Observing anyonization of bosons in a quantum gas

Abstract

Anyons^1,2 are low-dimensional quasiparticles that obey fractional statistics, hence interpolating between bosons and fermions. In two dimensions, they exist as elementary excitations of fractional quantum Hall states^3-5 and are believed to enable topological quantum computing^6,7. One-dimensional anyons have been theoretically proposed, but their experimental realization has proven to be difficult. Here we observed emergent anyonic correlations in a one-dimensional strongly interacting quantum gas, resulting from the phenomenon of spin-charge separation^8-10. A mobile impurity provides the necessary spin degree of freedom to engineer anyonic correlations in the charge sector and simultaneously acts as a probe to reveal these correlations. Starting with bosons, we tune the statistical phase to transmute bosons through anyons to fermions and observe an asymmetric momentum distribution^11-14, a hallmark of anyonic correlations. Going beyond equilibrium conditions, we observed dynamical fermionization of the anyons¹⁵. This study opens the door to the exploration of non-equilibrium anyonic phenomena in a highly controllable setting^15-17.

Fig. 1. Experimental realization of 1D anyons.

a, Emergence of anyons from spin–charge separation. For strong interactions in one dimension, the wavefunction factorizes into a charge part and a spin part. Note that, although the illustration depicts a localized spin impurity, we created a delocalized impurity. The charge sector (left) depicts one possible positional arrangement of the particles, whereas the spin sector (right) is one possible spin distribution in the squeezed space, both corresponding to the arrangement of the spinful particles in the top line. In the finite momentum ground state of the system, all the momentum is carried by the spin sector in the form of spin waves. Integrating out the spin degrees of freedom realizes an effective system of 1D hardcore anyons in the charge sector. The statistical phase θ of these emerging anyons is given by the momentum of the spin waves (see ‘Exchange symmetry engineering’ and ‘Emergence of anyons via spin–charge separation’ in Methods). b, Edge of the excitation spectrum of a 1D Bose gas for charge excitation (dashed line) and spin excitation (solid line). c, Expected momentum distributions of anyons for different values of the statistical phase θ as set by the momentum ħQ and indicated in b. d, Experimental realization of an ensemble of 1D Bose gases in tubes formed by two retro-reflected laser beams. On average, each tube contained one impurity particle (blue sphere). The interaction between the impurity and the Tonks–Girardeau host gas (red spheres) can be tuned by means of a Feshbach resonance. e, Host–host (dashed curve) and host–impurity (solid curve) scattering lengths a_↑↑ and a_↑↓ as a function of the magnetic field B.

Fig. 2. Momentum distribution of anyonized bosons.

a, Evolution of the measured impurity momentum distribution n_↓(k) for variable statistical phases θ determined by the injected momentum ħQ, as indicated. Each distribution is the average of seven experimental realizations. b, Numerical results of the anyonic momentum distribution n_a(k) using AHM. c–f, Example distributions for θ/π equal to 0 (c), 0.53(2) (d), 0.72(3) (e) and 0.98(4) (f). The error bars are smaller than the symbol sizes. The data were compared to the numerical results of the ground states of AHM (dotted lines), swap model (solid lines) and time evolution governed by the sBHM (dashed lines).

Fig. 3. Characterization of the anyonic momentum distribution.

a, Measured peak momentum k^*. b, Peak occupation of the impurity momentum distribution n_↓(k) as a function of statistical phase θ. The experimental data (dots) were compared with the results of the simulations on the basis of various models, as indicated. The error bars reflect the standard error.

Fig. 4. Dynamical fermionization of hardcore anyons.

a–c, Evolution of the impurity momentum distribution n_↓(k) after quenching the confinement to a flat-bottom trap and allowing 1D expansion for t_1D = 0 ms (a), 2 ms (b) and 5 ms (c) for three different values of statistical phase θ, as indicated. Each distribution is the average of ten experimental realizations. d–f, Theoretical prediction for the evolution of the momentum distribution of hardcore anyons (N = 10) during 1D free expansion for t_1D = 0 ms (d), 2 ms (e) and 5 ms (f).

Extended Data Fig. 1. Role of finite force and finite interaction.

a Measured n_↓(k) at fixed total momentum Q and fixed interaction strength γ_↑↓ for two different values of the force F_↓ as indicated. Each distribution is the average of 7 experimental realizations. The experimental data is compared to the results of the simulations based on the sBHM. b Measured n_↓(k) at fixed Q for different γ_↑↓. The solid line is the prediction from AHM.