R ES E A RC H

QUANTUM SIMULATION

Berry curvatures through nonadiabatic responsesof the system.

Second Chern number of aquantum-simulated non-Abelian
Yang monopole

Monopole fields and Chern numbers

Seiji Sugawa*†, Francisco Salces-Carcoba, Abigail R. Perry‡, Yuchen Yue, I. B. Spielman†
Topological order is often quantified in terms of Chern numbers, each of which classifies
a topological singularity. Here, inspired by concepts from high-energy physics, we use
quantum simulation based on the spin degrees of freedom of atomic Bose-Einsteincondensates to characterize a singularity present in five-dimensional non-Abelian gauge
theories—a Yang monopole. We quantify the monopole in terms of Chern numbers
measured on enclosing manifolds: Whereas the well-known first Chern number vanishes,
the second Chern number does not. By displacing the manifold, we induce and observea topological transition, where the topology of the manifold changes to a trivial state.

Joint Quantum Institute, National Institute of Standards andTechnology and the University of Maryland, Gaithersburg,
MD 20899-8424, USA.
*Present address: PRESTO, Japan Science and Technology Agency(JST), Saitama 332-0012, Japan, and Graduate School of Science,
Kyoto University, Kyoto 606-8502, Japan.
†Corresponding author. Email: sugawa@yagura.scphys.kyotou.ac.jp (S.S.); ian.spielman@nist.gov (I.B.S.) ‡Present address:
Georgia Tech Research Institute, Atlanta, GA 30318, USA.

Sugawa et al., Science 360, 1429–1434 (2018)

analog was synthesized inside a spinor condensate where the associated spin texture was observed(5). In quantum mechanical systems, gauge fields
such as the electromagnetic vector potential A
take central stage (in classical electromagnetism
B ¼ ∇  A) and are required to understand nature at the most fundamental level (8). The YangMills theory is a non-Abelian extension of Dirac’s
magnetic monopole (9) and requires higher-order
Chern numbers (higher-order than the first) for
its topological characterization.
Here we report on the quantum simulation ofa Yang monopole in a 5D parameter space built
from an atomic quantum gases’ internal states
and the measurement of its topological charges
by characterizing the associated non-Abeliangauge fields (often called curvatures). To extract
the second and higher Chern numbers that result from non-Abelian gauge fields, we developed a method to evaluate the local non-Abelian

Fmn ðqÞ ¼@An @Am i½Am ; An 
@qm @qnð1Þ

where i is an imaginary unit; in three spatial
dimensions, the components of the magnetic
field Bm ¼ Dmnl Fnl =2, where l is an integer, canbe determined from the elements of the Fnl
matrices (Dmnl is the rank-3 Levi-Civita symbol,
and we used Einstein’s implied summation convention for repeated indices). In analogy to theGauss’s law with electric charges (monopoles), thefirst Chern number is equivalently the integral
C1 ¼

1
1
∫S B  dS ¼ 4p
∫S2 Fmn dqm ∧dqn
2p 2

ð2Þ

of the magnetic field B or the Abelian fieldstrength Fnl over a closed 2D manifold S2, where∧ is the wedge product. The general nth Chern
number of a non-Abelian gauge field is then-wedge product of the non-Abelian curvature
an Cn ¼ ∫S2n tr½F ∧ F ∧ F ∧ … ∧ F  d2n S

ð3Þ

where an (a1 = 4p, a2 = 32p2, …) is a normalization factor and S2n is a closed 2n-dimensional
manifold (10).
Chern numbers provide a topological classification of monopoles in gauge field theories.
The monopoles are generally associated with

Fig. 1. Non-Abelian monopole and the appearance of nontrivial topology. (A) The 5D parameter
space. The system has a topological defect at the origin, a Yang monopole, providing a source of non-Abeliangauge field. The topological invariant associated with the monopole is the second Chern number,defined on an enclosing 4D manifold. (B) Quantum states can be mapped onto generalized Bloch
spheres. An additional Bloch sphere, which defines the wave function within each DS, is required to fully
define our systems eigenstates. The 5D generalized magnetization vector G remains parallel with q at
adiabaticity, and the leading order correction to the adiabatic changes to q, is a small deflection in G.

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he Yang-Mills theory is a non-Abelian gauge
field theory that includes a higher gauge
symmetry than quantum electrodynamics
and now forms a cornerstone of the standard model of particle physics (1, 2). In the
Yang-Mills theory, soliton solutions that include
monopoles and instantons play a key role, theoretically describing phenomena in high-energy
physics (3). The monopole solutions are sourcesof non-Abelian gauge fields and give rise to a
nontrivial topology.
The physical importance of magnetic monopoles was captured in the seminal work by
P. A. M. Dirac (4). Dirac considered a phase,
now known as the Aharonov-Bohm phase, acquired by an electron with charge qe movingaround a magnetic monopole and showed thatthe monopole charge must be qm = nh/qe, where
n is an integer and h is Planck’s constant. Following from this quantization condition, Gauss’s
law for the magnetic field B must take a quantized value nh=qe ¼ ∫S2 B  dS, which essentiallycounts the number of magnetic charges inside the
manifold S2 [here S2 is a closed two-dimensional(2D) surface and dS = n dS (n is a unit vector normal to the surface)]. The integral is topologicallyrobust against deformation of the enclosing manifold as long as the number of monopoles enclosed
is unchanged. The field from Dirac monopoles
has been observed in a range of physical systems, and the associated topological charge—
the first Chern number, often referred to as “theChern number”—has been measured (5–7). The
first Chern number and Abelian monopole field
were measured in the parameter space of a spin1/2 artificial atom (6, 7), and the Dirac monopole

