Dynamics of Interacting Fermions in Spin-Dependent Potentials

Recent experiments with dilute trapped Fermi gases observed that weak interactions can drastically modify spin transport dynamics and give rise to robust collective effects including global demagnetization, macroscopic spin waves, spin segregation, and spin self-rephasing. In this Letter, we develop a framework for studying the dynamics of weakly interacting fermionic gases following a spin-dependent change of the trapping potential which illuminates the interplay between spin, motion, Fermi statistics, and interactions. The key idea is the projection of the state of the system onto a set of lattice spin models defined on the single-particle mode space. Collective phenomena, including the global spreading of quantum correlations in real space, arise as a consequence of the long-ranged character of the spin model couplings. This approach achieves good agreement with prior measurements and suggests a number of directions for future experiments.

Figure 1

(a) Atoms spin-polarized along $X$ occupy single-particle eigenstates, labeled by mode number $n$. The potential is quenched to a spin-dependent form, and dynamics result from a spin model with long ranged interactions (green wavy lines) in energy space. (b) The state $|\psi ⟩$ is a coherent superposition of spins in many mode configurations (unoccupied modes are represented by open circles). In each configuration, particles are localized in mode space, with spin model Hamiltonian ${\widehat{H}}_i^{\mathrm{sm}}$. Coherences between the configurations capture motional effects.

Figure 2

Magnetization dynamics for a constant gradient. Collective $⟨{\widehat{\mathcal{S}}}^X⟩$ for a $x_0=0.1a_H$ (a) [and $x_0=0.3a_H$ (e)] displays global interaction-induced demagnetization, which damps single-particle oscillations. Collective [generic] Ising solutions, black lines, give the demagnetization envelopes. Local magnetizations $⟨{\widehat{\mathcal{S}}}^{X,Y,Z}(x)⟩$ with $x_0=0.1a_H$ (b)–(d) [and $x_0=0.3a_H$ (f)–(h)] reflect similar behavior, both shown with $u_{\uparrow \downarrow }=0.35\omega$.

Figure 3

Dynamics for a linear gradient. (a) Spin self-rephasing for ${\omega }_B=0.1\omega$: as interactions increase, demagnetization is suppressed and $⟨\widehat{\overrightarrow{\mathcal{S}}}⟩$ precesses collectively in the $XY$ plane (inset). (b) Simulation of a one dimensional gas at zero temperature with parameters from Ref. [7], showing $(n^{\uparrow }-n^{\downarrow })/n_0$ at the cloud center (blue solid line) with analytic prediction (red dashed line), and (c) segregated spin density profiles. (d) Data from Ref. [7], and prediction (red dashed line) based on a thermal average of Rabi oscillations between the Dicke and spin-wave manifolds.

Figure 4

(a) Real part of the connected correlation function $\mathrm{Re}[G^{++}(x,0;t)]$ for a weak gradient ($x_0=0.1a_H,u_{\uparrow ,\downarrow }=0.35\omega$). Correlations grow collectively due to the long-ranged nature of the interactions in energy space, and peak when the gas is demagnetized. (b) For a linear gradient in the self-rephasing regime (${\omega }_B=0.1\omega ,u_{\uparrow \downarrow }=0.45\omega$), the connected correlator $\mathrm{Re}[G^{++}(x,0;t)]$ rotates collectively in the $XY$ plane.