# Herd’s Eye View: Improving Game AI Agent Learning with Collaborative
Perception

Andrew Nash,1 Andrew Vardy, 1 2 David Churchill 1

###### Abstract

We present a novel paradigm named Herd’s Eye View (HEV) that adopts a global
perspective derived from multiple agents to boost decision-making capabilities
of reinforcement learning (RL) agents in multi-agent environments,
specifically in the context of game AI. The HEV approach utilizes cooperative
perception to empower RL agents with a global reasoning ability, enhancing
their decision-making. We demonstrate the effectiveness of the HEV within
simulated game environments and highlight its superior performance compared to
traditional ego-centric perception models. This work contributes to
cooperative perception and multi-agent reinforcement learning by offering a
more realistic and efficient perspective for global coordination and decision-
making within game environments. Moreover, our approach promotes broader AI
applications beyond gaming by addressing constraints faced by AI outside of
this field. The code is available on
GitHub.111https://github.com/andrewnash/Herds-Eye-View

## Introduction

Game environments traditionally grant AI agents access to extensive global
information from the game engine. While this configuration assists in
efficient decision-making, it doesn’t accurately represent the restrictions
encountered by AI applications outside of gaming, where comprehensive access
to a system’s software or engine isn’t feasible. Consequently, game AI
techniques that rely predominantly on game engine data may limit their
potential contribution to broader AI applications, as their dependency on
perfect information and global environmental data is often unrealistic in
other contexts.

In response to these challenges, our work delves into the application of more
constrained, realistic perception models for game AI. We take inspiration from
publications like the ViZDoom platform (Wydmuch, Kempka, and Jaśkowski 2019)
and the Obstacle Tower Challenge (Juliani et al. 2019) that have embraced this
shift towards real-world constraints. Both ViZDoom and Obstacle Tower have
utilized visual data as the primary input for AI agents, enabling them to
navigate complex 3D environments and reinforcing the importance of perception-
based game AI models without game engine access.

Research in autonomous vehicles has made extensive strides in AI perception
models, particularly using intermediary environmental representations like the
Bird’s Eye View (BEV). The BEV model provides an overhead perspective of the
environment, often in the form of a semantic obstacle grid, from a single
vehicle’s standpoint. This concept has become a key component in many self-
driving systems (Ma et al. 2022).

Drawing on this, we propose a similar intermediary representation for game AI:
the Herd’s Eye View (HEV) model. Differing from the BEV’s individual-centric
perspective, the HEV model offers a shared world-centric perception derived
from multiple agents. This shared perception model aligns closer to real-world
AI applications, where multiple systems often work together to understand and
navigate their environment.

The HEV model presents dual advantages. First, it mirrors the constraints
faced by AI outside of gaming, contributing to the development of more
believable AI behavior in games. Second, it alleviates the computational
demands associated with the BEV model, where each agent maintains its own
unique view of the environment, instead only a single shared global view is
utilized.

Emulating the successful methodologies of the ViZDoom project and the Obstacle
Tower paper, we also incorporate reinforcement learning (RL) in our approach.
RL enables our agents to tackle both low-level control tasks and high-level
planning challenges concurrently in complex, procedurally generated
environments. Importantly, similar to the Obstacle Tower approach, our agents
are assessed not solely on their ability to navigate familiar environments,
but also on their adeptness in handling unique variations of these
environments. This underscores the importance of generalization in adapting to
novel scenarios within the same environment.

To assess the effectiveness of the HEV model and RL methodology in game AI, we
conduct two sets of experiments in three simulated Multi-Agent Reinforcement
Learning (MARL) game environments. The first experiment compares the accuracy
of world-centric predictions using HEV with ego-centric predictions using BEV.
The second experiment evaluates the efficiency of policies learned by RL
agents trained on HEV views compared to those trained on BEV views.

Our work makes the following contributions:

  1. 1.

We propose a baseline model for performing semantic segmentation in a fixed
”HEV” world-view.

  2. 2.

We demonstrate the effectiveness of the HEV fixed world viewpoint in improving
collaborative perception and MARL in games.

Our exploration of more realistic perception models and the application of
reinforcement learning provides significant insights for game AI development,
stressing the wider applicability of these techniques beyond the gaming
industry.

## Related Works

### Birds Eye View Semantic Segmentation

In autonomous vehicle research, the bird’s-eye view semantic segmentation task
involves predicting pixel-level semantic labels for a top-down ego-centric
view of an environment. Segmentation classes are typically dedicated to
vehicles, driveable areas, and obstacles. In prior BEV research, a significant
point of distinction lies in the method used for transforming 2D perspective-
view features into 3D space or directly onto the BEV plane. Many previous
works have leveraged explicit geometric reasoning in their perspective
transformation (Reiher, Lampe, and Eckstein 2020; Philion and Fidler 2020; Hu
et al. 2021). An approach that has recently gained popularity is the Cross-
View Transformer (CVT) (Zhou and Krähenbühl 2022) model, which implicitly
models scene geometry. The CVT model leverages a camera-aware cross-view
attention mechanism to implicitly learn a mapping from individual camera views
to a canonical map-view representation for map-view semantic segmentation. The
model consists of a convolutional image encoder for each view and cross-view
transformer layers for inferring the segmentation in a simple, easily
parallelizable, and real-time manner. BEVFormer (Li et al. 2022c) uses a
similar cross-attention model to extract spatial and temporal BEV information.

The HEV semantic segmentation task poses a unique challenge compared to the
BEV task since the agent translations are unknown; this requires the model to
geometrically reason about multiple camera views to localize. For our baseline
approach, we leverage the CVT model proposed by (Zhou and Krähenbühl 2022).
The CVT model is well suited for the HEV task because of it’s global attention
mechanism. Many BEV publications aim to optimize this global attention
mechanism since in ego-centric tasks, a camera view only overlaps with a
consistent subsection of the map-view. Conversely, in our HEV world-centric
use case, global attention is an advantage because a camera view can overlap
with any part of the map-view. Additionally, we expect that the model’s
performance can be further improved by incorporating additional information
from other sensors, such as lidar and radar, as demonstrated by recent works
(Harley et al. 2022).

### Collaborative Perception Datasets

Figure 1: A visualization of the different map-view coordinate frames tested,
ego-centric views move in accordance to translation and rotation of the agent,
whereas the world-centric view does not. Figure 2: A visualization of the
proposed HEV approach in the Dungeon Escape environment: Agent camera views
are extracted via a backbone model, then combined in a cross attention module,
then decoded into a world-view semantic segmentation. The resulting semantic
segmentation can be used as a observation for a swarm of robots.

Autonomous vehicle datasets have been widely used in collaborative perception
research, comprising various sensory inputs, including cameras, lidar, and GPS
(Han et al. 2023), from multiple vehicles in a vehicle-to-vehicle (V2V)
environment (Xu et al. 2022c; Yu et al. 2022). Some datasets, such as those
proposed in (Li et al. 2022a; Mao et al. 2022), include infrastructure
sensors, resulting in a vehicle-to-infrastructure (V2I) data model. Others,
such as the dataset presented in (Xu et al. 2022b), employ a vehicle-to-
everything (V2X) model. The CoPerception-UAVs dataset (Hu et al. 2022) employs
five images from five drones flying in formation. It is worth noting that
these datasets are all sourced from CARLA (Dosovitskiy et al. 2017) in Unreal
Engine, a widely used open source platform for simulating autonomous vehicles.

The HEV datasets sourced from our simulated environments are uniquely
challenging in the field of collaborative perception, as the agenfts are
equipped with only one or two cameras. Unlike previously proposed
collaborative perception problems, the HEV task does not provide the agents
with the transformation component of their pose. The unknown position of each
camera view within the global coordinate frame adds a significant challenge to
the semantic segmentation prediction task and other downstream tasks.

