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arXiv:1212.2082 · Physical Review A

Continuous-wave solutions in spinor Bose-Einstein condensates

Richard S. Tasgal, Y. B. Band

Published Feb 25, 2013• 7 citations
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Abstract

We find analytic continuous-wave (cw) solutions for spinor Bose-Einstein condensates in a magnetic field that are more general than other published solutions. For particles with spin $F=1$ in a homogeneous one-dimensional trap, there exist cw states in which the chemical potential and wave vectors of the different spin components are different from each other. We include linear and quadratic Zeeman splitting. Linear Zeeman splitting, if the magnetic field is constant and uniform, can be mathematically eliminated by a gauge transformation, but quadratic Zeeman effects modify the cw solutions in a way similar to nonzero differences in the wave numbers between the different spin states. The solutions are stable fixed points within the continuous-wave framework, and the coherent spin mixing frequencies are obtained.

DOI
10.1103/physreva.87.023626

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We find analytic continuous-wave (cw) solutions for spinor Bose-Einstein condensates in a magnetic field that are more general than other published solutions. For particles with spin F=1F=1F=1 in a homogeneous one-dimensional trap, there exist cw states in which the chemical potential and wave vectors of the different spin components are different from each other. We include linear and quadratic Zeeman splitting. Linear Zeeman splitting, if the magnetic field is constant and uniform, can be mathematically eliminated by a gauge transformation, but quadratic Zeeman effects modify the cw solutions in a way similar to nonzero differences in the wave numbers between the different spin states. The solutions are stable fixed points within the continuous-wave framework, and the coherent spin mixing frequencies are obtained.
We analyze sound waves (phonons, i.e. Bogoliubov excitations) propagating on continuous-wave (cw) solutions of repulsive F=1F=1F=1 spinor Bose-Einstein condensates (BECs) such as 23Na^{23}\mathrm{Na}23Na (which is antiferromagnetic or polar) and 87Rb^{87}\mathrm{Rb}87Rb (which is ferromagnetic). Zeeman splitting by a uniform magnetic field is included. All cw solutions to ferromagnetic BECs with vanishing MF=0{M}_{F}=0MF​=0 particle density and nonzero components in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ fields are subject to modulational instability (MI). Modulational instability increases with increasing particle density. Modulational instability also increases with differences in the components' wave numbers; this effect is larger at lower densities but becomes insignificant at higher particle densities. Continuous-wave solutions to antiferromagnetic (polar) BECs with vanishing MF=0{M}_{F}=0MF​=0 particle density and nonzero components in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ fields do not suffer MI if the wave numbers of the components are the same. If there is a wave-number difference, MI initially increases with increasing particle density and then peaks before dropping to zero beyond a given particle density. The cw solutions with particles in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ components and nonvanishing MF=0{M}_{F}=0MF​=0 components do not have MI if the wave numbers of the components are the same, but do exhibit MI when the wave numbers are different. Direct numerical simulations of a continuous wave with weak white noise confirm that weak noise grows fastest at wave numbers with the largest MI and show some of the results beyond small-amplitude perturbations. Phonon dispersion curves are computed numerically; we find analytic solutions for the phonon dispersion in a variety of limiting cases.
Abstract We obtain general continuous-wave (CW) solutions in the model of a spinor positronium condensate in the absence of magnetic field. The CW solutions with both in-phase ( n = 0) and out-of-phase ( n = 1) spin components exist, with their ranges limited by the total particle density, ρ . In the limit of negligible population exchange between the spin components, the CW solutions are found to be stable or unstable, depending on the particle density of the para-positronium. Ortho-positronium, in the F = 1 spinor state, forms a ferromagnetic condensate with stable in-phase CW solutions only. Subsequent examination of the modulational instability is carried out both in the limit case of identical wavenumbers in the spin components, Δ k ≡ k 1 − k − 1 = 0 , and in the more general case of Δ k ≠ 0 too. The CW solutions with n = 0 and 1 solutions, which are stable in the case of Δ k = 0 , are unstable for Δ k ≠ 0 , for the natural repulsive sign of the nonlinearities. The total particle density, ρ , in the limit of Δ k = 0 is found to have a significant role in the stability of the condensate, which is determined by the sign of the self-interaction nonlinearity.
We propose a method to analytically solve the one-dimensional coupled nonlinear Gross-Pitaevskii equations which govern the motion of the spinor Bose-Einstein condensates. In a uniform external potential, the Hamiltonian comprises the kinetic energy, the linear and the quadratic Zeeman energies. Several classes of exact periodic and solitonic solutions, either in real or in complex forms, are obtained for both the F=1 and F=2 condensates. These solutions are general that contain neither approximations nor constraints on the system parameters.
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.
Spin-1 Bose-Einstein condensates provide a fundamental platform for exploring quantum magnetism and coherent spin dynamics. Their three Zeeman components are coupled by spin-exchange interactions under conserved particle number and magnetization, giving rise to rich magnetic phases and nonlinear spin structures; a rigorous classification of stationary states therefore clarifies the possible spin configurations and their physical origins. In this paper, as a continuation of our previous work \cite{LLWZ23,LLWZ24,LLWZ26}, we study the classification of solution for the spin-1 Bose-Einstein condensates in R\mathbb{R}R, by applying the non-standard Hirota's bilinear method, the determinant formula and the general left-uniqueness strategy, we present some novel explicit solution and give the complete classification of the solutions. In comparison with the classical Hirota's method, the non-standard Hirota's method needs to introduce auxiliary functions to construct Hirota's bilinear form, which essentially adds additional constraint on the wave functions, so that it possible to deal with more complex equations. As far as we know, this is the first systematic study the complete classification of solutions to the spin-1 Bose-Einstein condensates. Our classification results yields experimentally testable predictions for spin populations, relative phases, and magnetic textures
We present measurements and a theoretical model for the interplay of spin-dependent interactions and external magnetic fields in atomic spinor condensates. We highlight general features such as quadratic Zeeman dephasing and its influence on coherent spin mixing processes by focusing on a specific coherent superposition state in a F=1F=1F=1 87Rb^{87}\mathrm{Rb}87Rb Bose-Einstein condensate. In particular, we observe the transition from coherent spinor oscillations to the thermal equilibration.