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SPIN ÇALKANTILARIYLA KARARLI HALE GELEN SPIN-1 DAMLACIKLARI

YOĞURT, TAHA ALPER

Published Sep 13, 2023• 0 citations
OpenAlex

Abstract

The interacting Bose-Einstein condensates are mechanically stabilized by an external confinement such as magnetic, or optical traps. When the confinement is removed, they exhibit an expansion or collapse behavior depending on the interaction being repulsive, or attractive, respectively. The Gross-Pitaevskii mean-field theories are generally sufficient to provide a qualitative understanding of the condensate equilibrium properties and dynamics, therefore, beyond mean-field fluctuations are considered as minor quantitative corrections. The quantum droplet phases are self-stabilized phases of the condensate, in which the attractive mean-field interaction is stabilized by the repulsive Lee-Huang-Yang fluctuations without any external trap. The droplets are studied theoretically for binary Bose mixture and dipolar condensates, and both are recently realized in the ultracold atomic gas experiments. Here we propose the spin-1 condensate as another candidate to exhibit a quantum droplet phase. We argue that the spin-1 condensates in the polar and anti-ferromagnetic order parameters can exhibit a similar self-stabilizing mechanism. We also realize an analogy between the polarized spin-1 and Rabi-coupled binary mixture condensates. The Rabi frequency $\omega_R$ and quadratic Zeeman energy $q$ can tune the Lee-Huang-Yang energy for the Rabi-coupled mixture and spin-1 gas, respectively. Furthermore, the detuning $\delta$ and linear Zeeman energy $p$ are external parameters to tune the polarization in Rabi-coupled mixture, and spin-1 gas, respectively, which can significantly change the mean-field energy. We create the phase diagram for the mechanical stability of the Rabi-coupled mixture and spin-1 gas on the parameter planes of $\omega_R-\delta$ and $q-p$, respectively. We finally address how the Bose mixture quantum droplets respond to the rotation. We find that the strongly confined quantum droplets exhibit triangular vortex lattices under rapid rotation. The properties of the vortex lattices differ from the repulsive condensates in three points. First, the change in the droplet size is only fractional at the rapid rotation limit, unlike the diverging radius of the repulsive condensate. This property can lead the way to realizing the lowest Landau level states and strongly correlated phases in the rotating ultracold gases. Secondly, the change in the core size throughout the vortex lattice is more tractable due to the finite density at the edge of the condensate. Finally, the vortex lattices are slightly distorted, which agrees with the expected vortex density profile due to the nonuniform superfluid density. In the rapid rotation limit, the droplet approaches the flat-top density profile, in which the vortex lattice is closest to a perfect periodicity.

