gradient magnetic field is considered. It is worth noting that the skyrmion chain here is arrayed along the diagonal direction. Next we examine the effect of the gradient magnetic field on spin-orbit coupled two-component Bose-Einstein condensate. For the case of weak spin-orbit coupling and the slow rotation, a phase transition from a single plane-wave to half-skyrmion is found through increasing magnetic field gradient strength. For the case of strong spin-orbit coupling and the fast rotation, the nature of the ground state is shown to support the formation of a hidden vortex as the gradient magnetic field is enhanced. These hidden vortices have no visible cores in density distributions but have phase singularities in phase distributions, which are arrayed along the diagonal direction. This result confirms a new method of creating the hidden vortices in the two-component Bose-Einstein condensate. These topological structures can be detected by using the time-of-flight absorption imaging technique. Our results illustrate that the gradient magnetic field not only provides an opportunity to explore the exotic topological structures in spin-orbit coupled spinor Bose-Einstein condensate, but also is crucial for realizing the phase transitions among different ground states. This work paves the way for the future exploring of topological defect and the corresponding dynamical stability in quantum systems subjected to a gradient magnetic field.

Large-scale Bose-Einstein condensation (BEC) of cesium atoms has been observed (T=343K). The technical bottleneck of BEC is very small trapping volume (10-8cm3), which made the number of condensed atoms still stagnant (less than 107), much smaller than normal condensation (more than 1013), large-scale BEC has never been observed. In BEC experiment, scientists have applied magnetic field (used to trap atoms) and laser (used to cool atoms), but never considered applying electric field, because they think that all kinds of atoms are non-polar atoms. The breakthrough of the bottleneck lies in the application of electric field. In theory, despite 6s and 6p states of cesium are not degenerate, but Cs may be polar atom doesn't conflict with quantum mechanics because it is hydrogen-like atom. When an electric field was applied, Cs atoms become dipoles, therefore large-scale BEC can be observed. BEC experiment of cesium has been redone. From the entropy S=0, critical voltage Vc=78V. When V 0; when V > Vc, S Vc, almost all Cs atoms (bosons) are in exactly the same state，according to Feynman, “the quantum physics is the same thing as the classical physics”, so our classical theory can explain BEC experiment satisfactorily. Ultra-low temperature is to make Bose gas phase transition, we used critical voltage to achieve phase transition, ultra-low temperature is no longer necessary. Five innovative formulas were first reported in the history of physics, the publication of this article marking mankind will enter a new era of polar atoms.

We report Dirac monopoles with polar-core vortex induced by spin-orbit coupling in ferromagnetic Bose-Einstein condensates, which are attached to two nodal vortex lines along the vertical axis. These monopoles are more stable in the time scale of experiment and can be detected through directly imaging vortex lines. When the strength of spin-orbit coupling increases, Dirac monopoles with vortex can be transformed into those with square lattice. In the presence of spin-orbit coupling, increasing the strength of interaction can induce a cyclic phase transition from Dirac monopoles with polar-core vortex to those with Mermin-Ho vortex. The spin-orbit coupled Bose-Einstein condensates not only provide a new unique platform for investigating exotic monopoles and relevant phase transitions, but also can preserve stable monopoles after a quadrupole field is turned off.

Bose–Einstein condensation has now been observed in diverse physical systems, starting from liquid Helium, excitons, to alkali atoms at nanokelvin temperature. The trapped cold atoms have provided an ideal venue for exploring fascinating ideas, ranging from Kosterlitz–Thouless (KT) phase transition, metal-insulator quantum phase transition to the realization of Abelian and non-Abelian gauge fields and solitonic excitations, in a controlled environment. Here, after a brief introduction to condensation phenomena in free space and trap, we explicate the working of the magneto optical trap, the work horse of the cold-atom laboratories. Subsequently, we illustrate the properties of experimentally realized dark, bright and grey solitons in the cigar shaped Bose–Einstein condensate (BEC). Focusing on a pan-cake type BEC in two dimensions, the basic aspects of the unique vortex excitations on a plane is elaborated, from which the Kosterlitz–Thouless phase transition follows, when the bound vortex–anti-vortex pairs unbind at \(T_{KT}\). We then describe the recent realization of Bose–Einstein condensation of the ubiquitous photons at room temperature. Parametric condition for occurrence of KT phase transition is then obtained for the photon gas.

FORTRESS: FORTRAN programs to solve coupled Gross-Pitaevskii equations for spin-orbit coupled spin-f Bose-Einstein condensate with spin f = 1 or 2