Universal Family-Vicsek scaling in quantum gases far from equilibrium

Abstract

Fluctuations in the growing surfaces of classical systems can exhibit universal scaling behavior, known as Family-Vicsek (FV) scaling. Although this phenomenon was originally discovered in classical stochastic models, recent theoretical studies have demonstrated the presence of FV scaling in quantum many-body systems as well. Here, we observe the universal FV scaling in a one-dimensional Bose gas in an optical lattice. By monitoring the fluctuations of particle number in half of the system, which corresponds to the surface roughness, we extract all scaling exponents and demonstrate that the entire relaxation-from the growth of quantum fluctuations to their saturation-is captured by a single universal scaling function. Our results demonstrate that universal scaling laws of classical surface growth extend to quantum many-body systems, establishing a unified framework for nonequilibrium universality across classical and quantum systems.

Figures (13)

• Figure 1: Family-Vicsek scaling and experimental scheme.a, Schematic illustration of surface growth dynamics in a one-dimensional system with a length $L$. As time $t$ evolves, the standard deviation of the surface height, called surface roughness $w(L,t)$, increases. b, Time evolution of the surface roughness $w(L,t)$ at different system size. When we normalize the roughness and time by $L^{\alpha}$ and $L^z$, all curves follow a single universal function. c, Quantum gas microscope setup for studying Family-Vicsek scaling in optical lattices. Using a high-resolution objective lens, we are able to image and manipulate atomic distribution with single-site resolution. The programmable potential is created by using a digital micro-mirror device (DMD) and is combined with the imaging beam path via a dichroic. d, Cross-section of the potential in the gray shaded region in (c). The repulsive addressing beam can create a barrier wall at the two ends with its maximum height $54J$. The experiments are carried out in the central region of the length $L$. e, Patterns in the addressing beam, where each square corresponds to an individual lattice site and shaded regions indicate sites where the light is turned on. The staggered pattern is used to prepare the initial charge density wave state (top), while light is applied only to the outer edges during the subsequent dynamics (bottom).
• Figure 2: Quantum surface height and roughness of 1D spin chains.a, Fluorescence image of the initial CDW state and the site occupation. Our system of hardcore bosons is mapped to a spin-1/2 system, where the surface height is defined as the total spin of the subsystem up to site $i$. b, Atom distribution after 10$\tau$ the quench, where $\tau=\hbar/J=0.53$ ms. Different surface height patterns appear as a result of quantum tunneling. By counting the total atom number in each chain, we exclude data corresponding to imperfectly prepared initial states. c, Histogram of the center surface height at $t=0,1,10\tau$ is shown for $L=8$ (light blue) and $L=16$ (dark blue).
• Figure 3: Ballistic quench dynamics of the XX model.a, Surface roughness at various hold times $t\in [0,19]\tau$ for system size $L=8,10,12,14,16$. The solid lines represent numerical calculations that account for errors in the initial state. b, The curves collapse into a single curve when rescaled with the exponents $\alpha=0.49(1)$ and $z=1.00(4)$. The exponents and their errors are determined by optimizing the collapse of $10^4$ bootstrapped datasets. The joint probability of the exponents $P(\alpha,z)$ is shown in the inset.
• Figure 4: Diffusive quench dynamics of the XX model with dynamic disorder potentials.a, Schematic diagram of the temporal disorder. After reducing the lattice depth, the on-site potential barrier is temporarily superimposed on the lattice. The timing of the potential barrier is controlled by a trigger pulse (square-wave train), and the barrier potential is kept on (light-red area) for the time interval of the pulse $\Delta T$. The $\Delta T$ follows an exponential distribution (see the bottom panel). b, Scaled surface roughness at various hold times $t\in [0,380]\tau$ and different system sizes $L=12,16,20,24$ under the influence of temporal disorder potentials (see main text). The curves best collapse when rescaled with the exponents $\alpha=0.49(1)$ and $z=1.9(2)$. The solid lines correspond to the numerical simulations. The joint probability of the exponents $P(\alpha,z)$ is shown in the inset.
• Figure 5: Dynamics of two-point correlations.a, Time evolution of two-point correlations $C(d)$ is studied for the static XX model at system size $L=24$. The envelope of negative correlations spreads linearly with time. The dashed line represents the ballistic expansion with the characteristic velocity $v = 4J a_{\rm lat}/\hbar$ ($a_{\rm lat}=0.752$ nm), which is derived for the equal-time spin correlation based on the Lieb-Robinson velocity $2J a_{\rm lat}/\hbar$Bravyi2006. b, Time evolution of correlations with dynamic disorder potentials at system size $L=24$. The envelope is fit to a diffusively spreading Gaussian function $C(d)\sim \exp(-d^2/8Dt)$ where $D$ is the diffusion constant. The dashed-dot curve represents the $1/e^2$ envelope for the fitted diffusion constant $D=0.9(1)Ja_{\rm lat}^2/\hbar$.