Fate of coherent-beam-driven spin symmetry breaking and entropy production in a spinor Bose-Einstein condensate beyond the fully decoherent limit

A spin-1 ferromagnetic spinor Bose-Einstein condensate driven by a coherent beam of identical atoms was shown in 2020 (Xu, Zeng, Domanski, Li, Phys. Rev. Research 2, 023028) to relax to a steady spin distribution controlled entirely by the beam's incoming spin state and, in a spin-reversal-symmetric configuration, to undergo spontaneous spin-reversal symmetry breaking via a bifurcation at a mesoscopic time of order the atom number, with the entropy production rate peaking at the symmetry-breaking instance. These results hold only in an idealized limit: complete decoherence of the condensate between successive scatterings ($t_d < t_s$), a frozen spatial mode (single-mode approximation), neglected spin-dependent interactions, spin-1 ferromagnetic atoms, and near-zero temperature. The paper has received no citations and no follow-up theoretical or experimental work exists. The problem is to determine the nonequilibrium spin dynamics, steady states, and symmetry-breaking behavior of the beam-driven condensate in the coherent regime $t_d > t_s$, where quantum coherences between successive scattering events must be retained — the paper's own discussion notes that its study "can be generalized to the reverse case" $t_d > t_s$ — and to establish whether the beam-controlled steady distributions, the symmetry-breaking bifurcation, and the entropy-production phenomenology persist, so that the proposed beam-based control and readout of condensate spin distributions can be judged viable.

Background

A Bose-Einstein condensate (BEC) formed from atoms with nonzero internal angular momentum — a spinor BEC — carries an internal spin degree of freedom on top of its spatial order. In an optical trap with no applied magnetic field, the magnetic sublevels of, say, spin-1 $^{87}\mathrm{Rb}$ are degenerate, and for a ferromagnetic spin-spin interaction the $N$-atom ground state is a multiplet of total spin $s=N$ with $2N+1$ magnetic sublevels labeled by the collective magnetization quantum number $m$ from $-N$ to $N$. The equilibrium phases and the internal spin dynamics (spin mixing, the coherent internal conversion $2\,m{=}0 \leftrightarrow m{=}+1 + m{=}-1$) of such gases are reviewed in Kawaguchi and Ueda (2012) and Stamper-Kurn and Ueda (2013). Most theory in this area uses the single-mode approximation (SMA), which freezes the spatial wave function so that only the collective spin evolves.

Xu et al. (2020) studied a different drive: the condensate is probed by a coherent beam of identical atoms prepared in a pure spin state $|\xi\rangle_a$. Although the atom-atom interaction is spin-independent, the identity of the bosons makes it exchange spin: in the scattering Hamiltonian $H_i = g_0 \sum_{\mu_1\mu_2} a^\dagger_{\mu_1} c^\dagger_{\mu_2}(a_{\mu_1} c_{\mu_2} + a_{\mu_2} c_{\mu_1})$ (with $a_\mu$, $c_\mu$ annihilating a propagating and a condensed atom in sublevel $\mu$), the second, spin-exchange term exists only because of particle identity. Treating scattering in the Born approximation (amplitudes to first order in the short-range interaction) while keeping the many-body spin algebra exact via fractional parentage coefficients (Clebsch-Gordan-like coefficients linking the $N$-atom and $(N-1)$-atom spin states, i.e. going beyond mean field), they followed the condensate spin distribution $\lambda_m$ scattering by scattering.

The analysis rests on a separation of two timescales: the scattering time $t_s$ between successive beam atoms, and the decoherence time $t_d$ on which stray fields destroy the off-diagonal elements of the condensate's spin density matrix. Assuming $t_d < t_s$, the condensate is fully decoherent — its spin state diagonal in $m$ — before every scattering, so the dynamics reduces to a classical iteration on the populations $\lambda_m$. Each scattering entangles the outgoing atom with the condensate, producing an entanglement entropy $S_e$ (the von Neumann entropy of the scattered atom's reduced spin state); decoherence generates thermal entropy $S_{\mathrm{th}} = -\sum_m \lambda_m \ln \lambda_m$; and the entropy production per scattering is $S_p = S_e + S_{\mathrm{th}}(t_n) - S_{\mathrm{th}}(t_{n-1})$.

