Realization of an atomic quantum Hall system in four dimensions Jean-Baptiste Bouhiron Aur elien Fabre Qi Liu Quentin Redon Nehal Mittal Tanish Satoor Raphael Lopes and Sylvain Nascimbene

2025-04-26 0 0 3.4MB 14 页 10玖币
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Realization of an atomic quantum Hall system in four dimensions
Jean-Baptiste Bouhiron, Aur´elien Fabre, Qi Liu, Quentin Redon,
Nehal Mittal, Tanish Satoor, Raphael Lopes, and Sylvain Nascimbene∗
Laboratoire Kastler Brossel, Coll`ege de France, CNRS, ENS-PSL University,
Sorbonne Universit´e, 11 Place Marcelin Berthelot, 75005 Paris, France
(Dated: May 14, 2024)
Modern condensed matter physics relies on the concept of topology to classify matter, from
quantum Hall systems to topological insulators. Engineered systems, benefiting from synthetic
dimensions, can potentially give access to novel topological states predicted in dimensions D > 3.
We report the realization of an atomic quantum Hall system evolving in four dimensions (4D), with
two spatial dimensions and two synthetic ones encoded in the large spin of dysprosium atoms. The
non-trivial topology is evidenced by measuring a quantized electromagnetic non-linear response and
observing anisotropic hyperedge modes. We also excite non-planar cyclotron motion, contrasting
with its circular equivalents in D≤3. Our work opens to the investigation of strongly-correlated
topological liquids in 4D generalizing fractional quantum Hall states.
Topological order plays a central role in the classifi-
cation of states of matter beyond Landau’s paradigm
of symmetry breaking. It was originally introduced to
explain the Hall conductance of two-dimensional (2D)
quantum Hall systems, quantized by the first Chern num-
ber [1, 2]. Since then, various forms of topological sys-
tems, such as 3D topological insulators [3] and Weyl semi-
metals [4, 5], have been explored in condensed matter
[6]. A larger variety of topological systems have been
predicted, and organized in a ten-fold-way classification
scheme according to the system symmetries and dimen-
sionality [7]. In particular, higher-dimensional systems,
which can be explored in engineered systems with syn-
thetic dimensions [8, 9], can host special classes of topo-
logical matter [7], in particular a generalization of the
quantum Hall effect in 4D [10–12]. Different protocols
have been proposed [13–15] to realize 4D Hall insulators,
both for time-reversal invariant systems (classes AI and
AII) and in the absence of discrete symmetry (class A). In
those systems, the non-trivial topology leads to specific
behaviour, such as the quantization of the non-linear re-
sponse to both electric and magnetic perturbations, char-
acterized by the second Chern number C2[11, 14, 16] – a
topological invariant also relevant for tensor monopoles
in high dimensions [17–20]. The topology of a 4D quan-
tum Hall insulator (class A) also gives rise to anisotropic
motion close to a 3D hyperedge of the system, ballistic
along a given orientation, and still prohibited along the
two remaining directions of the hyperedge [21, 22].
So far, topological properties linked with the 4D Hall
effect have been revealed via geometrical charge pump ex-
periments in 2D systems [23, 24]. A truly 4D Hall system
has also been realized using electronic circuits – however,
no direct evidence of topological quantization has been
reported [25]. Here, we engineer an atomic quantum Hall
system evolving in 4D, by coupling with light fields two
spatial dimensions xand zand two synthetic ones en-
coded in the electronic spin J= 8 of dysprosium atoms
∗sylvain.nascimbene@lkb.ens.fr
ωx
ω0
x
ωz
ω0
z
x
z
atom motion
spatial dimensions x,z
atom
spin J= 8
synthetic dimensions n,m
n
m
ωx
ω0
x
ωz
ω0
z
magnetic projection m
energy
-4 -3 -2 -1 0 1 2 3 4
a
b
−12 −8−4 0 4 8 12
−10
0
10
p[k]
E[Er]
c
∆n∆m M∆v
x1 1 −2kˆ
x
z1 -2 −2kˆ
z
FIG. 1. Scheme of the 4D atomic system. (A) The
atomic motion in the xz plane is coupled to the internal spin
J= 8 using two-photon optical transitions along xand z
(blue and red arrows, respectively). The spin encodes two
synthetic dimensions given by the magnetic projection mand
its remainder n=m(mod 3) of its Euclidian division by
3, leading to a synthetic space of cylindrical geometry [28].
