Quantum gas microscopy of a geometrically frustrated Hubbard system Jirayu Mongkolkiattichai1Liyu Liu1Davis Garwood1Jin Yang1yand Peter Schauss1z 1Department of Physics University of Virginia Charlottesville Virginia 22904 USA

2025-04-29 0 0 7.98MB 13 页 10玖币

侵权投诉

Quantum gas microscopy of a geometrically frustrated Hubbard system

Jirayu Mongkolkiattichai,1, ∗Liyu Liu,1, ∗Davis Garwood,1Jin Yang,1, †and Peter Schauss1, ‡

1Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA

Geometrically frustrated many-particle quantum systems are notoriously hard to study numeri-

cally but are of profound interest because of their unusual properties and emergent phenomena. In

these systems energetic constraints cannot be minimized simultaneously, leading to large ground-

state degeneracy and a variety of exotic quantum phases. Here, we present a platform that enables

unprecedentedly detailed experimental exploration of geometrically frustrated electronic systems on

lattices with triangular geometry. We demonstrate the ﬁrst realization of triangular atomic Hubbard

systems, directly image Mott insulators in the triangular geometry with single-atom and single-site

resolution, and measure antiferromagnetic spin-spin correlations for all nearest neighbors allowing

for thermometry. This platform provides a powerful new approach for studying exotic quantum

magnetism and direct detection of quantum spin liquid signatures in Hubbard systems.

I. INTRODUCTION

Electronic systems typically establish long-range or-

der at zero temperature. Surprisingly, there are systems

that do not have this fundamental property. For exam-

ple, quantum spin liquids [1,2] form in the presence of

conﬂicting energetic constraints that prevent long-range

ordering. Interestingly, the absence of ordering opens

the door to a variety of exotic phenomena. For example,

quantum spin liquids can show fractional quasi-particle

statistics analogous to those underlying the quantum Hall

eﬀect [3].

Time-reversal symmetry breaking has been predicted

in numerical studies on frustrated systems and kinetic

constraints caused by the frustration lead to complex

time-evolution [4–6]. While frustrated systems with

small number of particles can be accurately simulated

with tremendous computational resources, predictions

for the low-temperature phases in the thermodynamic

limit are scarce and often debated [7–9]. Existing con-

densed matter realizations are complicated materials [4],

making well-controlled model systems a valuable alterna-

tive for gaining insight into the physics of frustration.

Ultracold atoms provide a unique way to explore quan-

tum many-body physics through quantum simulations

of frustrated quantum systems based on ﬁrst principles.

Prominent examples of quantum simulation with ultra-

cold atoms include the realization of Hubbard models

[10] and the observation of many-body localization [11].

While there is widespread evidence for insulating phases

without magnetic ordering in frustrated Hubbard mod-

els, their existence and properties are still controversial

on many lattice geometries, including the triangular lat-

tice. Ultracold atoms in optical lattices implement Hub-

bard models [10,12,13], where neighboring sites are cou-

∗Contributed equally

†Corresponding author: dypole jin@mit.edu; Present address:

Department of Physics, Research Laboratory of Electronics,

MIT-Harvard Center for Ultracold Atoms, Massachusetts Insti-

tute of Technology, Cambridge, Massachusetts 02139, USA

‡Corresponding author: ps@virginia.edu

pled by hopping, and atoms on the same lattice site inter-

act. Atomic Fermi-Hubbard systems were ﬁrst realized

with ultracold atoms in square lattices [14,15]. With the

realization of quantum gas microscopes for fermions, it

became possible to image fermionic atoms on the single-

atom level [16–20]. Later, two-dimensional (2d) fermionic

Mott insulators (MI) were detected with quantum gas mi-

croscopes using 6Li [21] and 40K [22]. In particular, the

characteristic antiferromagnetic correlations in the repul-

sive Hubbard model have been studied in detail [23–29].