An N dimensional vector gauge field AðqÞ ¼
ðA1 ; A2 ; …; AN Þ, where q ¼ ðq1 ; q2 ; …; qN Þ is the
position, is said to be non-Abelian when thevector components Am(q) fail to commute, i.e.,[Am, An] ≠ 0 (m; n∈ f1; …; N g), where m and n label
the different vector components. The resulting
curvature is given by

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Experimental Hamiltonian
We realized a non-Abelian gauge field by cyclically coupling four levels within the hyperfine groundstates of rubidium-87 using radio-frequency andmicrowave fields (Fig. 2, A and B), essentiallyforming a square plaquette. The four couplings
were parameterized by two Rabi frequencies
WA and WB and two phases fA and fB arranged sothat the sum of the phases around the plaquette
was p. This configuration of control fields, alongwith a detuning d ¼ jgF jmDB =ħ, where gF is the
Landé g factor, m is the Bohr magneton, DB is theshift in the magnetic field from resonant coupling condition, and ħ = h/2p, gave us an experimentally controllable 5D parameter space labeledby the Cartesian coordinates q = (−WB cos fB,−WA cos fA, −WA sin fA, d, −WB sin fB). In muchthe same way that a two-level atom in a magnetic
field can be understood in terms of three Paulimatrices, our four-level system is governed by the
Hamiltonian^ ¼ ħq  G^ ¼  ħ ðq1 G^ 1 þ q2 G
^ 2þ
H
2
2
^ 3 þ q4 G
^ 4 þ q5 G
^ 5Þ
q3 G

ces commutes with the time-reversal operator, thesystem has time-reversal symmetry (15); Kramers
theorem then implies that the system has twopairs of degenerate energy states, here with energies ET ¼ Tħjqj=2. Thus, each energy, labeled by+ or −, has two independent eigenstates j↑T ðqÞiand j↓T ðqÞi; each of these pairs define a degenerate subspace (DS). As shown in Fig. 2B, theseDSs are characterized by a generalized magneti^ 2 i; hG^ 3 i; hG^ 4 i; hG^ 5 iÞ
^ 1 i; hGzation vector hGi ¼ ðhG
on a unit 4-sphere in our 5D space. Different
configurations within each DS share the samemagnetization vector, which can be pictured interms of an additional 3D Bloch sphere (green
sphere in Figs. 1 and 2B). An eigenstate is fully
depicted by assigning the two such “Bloch” vectors. The Yang monopole (16, 17) resides at the
Hamiltonian’s degeneracy point at q = 0, a singularpoint where the non-Abelian Berry’s connection
diverges. The non-Abelian Berry’s curvatures fromour experimental Hamiltonian (Eq. 4) quantum
simulates the fields of a Yang monopole.
Quantum control and measurement

ð4Þ^ i (i = 1, 2, …, 5) are the ith comwhere qi and G^ and G^ i is represented as theponents of q and G
four-by-four Dirac matrices with the hyperfine
ground states shown in Fig. 2A taken as the basis. Furthermore, because each of the Dirac matri-

We begin by demonstrating the control and measurement capabilities of our system. We firstprepared the system in its groundat the popﬃﬃstate
ﬃ
sition q 0 ¼ q0 ð1; 1; 0; 0; 0Þ= 2 in parameter
space, where the generalized
magnetization ispﬃﬃﬃ
hGi ¼ ð1; 1; 0; 0; 0Þ= 2. Then, by ramping fA,
we slowly moved the control vector around theFig. 2. Schematics of theexperiment. (A) Schematicof our implemented coupling
using four hyperfine groundstates of rubidium-87. The fourstates were cyclically coupledwith radio-frequency (rf) andmicrowave fields. The rightpanel shows the resulting
plaquette and the associatedcoupling parameters. (B) At
any point in the 5D parameter
space, the energy spectrum
forms a pair of twofold
degenerate manifolds withthe energy gap equal to ħjqj,where q is the control field.
Each degenerate subspacecan be represented by a
Bloch sphere.

