Strongly interacting quantum gases in one-dimensional traps

Under second-order degenerate perturbation theory, we show that the physics of $N$ particles with arbitrary spin confined in a one-dimensional trap in the strongly interacting regime can be described by superexchange interaction. An effective spin-chain Hamiltonian (non-translationally-invariant Sutherland model) can be constructed from this procedure. For spin-1/2 particles, this model reduces to the non-translationally-invariant Heisenberg model, where a transition between Heisenberg antiferromagnetic (AFM) and ferromagnetic (FM) states is expected to occur when the interaction strength is tuned from the strongly repulsive to the strongly attractive limit. We show that the FM and the AFM states can be distinguished by two different methods: the first is based on their distinct responses to a spin-dependent magnetic gradient, and the second is based on their distinct momentum distributions. We confirm the validity of the spin-chain model by comparison with results obtained from several unbiased techniques.

Figure 1

Energy spectrum of the relative motion as a function of $1/g$ for three fermions with $(N_{\uparrow },N_{\downarrow })=(1,2)$, without (a) and with (b) the spin-dependent magnetic gradient. For (b), we have $G=0.05$. The main figures are obtained using the Green's function method. The red dotted lines in the negative-$g$ area represent the tightly bound molecular states. The inset figures show the comparison between the spectrum obtained from the Green's function method (dots) and that from the effective spin-chain model (solid lines) near $1/g=0$. In all the figures presented in this paper, we have adopted trap units with $\hslash =m=\omega =1$. Consequently, the energy $E$ is in units of $\hslash \omega$, and the interaction strength $g$ is in units of $\sqrt{{\hslash }^3\omega /m}$.

Figure 2

Real-space density profiles (upper panel) and momentum-space density profiles (lower panel) for $(N_{\uparrow },N_{\downarrow })=(1,1)$ versus $1/g$ for AFM (red solid lines) and FM (black dotted lines) states. In our trap units, position $x$ is in units of $\sqrt{\hslash /\left(m\omega \right)}$, the real-space density $\rho \left(x\right)$ is in units of $\sqrt{m\omega /\hslash }$, the momentum $p$ is in units of $\sqrt{\hslash m\omega }$, and the momentum-space density $\rho \left(p\right)$ is in units of $1/\sqrt{\hslash m\omega }$.

Figure 3

Momentum distribution for $(N_{\uparrow },N_{\downarrow })=(4,4)$. The black dashed curve is for the fully spin symmetric FM state, which has the same momentum distribution as $N$ spinless fermions. The red solid curve is for the AFM state. The blue dash-dotted curve is for the fully spin antisymmetric state, which has the same momentum distribution as $N$ spinless Tonks-Girardeau bosons. The inset shows the momentum distribution for the AFM state in the large-momentum limit, in comparison to the theoretical prediction $K/\left(2\pi p^4\right)$ (green dotted curve), where $K$ is the Tan contact.

Figure 4

Separation between the two spin species as a function of $G\left|g\right|$ for (a) $(N_{\uparrow },N_{\downarrow })=(1,1)$, (b) $(N_{\uparrow },N_{\downarrow })=(1,2)$, and (c) $(N_{\uparrow },N_{\downarrow })=(2,2)$. The black dashed curves are for the ground state with negative $g$, and the red solid curves are for the ground state with positive $g$. The symbols in (a) are obtained from the analytic solution detailed in Appendix pp5. The symbols in (c) are TEBD results. (d) The susceptibility $\frac{d\mathrm{\Delta }}{\left|g\right|dG}{|}_{G=0}$ as a function of $N$ for $N_{\uparrow }=N_{\downarrow }=N/2$. In our trap units, $\mathrm{\Delta }$ is in units of $\sqrt{\hslash /\left(m\omega \right)}$, and $Gg$ in units of ${\hslash }^2{\omega }^2$.

Figure 5

Adiabatic preparation of the FM state. At $t=0$, the system is prepared in the ground state with $1/g=0.01$ and $G=0.1$. (a) The value of the experimentally controlled parameters $1/g\left(t\right)$ and $G\left(t\right)$ for a total adiabatic evolution time $300T_{\mathrm{ho}}$. (b) The solid lines represent $\mathrm{\Delta }\left(t\right)$ obtained by solving the time-dependent Schrödinger equation under the effective Hamiltonian $H_{\mathrm{eff}}$. The dashed lines represent Eq. (11), which is the $\mathrm{\Delta }\left(t\right)$ of the instantaneous ground state for the given values of $g\left(t\right)$ and $G\left(t\right)$. Three different total adiabatic evolution times are calculated, $100T_{\mathrm{ho}},200T_{\mathrm{ho}}$, and $300T_{\mathrm{ho}}$. The inset shows the fidelity of the adiabatically prepared state for the total evolution time $300T_{\mathrm{ho}}$.

Figure 6

Quench dynamics for $(N_{\uparrow },N_{\downarrow })=(2,2)$. The initial state is prepared as the ground state with $1/g=0.01$ and $G=0.05$. At $t=0,G$ is set to zero, and the system starts to evolve in time. The red solid line is $\mathrm{\Delta }\left(t\right)$ calculated using the TEBD method, and the blue dashed line is calculated using the effective spin-chain model. The inset figure shows the evolution for a much longer time under the spin-chain model. In our trap units, $t/g$ is in units of $\sqrt{m/\left({\hslash }^3{\omega }^3\right)}$.

Figure 7

Dimensionless coefficients $C_i$ for $N=8$ and 13, calculated using the Monte Carlo integral method (Veges algorithm [46]). The solid lines are obtained using the approximate expression (12).