The Pauli exclusion principle is a cornerstone of quantum physics: it governs the structure of matter. Extensions of this principle, such as Haldane's generalized exclusion statistics, predict the existence of exotic quantum states characterized by fractional Fermi seas (FFS), i.e. momentum distributions with uniform but fractional occupancies. Here, we report the experimental realization of fractional Fermi seas in an excited one-dimensional Bose gas prepared through ramping cycles in the interaction strength. The resulting excited yet stable Bose-gas states exhibit Friedel oscillations, smoking-gun signatures of the underlying FFS. The stabilization of these states offers an opportunity to deepen our understanding of quantum thermodynamics in the presence of exotic statistics and paves the way for applications in quantum information and sensing.
Quantum statistics determines the behavior of matter at sufficiently low temperatures. Bosons may condense into a single macroscopic wavefunction, while fermions fill all states up to the Fermi energy, forming a Fermi sea [1][2][3][4][5]. This simple yet profound distinction underlies phenomena as diverse as superconductivity and superfluidity on the one hand, and metallic conductivity, neutron-star stability, and fractional quantum Hall response on the other. However, in one dimension, statistics and interactions are inherently linked. Exchanging particles necessarily involves scattering, and bosons can acquire fermionic traits [6]. Tomonaga-Luttinger liquid theory captures this universal behavior, predicting that interacting bosons may exhibit emergent Fermi seas characterized by power-law correlations and Friedel oscillations (FO) [7][8][9][10].
It is possible to generalize quantum statistics beyond the dichotomy of bosons and fermions. Inspired by the work on anyons [11], Haldane pioneered generalized exclusion statistics (GES) [12], where the addition of a particle to the system reduces the available number of states by α, interpolating between bosons α = 0 and fermions α = 1. The celebrated anyonic statistics is described by GES with α <1 [13]. Extending the GES idea, one could imagine fractional Fermi seas (FFS) as many-body systems in which each particle occupies an α > 1 number of states. One might call the resulting statistics “superfermionic”.
Here we report the experimental realization of FFS. Using ultracold one-dimensional (1D) cesium (Cs) gases, we prepare FFS states and measure their momentum distribution for α = 2 and 4. In the first-order correlation functions, we observe Friedel oscillations, enhanced by the non-equilibrium nature of the engineered FFS. We model the experiment using a hydrodynamic approach specifically tailored to nearly integrable models known as generalized hydrodynamics (GHD) [14][15][16][17][18][19][20][21][22][23][24][25][26].
We base our description of the system on the celebrated Lieb-Liniger (LL) model [28][29][30]. It describes interacting bosons in 1D with contact interaction parametrized by the coupling constant g 1D . In our cold-atom experiment, g 1D can be freely and dynamically tuned by means of Feshbach resonances in conjunction with a confinementinduced resonance (CIR) [30,31] to values from zero to +∞ and -∞. The coupling constant is given by g 1D = 2 ̵ h 2 a 3D /[ma 2 ⊥ (1 -1.0326a 3D /a ⊥ )] [29], where m is the atoms’ mass and a ⊥ is the transverse harmonic oscillator length a ⊥ = √ ̵ h/mω ⊥ of our 1D trap with transverse frequency ω ⊥ , and a 3D =a 3D (B) is the 3D scattering length, which we tune by means of a magnetic field B. FFS have recently been predicted to exist for the LL model as excited many-body states [14,27]. It was proposed that repeated holonomy cycles [32] would give access to a succession of excited states. In these cycles [27] the coupling g 1D is ramped from the non-interacting value g 1D = 0 to the strongly repulsive regime (g 1D = +∞), where the system realizes the Tonks-Girardeau (TG) phase [33,34]. Then, the coupling is switched to strongly attractive values (g 1D = -∞), where the gas is in the super-Tonks-Giradeau (sTG) state [31,35]. Finally, g 1D is ramped back to g 1D =0. The state of the system is parameterized by the charge parameter ℓ, as shown in Fig. 1 (A). As we shall see in the following, due to the collective nature of our quantum system, our experiment is better understood in terms of taking the system into thermodynamic states of progressively higher energy, described by generalized Gibbs ensembles (GGE) [36][37][38]. In view of this, we refer to our experimental protocol as interaction cycles. The resulting quasi-adiabatic dynamics is expected to induce the formation of the FFS at ℓ = 2, 4, …. As depicted in Fig. 1 (B), the initial delta-function-like momentum distribution for ℓ=0 transmutes into a box-type distribution n(k) = 1/2 for |k| < 2k F for ℓ = 2 and then to n(k) = 1/4 for |k| < 4k F for ℓ = 4. Here, k F is the Fermi wave-vector of the TG state. With this parameterization, we can identify α = ℓ. Even though the FFS are highly excited many-body states, they are stabilized by the integrability of the LL model [14,27]. Note that the FFS are realized at the non-interacting points.
Our experiment begins by preparing a pure Cs Bose-Einstein condensate (BEC) in the Thomas-Fermi regime with no detectable thermal fraction and an atom number of 5×10 4 with ±5% shot-to-shot fluctuations [39]. The scattering length is initially set to a 3D = 195(1) a 0 . Here, a 0 is Bohr’s radius. We adiabatically load the BEC into a 2D optical lattice with a spacing of 532.35 nm and depth of 30 E r , where E r = ̵ h 2 k 2 L /(2m) is the photon recoil energy with the lattice wavevector k L . As part of this procedure, a 3D is ramped to a 3D =517(1)a 0 during lattice loading. The atoms then populate a non-uniform array of abo
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