### Collaborative Perception Methods

Collaborative perception has been explored in recent years, improving the
capability of single-agent perception models (Li et al. 2022b; Hu et al. 2022;
Lei et al. 2022; Su et al. 2023; Zhou et al. 2022). In conventional
collaborative perception, intermediate representations produced by sensors or
neural networks from multiple viewpoints are propagated among a team of
robots, such as a group of vehicles (Hu et al. 2022; Xu et al. 2022a) or a
swarm of drones (Zhou et al. 2022; Hu et al. 2022). The existing works
commonly learn a collaboration module, produced by a Graph Neural Network
(GNN) (Zhou et al. 2022; Li et al. 2021), Convolutional Neural Network (CNN)
(Li et al. 2022b; Qiao and Zulkernine 2023), or a Transformer (Xu et al.
2022a; Hu et al. 2022) to combine multiple robot intermediate representations.

Prior research has focused on robots equipped with multiple sensors, requiring
sensor data fusion on a per-agent basis before information exchange among
agents (Han et al. 2023). However, in this work, we focus on robots with only
one or two cameras and no additional sensors, making our approach more
amenable to smaller, simpler robot swarms. Since we focus on simpler robots,
we do not utilize a collaboration module, and instead fuse all camera views
together in a single cross-attention module.

## Methodology

### Herd’s Eye View

In the Herd’s Eye View (HEV) semantic segmentation task, we are given a set of
$n$ monocular camera views, $(I_{k},K_{k},R_{k})^{n}_{k=1}$ consisting of an
input image $I_{k}\in\mathbb{R}^{H\times W\times 3}$, camera intrinsics
$K_{k}\in\mathbb{R}^{3\times 3}$, and extrinsic rotation
$R_{k}\in\mathbb{R}^{3\times 3}$ with respect to the robot base . The goal of
the HEV task is to predict a binary semantic segmentation mask
$y\in\\{0,1\\}^{h\times w\times C}$ in the global coordinate frame, where $C$
is the desired number of segmentation classes. The HEV task adds additional
ambiguity to the well-studied BEV task as each camera view is at an unknown
translation within the global coordinate frame.

(a)

(b)

(c)

Figure 3: Images of the three environments used to test the HEV collaborative
perception and reinforcement learning algorithms. From Left-to-right: (a)
Collaborative Push Block, (b) Dungeon Escape, (c) Planar Construction

We define a BEV as a single agent perception to an ego-view, whereas the HEV
is a collaborative perception to a fixed world-view. A comparison of the ego-
centric views tested and the fixed word-view can be seen in Figure 1.

Our approach, seen in Figure 2, follows three steps:

  1. 1.

We collect multiple views of the environment from robot cameras.

  2. 2.

We use a collaborative perception model to obtain the HEV, the world-view
semantic segmentation of the environment.

  3. 3.

We input the HEV to a Reinforcement Learning (RL) control policy to obtain
agent control commands.

Our goal is to establish a baseline HEV perception model to extract
information from the multiple camera views and project them into a fixed
world-view. We propose a baseline perception model using the Cross-View
Transformer (CVT) (Zhou and Krähenbühl 2022) and use semantic segmentation as
our downstream task. The Cross-View Transformer is a recent approach that uses
a cross-view attention module, first proposed by (Zhou and Krähenbühl 2022),
enabling the agents to reason about their environment in a ego-centric
coordinate frame. We extend the CVT model to further improve its accuracy and
speed for the HEV use case. We name our baseline model the Herd’s Eye View
Cross-View Transformer (HEV-CVT). We use a world-centric map embedding and
tune positional embeddings, output sizes, and the number of transformer layers
to fit our proposed HEV environments.

### Data Collection

Table 1: HEV simulated environment parameters. | Robots | Robot Cameras | Grid Size
---|---|---|---
Collaborative Push Block | 3 | 1-left, 1-right | 32x32
Dungeon Escape | 2-3 | 1-left, 1-right | 32x32
Planar Construction | 6 | 1-forward | 32x64

We use identical Unity simulation environments to source the datasets for
training the HEV semantic segmentation task and MARL task. To collect the HEV
ground truth for both tasks, we use our custom fork of the Unity Grid-Sensor
Library (Baske 2021) that allows the collection HEV world-centric grid-sensors
that were not previously possible. The only difference between ego-centric
based agents and world-centric based agents is the location of their grid-
sensor and the perspective at which they take their actions (e.g., forward for
the word-centric agent is always north, but forward for the ego-centric agent
is with respect to their current orientation). All agents are trained on
ground-truth sensors, calculated using the bounding boxes that are
individually tuned to each object. The resolution of the grid-sensor is
adjusted to accommodate the complexity and size of the environment as seen in
Table 1.

Our simulations are conducted in three different ML-Agents environments:

Collaborative Push Block: Agents are required to push white blocks to the
green area, with larger blocks requiring more agents to push (Cohen et al.
2022).

Dungeon Escape: Agents must sacrifice one of their own to take down a green
dragon and obtain a key, which the rest of the team must then use to escape
the dungeon (Cohen et al. 2022).

Planar Construction: Six agents collaborate to push red pucks into desired
positions. Desired positions can be random or static, and are observed via a
Grid-Sensor, as in the push block environment (Strickland, Churchill, and
Vardy 2019).

We utilize the open-source Collaborative Push Block and Dungeon Escape
environments from ML-Agents (Cohen et al. 2022), and only change the sensor
input of agents. We recreate the Planar construction task (Vardy 2018;
Strickland, Churchill, and Vardy 2019; Vardy 2020, 2022, 2023) based on
previous work (Strickland, Churchill, and Vardy 2019) in ML-Agents. All three
environments can be seen in Figure 3. For MARL training, we use the HEV ground
truth as model input and identical reward functions to the original
implementations. Specifically, the agents are trained using the Multi-Agent
POsthumous Credit Assignment (MA-POCA) algorithm (Cohen et al. 2022) in Unity
ML-Agents. By using identical reward functions, we aim to create a fair
comparison between the performance of agents using HEV and those using
traditional sensor frames in cooperative scenarios.

The MARL task enables us to train the HEV-CVT model, which can perform
semantic segmentation in an ego-centric or world-centric view. To collect the
data necessary for training the HEV-CVT model, we run the trained MA-POCA
models and collect the camera view, camera intrinsics, and rotation extrinsic
from each agent at each step of the simulation, along with the ground truth
HEV. By collecting data from various environments and introducing variations,
we aim to create a diverse and robust dataset for training the HEV-CVT model.

## Experiments and Results

### Collaborative Perception

Table 2: HEV-CVT validation IoU results per coordinate frame in each environment (higher is better). | World-Centric | Ego-Centric | Ego-Forward
---|---|---|---
Collaborative Push Block | 96.94% | 63.87% | 64.22%
Dungeon Escape | 43.53% | 1.82% | 26.07%
Planar Construction | 48.37% | 35.45% | 10.16%
Table 3: MA-POCA mean episode length per coordinate frame in each environment (lower is better). | World-Centric | Ego-Centric | Ego-Forward
---|---|---|---
Collaborative Push Block | 222.6 | 246.6 | 230.5
Dungeon Escape | 14.15 | 16.74 | 23.27
Planar Construction | 110.5 | 120.3 | 149.9

The HEV-CVT model must accurately localize the position of each agent based on
the overlap of six camera frames, which are located at unknown rotations and
positions. The cameras are recorded at resolution $480\times 224$, and we use
the camera intrinsics of a Raspberry Pi Camera Module 3. Consistent with prior
works (Ma et al. 2022), we show the result Intersection over Union (IoU)
metric for the HEV-CVT model trained on each environment in Table 2. We
compare the performance of the baseline CVT model on world-centric, ego-
centric, and ego-centric forward coordinate frames.

