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arXiv:2510.10050 · Physical Review A

Quantum many-body analysis of spin-2 bosons with two-body inelastic decay

Takeshi Takahashi, Hiroki Saito

Published Apr 6, 2026• 0 citations
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Abstract

Bose-Einstein condensates of $^{87}\mathrm{Rb}$ atoms with a hyperfine spin of 2 are open quantum systems, where the atoms are lost through two-body inelastic collisions. In this dissipation process, a collision channel with total spin of 4 is forbidden by angular momentum conservation, which results in magnetization of the atoms remaining in the condensate. Here, we investigate the quantum many-body properties of spin-2 bosons that undergo two-body atomic loss. We show that the system finally reaches a steady state, which is a mixture of the states with maximum total spins. In addition, we find that a nonclassical steady state can be obtained by applying and quenching the quadratic Zeeman field.

DOI
10.1103/7vzn-kmgv

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Bose-Einstein condensates of 87Rb^{87}\mathrm{Rb}87Rb atoms with a hyperfine spin of 2 are open quantum systems, where the atoms are lost through two-body inelastic collisions. In this dissipation process, a collision channel with total spin of 4 is forbidden by angular momentum conservation, which results in magnetization of the atoms remaining in the condensate. Here, we investigate the quantum many-body properties of spin-2 bosons that undergo two-body atomic loss. We show that the system finally reaches a steady state, which is a mixture of the states with maximum total spins. In addition, we find that a nonclassical steady state can be obtained by applying and quenching the quadratic Zeeman field.
We studied spin-dependent two-body inelastic collisions in F=2F=2F=2 87Rb^{87}\text{R}\text{b}87Rb Bose-Einstein condensates both experimentally and theoretically. The 87Rb^{87}\text{R}\text{b}87Rb condensates were confined in an optical trap and selectively prepared in various spin states in the F=2F=2F=2 manifold at a magnetic field of 3.0 G. The measured atom loss rates depend on the spin states of colliding atoms. We measured two fundamental loss coefficients for two-body inelastic collisions with total spins of 0 and 2. The loss coefficients determine the loss rates of all the spin pairs. The experimental results for mixtures of all spin combinations are in good agreement with numerical solutions of the Gross-Pitaevskii equations that include the effect of a magnetic field gradient.
We experimentally studied the spin-dependent collision dynamics of 87Rb^{87}\mathrm{Rb}87Rb spin-2 Bose-Einstein condensates confined in an optical trap. The condensed atoms were initially populated in the $\ensuremath{\mid}F=2,{m}_{F}=0⟩$ state, and their time evolutions in the trap were measured in the presence of external magnetic field strengths ranging from 0.1 to 3.0G3.0\phantom{\rule{0.3em}{0ex}}\mathrm{G}3.0G. The atom loss rate due to inelastic two-body collisions was found to be $1.4(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{3}\phantom{\rule{0.3em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$. Spin mixing in the F=2F=2F=2 manifold developed dramatically for the first few tens of milliseconds, and the oscillations in the population distribution between different magnetic components were observed over a limited range of magnetic field strengths. The antiferromagnetic property of this system was deduced from the magnetic field dependence on the evolution of relative populations for each mF{m}_{F}mF​ component.
The ground states of Bose-Einstein condensates (BECs) of spin-2 bosons are classified into three distinct phases (ferromagnetic, ``antiferromagnetic,'' and cyclic) depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while low-lying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet ``pairs'' undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson ``trios'' condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one single-particle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.
We experimentally investigate and analyze the rich dynamics in F=2F=2F=2 spinor Bose-Einstein condensates of 87Rb^{87}\mathrm{R}\mathrm{b}87Rb. An interplay between mean-field driven spin dynamics and hyperfine-changing losses in addition to interactions with the thermal component is observed. In particular, we measure conversion rates in the range of ${10}^{\ensuremath{-}12}\text{ }\text{ }{\mathrm{c}\mathrm{m}}^{3}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ for spin-changing collisions within the F=2F=2F=2 manifold and spin-dependent loss rates in the range of ${10}^{\ensuremath{-}13}\text{ }\text{ }{\mathrm{c}\mathrm{m}}^{3}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ for hyperfine-changing collisions. We observe polar behavior in the F=2F=2F=2 ground state of 87Rb^{87}\mathrm{R}\mathrm{b}87Rb, while we find the F=1F=1F=1 ground state to be ferromagnetic. We further see a magnetization for condensates prepared with nonzero total spin.
This paper presents quantitative calculations of collisional loss rates for pure 23Na^{23}N\mathrm{a}23Na or 87Rb^{87}R\mathrm{b}87Rb atoms in a binary mixture of hyperfine states ∣F,M〉|F,M〉∣F,M〉. We find that the recent observation of a dual Bose condensate consisting of 87Rb^{87}R\mathrm{b}87Rb $|1,\ensuremath{-}1〉$ and ∣2,2〉|2,2〉∣2,2〉 atoms is unique and will not occur in Na. We attribute this to the unexpectedly small inelastic spin-exchange rate associated with 87Rb^{87}R\mathrm{b}87Rb s-wave hyperfine collisions. This fortuitous result arises from nearly equal $(5.45\ifmmode\pm\else\textpm\fi{}0.26\mathrm{nm})$ scattering lengths for two colliding 87Rb^{87}R\mathrm{b}87Rb atoms in their $|1,\ensuremath{-}1〉$ or ∣2,2〉|2,2〉∣2,2〉 states.
Recently it has been predicted that if a gas of spin-1/2 bosonic atoms is cooled down while conserving its total spin, it can form a Bose-Einstein condensate in a new type of fragmented state. In that fragmented state two orbital wave functions are macroscopically occupied. We derive a set of Gross-Pitaevskii equations that describe the two occupied wave functions, taking into account the effects of interatomic interactions. We then analyze the solutions of those equations in some special cases. In particular, we find that if the fragmented state is realized in a toroidal trap, under certain conditions the system will spontaneously acquire macroscopic orbital angular momentum. Finally, we discuss the nature of the phase transition from an uncondensed gas to a fragmented-state condensate.