Here, we expand these capabilities to a triangular lat-

tice structure as a paradigm for studies of geometric

frustration [2], and report on the site-resolved imaging

of atomic Mott insulators in a triangular lattice. Geo-

metric frustration does not preclude short-range correla-

tions, and we measure these correlations to study Hub-

bard physics on the triangular lattice.

II. TRIANGULAR-LATTICE HUBBARD

MODEL

The Hamiltonian of the fermionic system in a two-

dimensional lattice at half-ﬁlling is

H=−tX

hrr0i,σ

(c†

r,σcr0,σ +c†

r0,σcr,σ) + UX

nr,↑nr,↓

−µ(r)X

(nr,↑+nr,↓)

(1)

where tis the tunneling strength between nearest-

neighbor lattices, Uis the on-site interaction, cr,σ(c†

r0,σ)

is the annihilation (creation) operator for a fermion with

spin σon site r,nr,σ =c†

r,σcr,σ is the number opera-

tor, and µ(r) is the chemical potential. This model de-

scribes the transition from a metal to a fermionic Mott

Insulator, a prototypical example of a quantum phase

transition. The insulating behavior originates from the

electron-electron correlations and cannot be explained in

a non-interacting electron picture. At temperatures be-

low U/kB, double occupation of sites is suppressed. Sin-

gle occupation is energetically preferred at µ∼U/2 and

the density variance vanishes, leading to a MI. When the

arXiv:2210.14895v1 [cond-mat.quant-gas] 26 Oct 2022

Signal (arb. u.)

A B

FIG. 1. Triangular-lattice quantum gas microscope. (A) A triangular optical lattice is realized by interfering three

circularly polarized laser beams (T1, T2, and T3) using 1064 nm light in the center of a vacuum chamber. The conﬁnement

of the atoms into two dimensions is achieved by a 1d accordion lattice in vertical direction, formed by 532 nm laser beams.

A combination of a beam splitter and mirrors allows us to vary the distance between G1 and G2 via the height of the input

beam, therefore forming a lattice with a variable spacing between 3 µm and 8 µm. A high-resolution objective enables single-site

resolved imaging of the atoms in the triangular lattice. The inset demonstrates 120◦order which is the classical analog of the

spin ordering expected at large interactions. (B,C) Triangular-lattice Mott insulators at U/t = 10(1) with 109 atoms (top

right) and U/t = 26(3) with 203 atoms (bottom right). The ﬁeld of view is 40 µm×40 µm.

chemical potential is larger than the energy gap, doublons

(two atoms on a site) are formed. They ﬁrst appear at

the center of the trap because of the lower harmonic po-

tential forming a band insulating core. More than two

atoms per site are forbidden by the Pauli exclusion prin-

ciple. Antiferromagnetic ordering can be observed in MIs

when the temperature is comparable to the exchange en-

ergy J= 4t2/U [30]. In the following, we present experi-

mental data in this temperature regime and the observa-

tion of antiferromagnetic correlations on the triangular

lattice.

III. EXPERIMENTAL SYSTEM

We prepare a spin-balanced Fermi gas in a single layer

of a one-dimensional (1d) accordion lattice (Fig. 1A) with

a variable spacing. The gas is a mixture of the two lowest

hyperﬁne ground states |↑i =|F= 1/2, mF= 1/2iand

|↓i =|F= 1/2, mF=−1/2iof 6Li, where Fand mF

are the hyperﬁne quantum numbers (for details see [31]).

Next, the atoms are adiabatically loaded into the trian-

gular lattice of depth 9.7(6)ERusing an s-shaped ramp.

Here, ER=~2π2/(2ma2

latt) = h×8.2 kHz is the recoil

energy where his Planck’s constant, mis the atomic

mass, and alatt = 1003 nm. The tunneling parameter is

t=h×436(40) Hz [31]. The atom number and density

in the lattice is adjustable by varying evaporation pa-

rameters. Once the atoms are in the lattice, we set the

scattering length to 525(4)a0where a0is the Bohr radius,

thereby adjusting the interaction to U/t = 10(1). To de-

tect the singles density (ns=n−n↑n↓), the atom motion

is frozen by linearly increasing the lattice depth to 100ER

within 8 ms. For imaging, we turn oﬀ all magnetic ﬁelds

and switch to maximum lattice depth. Images of MI for

diﬀerent interaction strengths are shown in Figs. 1B and

1C.

