Dynamical Fractal in Quantum Gases with Discrete Scaling Symmetry Chao Gao,1, ∗ Hui Zhai,2 and Zhe-Yu Shi3, y 1Department of Physics, Zhejiang Normal University, Jinhua, 321004, China 2Institute for Advanced Study, Tsinghua University, Beijing, 100084, China 3School of Physics and Astronomy, Monash University, Victoria 3800, Australia (Dated: January 23, 2019) Inspired by the similarity between the fractal Weierstrass function and quantum systems with discrete scaling symmetry, we establish general conditions under which the dynamics of a quantum system will exhibit fractal structure in the time domain. As an example, we discuss the dynamics of the Loschmidt amplitude and the zero-momentum occupation of a single particle moving in a scale invariant 1=r2 potential. In order to show these conditions can be realized in ultracold atomic gases, we perform numerical simulations with practical experimental parameters, which shows that the dynamical fractal can be observed in realistic time scales. The predication can be directly verified in current cold atom experiments. In the lecture presented to K¨oniglicheAkademie der 2 Wissenschaften in 1872 [1,2], Karl Weierstrass intro- duced an intriguing function series, 0 1 X -2 W (x) = an cos(bnπx); 0 < a < 1; (1) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 n=0 2 which is known as the Weierstrass function nowadays. 0 The original intent of Weierstrass's work is to construct an example of a real function being continuous every- -2 where while differentiable nowhere. After its publication, -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 the remarkable function has intrigued many mathemati- 2 cians and physicists, who have made substantial contri- butions to the understanding of Weierstrass's function [3{ 0 8]. Among these works, probably the most important -2 discovery is that the Weierstrass function can have non- -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 integer Hausdorff dimensions, indicating that it is not a regular curve but a fractal [4{6], although the term `frac- tal' was invented over a hundred years later by Mandel- FIG. 1. The Weierstrass function W (x). Top: b = 1 and ab < brot in 1975 [5]. 1, regular curve; Middle: b = 2 and ab = 1, the transition Here we review two crucial properties of the Weier- point; Bottom: b = 3 and ab > 1, fractal curve. a = 1=2 is strass function that are closely related to the following fixed for all three cases. The inset is a zoom in of the detailed structure around the red point, which shows the self similarity discussions. As plotted in Fig.1, the fractal behav- behavior of W (x). ior of the Weierstrass function greatly depends on the parameter ab. For ab < 1, W (x) is a regular one di- mensional curve with continuous derivative. While the In physical systems, the DSS or self similarity emerges function becomes `pathological' and an fractal structure in a quantum system if its renormalization group (RG) emerges when ab is greater than 1. In this regime, it is flow shows a limit cycle behavior [10]. In this case, believed that the Weierstrass function has fractal dimen- the RG flow of the quantum system becomes periodic sion, [6,8,9] when the cutoff changes by a scaling factor λ. Proba- bly the most celebrated example of a RG limit cycle is log a arXiv:1901.06983v1 [cond-mat.quant-gas] 21 Jan 2019 DH = 2 + : (2) the so-called Efimov effects discovered by Vitaly Efimov log b in 1970 [11, 12]. Efimov showed that in a three-particle Secondly, like other fractals, the Weierstrass function system with resonant two-body interaction, there can ex- displays a self-similar graph with infinitely fine details. ist an infinite tower of quantum mechanical three-body This property can be directly observed from Fig.1, and bound states. These bound states are self-similar in the mathematically, it is related to the discrete scaling sym- sense that their wave functions n(r) satisfy, metry (DSS) of W (x), n+1(r) / n(λr); (4) −1 W (bx) ' a W (x): (3) where λ > 1 is the scaling factor. Their binding energies 2 2 En also follow similar DSS, moving in a D-dimensional 1=r attractive potential like Eq.(6). This Hamiltonian is able to describe a wide vari- 2 En+1 ' λ En: (5) ety of systems, including the conventional Efimov bound states of three three-dimensional particles at resonance For the convenience of the later discussion, we have la- [11{13]. beled the bound states in a reverted way comparing to Inserting a complete basis of eigenstates of H^ into the conventional Efimov labeling [13]. We choose an ar- E Eq.(7), we obtain bitrary shallow bound state to be n = 0, deeper bound i states will be n = 1; 2; 3 :::N with N being the deep- X − HE t L(t) = hφ0je ~ jnihnjφ0i + :::; (8) est bound state. Shallower bound states are labelled by n n = −1; −2; −3;:::. These scaling behavior can be explained by an effec- where jni is the bound state with eigenenergy En. The tive Schr¨odingerequation which describes a single parti- terms denoted by ::: correspond to the contribution from cle moving in an 1=r2 attractive potential [13], the scattering states, which do not possess the DSS. Therefore, in order to obtain a dynamic fractal with DSS, 2 1 s2 + 1=4 the system need to satisfy Condition 1: the contri- − ~ @ (rD−1@ ) + 0 (r) = E (r); 2m rD−1 r r r2 bution of the scattering states is negligible in the (6) time interval of interests. For sufficiently long time, this requirement should always be satisfied if the initial where D is the spatial dimension, s0 is a dimension- state φ0 is a square-integrable wave packet. This is be- less parameter that controls the strength of the poten- cause the scattering states will always scatter an initial tial. It can be shown that, after imposing a proper wave packet far away from the interaction center, which short-range regularization, the zero-energy solution of leads to a negligible overlap with the initial wavefunction the Schr¨odingerequation shows a log-periodic behavior after long time. (1−D)=2 (r) ' r cos(s0 log r + '), which is the origin of Using the energy scaling relation of Eq.(5), we find the DSS of Eq.(4) and Eq.(5)[14{16]. The scaling pa- π=s0 N rameter λ is also determined by s through λ = e . i 2n 0 X − λ E0t Inspired by the similarity between the DSS in fractal L(t) ' αne ~ ; (9) Eq.(3) and in quantum system Eq.(4) and (5), in this n=−∞ work we explore the connection between these two. No- 2 R D ∗ 2 where αn = jhφ0jnij = j d rφ0(r) n(r)j . To connect tice that the time evolution of a quantum system natu- i L(t) to a Weierstrass function, we need Condition 2: αn − Ent n rally involves oscillation terms like e ~ . This suggests can be expressed as a with a properly chosen a. that the dynamics of a quantum system with eigenener- Indeed, because of the scaling Eq.(4), we have n+1(r) ' n D=2 gies En ' b (i.e. a system with DSS) makes a perfect λ n(λr), which indicates that the sizes of the bound −1 candidate for realizing fractal structures in the time do- states satisfy Rn+1 ' λ Rn. Now if we assume that main. Thus we discuss two post-quench dynamical mea- the initial state φ0(r) is a wave packet with radius L, for surements in quantum systems with DSS, the Loschmidt deep bound states with n ≥ 0 [21], we have Rn . L and amplitude and the zero-momentum occupation. Through thus the following discussion, we will reveal, one by one, the Z 2 general conditions under which the dynamics of these D ∗ αn+1 ' d rφ (0) n+1(r) systems can be expressed by a Weierstrass function and 0 display fractal behavior. We argue that these conditions Z 2 D=2 D ∗ can be satisfied with realistic parameters in cold atoms ' λ d rφ (0) n(λr) 0 experiments and verify this by numerical simulation. −D Loschmidt amplitude. First we consider the Loschmidt ' λ αn; (10) amplitude [17] of a quantum system, ∗ ∗ where we have assumed φ0(r) ' φ0(0) because n(r) is i highly localized around the potential center. For other − H^E t L(t) ≡ hφ je ~ jφ i = hφ jφ i; (7) 0 0 0 t shallow bound states with n < 0, on one hand, the size mismatch leads to very small wavefunction overlaps, and where H^ is a Hamiltonian with DSS. The Loschmidt E on the other hand, these terms correspond to low fre- amplitude L(t) is the wave function overlap between a quency oscillations which can be regarded as a constant time evolved quantum state and its initial state, which in the time scale of interests. Ignoring these shallow can be measured by a standard Ramsey interferometry states, we finally obtain a Weierstrass-like function, protocal in ultracold atom experiments [18{20]. In principle, H^E can be a complicated many-body N i 2n X −nD − λ E0t Hamiltonian. However, to illustrate the basic idea, we L(t) / λ e ~ : (11) shall first consider the simplest case of a single particle n=0 3 Zero Momentum Occupation. It is possible to show 4 that the dynamics of other observables under HE can 10 also be related to the Weierstrass function.