Bose-Fermi mapping and a multibranch spin-chain model for strongly interacting quantum gases in one dimension: Dynamics and collective excitations

We show that the wave function in one spatial sector $x_1<x_2<\cdots <x_N$ (with $x_i$ being the coordinate of the $i\mathrm{th}$ particle) of a one-dimensional spinor gas with contact $s$-wave interaction, either bosonic or fermionic, can be mapped to the direct product of the wave function of a spinless Fermi gas with short-range $p$-wave interaction and that of a spin system governed by spin-parity projection operators. Applying this mapping to strongly interacting spinor gases, we obtain a generalized spin-chain model that captures both the static and dynamics properties of the system. Using this spin-chain model, we investigate the breathing-mode frequency and the quench dynamics of strongly interacting, harmonically trapped spinor gases.

Figure 1

The red dots represent results from the LDA approximation for $C_i^{\left(0\right)}$ using Eqs. (13) and (14). The blue solid lines are the exact integral of $C_i^{\left(0\right)}$ by a method similar to calculating ${\rho }^{\left(i\right)}\left(z\right)$ in [14]. The comparison is for particle numbers that run from 2 (bottom) to 15 (top).

Figure 2

Schematic representation of (a) the ground state, (v) the first excited state, and (c) the second excited states of a harmonically trapped ideal spinless Fermi gas.

Figure 3

(a),(b) Ground-state energy and (c),(d) breathing-mode frequency shift as functions of $N$. In (a) and (c), we present results for bosons and spin-$1/2$ fermions with various $N_{\uparrow }/N$. In (b) and (d), we present results for bosons, and for fermions with different spin $s$ and equal population in each spin component. For bosons, the ground-state energy and the breathing-mode frequency shift are independent of spin. The black solid lines represent the analytic LDA results for bosons given in Eqs. (21) and (22).

Figure 4

Evolution after a sudden quench of the spin-dependent magnetic gradient for a harmonically trapped spin-1/2 Fermi gas. Upper panel: $N_{\uparrow }=N_{\downarrow }=2$ and $g=100$. The left panel shows the spin density profiles for spin-up (dashed line) and spin-down (solid line) component before the quench. The right panel shows the after-quench evolution of $\mathrm{\Delta }$, the center-of-mass separation of the two spin species. The blue dashed line is obtained from the multibranch spin-chain model by including eight excited branches. The red solid line is the TEBD result. The lower panel is the same as the upper panel, except that $N_{\uparrow }=1$ and $N_{\downarrow }=3$.