(B) Scheme of the light-induced spin transitions, of first- and
second-order along xand z, respectively. They induce corre-
lated spin-orbit dynamics, with distinct hopping along nand
maccording to the rules given in the table. (C) Dispersion re-
lation plotted as a function of the momentum p, for 6 values
of the quasi-momentum quniformly spanning the Brillouin
zone. The ground band is pictured as blue lines.
[8, 26, 27] (Fig. 1A). As a signature of non-trivial topol-
ogy, we measure a quantized electromagnetic non-linear
response and observe anisotropic hyperedge modes. We
also probe low-lying excitations, revealing complex cy-
clotron orbits beyond the planar circular paradigm oc-
curring in lower dimensions D≤3.
The coupling between motion and spin degrees of free-
arXiv:2210.06322v2 [cond-mat.quant-gas] 13 May 2024
2
dom is generated by a pair of lasers counter-propagating
along x(resp. z) and resonantly driving spin transitions
m→m+ 1 (resp. m→m−2), while imparting a mo-
mentum kick −2kˆ
x(resp. −2kˆ
z) [29] (Fig. 1B). Here, m
is the spin projection along z(−J≤m≤J,minte-
ger), k= 2π/λ is the light momentum for a wavelength
λ= 626.1 nm, and we assume a unit reduced Planck con-
stant ℏ= 1. The atom dynamics is described by the
Hamiltonian
H=Mv2
2−txeiϕxJ+
J+tzeiϕzJ2
−
J2+ hc+βJ2
z
J2,(1)
where vis the atom velocity and ϕα=−2kα is the rela-
tive phase of the two laser beams involved in each Raman
process α=x, z. The laser intensities and polarisations
control the amplitudes tαand the quadratic Zeeman shift
β=−2tz[30].
The laser-induced spin transitions can be interpreted
as hopping processes in a two-dimensional synthetic
space (m, n) involving the spin projection mand the re-
mainder n=m(mod 3) of its Euclidian division by 3
(with n= 0,1,2) [28]. While the first-order spin cou-
pling J+acts on these two dimensions in a similar man-
ner (hopping ∆n(x)= ∆m(x)= 1), the second-order cou-
pling J2
−induces hoppings ∆n(z)= 1 and ∆m(z)=−2
leading to differential dynamics along mand n. The
complex phases ϕα(with α=x, z) can be interpreted as
Peierls phases upon the hopping of a charged particle on
a lattice subjected to a magnetic field. Assuming unit
charge, we write ϕα=RAβdrβ=An∆n(α)+Am∆m(α),
leading to the explicit expression for the vector potential
A=1
3(0,0,2ϕx+ϕz, ϕx−ϕz)x,z,n,m.(2)
The magnetic field is then defined by the anti-symmetric
tensor Bαβ =∂αAβ−∂βAα, as
B=2k
3
0 0 −2−1
0 0 −1 1
2 1 0 0
1−1 0 0
.(3)
Similarly to the 2D quantum Hall effect, this magnetic
field gives rise to an energy separation between quasi-
flat magnetic Bloch bands. Within each band, motion
becomes effectively two-dimensional, with the guiding
center coordinate along ncanonically conjugated to the
position along ˆ
ν= (2ˆ
x+ˆ
z)/√5, while mis conju-
gated to the projection on ˆ
µ= (ˆ
x−ˆ
z)/√2. The en-
ergy levels are indexed by the canonical momentum p=
Mv+ 2kmˆ
x(mod K), which is conserved in the absence
of external force. Here, the reciprocal lattice vector K=
2k(2ˆ
x+ˆ
z)∥ˆ
νcorresponds to the momentum kick im-
parted on a non-trivial cycle mz
→m+ 2 x
→m+ 1 x
→m
involving one transition along zand two along x. In the
following, we decompose the momentum as p=pˆ
ξ+qˆ
ν,
with ˆ
ξ= (ˆ
x−2ˆ
z)/√5⊥ˆ
ν, such that the first Brillouin
−4−2 0 2 4
-8
-4
0
4
8
p[k]
m
a0 0.25
Πm
−4−2 0 2 4 0
1
2
p[k]
n
0.3 0.37
Pn
−2−10 1 2
-8
-4
0
4
8
q[k]
m
b0 0.23
Πm
−2−1 0 1 2 0
1
2
q[k]
n
0.3 0.37
Pn
ba
−2
0
2
q[k]
−8
0
8
hmi
−12 −8−4 0 4 8 12
−2
0
2
p[k]
q[k]
0
1
2hni
c
FIG. 2. Hall drift along the synthetic dimensions.