Two findings define the problem. First, the condensate relaxes to a steady spin distribution that is independent of its initial state and entirely determined by the incoming beam spin state — neither microcanonical nor canonical, Gaussian-like with width $\sim 1/\sqrt{N}$ — so the beam is a handle to prepare and read out the condensate's spin distribution. Second, for an initially unmagnetized condensate driven by the spin-reversal-symmetric input state $\tfrac{1}{\sqrt{2}}(|-1\rangle_a + |1\rangle_a)$, the magnetization fluctuation diverges within a large-$N$ random-walk scaling argument, the spin distribution bifurcates at a mesoscopic time of order $N$ to $2N$ scattering times, and the condensate ends in one of two fully ordered states of opposite magnetization: a spontaneous breaking of spin-reversal symmetry, with $S_e$, $S_{\mathrm{th}}$, and $S_p$ all extremal near the bifurcation. This mechanism differs from the established spontaneous symmetry breaking in spinor condensates — quench-driven formation of spatial spin domains (Sadler et al., 2006) — in being driven by entanglement with a scattered probe rather than by a parameter quench, and in living purely in the internal spin space. It connects to the measurement of irreversible entropy production in mesoscopic quantum systems (Brunelli et al., 2018) and to experiments on spin-correlated matter-wave jets from ferromagnetic spinor condensates (Kim et al., 2021), which demonstrate spin-correlated atom-condensate scattering but with internally generated jets rather than an external coherent probe beam. A precursor study (Li, Bao, and Zheng, 2018) established the hyperfine-state entanglement between one scattering atom and the spinor condensate on which the entropy transfer rests.

Problem Statement

Within the model of Xu et al. (2020) — $N$ spin-1 bosons with ferromagnetic interaction in an optical trap, the condensate restricted to its degenerate ground multiplet of total spin $s=N$, driven by a monochromatic coherent beam of the same atomic species in a pure spin state, with scattering treated in the Born approximation while the many-body spin algebra is kept exact — determine the nonequilibrium spin dynamics, steady states, and symmetry-breaking behavior in the coherent regime $t_d > t_s$, the generalization explicitly flagged by the paper's discussion when it notes that the fully decoherent treatment "can be generalized to the reverse case":

Derive the evolution of the full condensate spin density matrix, including the off-diagonal coherences in the magnetization basis that the $t_d < t_s$ assumption discarded, when the condensate is not fully decoherent between successive scatterings; characterize the resulting steady states and the fate of the spin-reversal-symmetry-breaking bifurcation and its entropy-production maximum; and determine the conditions on the ratio $t_d/t_s$ (and on the incoming beam state) under which the beam still controls the steady spin distribution.

The objective is the fate of the beam-driven symmetry breaking, entropy production, and beam-based control/readout of the condensate spin distribution outside the fully decoherent limit: an answer establishes whether the beam-controlled steady distributions and the bifurcation-driven symmetry breaking persist, weaken, or disappear when the coherences between successive scattering events are retained. The further idealizations of the 2020 analysis — the single-mode approximation, the neglected spin-dependent interaction, the restriction to spin-1 ferromagnetic atoms, and the near-zero temperature with the condensate confined to its degenerate ground multiplet — are modeling assumptions of the same framework rather than openings stated by the source; relaxing them poses further generalizations of the same question but is not part of the target here. The experimental counterpart — realizing the scheme with a sub-nanoKelvin condensate and a characterized coherent incident beam — belongs to the source's outlook ("The required sub-nano Kelvin temperature would be realized in the future") and remains unachieved.

References

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