In the Collaborative Push Block environment, three agents are equipped with
two forward-facing cameras and are tasked with predicting the occupancy of all
push blocks, agents and the goal area. In the Dungeon Escape environment,
three agents are equipped with two forward-facing cameras and are tasked with
predicting the occupancy of the dragon, agents and key. In the Planar
Construction environment, six agents are equipped a single forward-facing
camera and are tasked with predicting the occupancy of all pucks, agents and
walls.

To evaluate the robustness of the HEV-CVT model, we conduct extensive
experiments in the Planar Construction environment. The standard ML-Agents
environments, were not as challenging for the HEV-CVT model due to the
availability of objects in predictable locations. The Planar Construction
environment presents a more complex challenge as we randomly change the
colouring of six wall and floor components at every time step of the
environment during data collection.

Our results shown in Table 2 demonstrate the world-centric coordinate frame
consistently outperforms the ego-centric coordinate frames in all
environments. The Collaborative Push Block and Dungeon Escape environments
show the largest performance improvements, with up to 32.72% and 43.71%
improvement in IoU, respectively. These results suggest that the world-centric
HEV approach is effective in addressing the challenges of collaborative
perception in multi-agent environments. This result is especially apparent in
the Collaborative Push Block environment, where HEV-CVT model easily localizes
itself based on the large goal location seen in most camera views for a near
perfect 96.94% IoU score.

### Multi-Agent Reinforcement Learning

In order to compare the performance of the fixed world-view coordinate frames
with other commonly used coordinate frames, we conduct experiments in all
three proposed environments. To ensure a fair comparison between the
performance of agents using different coordinate frames, we use identical
reward functions to each environment’s original implementation and identical
grid sizes.

Table 3 compares the performance of agents using different coordinate frames
to those using traditional sensor frames in all three proposed environments.
We find consistently lower episode lengths with world-centric based agents
compared to ego-centric. We report episode length as the indicator of how
efficiently the learned policy acts, all agents achieve near maximum possible
reward with little variation.

Furthermore, our experiments highlighted a common challenge faced by BEV-based
agents in all three environments, where sometimes the object necessary to take
the optimal action was missing from their view, leading to sub-optimal
decision-making and increased episode lengths. The HEV-based agents were able
to leverage the multiple viewpoints available to them, enabling them to better
perceive their environment and take more optimal actions. This issue was
particularly evident in the Planar Construction environment, where the world-
centric agents’ improved perception resulted in significantly lower episode
lengths compared to the ego-based agents.

Overall, these findings suggest that the HEV framework offers a superior
perception model in MARL environments, providing agents with a more
comprehensive understanding of their surroundings, leading to improved
decision-making and better overall performance.

## Conclusion

We have proposed a new perception framework called Herd’s Eye View that
provides a global view of the environment, enabling better global coordination
and cooperation in MARL scenarios. Our results demonstrate that the world-view
coordinate frame outperforms traditional ego-centric frames in all three
proposed environments, resulting in more efficient coordination, better
decision-making, and improved overall performance of MARL agents. By using the
HEV world-view collaborative perception problem, we show that the accuracy of
collaborative perception models is significantly improved, leading to better
IoU scores. Our work opens up new possibilities for advanced perception models
in MARL, which can greatly enhance the performance of multi-robot systems by
enabling better collaboration and coordination.

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# Observation of Universal Expansion Anisotropy from Cold Atoms to Hot Quark-
Gluon Plasma

Ke Li<EMAIL_ADDRESS>College of Science, Huzhou University, Huzhou,
Zhejiang 313000, China  Strong-Coupling Physics International Research
Laboratory (SPiRL), Huzhou University, Huzhou, Zhejiang 313000, China  Hong-
Fang Song  College of Science, Huzhou University, Huzhou, Zhejiang 313000,
China  Strong-Coupling Physics International Research Laboratory (SPiRL),
Huzhou University, Huzhou, Zhejiang 313000, China  Yu-Liang Sun  College of
Science, Huzhou University, Huzhou, Zhejiang 313000, China  Strong-Coupling
Physics International Research Laboratory (SPiRL), Huzhou University, Huzhou,
Zhejiang 313000, China  Hao-Jie Xu  College of Science, Huzhou University,
Huzhou, Zhejiang 313000, China  Strong-Coupling Physics International Research
Laboratory (SPiRL), Huzhou University, Huzhou, Zhejiang 313000, China  Fuqiang
Wang<EMAIL_ADDRESS>College of Science, Huzhou University, Huzhou,
Zhejiang 313000, China  Strong-Coupling Physics International Research
Laboratory (SPiRL), Huzhou University, Huzhou, Zhejiang 313000, China
Department of Physics and Astronomy, Purdue University, West Lafayette,
Indiana 47907, USA

###### Abstract

Azimuthal anisotropy has been ubiquitously observed in high-energy proton-
proton, proton-nucleus, and nucleus-nucleus (heavy-ion) collisions, shaking
the early belief that those anisotropies must stem from utterly strong
interactions. This work reports a study of anisotropic expansion of cold 6Li
Fermi gases, initially trapped in an anisotropic potential, as a function of
the interaction strength that can be readily tuned by an external magnetic
field. It is found that the expansion anisotropy builds up quickly at small
interaction strength, without the need of utterly strong interactions. A
universal behavior of the expansion anisotropy is quantitatively observed
between cold-atom and heavy-ion systems, despite their vast differences in
physics. This universality will potentially unify a variety of disciplines in
nature, from the weakly interacting dilute systems of gases to the strongly
interacting quark-gluon plasma of the early universe.

##  I Introduction

It is believed that the universe started with a big bang singularity—a vacuum
where all charge quantum numbers are zero. The universe expanded, cooled down,
and turned from a soup of quarks and gluons into particles like protons and
neutrons at a temperature around $10^{12}$ Kelvin (or $\sim 200$ MeV) at a
time of approximately $10~{}\mu$s after the big bang [1, 2]. Protons and
neutrons make atomic nuclei and all visible matter of stars and galaxies we
see today. To study the state of the early universe, physicists collide heavy
nuclei at speeds over $99.99\%$ of the speed of light, at the Relativistic
Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory, New York and the
Large Hadron Collider (LHC) at CERN, Geneva, to create the state similar to
that in the early universe, called the quark-gluon plasma (QGP) [3, 4, 5].
Much like the early universe, the QGP expands, cools down, and undergoes a
phase transition to a system of hadrons. Contrary to the early expectation of
a free gas of quarks and gluons, the QGP was found to be strongly interacting
[6]. One of the evidence for this finding is the large anisotropy in final-
state particle momentum distribution, called anisotropic flow [7], observed in
non-head-on collisions [3, 4, 5]. In those collisions, the nuclei are off
center from each other, and the overlap portions of the nuclei on each other’s
path form a region of an almond shape with extremely high temperature and
energy density. The almond-shape region expands anisotropically because of
interactions among the constituents, converting the initial spatial anisotropy
into final-state momentum anisotropy [7]. The interaction must be strong,
close to the hydrodynamic limit with minimal viscosity to entropy density
ratio ($\eta$/$s$) [8], in order to attain the observed large anisotropy. In
fact, the $\eta$/$s$ of the QGP was estimated to be 0.1 in the unit of
$\hbar$/$k_{B}$ ($\hbar$ is the reduced Planck constant and $k_{B}$ is the
Boltzmann constant), close to the conjectured quantum limit of $1/4\pi$ by
string theory [8]. In other words, the QGP behaves like a nearly perfect fluid
[6].

More recently, large anisotropies have also been observed in small systems,
like proton-proton and proton-nucleus collisions, where the interactions were
not perceived to be strong [9, 10]. This prompted several authors [11, 12, 13]
to suggest that strong interactions may not be a necessary prerequisite for
anisotropy generation – weak interactions or even a single collision may
already be sufficient to cause enough anisotropic escape of particles
resulting in a significant momentum anisotropy. This casts doubt on the
robustness of the nearly perfect fluid conclusion and begs the question how
anisotropy builds up as a function of the interaction strength. This is,
unfortunately, a difficult question to answer with nuclear experiments beyond
changing their beam species and collision impact parameter.