By varying the atom number in the trap before loading

atoms into the lattice, we observe MI and band insulators

(BI) at U/t = 10(1) (Fig. 2). The MI region (Fig. 2B)

has nearly unit ﬁlling and atom number ﬂuctuations are

suppressed. When the chemical potential µexceeds the

value of U/2 (approximately half-ﬁlling), doubly occu-

pied sites are formed, therefore a BI region in the center

of the trap forms as shown in Figs. 2C and 2D. Doubly

occupied sites are detected as empty sites due to light-

induced collisions at the imaging stage [21].

IV. TRIANGULAR-LATTICE MOTT

INSULATORS

To access the singles density proﬁle, we perform a de-

convolution to determine the site occupation numbers

and obtain singles density (ns) and variance (σ2

ns) via az-

imuthal averaging (bottom panel of Fig. 2). We ﬁt the ex-

perimental density proﬁle using determinantal quantum

Monte Carlo (DQMC) and Numerical Linked Cluster Ex-

pansion (NLCE) calculations [31]. The temperature and

0246810

0.5

Lattice sites

Singles density (ns)

0246810

0 -1.0 -2.0

µ/U

Raw images

5µm

0246810

Lattice sites

0246810

0.25

Variance (σ2

ns)

0.5 0 -1.0

µ/U

0246810

0.5

Lattice sites

Singles density (ns)

0246810

0.9 0 -1.0

µ/U

Raw images

0 2 4 6 8 10 12 14

Lattice sites

0 2 4 6 8 10 12 14

0.25

Variance (σ2

ns)

1.9 1.0 0 -1.0

µ/U

FIG. 2. Triangular-lattice Mott insulators. (A-D) top Site-resolved ﬂuorescence images of fermionic Mott insulators with

increasing atom number integrated from ﬁt, 77, 119, 175, and 183 at interaction U/t = 10(1). (A-D) bottom Comparison of

azimuthally averaged singles density (dots) and variance (triangles) with theory calculations, QMC (red) and NLCE (orange).

The data points of the variance are horizontally oﬀset by 0.3 lattice sites for clarity. Both singles density nsand variance σ2

are ﬁt with QMC and NLCE theory using the local density approximation [31]. The detected variance is the square of the

standard deviation of the sample within a radial bin. The ﬁts yield temperatures kBT/t = 0.9(2),0.9(1),1.5(1), and 2.4(1)

with chemical potentials µ0/U = 0.24(10),0.5(4),0.91(3), and 1.94(1), respectively, at the trap center for increasing atom

number in both QMC and NLCE calculations. Error bars on nsare standard error of the mean and error bars on σ2

nsare

determined by error propagation from σ2

ns=ns−(ns)2.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

QuantumgasmicroscopyofageometricallyfrustratedHubbardsystemJirayuMongkolkiattichai,1,LiyuLiu,1,DavisGarwood,1JinYang,1,yandPeterSchauss1,z1DepartmentofPhysics,UniversityofVirginia,Charlottesville,Virginia22904,USAGeometricallyfrustratedmany-particlequantumsystemsarenotoriouslyhardtostudynumeri-cal...

展开>> 收起<<

Quantum gas microscopy of a geometrically frustrated Hubbard system Jirayu Mongkolkiattichai1Liyu Liu1Davis Garwood1Jin Yang1yand Peter Schauss1z 1Department of Physics University of Virginia Charlottesville Virginia 22904 USA.pdf

共13页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Quantum gas microscopy of a geometrically frustrated Hubbard system Jirayu Mongkolkiattichai1Liyu Liu1Davis Garwood1Jin Yang1yand Peter Schauss1z 1Department of Physics University of Virginia Charlottesville Virginia 22904 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: