Observation of the Efimovian expansion in scale-invariant Fermi gases

Abstract

Scale invariance emerges and plays an important role in strongly correlated many-body systems such as critical regimes nearby phase transitions and the unitary Fermi gases. Discrete scaling symmetry also manifests itself in quantum few-body systems such as the Efimov effect. Here we report both theoretical predication and experimental observation of a novel type expansion dynamics for scale invariant quantum gases. When the frequency of the harmonic trap holding the gas decreases continuously as the inverse of time t, surprisingly, the expansion of cloud size exhibits a sequence of plateaus. Remarkably, the locations of these plateaus obey a discrete geometric scaling law with a controllable scale factor and the entire expansion dynamics is governed by a log-periodic function. This striking expansion of quantum Fermi gases shares similar scaling laws and same mathematical description as the Efimov effect. Our work demonstrates the first expansion dynamics of a quantum many-body system with the temporal discrete scaling symmetry, which reveals the underlying spatial continuous scaling symmetry of the many-body system.Interaction between dilute ultracold atoms is described by the s-wave scattering length. For a spin-1/2 Fermi gas, when the scattering length diverges at a Feshbach resonance, there is no length scale other than the interparticle spacing in this many-body system, and therefore the system, known as the unitary Fermi gas, becomes scale invariant. The spatial scale invariance leads to universal thermodynamics and transport properties as revealed by many experiments [1][2][3][4][5][6][7][8][9][10][11][12][13]. On the other hand, in a boson system with an infinite scattering length, threebody bound state can form, where an extra length scale of the three-body parameter sets a short-range boundary condition for all three bosons being very close. It turns the continuous scaling symmetry into a discrete scaling symmetry, and gives rise to infinite number of three-body bound states whose energies obey a geometric scaling symmetry. This is well known as the Efimov effect [14,15], which has been observed in quite a few cold atom experiments [16][17][18][19][20][21][22][23][24], and recent experiments have also confirmed the geometric scaling of the energy spectrum [25][26][27][28]. Both the continuous and the discrete scaling symmetry are interesting emergent phenomena in a strongly interacting system. For a harmonic trapped gas, the expansion dynamics offers great insight to the property of the gas. Well known example is the anisotropic expansion that proves hydrodynamics behavior due to the Bose condensation [29,30] or strong interactions of Fermi gas [31]. Other examples are, for instance, slowing down of expansion in a disorder potential provides evidence for localization behaviors [32,33] and expansion in the presence of optical lattice reveals correlation effects [34]. In this work, we ask a question that, considering a scale invariant quantum gas hold by a harmonic trap, when the trap is gradually opened...

Dynamic super Efimov effect

“…This alikeness is due to a profound connection we discovered between the expansion dynamics of a scale invariant quantum gas and a one-dimensional Schrödinger equation [15]. This Efimovian expansion has recently been observed experimentally [15].…”

“…(3), it becomes clear that the double exponential scaling of the binding energies corresponds to a double-log periodicity of the bound state wave-functions in space [13,14]. In a recent separate development, a dynamic Efimov effect for a scale invariant many-body quantum gas has been proposed [15], in which we considered the situation that the trap frequency ω(t) decreases as 1/( √ λt) with time t. We have shown that the expanding cloud size as a function of time bears the same geometric scaling behavior as the three-body Efimov effect. This alikeness is due to a profound connection we discovered between the expansion dynamics of a scale invariant quantum gas and a one-dimensional Schrödinger equation [15].…”

“…This alikeness is due to a profound connection we discovered between the expansion dynamics of a scale invariant quantum gas and a one-dimensional Schrödinger equation [15]. This Efimovian expansion has recently been observed experimentally [15].In this paper, we show that the hallmark of the super Efimov effect, the double-log periodicity, emerges in the cloud size of the expanding scale invariant gas when the trap frequency ω(t) is specifically engineered to vary with time t as ω SE (t) ≡ 1 4t 2 + 1 λt 2 log 2 (t/t * ) ,where t * and λ (< 4) are tunable parameters. Our predicted dynamic super Efimov effect applies to all scale invariant quantum gases, including i) non-interacting bosons or fermions in any dimension, ii) bosonic or arXiv:1608.05799v1 [cond-mat.quant-gas]…”

“…i , satisfies the dynamic equation (5) which can be derived by calculating directly the equation of motion for R 2 (t) [15,16]. The absence of interactions in Eq.…”

“…Based on the same general formalism (5-7), previously we have successfully realized the dynamic Efimov effect by implementing ω(t) = ω E (t) ≡ 1/( √ λt) with λ < 4 to match the effective hyper-spherical potential V E (ρ) ≡ −(s 2 0 + 1/4)/ρ 2 [15], which is known to be responsible for the three-body Efimov effect [1,2]. Equation (7) indicates an intriguing underlying connection between the super Efimov effect and the Efimov effect, and suggests interesting effects beyond.…”