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Interfaces in spinor Bose-Einstein condensates

J. Corver

Published Aug 5, 2015• 0 citations
OpenAlex

Abstract

Bose-Einstein condensates - a state of matter at low temperature in which a macroscopic amount of bosons occupy the quantum-mechanical ground state- were originally trapped magnetically. The drawback of this trapping method is that only the weak-field seeking spin state is trapped and we therefore only have a one component i.e. scalar condensate. In contrast, optical trapping confines atoms of all spin state and therefore allows for spin degrees of freedom, resulting in spinor Bose Einstein condensates. In this thesis, density profiles of these spinor condensates have been calculated in Thomas-Fermi approximation. The profiles show areas with strict phase separation, but also areas with spin mixtures. For sodium atoms we get a strict separation between the mF = ±1 and the mF = 0 spin state in Thomas-Fermi approximation. This however causes discontinuities in the transition between spin domains and thus these interfaces were studied further. We found that the density profile of a spin state behaves roughly like a hyperbolic tangent at such an interface. Finally, a differential equation is proposed that could be solved as a starting point of follow-up research.

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In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field $B$. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field $B_c$, asymptotic values of $n_0$ (relative atom number in $m_F=0$ state) at large $B$ and the phase diagram of $n_0$. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of $n_0$. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the ``semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the ``spin temperature" $T_s$ and the ``kinetic temperature" $T_k$ with $T_s\ll T_k$, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.

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Spin domains in ground state of a trapped spin-1 condensate: a general study under Thomas–Fermi approximation

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In the presence of a magnetic field, multicomponent ground states appear in trapped spin-1 Bose-Einstein condensates for both ferromagnetic and antiferromagnetic types of spin-spin interaction. We aim to produce an accurate analytical description of the multicomponent states which is of fundamental importance. Despite being in the so-called regime of the Thomas-Fermi approximation (condensates with large particle number), the scenario of multicomponent states is problematic under this approximation due to the large variation in densities of the subcomponents. We generalize the variational method that we have introduced [P. K. Kanjilal and A. Bhattacharyay, Eur. Phys. J. Plus 137, 547 (2022)] to overcome the limitations of the Thomas-Fermi approximation. We demonstrate that the variational method is crucial in identifying multicomponent ground states. A comparison of the results of the variational method, which is multimodal by construction, with that of the single-mode approximation is also presented in this paper to demonstrate a marked improvement in accuracy over the single-mode approximation. We have also looked into the phase transition between phase-matched and polar states in a trapped condensate using the variational method and have identified substantial change in the phase boundary. The correspondence of the phase diagram of the trapped case with the homogeneous one identifies other limitations of the Thomas-Fermi approximation as opposed to the more accurate variational method.

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Bose-Einstein condensates - a state of matter at low temperature in which a macroscopic amount of bosons occupy the quantum-mechanical ground state- were originally trapped magnetically. The drawback of this trapping method is that only the weak-field seeking spin state is trapped and we therefore only have a one component i.e. scalar condensate. In contrast, optical trapping confines atoms of all spin state and therefore allows for spin degrees of freedom, resulting in spinor Bose Einstein condensates. In this thesis, density profiles of these spinor condensates have been calculated in Thomas-Fermi approximation. The profiles show areas with strict phase separation, but also areas with spin mixtures. For sodium atoms we get a strict separation between the mF = ±1 and the mF = 0 spin state in Thomas-Fermi approximation. This however causes discontinuities in the transition between spin domains and thus these interfaces were studied further. We found that the density profile of a spin state behaves roughly like a hyperbolic tangent at such an interface. Finally, a differential equation is proposed that could be solved as a starting point of follow-up research.
In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field BBB. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field BcB_cBc​, asymptotic values of n0n_0n0​ (relative atom number in mF=0m_F=0mF​=0 state) at large BBB and the phase diagram of n0n_0n0​. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of n0n_0n0​. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the ``semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the ``spin temperature" TsT_sTs​ and the ``kinetic temperature" TkT_kTk​ with Ts≪TkT_s\ll T_kTs​≪Tk​, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.
Abstract Investigation of ground state structures and phase separation under confinement is of great interest in spinor Bose–Einstein condensates. In this paper we show that, in general, within the Thomas–Fermi approximation, the phase separation scenario of stationary states can be obtained including all the mixed states on an equal footing for a spin-1 condensate under any confinement. According to the process used here, the density of each and every stationary states can be determined exactly in terms of the system parameters. This allows to write the expression for energy density of all states in a form which are not an explicit function of densities of those states. An energy density comparison therefore, among all the allowed stationary states can be done which will reveal domain structures depending on the system parameters. We study here, in details, a particular case of spherically symmetric harmonic confinement as an example and show a wide range of potential phase separation scenario for anti-ferromagnetic and ferromagnetic interactions.
The spatial structure of the spinor Bose-Einstein condensates with the spin degrees of freedom is analyzed based on the generalized Gross-Pitaevskii (GP) equation in light of the present spin-domain experiment on ${m}_{F}=\ifmmode\pm\else\textpm\fi{}1,$ and 000 of the hyperfine state F=1F=1F=1 of 23Na{}^{23}\mathrm{Na}23Na atom gases. The GP solutions in three- and one-spatial dimensional cases reproduce the observed spin-domain structures, revealing the length scale associated with the existence of the weak interaction of the spin-spin channel, other than the ordinary coherence length related to the density-density channel. The obtained domain structure in GP is compared with the result in the Thomas-Fermi approximation. The former solution is found to better describe the observed features than the latter.
Bose-Einstein condensation (BEC) is a phenomenon in which identical bosons occupy the same quantum state below a certain critical temperature. A hallmark of BEC is the coherence between particles every particle shares the same quantum wavefunction and phase. This coherence has been demonstrated for the external (motional) degrees of freedom of the atomic condensates by interfering two condensates. In this thesis, the coherence is shown to extend to the internal spin degrees of freedom of a spin-1 Bose gas evidenced by the observed coherent and reversible spin-changing collisions. The observed coherent dynamics are analogous to Josephson oscillations in weakly connected superconductors and represent a type of matter-wave four-wave mixing. Control of the coherent evolution of the system using magnetic fields is also demonstrated. The studies on spinor condensates begin by creating spinor condensates directly using all-optical approaches that were first developed in our laboratory. All-optical formation of Bose-Einstein condensates (BEC) in 1D optical lattice and single focus trap geometries are developed and presented. These techniques offer considerable flexibility and speed compared to magnetic trap approaches, and the trapping potential can be essentially spin-independent and are ideally suited for studying spinor condensates. Using condensates with well-defined initial non-equilibrium spin configuration, spin mixing of F = 1 and F = 2 spinor condensates of rubidium-87 atoms confined in an optical trap is observed. The equilibrium spin configuration in the F = 1 manifold confirms that 87Rb is ferromagnetic. The coherent spinor dynamics are demonstrated by initiating spin mixing deterministically with a non-stationary spin population configuration. Finally, the interplay between the coherent spin mixing and spatial dynamics in spin-1 condensates with ferromagnetic interactions is investigated.
In this thesis we investigate the static, dynamic, and thermodynamic properties\nof atomic spin-1 Bose gases in external magnetic fields. At low magnetic fields\nthe properties of single-component, or scalar condensates, are essentially\nunaffected but can become significantly altered for spinor Bose condensates as\nshown by our studies.\n\nWe first study the Bose-Einstein condensation of trapped spin-1 Bose gases by\nemploying the Hartree-Fock approximation and the two-fluid model within a mean\nfield approximation. Our detailed investigation reveals that the\nferromagnetically interacting spin-1 condensates exhibit triple condensations\nwhile the antiferromagnetically interacting ones show double condensations.\n\nThe ground state structure of homogeneous and trapped spin-1 Bose condensates\nwith ferromagnetic and antiferromagnetic interactions at zero temperature in\nmagnetic fields are then investigated systematically. We further illuminate the\nimportant effect of quadratic Zeeman shift which causes a preferred occupation\nof the $|m_F=0\nangle$ state through spin exchange collisions, $2|m_F=0\nangle\nleftrightarrow |m_F=1\nangle + |m_F=-1\nangle$.\n\nWe also present detailed studies of the off-equilibrium coherent dynamics of\nspin-1 Bose condensates in magnetic fields within the single spatial mode\napproximation. Dynamical instabilities of the off-equilibrium oscillations are\nshown to be responsible for the formation of multiple domains as recently\nobserved in several 87^{87}87Rb experiments.\n\nFinally, we discuss briefly excited condensate states, or soliton-like states,\nin cigar-shaped spin-1 Bose condensates with an effective quasi-1D description,\nusing the developed nonpolynomial Schr"odinger equation.
In the presence of a magnetic field, multicomponent ground states appear in trapped spin-1 Bose-Einstein condensates for both ferromagnetic and antiferromagnetic types of spin-spin interaction. We aim to produce an accurate analytical description of the multicomponent states which is of fundamental importance. Despite being in the so-called regime of the Thomas-Fermi approximation (condensates with large particle number), the scenario of multicomponent states is problematic under this approximation due to the large variation in densities of the subcomponents. We generalize the variational method that we have introduced [P. K. Kanjilal and A. Bhattacharyay, Eur. Phys. J. Plus 137, 547 (2022)] to overcome the limitations of the Thomas-Fermi approximation. We demonstrate that the variational method is crucial in identifying multicomponent ground states. A comparison of the results of the variational method, which is multimodal by construction, with that of the single-mode approximation is also presented in this paper to demonstrate a marked improvement in accuracy over the single-mode approximation. We have also looked into the phase transition between phase-matched and polar states in a trapped condensate using the variational method and have identified substantial change in the phase boundary. The correspondence of the phase diagram of the trapped case with the homogeneous one identifies other limitations of the Thomas-Fermi approximation as opposed to the more accurate variational method.