Emergence of magnetic excitations in one-dimensional quantum mixtures under confinement

Abstract

We obtain an exact solution for the spectral function for one-dimensional Bose-Bose and Fermi- Fermi mixtures with strong repulsive interactions, valid in arbitrary confining potentials and at all frequency scales. For the case of harmonic confinement we show that, on top of the ladder structure of the density excitations imposed by the external confinement, spin excitations emerge as sideband peaks, with dispersion related to the one of ferromagnetic or antiferromagnetic spin chains and a width fundamentally larger for fermionic mixtures than for bosonic ones, as determined by the different symmetry of spin excited states. The observation of spin excitation branches can provide a univocal probe of interaction-induced magnetism in ultracold atoms.

Figures (4)

• Figure 1: (a) Sketch of the excitation processes in $\mathcal{A}(k,\omega)$. (b) Sketch of the mapping of the trapped continuous system on a spin chain. (c) Ground and excited state energy manifolds for $N=4$ two-component bosons trapped in a harmonic potential as a function of the inverse interaction strength ($1/g >0$). The splitting at finite interactions is related to the spin degeneracy and the corresponding Hamiltonian can be mapped to a single or coupled spin chain, as illustrated by the sketches inside the figure. The energies at $1/g = 0$ are the Fermi energies of the corresponding non-interacting system of fermions trapped in the same external potential, whose excitations are illustrated on the right.
• Figure 2: Top panels: spectral function of bosonic and fermionic gases in harmonic potential, in the $k-\omega$ plane (a) $N=10$ Tonks-Girardeau bosons (b) $N=5+5$ strongly repulsive BB mixture $(g = 100\hbar\omega_0\,a_{\mathrm{ho}})$ . (c) $N=5$ spinless, noninteracting fermions (d) $N=5+5$ strongly repulsive FF mixture $(g = 100\hbar\omega_0\,a_{\mathrm{ho}})$. The dashed lines in panels (b) and (d) are the theoretical prediction for ferromagnetic (antiferromagnetic) spin chains for bosonic (fermionic) mixtures respectively. Bottom panels: cuts of $\mathcal{A}_\sigma(k,\omega)$ at fixed $k$ values ($k a_\mathrm{ho} = 0$ top and $k a_\mathrm{ho} = 4$ bottom), indicated by the arrows in (b) and (d). The Young diagrams indicate the symmetry of the excited states. The zero of the frequency is set to $\mu_0 = E_{\mathrm{F}}^{(0)}(N) - E_{\mathrm{F}}^{(0)}(N-1) = \hbar \omega_0(N\!-\!\frac{1}{2})$, chemical potential of the mapped Fermi gas.
• Figure 3: Energy widths $\Delta\omega_{B,F\pm}^{(1st)}$ of the lowest spin excitation bands of bosonic (B) and fermionic (F) mixtures in a harmonic trap, where in the latter case $+(-)$ refer to excitations with energies above (below) $\mu$, respectively, as functions of particle number $N$, from the exact spectral function (symbols) and large-$N$ analytical predictions (dashed lines).
• Figure 4: Schematic representation of the subsets in Eq. \ref{['eq:g1l_AB_1']}. The starting ordered set of indices $\{2, \dots, N\}$ is represented by the horizontal line. When we apply the permutation $Q$, the indices $l \in [2, j]$ go to the subset $L$, instead the indices $r \in [j+1, N]$ go to the subset $R$. The subset $L$ contains two disjoint subsets, $L_1$ and $L_2$, such that if $Q(l) \in L_1$ if $Q"(Q(l)) = Q(l)$ and all the remaining indices belong to $L_2$.