One can, however, tune interactions in cold-atom systems exploiting Feshbach
resonances [14]. It has been shown that cold-atom gases in the strong
interaction regime behave like a nearly perfect fluid with $\eta$/s estimated
to be approximately 0.3 [15, 16]. Large anisotropic expansion has also been
observed in strongly interacting cold Fermi gas [17]. These findings suggest
that cold-atom gases and the QGP may share some commonalities in the limit of
strong interactions. With tunability of the interaction strength, one may
examine not only the strong interaction regime but also the weak and
intermediate interaction regimes. Together with manufacturability of the
geometry, cold-atom systems may offer a viable means to emulate the full range
of nuclear collisions.

In this article, we perform a cold-atom expansion experiment, with systematic
tuning of the interaction strength from zero to maximum and with two initial
geometries of the gas cloud. Expansion anisotropies are measured as a function
of the interaction strength. Comparisons are made to findings from
relativistic heavy-ion collisions.

##  II Experiment

The experimental setup, methods for preparing and detecting cold-atom systems
are described in Appendix A. A cold Fermi gas of neutral 6Li atoms is prepared
in the two lowest energy states with opposite spins. The gas is trapped in an
optical dipole trap (ODT) [18] formed by crossing laser beams and
evaporatively cooled down by lowering the trap depth. The interaction strength
(between atoms of opposite spins) of cold atom system is tuned over a wide
range with a homogenous external magnetic field $B$. During evaporative
cooling process, the $B$ is set to 841 G, near the Feshbach resonance value of
834 G to maximize interatomic interaction for efficient cooling [14]. At the
end of this process, the ODT depth is stabilized at $\sim 50~{}\mu$K. The beam
cross angle is $10^{\circ}$, and the anisotropic trapping frequencies are
calculated to be $\omega_{r}/2\pi$ = 2140 Hz and $\omega_{z}/2\pi$ = 187 Hz
along the radial and axial directions, respectively. The aspect ratio of the
confined atom gas is given by $\beta=\omega_{z}/\omega_{r}=1/11.4$. The
magnetic field is then ramped down to the desired value $B$, and the ODT trap
is turned off abruptly (in less than $1~{}\mu$s) to let the gas expand.

The standard resonant absorption imaging (RAI) technique [19] is employed to
evaluate the properties of expanding Fermi gas. The population in each spin
state is observed to be $N\approx 5.2\times 10^{5}$, and the quantum
degeneracy parameter is found to be $T/T_{F}=0.72$. Here, the gas temperature
$T=4.6~{}\mu$K is determined by observing the time-of-flight (TOF) of
interaction-free ballistic expansion at $B$ = 527 G. The Fermi temperature is
given by $T_{F}=\hbar(6\omega_{r}^{2}\omega_{z}N)^{1/3}/k_{B}=6.4~{}\mu$K.
Since the degeneracy parameter is higher than the quantum degeneracy criterion
($T/T_{F}~{}=~{}0.5$) [20, 21], the prepared cold-atom Fermi gas is in normal
phase and can be treated as a thermal cloud.

When $B$ is tuned to the vicinity of the Feshbach resonance point, the gas
expands anisotropically. The gas is imaged with the RAI technique at a given
time. Since the imaging process is destructive, the gas is prepared with the
same condition and the expansion experiment is repeated and imaged at each of
several time instances. Figure 1 shows the absorption images of the expanding
gas with $B$ = 831 G at several time instances. The trapped gas is initially
of pencil shape; once released, it expands and reverses its aspect ratio at
$\sim 0.8$ ms; the aspect ratio is $\beta\approx 1.95$ at 2.0 ms.

Figure 1: Absorption images of the strongly interacting Fermi gas with $B$ =
831 G at several time instances (indicated at top right of each image) after
released from an anisotropic potential trap of aspect ratio of $\beta$ =
1/11.4 (crossing angle 10∘). The field of view of each image is 0.94 mm
$\times$ 0.94 mm.

We have also build a second version of ODT with the laser beam crossing angle
changed to $35^{\circ}$, so that the aspect ratio of trapped atom gas is
reduced to $\beta$ = 1/3.17. The prepared Fermi gas system has a total atom
number $\sim 4.8\times 10^{5}$ and a degeneracy parameter $T/T_{F}$ = 0.68.
The expansion experiment is repeated.

##  III Data Analysis

As described in Appendix C, the $z$ and $x$ projections of the absorption
images are fitted with Gaussian function to extract the axial ($\sigma_{z}$)
and transverse ($\sigma_{x}$) root-mean-square (RMS) size of the expanding
Fermi gas, respectively. Figure 2(a) shows $\sigma_{z}$ and $\sigma_{x}$ as
functions of the expansion time for $B$ = 831 G. Jitters in the data points
reflect the size of statistical uncertainties, which becomes relatively large
at long expansion time because of lower signal-to-noise ratio. The expansion
in the axial direction is slow while it is rapid in the transverse (radial)
direction; this causes the reversion of the aspect ratio. For comparison, the
result with $B$ = 527 G is shown in Fig. 2(b) where the interaction vanishes
and the expansion is ballistic; the aspect ratio does not reverse. Note that
the measured axial size of the gas decreases initially because of absorption
saturation; those axial data points are not used in subsequent analysis.

Figure 2: The RMS size of the cold-atom gas in the $z$ (axial) and $x$
(transverse) directions as functions of the expansion time. The external
magnetic fields are $B$ = 831 G (a) and 527 G (b). The curves in (a) and (b)
are fits to Eq.(1) within the range 1.0 – 2.0 ms and 0.3 – 1.0 ms,
respectively; fitting parameters are written on the plots.

The anisotropic expansion is caused by redistribution of momentum among
particles. It happens at the initial stage of expansion when interactions are
strong. When the gas becomes dilute and interactions become negligible, the
atoms stream freely. The late-time expansion of the cold gas cloud is treated
as ballistic and can be described by [22, 23],

$\sigma_{i}(t)^{2}=\sigma_{i}(0)^{2}+\left\langle\textit{v}_{i}^{2}\right\rangle\textit{t}^{2}\,,$
(1)

where $i=x,y,z$ are cartesian components. Practically, free streaming starts
to set in when the mean free path $\lambda=\left(\left\langle
n\left(t\right)\right\rangle\sigma_{s}\right))^{-1}$ becomes larger than the
average size of the expanding gas [24, 25], where $\left\langle
n\left(t\right)\right\rangle$ is the mean atom density at expansion time $t$,
and $\sigma_{s}$ is the s-wave scattering cross-section. At small $B$, this
happens early, while for $B$ close to the Feshbach resonance point, it happens
late. For $B$ = 831 G the time is found to be 0.8 ms, which is safely beyond
the region of absorption saturation aforementioned. On the other hand, at long
expansion time (after $\sim 2$ ms) the signal-to-noise ratio of the absorption
image becomes too poor to yield a reliable RMS size. We therefore fit the 831
G data to Eq. (1) within expansion time 1.0 – 2.0 ms. The fitted $\left\langle
v_{i}^{2}\right\rangle$ values along $z$ and $x$ directions are written on the
plots. For $B$ = 831 G, $\left\langle v_{x}^{2}\right\rangle$ is significantly
larger than $\left\langle v_{z}^{2}\right\rangle$, indicating stronger
expansion in the $x$ direction than in the $z$ direction, a result of strong
interactions among the atoms. Similarly, we fit the 527 G data within 0.3 –
1.0 ms. The fitted $\left\langle v_{x}^{2}\right\rangle$ and $\left\langle
v_{z}^{2}\right\rangle$ values are comparable, indicating an isotropic and
unchanged momentum distribution during the whole ballistic expansion process.