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Self-bound quantum droplets form when the mean-field tendency of the gas to collapse is stabilized by the effectively repulsive beyond-mean-field fluctuations. The beyond-mean-field effects depend on Rabi frequency ${\ensuremath{\omega}}_{R}$ and quadratic Zeeman effect qqq for the Rabi-coupled Bose mixtures and the spinor gases, respectively. For a quantum droplet, the effects of varying ${\ensuremath{\omega}}_{R}$ and qqq have recently been examined only for unpolarized Rabi-coupled Bose mixtures and unpolarized spinor gases. In this paper, we theoretically explore the stability of the droplet phase for polarized Rabi-coupled Bose mixtures and polarized spinor gases. We calculate the beyond-mean-field corrections for both gases with polarized order parameters and obtain the phase diagram of the droplets on the parameter space of Rabi frequency ${\ensuremath{\omega}}_{R}$ and detuning $\ensuremath{\delta}$ for Rabi-coupled mixtures and quadratic Zeeman energy qqq and linear Zeeman energy ppp for spinor gases. Finally, we highlight the similarities and differences between the two systems and discuss their experimental feasibility.
Bose mixture quantum droplets display a fascinating stability that relies on quantum fluctuations to prevent collapse driven by mean-field effects. Most droplet research focuses on untrapped or weakly trapped scenarios, where the droplets exhibit a liquidlike flat density profile. When weakly trapped droplets rotate, they usually respond through center-of-mass motion or splitting instability. Here, we study rapidly rotating droplets in the strong external confinement limit where the external potential prevents splitting and the center-of-mass excitation. We find that quantum droplets form a triangular vortex lattice as in single-component repulsive Bose-Einstein condensates (BECs), but the overall density follows the analytical Thomas-Fermi profile obtained from a cubic equation. We observe three significant differences between rapidly rotating droplets and repulsive BECs. First, the vortex core size changes markedly at finite density, visible in numerically obtained density profiles. We analytically estimate the vortex core sizes from the droplets' coherence length and find good agreement with the numerical results. Second, the change in the density profile gives a slight but observable distortion to the lattice, which agrees with the distortion expected due to nonuniform superfluid density. Lastly, unlike a repulsive BEC, which expands substantially as the rotation frequency approaches the trapping frequency, rapidly rotating droplets show only a fractional change in their size. We argue that this last point can be used to create clouds with lower filling factors, which may facilitate reaching the elusive strongly correlated regime.
Self-trapped droplets stabilized by quantum fluctuations have been experimentally realized in dipolar gases and binary boson mixtures. In this paper, we propose spinor Bose gases as another candidate for droplet formation. For spin-1 gas, we find that spin fluctuations give a dilute but self-trapped state for two different order parameters where the mean-field picture predicts collapse. A polar droplet phase can be stabilized by spin fluctuations for both antiferromagnetic and ferromagnetic spin-dependent coupling. An antiferromagnetic droplet phase can be stabilized similarly with a negative quadratic Zeeman shift. Furthermore, the beyond mean-field energy of the system depends on the quadratic Zeeman coupling, which provides a mechanism to tune the droplet formation and its density. We discuss the parameters necessary for the experimental realization of such spinor droplets.
Ultra-dilute self bound droplets in Bose-Einstein condensates have been of great interest since their first prediction in 2015.The self-bound nature of these bosonic gases combined with their extremely low densities make them a prime target of investigation for both experimentalists and theoreticians.In this thesis work dipole-dipole interacting gases are simulated using the Gross-Pitaevskii equation applying a fourth order, split-operator imaginary time method in two dimensions.First, the systems are studied at collapse for dipoles aligned with a magnetic field that tilts the dipoles in the two-dimensional plane.For some angles the dipole interaction becomes partly attractive which can lead to the collapse of the mean field solution.However, quantum fluctuations can stabilize the gas.Introducing the so-called Lee-Huang-Yang (LHY) correction, the system can be stabilized and self-bound droplets may form.Here, we investigate this stabilization for a quasi two-dimensional set-up. Popular science Ultra-cold droplets intrigue physicistsA hot topic within ultra-cold atom physics has been the recent discovery of droplets formed out of Bose-Einstein condensates, a specific form of a super cold and very dilute gas.The condensate is however very different from any gas encountered in everyday life.A normal gas can be thought of as atoms or molecules moving around in all directions.If you cool this gas, all the particles move slower until they either form a liquid or freeze and become a solid.This is not the case for the Bose-Einstein condensate.Instead they spread through space due to quantum mechanics and together they form one entity.This entity, the so called condensate, behaves like no classical gas, as it is mainly governed by quantum mechanics and not by the random chance of a lot of small particles moving about.To add to this strangeness, droplets now seem to form out of these condensates.The existence of these droplets was quite a surprise.Among experimentalists working with creating Bose-Einstein condensates the expected result was a collapse of the system if forces pushing the condensate together won against the forces pushing it apart.However, experimentalists revealed that some other internal force stabilized drops inside the condensate.Not only that, this force seemed to be able to create drops that could exist outside a "trap", which usually is a number of lasers and magnets that keep the condensate together.Some droplets could not be formed without a trap, but instead formed crystal structures, a very intriguing result.This has led to an excited mood within the field of cold atom physics.But what to expect from this anti-collapse force, mainly being described as quantum fluctuations, and the droplets it creates needs further study to find out.In this thesis work we simulate the anti-collapse force and possible droplet systems solving the underlying quantum-mechanical many-particle equations numerically.One system of interest is the very thin disc of condensate, or a quasi two-dimensional condensate.This system is excellent for rotating the condensate in.Why? Imagine having a piece of pizza dough that in this case represents a condensate.If you spin this dough as a skilled pizza maker does, the pizza will become flat due to rotational forces.The same goes for the condensate.As you rotate it it will become flat anyways and therefore having a model where our very cold gas is already flat makes it easier to simulate rotating systems.But before you start rotating your drops, you have to find the drops.And to find the drops you have to study the forces pushing the condensate together and the forces pushing it apart.Therefore we presently aim at studying possible droplet formation in these pizza crust shaped condensates, to make way for the study of rotational excitations.Initial results show some promise that droplets can be found in these quasi two-dimensional systems.
This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.
We study stability of a zero temperature mixture of attractively interacting degenerate bosons and spin-polarized fermions in the absence of confinement. We demonstrate that higher order corrections to the standard mean field energy of the system can lead to a formation of liquid droplets -- self-bound incompressible systems in a three-dimensional space. In addition to the quantum Lee-Huang-Yang correction to the boson-boson interaction energy, a higher order term in the Bose-Fermi coupling constant is crucial for stabilization of the system. We study a stability region of droplets for three different mixtures of species with different mass ratios, namely: 41^{41}41K \,-40^{40}40K, 87^{87}87Rb \,-40^{40}40K, and 133^{133}133Cs \,-6^{6}6Li as a function of the ratio of Bose-Fermi to Bose-Bose scattering length. The stability analysis is supported by a numerical simulations of droplet formation dynamics starting with a trapped system which is subsequently released from the trap. Our results indicate that Bose-Fermi droplets can be realized experimentally.