From Many-Body Oscillations to Thermalization in an Isolated Spinor Gas

The dynamics of a many-body system can take many forms, from a purely reversible evolution to fast thermalization. Here we show experimentally and numerically that an assembly of spin-1 atoms all in the same spatial mode allows one to explore this wide variety of behaviors. When the system can be described by a Bogoliubov analysis, the relevant energy spectrum is linear and leads to undamped oscillations of many-body observables. Outside this regime, the nonlinearity of the spectrum leads to irreversibility, characterized by a universal behavior. When the integrability of the Hamiltonian is broken, a chaotic dynamics emerges and leads to thermalization, in agreement with the eigenstate thermalization hypothesis paradigm.

Figure 1

Experimental observation of many-body oscillations. (a) Evolution of the mean number of pairs (circles) and its standard deviation (squares) in a sudden quench from $q_i=277(3)\mathrm{Hz}$ to $q_f=0.31(1)\mathrm{Hz}$. Here $N\approx 5400(740)$. The solid line is the result of a fit of the Bogoliubov prediction (6) to the experimental data, with $U_s=17.5(1.4)\mathrm{Hz}$ as a single free parameter. (b) Mean (circles) and standard deviation (squares) of the magnetization $S_z$. The shaded region indicates the detection noise level. (c) Oscillation frequency obtained from a fit to the measured ${\overline{N}}_p(t)$ (circles) for various $q$, compared to the prediction (5) (line). (d)–(f) Measured distribution of the number of pairs for $t=10$, 80, and 160 ms, with the solid lines corresponding to the prediction (7).

Figure 2

Experimental observation of relaxation near zero magnetic field. (a) Evolution of the population $n_0(t)$ following a fast quench of $q$ to a negligible value, for various atom numbers $N$. Disks: $N=107$, $U_s=17.2\mathrm{Hz}$; squares: $N=230$, $U_s=24.2\mathrm{Hz}$; lozenges: $N=835$, $U_s=64.7\mathrm{Hz}$. The initial state is $|m=0{⟩}^{\otimes N}$. For $\tau =9$, the “real” time spanned is $t=609$, 635, and 452 ms for the three atom numbers. The solid line is the universal prediction (10). (b)–(e) Distribution of the population $n_0$ at $t=30$, 100, 200, and 500 ms for $N=107$ atoms. In (b)–(d), the solid lines are the results of a numerical simulation. In (e), the green dotted line is the prediction from the microcanonical ensemble [51], and the red dashed line is the prediction from the GGE with the constraint $S_z=0$ [35].

Figure 3

Numerical exploration of chaotic behavior and thermalization. (a) Density of states (dashed line) for the Hamiltonian ${\widehat{H}}^{\prime }$ for $q/U_s=0.8$, $\mathrm{\Omega }/U_s=0.6$, and $N=100$. The histograms show the populations (in arbitrary units) of the states chosen in (d) and (e) to investigate the dynamics in the colored zones $A$ and $B$. (b),(c) Statistics of the splitting $\mathrm{\Delta }E$ between adjacent energy levels for zones $A$ and $B$, containing, respectively, 724 and 286 levels. The continuous and dashed lines show the Wigner-Dyson (chaotic motion) and Poisson (regular motion) distributions, respectively [66]. (d),(e) Evolution of $⟨{\widehat{N}}_0⟩$ and $⟨{\widehat{S}}_x^2⟩$ for the wave packet localized in zone $A$ or $B$ shown in (a). The dashed lines are the predictions from the microcanonical ensemble.