### Results and Discussions

The momentum anisotropy of the expanding gas can be evaluated by the $v_{2}$
parameter [7],

$\textit{v}_{2}=\frac{\left\langle v_{x}^{2}\right\rangle-\left\langle
v_{z}^{2}\right\rangle}{\left\langle v_{x}^{2}\right\rangle+\left\langle
v_{z}^{2}\right\rangle}\,.$ (2)

The $v_{2}$ for $B$ = 527 G is practically zero, whereas that for 831 G is
large. The interaction strength can be varied by tuning the magnetic field
$B$. Figure 3 shows the $v_{2}$ parameter (red spheres) of the expanding Fermi
gas as a function of the magnetic field $B$, which tunes the interaction
strength. After an initial slow rise, the $v_{2}$ parameter increases rapidly
with $B$ (hence the interaction strength) and apparently saturates around the
Feshbach resonant value. The saturation $v_{2}$ value is large, well above
0.5.

Besides the interaction strength, the $v_{2}$ parameter also depends on the
magnitude of the initial spatial anisotropy of the trapped Fermi gas. The
experiment is repeated with crossing angle $35^{\circ}$, and the resultant
$v_{2}$ is shown in Fig. 3 as the blue triangles. The $v_{2}$ parameter is
found to be smaller than those obtained with the $10^{\circ}$ cross angle. The
initial spatial anisotropy of the Fermi gas can be characterized by the
eccentricity, $\varepsilon_{2}\equiv(1-\beta^{2})$/$(1+\beta^{2})$. The
eccentricities of the two Fermi gases are $\varepsilon_{2}$ = 0.98 and 0.82,
respectively. The observed $v_{2}$ difference is due, in part, to the
difference in $\varepsilon_{2}$; one thus often divides $v_{2}$ by
$\varepsilon_{2}$ when presenting data because the initial-state
$\varepsilon_{2}$ is the root reason for the final-state $v_{2}$. The other
part that causes a difference in $v_{2}$ is the interaction strengths of the
gases that are slightly different between the two cross angles.

Figure 3: The $v_{2}$ parameter as a function of the magnetic field $B$. The
expanding cold-atom gases are released from two ODT potential traps of
differing shapes, one with beam cross angle of $10^{\circ}$ (red spheres) and
the other with $35^{\circ}$ (blue triangles).

The amount of interaction can be quantified by the average number of
collisions ($n$) an atom of the Fermi gas encounters during expansion. We term
it opacity. For a particle traversing in a medium of uniform density $\rho$
over distance $L$ with interaction cross-section $\sigma$, it is simply
$n=\rho\sigma L$. For our trapped atom gas, the opacity can be estimated by

$\textit{n}=\frac{\sigma_{s}}{4\pi}\oint{\rho_{0}\exp\left[-\left(\frac{r^{2}\sin\theta^{2}}{2\sigma_{x}^{2}}+\frac{r^{2}\cos\theta^{2}}{2\sigma_{z}^{2}}\right)\right]\sin\theta
drd\theta d\phi}\,,$ (3)

corresponding to the average number of collisions a test atom encounters
traversing out from the center of the trapped gas cloud. Here,
$\rho_{0}=\frac{N}{(2\pi)^{3/2}\sigma_{x}^{2}\sigma_{z}}$ is the center
density of the cold cloud of single spin. For initial RMS radii of
$\sigma_{z}$ and $\sigma_{x}$ (and $\beta=\sigma_{x}/\sigma_{z}$), the
integration gives
$n=\frac{N\sigma_{s}}{4\pi\sigma_{x}\sigma_{z}}\frac{\arctan(\sqrt{1-\beta^{2}}/\beta)}{\sqrt{1-\beta^{2}}}$.
Figure 4(a) shows $v_{2}$ (now divided by $\varepsilon_{2}$) as a function of
$n$. The $v_{2}/\varepsilon_{2}$ versus $n$ data points appear to fall onto a
common curve. Note that the $n$ values differ somewhat between the two cases
because the initial geometries are slightly different between the two cross
angles; the smaller $v_{2}$ for the $35^{\circ}$ cross angle is a combined
effect of the smaller values of both $\varepsilon_{2}$ and $n$.

As mentioned in the introduction, strong elliptic flow has been observed in
heavy-ion collisions. While the cold-atom gas is three-dimensional, the heavy-
ion collision system is effectively two-dimensional because the longitudinal
beam direction is approximately Lorentz boost invariant. The fireball created
in heavy-ion collisions can be assumed to equilibrate at the typical strong
interaction proper time $\tau\sim 1$ fm/$c$ with a longitudinal extent of
$c\tau\sim 1$ fm, where $c$ is the velocity of light in vacuum. The initial
Bjorken density [26] can be estimated by $\frac{dN_{ch}/dy}{S~{}c\tau}$ where
$dN_{ch}/dy$ is the speudorapidity density of particle multiplicity and $S$ is
the transverse overlap area of the two colliding nuclei. Again, taking a test
particle flying out from the center of the fireball, the opacity can be
estimated as $n=\frac{\sigma}{c\tau}\frac{dN_{ch}/dy}{\sqrt{\pi S}}$, where
$\sigma$ = 3 mb is the parton-parton interaction cross-section. Here we have
assumed isentropic evolution with entropy conservation, so one gluon turns
into one final-state pion. Figure 4(b) shows the $v_{2}/\varepsilon_{2}$ as a
function of $n$ in gold-gold (Au+Au) collisions over a wide range of collision
energy and impact parameter [27, 28, 29, 30]. All the data points collapse
onto a common curve similar to the cold-atom gas experiment. However, while
the opacities are comparable, the magnitude of $v_{2}/\varepsilon_{2}$ is
larger in cold-atom gas than that in heavy-ion collisions.

Figure 4: The eccentricity normalized $v_{2}/\varepsilon_{2}$ as a function of
opacity ($n$) in cold-atom gas (a) and relativistic heavy-ion collisions (b).
For the origin of heavy-ion data, please refer to Appendix G

.

There are at least two important differences between the observed $v_{2}$ from
our cold atom experiment and the heavy-ion data. The first is technical: in
heavy-ion experiment, the momentum is relativistic and measured particle by
particle, and the $v_{2}$ parameter is averaged over all detected particles as
$v_{2}=\left\langle\cos
2\phi\right\rangle=\left\langle\frac{p_{x}^{2}-p_{y}^{2}}{p_{x}^{2}+p_{y}^{2}}\right\rangle$,
where $p_{x}$ and $p_{y}$ are the particle momentum components on the
transverse $x$-$y$ plane [31]; in cold-atom systems which are non-
relativistic, the average squared velocities are extracted from the two-
dimensional density distributions obtained through absorption imaging, and the
$v_{2}$ parameter is calculated by Eq.(2). To use the same definition as for
cold atoms, the heavy-ion $v_{2}$ should be weighted by $p_{T}^{2}\equiv
p_{x}^{2}+p_{y}^{2}$. Since $v_{2}$ is approximately proportional to
transverse momentum $p_{T}$ in heavy-ion collisions, $v_{2}\propto p_{T}$
[31], and the $p_{T}$ distributions are typically exponential, $dN\propto
p_{T}e^{-bp_{T}}dp_{T}$ where $b$ is a parameter related to the effective
temperature [32], the $p_{T}^{2}$ weighting would amount to a factor of 2
difference: $\frac{\int p_{T}^{2}v_{2}dN}{\int p_{T}^{2}dN}$/$\frac{\int
v_{2}dN}{\int dN}=2$. In other words, the $v_{2}$ in heavy-ion collisions
would be a factor of 2 larger if it is calculated in the same way as in cold-
atom experiment. The other difference is physical, namely non-linearity
correction. The $\varepsilon_{2}$ of our cold-atom gases is close to the
maximum of unity, whereas that in heavy-ion collisions is relatively small.
According to hydrodynamic calculations of heavy-ion collisions [33], the
effect of non-linearity for close to unity $\varepsilon_{2}$ can be as large
as 50%. The linear and cubic responses are shown [33] to be approximately
independent of centrality (or opacity). In other words, one should plot
$v_{2}/\left[\varepsilon_{2}(1+\varepsilon_{2}^{2}/2)\right]$ in order to put
systems of vastly different $\varepsilon_{2}$ onto the same footing. Such a
plot is shown in Fig. 5. The cold-atom data and the heavy-ion data appear now
to follow a similar trend. Ideally, one would want to make the eccentricity of
the potential trap comparable to those in heavy-ion collisions by increasing
the cross angle. However, this is not possible because of the space
limitations around our experimental chamber. Note, for the two cold-atom gases
with different $\varepsilon_{2}$, the nonlinearity corrections are similar,
differing by only 10%. As such, the cold-atom gases appear consistent with
each other even without nonlinearity corrections as in Fig. 4(a).

