Relaxation Dynamics of an Isolated Large-Spin Fermi Gas Far from Equilibrium

A fundamental question in many-body physics is how closed quantum systems reach equilibrium. We address this question experimentally and theoretically in an ultracold large-spin Fermi gas where we find a complex interplay between internal and motional degrees of freedom. The fermions are initially prepared far from equilibrium with only a few spin states occupied. The subsequent dynamics leading to redistribution among all spin states is observed experimentally and simulated theoretically using a kinetic Boltzmann equation with full spin coherence. The latter is derived microscopically and provides good agreement with experimental data without any free parameters. We identify several collisional processes that occur on different time scales. By varying density and magnetic field, we control the relaxation dynamics and are able to continuously tune the character of a subset of spin states from an open to a closed system.

Popular Summary

At very, very cold temperatures rather close to absolute zero, an atomic gas is dominated by its quantum-mechanical properties: Not only must their spatial motion be described in terms of discrete wave functions, but their internal spin states also matter. At equilibrium, the occupations (or “distributions”) of those quantum states follow particular rules. Imagine, then, knocking the gas out of its equilibrium by changing the spin-state distribution and “watching” it regain its equilibrium: What does one see? As simple as this question sounds, the answer was far from being well established. In this combined experimental and theoretical paper, we bring that process to light and show that the spatial “collisions” of the atoms can flip their internal spins and lead to the relaxation of the spin-state distribution to its equilibrium.

In our experiment, we knock an ultracold gas of potassium atoms out of its equilibrium by putting the atoms in two of the ten available internal spin states only. We then monitor how the atoms redistribute themselves among all ten states, and we are able to keep track of that process for time spans of up to 10 seconds by “counting” the number of atoms in each spin state. Complementing such experimental data obtained at different gas densities and applied magnetic fields with theoretical modeling, we have unraveled many aspects of the relaxation process: There are a number of different processes of atomic collision taking place at different time scales that cause spin changes, but the collision-aided spin-redistribution process is the slowest one in the whole system.

Our results significantly advance the understanding of relaxation processes in quantum many-body systems.

Figure 1

Schematic description of the relaxation process in a large-spin Fermi gas involving spin and spatial degrees of freedom. (a) The ten spin states of ${}^{40}\mathrm{K}$. (b) A typical spin-changing collision in the center-of-mass frame and another collision forbidden by the Pauli exclusion principle. (c) Top: Initially, all atoms are prepared in a binary spin mixture $m=\pm 1/2$. Spin-changing collisions distribute atoms among all other spin states until an approximately balanced population is reached. The Fermi energies for each two-component subsystem are lower than the initial Fermi energy. Bottom: Time evolution of the spatial density for each spin component.

Figure 2

(a) Measurement of damped spin oscillations and (b) subsequent relaxation toward equilibrium, observed in a 3D fermionic quantum gas with large spin. Depicted is the time evolution of the relative populations of all spin components $\pm m$, starting from an initial superposition of all ten spin states. For the exact experimental configuration, see Appendix pp1-s1. Solid lines are guides to the eye. Note the three time scales of (i) the spin oscillations, (ii) their damping, and (iii) the subsequent relaxation of the total system. The redistribution among all spin states occurs on a time scale of 10 s. The magnetic field is $B=0.17\mathrm{G}$, particle number $N=4.9\times {10}^5$, and temperature $T/T_F=0.22$.

Figure 3

Numerical simulation of coherent oscillations, damping, and relaxation in the 1D case. The initial spin configuration is the same as in Fig. 2. The axial trapping frequency is ${\omega }_x=2\pi \times 84\mathrm{Hz}$, and radial frequencies are ${\omega }_{y,z}=2\pi \times 47\mathrm{kHz}$, particle number $N=100$ per tube at temperature $T/T_F=0.2$, and magnetic field $B=1.5\mathrm{G}$. As in Fig. 2, we observe three time scales related to oscillations, damping, and relaxation.

Figure 4

Comparison of spin relaxation for 1D and 3D. The initial spin configuration is a mixture of $m=\pm 1/2$. (a) Experimental data in a 1D geometry (circles) compared to numerical results (lines) from the 1D Boltzmann equation (4) and (dots) from a 1D version of the single-mode approximation [Eq. (9)]. The axial trapping frequency is ${\omega }_x=2\pi \times 84\mathrm{Hz}$, and radial frequencies are ${\omega }_{y,z}=2\pi \times 47\mathrm{kHz}$, particle number $N=100$ per tube at temperature $T/T_F=0.2$, and magnetic field $B=0.12\mathrm{G}$. Inset: The system approaches a steady state for longer times. (b) Experimental data (circles) in a 3D configuration compared to calculations (lines) in the single-mode approximation [Eq. (9)] $\overrightarrow{\omega }=2\pi \times (33,33,137)\mathrm{Hz}$, $N=1.3\times {10}^5$, and $T/T_F=0.15$ at $B=0.34\mathrm{G}$.

Figure 5

Density dependence of the spin-relaxation rate in 3D, with an initial mixture of atoms in $m=\pm 1/2$. The spin-changing rate is obtained by fitting the solution of coupled rate equations to experimental (points) and theoretical (lines) data. Theoretical values are obtained using the single-mode approximation Eq. (9). The magnetic field is $B=0.11\mathrm{G}$. We experimentally tune the density by changing the particle number, keeping the temperature constant at $T/T_F=0.26$.

Figure 6

Dependence of spin relaxation on magnetic field. (a) Experimental data, obtained from a 3D experiment (circles) and theoretical results from a single-mode approach (lines). Spin populations are measured after 2 s. (b) Spin populations after 2 ms, as obtained from full 1D simulations. The inset sketches how the interplay of the differential QZE and Fermi energy determines the probabilities for lateral spin-changing collisions.

Figure 7

Temperature increase due to spin redistribution. (a) Time evolution of the temperature difference between a closed (high magnetic field at $B=7.6\mathrm{G}$) and a maximally open system (low magnetic field at $B=0.12\mathrm{G}$). The shaded area serves as a guide to the eye. The particle number is $N=3.9\times {10}^5$ at an initial temperature $T=0.24T_F=65\mathrm{nK}$. (b) Results from a simulation of the 1D equation (4) at magnetic fields $B=0.1\mathrm{G}$ and $B=8\mathrm{G}$. The method used to obtain the temperature is discussed in Appendix pp8.

Figure 8

Comparison of the imaginary part of the $T\text{-matrix}$ [Eq. (d3)] (red line and region) with the expansion [Eq. (d4)] (green line and region) for a small coupling constant $g_S$. To avoid the singularity at $k^{\prime }=0$, we choose $T_S=0$ inside the region $|k^{\prime }|\le M{g}_S/2{\hslash }^2$ indicated by the black line. The wave vector is scaled in terms of the trapping frequency $k_{\text{trap}}=\sqrt{M\omega /\hslash }$.

Figure 9

Overlap between a noninteracting equilibrium distribution [Eq. (g7)] and the Wigner function during the simulations performed to obtain the temperatures in Fig. 7.