(A) Evolution of the measured spin projections Πmand Pn
as a function of pupon adiabatic driving along ˆ
µ– the spatial
direction conjugated with m. The mean values ⟨m⟩and ⟨n⟩
(computed as 3
2πarg⟨ei2πm/3⟩) are shown as red lines. (B)
Same quantities plotted for a driving along ˆ
ν– the spatial
direction conjugated with n. (C) Measurements of the mean
values ⟨m⟩and ⟨n⟩in the Brillouin zone. The green arrows
represent the driving directions considered in Aand B.
zone is defined for |q|< K/2 and arbitrary p. The energy
levels of the Hamiltonian (1) organize in Bloch bands
shown in Fig. 1d. We focus here on the ground band,
which is quasi-flat in the bulk mode region |p|≲7k[30].
Our experiments use ultracold dilute samples of ≃
3.0(3) ×104atoms of 162Dy, prepared in an optical
dipole trap at a temperature T= 260(10) nK. The
atoms are subjected to a magnetic field B= 221(1) mG
along z, and initially spin-polarized in the magnetic sub-
level m=−J. We adiabatically ramp up the laser
intensities to generate the spin couplings described in
(1) with tx= 5.69(6) Erec and tz= 5.1(1) Erec, where
Erec =k2/(2M) is the recoil energy. Starting in the
m=−Jedge mode region with p < 7k, we prepare ar-
bitrary momentum states of the ground energy band by
applying a weak force on a typical 1 ms timescale [30]. At
the end of the experiments, we probe the velocity distri-
bution by imaging the atomic sample after free expansion
in the presence of a magnetic field gradient, such that the
different mstates are spatially separated.
3
−12 −8−4 0 4 8 12
−2
0
2
p[k]
q[k]
hvi
hvξihvνi
5vrec
−12−8−40 4 8 12
−5
0
5
p[k]
hvαiq[vrec]
hvξiqhvνiq
ab
FIG. 3. Frustration of motion in the bulk and anisotropic ballistic edge modes. (A) Evolution of the mean velocity
⟨v⟩versus momentum. The arrow is scaled according the mean velocity modulus. (B) Measurements of the q-average velocity
components versus p. The solid lines are the expected variations for the ground band of the Hamiltonian (1).
We first investigate the anomalous Hall drift in spin
space upon the application of a weak force in the xz
plane. For a force oriented along ˆ
µ(spatial direction con-
jugated to m), the spin projection probabilities Πmreveal
a drift of the mean spin projection ⟨m⟩, while the mean
remainder ⟨n⟩remain approximately constant (Fig. 2a).
An opposite behavior is observed when applying a force
along ˆ
ν(direction conjugated to n), with a quasi-linear
variation of ⟨n⟩while ⟨m⟩remains constant (Fig. 2b).
More generally, in the bulk of the system, where the band
dispersion can be neglected, the variation with momen-
tum of the mean values ⟨n⟩and ⟨m⟩can be expressed as
an anomalous Hall drift governed by the antisymmetric
Berry curvature tensor Ωbulk, as [31, 32]
d⟨rα⟩= Ωαβ
bulkdpβ,(4)
Ωbulk =B−1=1
2k
0 0 1 1
0 0 1 −2
−1−1 0 0
−1200
,(5)
where ris the position vector and dpis the momen-
tum variation due to the external force. We con-
front this prediction to our measurements of the mean
positions ⟨m⟩and ⟨n⟩as a function of p(Fig. 2c).
In the center of the Brillouin zone |p| ≤ 4k(bulk
mode region), we fit the measured Hall drift with the
linear function (4), yielding {Ωnx,Ωnz,Ωmx,Ωmz}=
{−1.00(2),−0.98(2),−0.98(2),1.96(2)}/(2k), consistent
with (5). We do not measure any significant spatial drift
upon the application of a force in the xz plane, consistent
with Ωxz = 0. Similarly, no Hall current along nis mea-
sured when applying a force along m(corresponding to
a perturbative Zeeman field), compatible with Ωnm = 0
[30].
Further insight on the ground band properties is pro-
vided by the mean velocity ⟨v⟩(Fig. 3), which reveals
distinct behaviours between bulk and the edge modes.
In the bulk |p|≲7k, the mean velocity remains much
smaller than the recoil velocity vrec =k/M (Fig. 3b).
Since ⟨v⟩=∇pE0, it confirms that the ground band
energy E0is quasi-flat in the bulk (Fig. 1d). This mea-
surement illustrates the frustration of motion induced by
the magnetic field, similarly to flat Landau levels in 2D
electron gases.
For p≳7k, the atoms mostly occupy the edge redge =
Jˆ
mof the synthetic dimension m. We measure a non-
zero mean velocity, whose ξcomponent increases with
p, while the νprojection remains small (Fig. 3b). This
observation is characteristic of an anisotropic edge mode
of a 4D quantum Hall system, which corresponds to a
collection of 1D conduction channels oriented along the
direction wmotion, with wα
motion ∝Ωαβ
bulkredge
β,here corre-
sponding to the direction ˆ
ξ[21]. Within the edge, the
motion in the plane orthogonal to wmotion remains in-
hibited, in agreement with the measured νvelocity. A
similar behaviour is found on the edge −Jˆ
m, albeit with
opposite orientation of velocity.
A hallmark of 4D quantum Hall physics is the peculiar
nature of excitations above the ground band, which can
be linked to classical cyclotron trajectories. While cy-
clotron motion in 2D and 3D always corresponds to pla-
nar circular orbits, we expect more complex trajectories
in 4D, involving two planar rotations occurring at differ-
ent rates. In our system, each of these two elementary
excitations is generated by the Raman coupling along x
or z, of corresponding frequencies ωxand ωzindepen-
dently set by the amplitudes txand tz. We excite the
atoms by applying a diabatic velocity kick, and measure
the subsequent time evolution of the center of mass. We
show in Fig. 4a the orbit measured for ωz/ωx≃2, reveal-
ing a non-planar trajectory. For this integer frequency
ratio, the orbit is almost closed, and is reminiscent of a
Lissajous orbit (Fig. 4b). We also studied the case of de-
generate frequencies ωz≃ωx, which correspond to the
coupling amplitudes used for our study of the ground
band. In this ‘isoclinic’ case, we recover a planar cy-
clotron motion akin to lower-dimensional cyclotron orbits
(Fig. 4c,d).
The non-trivial topology of a quantum Hall system
manifests in the quantization of its electromagnetic re-
sponse. While involving the linear Hall conductance in
2D, it requires in 4D considering the non-linear response
to both an electric force fand a magnetic field b. These
two perturbations induce a current
jα
non-linear =C2
4π2ϵαβγδfδbβγ ,
where ϵαβγδ is the 4D Levi-Civita symbol and C2is the
integer second Chern number [14]. In other words, the
magnetic field bβγ induces a Hall effect in the perpendic-
ular plane.
We first demonstrate the quantization of the non-linear
摘要：

RealizationofanatomicquantumHallsysteminfourdimensionsJean-BaptisteBouhiron,Aur´elienFabre,QiLiu,QuentinRedon,NehalMittal,TanishSatoor,RaphaelLopes,andSylvainNascimbene∗LaboratoireKastlerBrossel,Coll`egedeFrance,CNRS,ENS-PSLUniversity,SorbonneUniversit´e,11PlaceMarcelinBerthelot,75005Paris,France(Da...

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