Figure 5: The $v_{2}/\left[\varepsilon_{2}(1+\varepsilon_{2}^{2}/2)\right]$ as
functions of opacity ($n$). The cold-atom $v_{2}$ is first divided by 2
because of the different means to measure the $v_{2}$ from that in heavy-ion
collisions (see text).

The similar trend shown in Fig. 5 suggests that the expansion dynamics is
universal in interacting systems, from weak to strong. This is remarkable
considering the vast differences between the two systems — the density of the
QGP is $\sim 10^{39}$ cm-3 [34] and that of a typical cold-atom gas is $\sim
10^{12}$ cm-3 (about 7 orders thinner than air); the temperature of the QGP is
$\sim 10^{12}$ K and that of a cold-atom gas is $\sim 10^{-6}$ K; the physics
governing the QGP is the strong interaction of quantum chromodynamics (QCD)
and that governing cold-atom gases is electromagnetic interaction of quantum
electrodynamics (QED).

The universal trend exhibits two distinct regions: a sharp rise at small $n$
up to $n\approx 2$ and a flattening increase at larger $n$. This may indicate
two phases with a phase transition at $n\approx 2$: a gas phase at small $n$
with minimal interactions where the anisotropy is primarily generated by the
escape mechanism [11], and a liquid phase at large $n$ where the system is
strongly interacting, asymptotically approaching hydrodynamics [6, 17]. It
suggests that 1-2 interactions per constituent may be sufficient for a system
to reach equilibrium.

Strongly interacting systems are common in nature, e.g. black holes [35],
neutron stars [36], strongly coupled Bose fluid [37], superfluid liquid helium
[38], and other condensed matter [39] and quantum systems [40], in addition to
the QGP and cold-atom Fermi gas. Studies of quantum cold-atom gases, with
their advantages of tunable interactions and variable geometries, may shed
lights on non-perturbative many-body interactions in a variety of disciplines
in the future [41, 42].

##  IV Summary

To summarize, we have carried out a cold-atom experiment with two trap
geometries to systematically study the expansion behavior as a function of the
interaction strength, tuned by an external magnetic field. The interaction
strength is characterized by the opacity variable $n$. It is found that the
anisotropy builds up quickly at small $n<2$, without the need of utterly
strong interactions. The anisotropy is found to increase with interaction
strength, more slowly at larger $n$. A universal behavior is quantitatively
observed between the vastly different systems of cold-atom gases and heavy-ion
collisions, where the non-linearity corrected eccentricity normalized elliptic
anisotropy parameter $v_{2}$ follows the same trend in opacity $n$. This
universality suggests that all interacting systems behave similarly in their
expansion dynamics, over a wide range in interaction strength from weakly
interacting systems to strongly interacting ones. The two apparently distinct
regions in $n$ may suggest a phase transition from weakly interacting gas
phase at small $n<2$ to strongly interacting liquid phase at large $n>2$. It
indicates that 1-2 interactions per constituent may be sufficient to drive a
system to equilibrium.

Our cold-atom experiment emulator can be improved in a number of ways. For
example, the temperature of the atom gas can be lowered to superfluid regime
to study effects of phase transition on expansion dynamics; the atom density
may be increased to reach higher opacity to investigate the expansion behavior
at even higher interaction strength; and the geometry of the gas cloud can be
varied and a triangular geometry may be manufactured to study triangular flow
expansion. The cold-atom experiment can also be extended for other tests. For
example, with addition of an ion trap, one may shoot an energetic ion through
a cold-atom gas to study their interactions. This would be similar to the jet
quenching phenomenon [43] observed in relativistic heavy-ion collisions,
another evidence for the strongly interacting QGP besides the anisotropic
flow. At ion speed higher than the speed of sound of the atom gas, the Mach-
cone shock wave phenomenon may be studied.

###### Acknowledgements.

This work was supported by Huzhou University Educational and Research Fund,
National Natural Science Foundation of China (12035006, 12075085, 12205095,
12275082), Ministry of Science and Technology of China (2020YFE020200).

##  Appendix A The cold Fermi gas of 6Li atoms

The laser source constructing the optical dipole trap (ODT) [18] is derived
from the infrared emission of a fiber laser (IPG Photonics, YLR-100-1064-LP)
which is centered at 1064 nm and has a maximum power of 100 W. The beam waist
of the ODT laser is $\sim 40~{}\mu$m, corresponding to a calculated trap depth
of $\sim 1.8$ mK at full power. The laser is split into two parts equally,
which propagate on the horizontal plane with orthogonal polarizations and are
finally focused and intersected at the focal points with a crossing angle
$\theta$. The crossed beams produce an anisotropic trapping potential, which
can be described by a three-dimensional harmonic oscillator,
$U=\frac{1}{2}m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})$,
where $m$ is the mass of 6Li atom, and $\omega_{i}$ is the trapping frequency
along $i$-direction ($i=x,y,z$) and $\omega_{x}=\omega_{y}=\omega_{r}$. The
trapping frequencies are determined mainly by the potential depth $U$, the
beam waist $w_{0}$, and the crossing angle $\theta$.

The ODT is loaded with a sample of cold 6Li atoms directly from a magneto-
optical trap (MOT) [44]. After loading, the laser power is ramped down from
full power to 30 W where the trap depth is $\sim 550~{}\mu$K. The atomic
population in each spin state of 6Li atoms, $\left|1/2,\pm 1/2\right\rangle$
[45], is resolved by using the standard RAI technique at high magnetic field
to be $\sim 8.2\times 10^{5}$. The population difference between the two
states is less than 12%.

The temperature of the atom gas is further lowered through two-stage
evaporative cooling [46, 47]. The first stage is the so-called free
evaporative cooling process, during which the external magnetic field $B$ =
841 G is switched on and the laser power is maintained at 30 W for 300 ms.
Then the second stage, a forced evaporative cooling process, is conducted by
lowering the laser power according to an optimized ramping trajectory. The
complete process of the forced evaporative cooling takes 900 ms. Finally the
laser power is reduced to 1 W, corresponding to a trap depth of $\sim
50~{}\mu$K.

Properties of the prepared Fermi gases are tabulated in Table 1. The trap
depth is based on theoretical calculation; $T$ is measured with TOF of
ballistic expansion; $T_{F}$ is the Fermi temperature, and $N$ is the number
of atoms in a single spin state. $N$ is smaller than the initial loaded
population because of evaporation during the cooling processes.

Table 1: Properties of the prepared Fermi gases Crossing-angle | Trap Depth | $N(\times 10^{5})$ | $T$ | $T_{F}$ | $T/T_{F}$
---|---|---|---|---|---
10∘ | 50 $\mu$K | 5.2 $\pm$0.3 | 4.6 $\mu$K | 6.4 $\mu$K | 0.72
35∘ | 67 $\mu$K | 2.4 $\pm$ 0.3 | 6.4 $\mu$K | 9.4 $\mu$K | 0.68

##  Appendix B Feshbach resonance

At zero magnetic field, the $2S_{1/2}$ ground state of 6Li atoms splits into
two hyperfine states with total angular momentum ($\hbar$/2 and 3$\hbar$/2).
In an external magnetic field $B\neq 0$, they split further into six Zeeman
states as shown in Fig. 6(a), labeled $\left|i\right\rangle$ (i=1,2,…,6) [45].
The ODT can trap atoms in all of these spin states. For simplicity, our
confined cold atoms are prepared in the two lowest energy states
$\left|1\right\rangle$ and $\left|2\right\rangle$.

Figure 6: (a) Energy splitting of the 6Li $2S_{1/2}$ ground state in external
magnetic field. (b) Theoretical s-wave scattering length $a_{s}$ between
states $\left|1\right\rangle$ and $\left|2\right\rangle$ a function of the
external magnetic field $B$; $a_{0}$ is the Bohr radius.

The interaction strength between 6Li atoms populated in spin states
$\left|1\right\rangle$ and $\left|2\right\rangle$ can be described with a
single parameter, the s-wave elastic scattering length $a_{s}$, which is
controllable with the external magnetic field $B$ as shown in Fig. 6 [48]. The
Feshbach resonant point between atom populated in spin states
$\left|1\right\rangle$ and $\left|2\right\rangle$ is measured with Radio-
Frequency (RF) Spectroscopy technique with high precision [49, 50]. In our
experiment, we employ the RF spectroscopy method to determine $B$ with an
accuracy of 10 mG. The s-wave scattering cross-section between atoms of
different spin states is given by [45]

$\sigma_{s}=\frac{4\pi a_{s}^{2}}{1+k^{2}a_{s}^{2}}\,,$ (4)

where $k=\sqrt{2mk_{B}T/\hbar^{2}}$ is the typical relative wave number of two
colliding atoms. For $\left|ka_{s}\right|\gg 1$, the scattering cross-section
becomes unitary limited, $\sigma_{s}=4\pi/k^{2}$; for $\left|ka_{s}\right|\ll
1$, $\sigma_{s}=4\pi a_{s}^{2}$.

As shown in Fig. 6(b), $a_{s}$ is tuned with the same magnetic field $B$ that
causes the splitting of hyperfine states. When the magnetic field is tuned
across 527 G, $a_{s}$ crosses zero and becomes positive. When $B$ is further
approaching the Feshbach resonance point of 834 G, $a_{s}$ grows to positive
infinity. For several representative magnetic field strengths of $B$ = 685,
763 and 831 G, the scattering lengths are $\sigma_{s}$ = 0.07, 0.236, and 18.6
$\mu$m, and the values of $k_{F}a_{s}$ are 0.87, 3.5, and 241, respectively.
Here, $k_{F}=\sqrt{2mk_{B}T_{F}/\hbar^{2}}$ is the Fermi wave number [24].

##  Appendix C Resonant absorption imaging (RAI)

In our experiment, $z$-$x$ plane is horizontal with the $z$-axis lies in the
axial direction and the $x$-axis in one of the radial directions, as shown
Fig. 7. The external magnetic field $B$ is along the vertical direction
($y$-axis, pointing upward). The imaging beam propagates in the vertical
direction and is $\sigma^{-}$ polarized (antiparallel to $B$). During imaging
process, only the ODT is switched off while $B$ is kept on. After expansion
time $t$, a 10 $\mu$s probe pulse is fired with light frequency finely tuned
to be on resonant with the specific spin state at the given $B$. The behavior
of the expanding Fermi gas can be assessed by imaging atoms populated in any
one of the spin states. In our experiment, we mainly detect atoms in spin
state $\left|1\right\rangle$.

Figure 7: (a) Setup of the Fermi gas expansion experiment. (b) Schematic
drawing of the detection method by absorption imaging. The magnetic field $B$
generated by a pair of Holmholtz coils (Feshbach coils) is in vertical
direction ($y$-axis). Axial direction lies in $z$-axis, and the horizontal
radial direction is $x$-axis. The $\theta$ refers to the crossing angle.

The center intensity of the probe light is $I_{\rm probe}=0.23$ mW/cm2,
corresponding to a saturation parameter $I_{\rm probe}/I_{\rm sat}\sim 0.09$,
which ensures the population in the spin state is not disturbed during
imaging. Here, $I_{\rm sat}=~{}2.54$ mW/cm2 is the saturation intensity of
D2-line transition of 6Li atom. The imaging system consists of two stages. The
first stage is a 1:1 image-relay, which is formed with an $f$ = 50 mm lens
pair. The image-relay has a numerical aperture of 0.2, corresponding to an
optical resolution $\sim 2~{}\mu$m. The relayed image is magnified and
projected onto the electron multiplying charge coupled device (EMCCD) camera.
The pixel size of the camera sensor is 13 $\mu$m and the overall magnification
of the imaging system is calibrated to be $\sim 4.15$, resulting in a spatial
resolution of 3.1 $\mu$m.

Figure 8 illustrates the time sequence of the resonance absorption imaging
(RAI). At the end of forced evaporative cooling, the magnetic field is ramped
down from $B$ = 841 G to the set value $B$ (527–841 G) at a rate of 10 G/ms
and stablized for 10 ms. The ODT laser power is then switched off and the
Fermi gas is released to expand. At a given expansion time $t$, the gas is
shined by a resonant light pulse of 10 $\mu$s and the absorption image $I_{\rm
atom}(x,z)$ is recorded by the EMCCD camera. Two more images are recorded,
each at 1500 ms later. Because the RAI technique is desctructive, the second
image $I_{\rm light}(x,z)$ is recorded as a reference with no absorption. The
third image $I_{\rm dark}(x,z)$ is recorded with all light blocked to serve as
the dark counts of the EMCCD camera.

Figure 8: Time sequence of the resonance absorption imaging (RAI). Absorption
imaging is conducted at non-zero magnetic field $B$ ranging from 527 G to 841
G. The ODT laser power is switched off by Acoustic Optical Modulator (AOM) in
less than 1 $\mu$s.

The recorded images are digitized into two-dimensional matrices to obtain the
optical density profile, $\ln\left[\frac{I_{\rm light}-I_{\rm dark}}{I_{\rm
atom}-I_{\rm dark}}\right]$. The atom number per pixel in the recorded spin
state is $A_{eff}/\sigma_{0}$ times the optical density, where $\sigma_{0}$ =
0.143 $\mu$m2 is the resonant absorption cross-section and $A_{eff}$ = 9.81
$\mu$m2 is the effective area of each pixel after considering magnification.
When the optical density drops below unity, signal-to-noise ratio of the
absorption image is too poor to yield reliable measurement.

##  Appendix D Gaussian fit

In our experiment ($T/T_{F}>0.5$), the density profile of the expanding gas
can be well described by Gaussian function,
$\frac{N}{(2\pi)^{3/2}\sigma_{x}\sigma_{y}\sigma_{z}}\exp\left[-\left(\frac{x^{2}}{2\sigma_{x}^{2}}+\frac{y^{2}}{2\sigma_{y}^{2}}+\frac{z^{2}}{2\sigma_{z}^{2}}\right)\right]$,
in both weak and strong interaction regime, as shown in Fig. 9. This is the
case described in [20].

Figure 9: Axial and radial projections of the optical density profiles of the
cold Fermi gases at time-of-flight (TOF) of 1 ms after their release from
potential traps, for the strongly interacting case at $B$ = 831 G in (a) and
(b), and for the non-interacting case at $B$ = 527 G in (c) and (d). Curves in
red are Gaussian fits.

##  Appendix E Ballistic expansion and temperature measurement

According to [23], the scaled RMS size $b_{i}=\sigma_{i}(t)/\sigma_{i}(0)$ of
ballistically expanding gas fulfills a simple differential equation,
$b_{i}^{{}^{\prime\prime}}(t)-\omega_{i}^{2}/b_{i}(t)^{3}=0$, where
$\omega_{i}$ is the trapping frequency along $i$-axis. This equation has an
analytical solution $b_{i}(t)=\sqrt{1+(\omega_{i}t)^{2}}$ at the given initial
condition, $b_{i}(0)=1$ and $b_{i}^{{}^{\prime}}(0)=0$. The radius
$\sigma_{i}(t)$ is thus described by
$\sigma_{i}(t)=\sigma_{i}(0)\sqrt{1+(\omega_{i}t)^{2}}$, where
$\sigma_{i}(0)=(k_{B}T/m\omega_{i}^{2})^{1/2}$ is the initial in-trap RMS
radius. This can be rewritten into Eq.(1) in the main text, where
$\left\langle v_{i}^{2}\right\rangle=k_{B}T_{i}/m$. Equation (1) is widely
used as a fitting function to extract temperature of non-interaction cold-atom
systems [22, 23].

In our experiment, the confining potential is ellipsoidal with
$\omega_{x}\sim\omega_{y}=\omega_{r}$ and $\beta=\omega_{z}/\omega_{r}<1$;
$\beta$ determines the in-trap aspect ratio. The aspect ratio of the
ballistically expanding Fermi gas is then given by [23]

$\frac{\sigma_{r}(t)}{\sigma_{z}(t)}=\frac{\omega_{z}}{\omega_{r}}\frac{\sqrt{1+(\omega_{r}t)^{2}}}{\sqrt{1+(\omega_{z}t)^{2}}}\,.$
(5)

According to Eq. 5,the aspect ratio of the expanding non-interacting Fermi gas
approaches unity asymptotically.

As shown in Fig. 6(b), the s-wave scattering length $a_{s}$ vanishes at $B$ =
527 G. At the end of evaporative cooling, the magnetic field is tuned to 527
G, and then the ODT laser light is switched off abruptly. The released gas
expands in the homogeneous magnetic field stabilized at 527 G. Since the gas
is non-interacting and follows ballistic expansion, the temperature can be
extracted by fitting Eq.(1) to the size of the expanding gas from a series of
TOF images, as shown in Fig. 2(b). The temperature parameters obtained from
expansion data in the $x$ and $z$ directions are similar, $T_{x}=4.56~{}\mu$K
and $T_{z}=4.71~{}\mu$K; the momentum distribution of the released Fermi gas
is indeed isotropic. The mean value $T=(2T_{x}+T_{z})/3$ is used as the
temperature $T$ in Table 1.

It should be noted that the expansion time range that can be used for
temperature fitting is limited. For short expansion time, center atom density
of the expanding gas is still high such that the absorption image is strongly
saturated, resulting in an artificially larger RMS size extracted from
Gaussian fitting. Moreover, the gas size is small at short expansion time,
comparable to the spatial resolution of the imaging system. As a result the
diffraction effect is significant in the radial direction ($x$-axis shown in
Fig. 7), such that the fitted RMS size is also artificially larger than truth.
Last, a strong loss of atoms is unavoidable when $B$ is tuned across 650 G; it
is found that the atom number drops by half after $B$ is swept to 527 G. This
puts a limitation on the longest applicable expansion time for imaging with
high enough signal-to-noise ratios. Based on the above considerations, only
data points within the range 0.3 – 1.0 ms are chosen for temperature fitting
at $B$ = 527 G.

##  Appendix F Anisotropic expansion

In our experiment ($T>0.5~{}T_{F}$), the gas is still in normal state. When
$B$ is tuned to the vicinity of the Feshbach resonance point of 834 G, an
anisotropic expansion can be observed after the gas is released. This can be
treated as hydrodynamic expansion where the positive s-wave scattering length
is extremely large and the collisional cross-section is unitary limited [24].
Whereas the anisotropic expansion in Ref.(17) is observed at large and
negative $s$-wave scattering length ($a_{s}\sim-10^{4}~{}a_{0}$) and the gas
is highly degenerate ($T\sim 0.1~{}T_{F}$). Here, $a_{0}=0.53\times 10^{-10}$
m is the Bohr radius. Fig. 10 shows one-dimensional optical density profile at
different expansion times. The external magnetic field is set to $B$ = 831 G,
where the $s$-wave scattering length is positive ($a_{s}>$ 0) and extremely
large and in the unitary limit regime. The gas expands rapidly in the radial
direction (Fig. 10(b)) while remains nearly stationary in the axial direction
(Fig. 10(a)) in the measured time range from 0.3 ms to 1.8 ms. The RMS sizes
of the atom Fermi gas are extracted from Gaussian fits to the density profile,
as shown in Fig. 2. Similar exercises are performed with other $B$ values.

Figure 10: Integrated one-dimensional optical density profile at different
expansion times, 0.3 ms (red), 0.8 ms (blue), 1.3 ms (green) and 1.8 ms
(black), along axial (a) and radial direction (b). Because of absorption
saturation, the axial profile of the center density at 0.3 ms is lower than
that at 0.8 ms and 1.3 ms. These profiles are obtained at $B$ = 831 G.

Figure 11 shows the aspect ratio as a function of expansion time for four $B$
values. At $B$ = 527 G, the expansion is ballistic and the aspect ratio
approaches unity and does not reverse. When the magnetic field is set at 685
G, a moderate reversion of the aspect ratio is observed. When the magnetic
field is tuned to values around the Feshbach resonance, namely 763 and 831 G,
the aspect ratio quickly exceeds unity at expansion time shorter than 1 ms.
The solid curves in Fig. 11 are theoretical calculations with a set of coupled
nonlinear equations as described in Ref (23) in the main text,

$b_{i}^{{}^{\prime\prime}}(t)-\frac{\omega_{i}^{2}}{b_{i}^{3}(t)}+\frac{3}{2}\chi\omega_{i}^{2}\left(\frac{1}{b_{i}^{3}(t)}-\frac{1}{b_{i}(t)V(t)}\right)=0\,,$
(6)

where $\chi$ quantifies the interaction strength, and
$V\left(t\right)=b_{x}\left(t\right)b_{y}\left(t\right)b_{z}\left(t\right)$ is
the scaled volume. For ballistic expansion at $B$ = 527 G, $\chi$ = 0; while
for $B$ = 685, 763 and 831 G, $\chi$ = 0.5, 0.63 and 0.66 is chosen
respectively. The high $\chi$ value represents a strong interaction strength
within the gas when $B$ is tuned to the vicinity of Feshbach resonance point
($B$ = 834 G). These comparisons support the collisional hydrodynamic picture
of anisotropic expansion of Fermi gas at large interaction strengths in normal
state.

Figure 11: Aspect ratio of the expanding cold atom 6Li Fermi gas as a function
of expansion time. The points are experimental data. The solid curves are
theoretical calculations of ballistic expansion (black) and collisional
hydrodynamic fits (blue, green, red) with different $\chi$ values. See text
for details.

##  Appendix G Heavy-ion collision data

For Au+Au collisions at the nucleon-nucleon center-of-mass energy
$\sqrt{s_{NN}}$ = 200 GeV and 62.4 GeV, the $v_{2}$ data are taken from Ref.
[34], and the charged hadron multiplicity $dN_{ch}/dy$ and collision geometry
parameters $S$ are taken from Ref. [32]. For Au+Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV, the $v_{2}$ data and the collision geometry
parameters are taken from Ref. [28], and the multiplicity data are taken from
Ref. [32]. For Au+Au collisions at $\sqrt{s_{NN}}$ = 39, 27, 19.6, 11.5, 7.7
GeV, the $v_{2}$ data and the collision geometry parameters are taken from
Ref. [29], and the multiplicity data are taken from Ref. [30]. The
eccentricities are calculated from nuclear collision geometry using the
Glauber model [51]. The systematic uncertainties are usually large in central
(small impact parameter) collisions, which is seen in the spread of the data
points in Fig. 4(b) and Fig. 5. The eccentricity for the most peripheral
(large impact parameter) data point at 130 GeV may also have a large
uncertainty as this was one of the earliest results at RHIC [28].

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