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arXiv:1707.00235 · Scientific Reports

Singularity in the matrix of the coupled Gross-Pitaevskii equations and the related state-transitions in three-species condensates

Y. M. Liu, Y. Z. He, Bao Cheng-guang

Published Jul 20, 2017• 8 citations
arXivHTMLPDF

Abstract

An approach is proposed to solve the coupled Gross-Pitaevskii equations (CGP) of the 3-species BEC in an analytical way under the Thomas-Fermi approximation (TFA). It was found that, when the strength of a kind of interaction increases and crosses over a critical value, a specific type of state-transition will occur and will cause a jump in the total energy. Due to the jump, the energy of the lowest symmetric state becomes considerably higher. This leaves a particular opportunity for the lowest asymmetric state to replace the symmetric states as the ground state. It was further found that the critical values are related to the singularity of either the matrix or a sub-matrix of the CGP. These critical values are not arising from the TFA but inherent in the CGP, and they can be analytically expressed. Furthermore, a model (in which two kinds of atoms separated from each other asymmetrically) has been proposed for the evaluation of the energy of the lowest asymmetric state. With this model the emergence of the asymmetric ground state is numerically confirmed under the TFA. The theoretical formalism of this paper is quite general and can be generalized for BEC with more than three species.

DOI
10.1038/s41598-017-06843-3

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An approach is proposed to solve the coupled Gross-Pitaevskii equations (CGP) of the 3-species BEC in an analytical way under the Thomas-Fermi approximation (TFA). It was found that, when the strength of a kind of interaction increases and crosses over a critical value, a specific type of state-transition will occur and will cause a jump in the total energy. Due to the jump, the energy of the lowest symmetric state becomes considerably higher. This leaves a particular opportunity for the lowest asymmetric state to replace the symmetric states as the ground state. It was further found that the critical values are related to the singularity of either the matrix or a sub-matrix of the CGP. These critical values are not arising from the TFA but inherent in the CGP, and they can be analytically expressed. Furthermore, a model (in which two kinds of atoms separated from each other asymmetrically) has been proposed for the evaluation of the energy of the lowest asymmetric state. With this model the emergence of the asymmetric ground state is numerically confirmed under the TFA. The theoretical formalism of this paper is quite general and can be generalized for BEC with more than three species.
Abstract Under the Thomas–Fermi approximation, an approach is proposed to solve the coupled Gross–Pitaevskii equations (CGP) for the two-species Bose–Einstein condensate analytically. The essence of this approach is to find out the building blocks to build the solution. By introducing the weighted strengths, relatively simpler analytical solutions have been obtained. A number of formulae have been deduced to relate the parameters when the system is experimentally tuned at various status. These formulae demonstrate the combined effect of the parameters, and are useful for the evaluation of their magnitudes. The whole parameter space is divided into zones, where each supports a specific phase. All the boundaries separating these zones have analytical expressions. Based on the division, the phase diagrams against any set of parameters can be plotted. In addition, by introducing a model for the asymmetric states, the total energies of the lowest symmetric and asymmetric states have been compared. Thereby, in which case the former will be replaced by the latter has been evaluated. The CGP can be written in a matrix form. For repulsive inter-species interaction V AB , when the parameters vary and cross over the singular point of the matrix, a specific state transition will happen and the total energy of the lowest symmetric state will increase remarkably. This provides an excellent opportunity for the lowest asymmetric state to emerge as the ground state. For attractive V AB , when the parameters tend to a singular point, the system will tend to collapse. The effects caused by the singular points have been particularly studied.
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Abstract The coupled Gross–Pitaevskii equations for two-species BEC have been solved analytically under the Thomas-Fermi approximation (TFA). Based on the analytical solution, two formulae are derived to relate the particle numbers N A and N B with the root mean square radii of the two kinds of atoms. Only the case that both kinds of atoms have nonzero distribution at the center of an isotropic trap is considered. In this case the TFA has been found to work nicely. Thus, the two formulae are applicable and are useful for the evaluation of N A and N B .
Abstract The coupled Gross–Pitaevskii equations for the ground state of the three-species condensates have been solved analytically under the Thomas–Fermi approximation. Six types of spatial configurations in the miscible phase are found. The whole parameter space has been divided into zones, each of which supports a specific configuration (miscible or immiscible). The boundaries of the zones are described by analytical formulae. Due to the division, the variation of the spatial configuration against the parameters can be visualized, and the effects of the parameters can thereby be understood. There are regions in the parameter space where the configuration is highly sensitive to the parameters. These regions are tunable and valuable for the determination of the parameters.
In this thesis we study a two-component Bose-Einstein Condensate system in 2D, using a combination of approximate analytical and exact numerical methods of solving the Gross-Pitaevskii equation (GPE). We discuss some of the ways of finding approximate ground state solutions to the GPE, one of them being the Thomas-Fermi approximation. Using the Thomas-Fermi approximation, one can separate the system into two regimes; one where both components are disks and one where one of the components is a disk and the other component is an annulus. However, the Thomas-Fermi approximation does not hold true beyond a critical value. We demonstrate a method using which one can get more precise solutions for the GPE. This involves using a method of approximation of the GPE near the critical boundaries, which has been previously used for a single component Bose gas. This can be adapted for a two-component system in order obtain explicit solutions for a BEC in a 2D harmonic trap with repulsive interactions. We show in detail how this approximation works for the different regimes suggested by the Thomas-Fermi analysis and also for more generalized cases where one has different masses, particle numbers and trapping frequencies for the two components. This thesis also talks about how one could analyse the system using this approximation in 1D and 3D, shedding more light on what might be the parameters affecting the precision of the solution.
We identify all possible classes of solutions for two-component Bose-Einstein condensates (BECs) within the Thomas-Fermi (TF) approximation and check these results against numerical simulations of the coupled Gross-Pitaevskii equations (GPEs). We find that they can be divided into two general categories. The first class contains solutions with a region of overlap between the components. The other class consists of non-overlapping wavefunctions and also contains solutions that do not possess the symmetry of the trap. The chemical potential and average energy can be found for both classes within the TF approximation by solving a set of coupled algebraic equations representing the normalization conditions for each component. A ground state minimizing the energy (within both classes of states) is found for a given set of parameters characterizing the scattering length and confining potential. In the TF approximation, the ground state always shares the symmetry of the trap. However, a full numerical solution of the coupled GPEs, incorporating the kinetic energy of the BEC atoms, can sometimes select a broken-symmetry state as the ground state of the system. We also investigate effects of finite-range interactions on the structure of the ground state.