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a divergence in the field strength and can contribute a unit of flux through any enclosing
manifold. This generalized flux is quantized andis given by the Chern numbers. In particular, for
Yang monopoles, the first Chern number is zero,but the second Chern number is either +1 or −1(Fig. 1).
Many quantum systems can be described by a
Hamiltonian Ĥ(q) that depends on position q inparameter space. At each position, the system ischaracterized by energies Ek(q) and eigenstatesjkðqÞi, where k∈f1; …; Kg is an index that identifies the eigenstate in ourK -dimensional Hilbert
space. A gauge potential called the non-Abelian
Berry connection Abg
m ðqÞ ¼ ihbðqÞj@=@qm jgðqÞi,where b; g∈f1; …; Kg , is encoded in the wave
functions; thus, for any position q, each vector
component Am is represented by a matrix. Chern
numbers and curvatures can be then defined by
Eqs. 1 to 3 for each well-separated energy level.
Because of these gauge fields, an initial quantum
state can acquire a geometric phase as the location in parameter space is adiabatically changed.
For nondegenerate quantum systems, the resulting geometric phase is called the Berry phase(11). A quantum state evolving within a degenerate subspace can acquire a Wilczek-Zee geometric phase, a matrix-valued generalization of
the Berry phase obtained as the path-orderedline integral of a non-Abelian gauge potential(12–14).

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circleqðtÞ
pﬃﬃﬃ ¼ q0 ½1; cosð2pt=T Þ;  sinð2pt=T Þ;
0; 0= 2 shown in Fig. 3A, where T is the full
ramp time, and q0 ¼ jq0 j ¼ 2p  2kHz.
After preparing the eigenstate j↑ ðq 0 Þi þ
 pﬃﬃﬃ pﬃﬃﬃ
pﬃﬃﬃ
ij↓ ðq0 Þi = 2 ¼ 2j1i  ð1 þ iÞj2i þ i 2j3iþ
 pﬃﬃﬃ
ð1  iÞj4i =ð2 2Þ in the ground DS by rotating
the control field (15) from the north pole qN =q0(0, 0, 0, 1, 0), we measured the state for different evolution times in this nearly adiabatic ramp
(Fig. 3B), and identified the orientation within the
DS by performing quantum state tomography,
giving the expectation values of the Pauli operators^ i ði ¼ x; y; zÞ in the ground DS at qN. As seen insFig. 3B, after the control field completed one cycle,
the orientation of the state vector within the DS
differed from its initial value. After one cycle, theBerry’s phase from an Abelian gauge fieldwould contribute only an overall phase, leaving the state vector otherwise unchanged. In
agreement with our numerical simulation
obtained by solving (15) the time-dependent
Schrödinger equation (TDSE) for the Hamiltonianin Eq. 4 (curves in Fig. 3B), this shows that theobserved evolution resulted instead from the
Wilczek-Zee phase derived from a non-Abeliangauge field.
^ 4 i during this ramp andWe then measured hG
noted a small deflection of the magnetization of
the state vector owing to remnant nonadiabatic
effects (Fig. 3C). In linear response theory, deviations from adiabaticity can be described in termsof the response of the state vector to a general-^ m ¼ ð@ H^ =@qm Þ=ħ acting on theized force M
state (Fig. 1B). For a conventional Abelian system, the local force at a fixed time (18, 19)^ m i ¼ vn Fmn þ constanthM

Sugawa et al., Science 360, 1429–1434 (2018)

Fig. 4. Generalized magnetization. (A) Schematic of the control field trajectory. The twophases (fA, fB) are ramped for T = 4 ms withthe laboratory parameters WA/2p = WB/2p =1.41 kHz and d = 0. (B) Quantum states were
measured by evaluating the expectation valuesof the five Dirac matrices. The red curves plot
the trajectories expected for adiabatic motion,
whereas the black curves are numerical simulations,
including our finite ramp time (15).

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ð6Þ

acting on the state, where the expectation valueon the right-hand side is taken for a pure stateat adiabaticity and F^ mn is the Berry curvature of
the associated degenerate subspace (26). In contrast to the Abelian case, where the generalized
geometric force is simply the product of the local
Berry curvature and the velocity, the force in
Eq. 6 also depends on the quantum state within
the DS. As we saw, even for adiabatic motion,
Wilczek-Zee phases can lead to considerable evolution within the DS, making Eq. 6 essential for
describing generalized geometric forces.
The sign change in Fig. 3C is now explainedby the dependence of the geometric force on thestate as it evolved within the DS. If the gauge
field is Abelian, independent of the state within
the DS, the force components should be constantin the spherical coordinate along the path forconstant ramp velocity. The sign change reveals
that the quantum state acquired a Wilczek-Zee
phase from a non-Abelian gauge potential, contributing to the geometric force. Indeed, the solid
curves depict the prediction of our TDSE simulations (15) and confirm that the geometric force
in our experiment cannot be derived from an
Abelian gauge potential.
In general, we can observe the full magnetization of the state vector by carefully measuring^ i.
the expectation values (15) of all five operators G
To demonstrate this capability, we moved along
the circle qðtÞ ¼ q0 ½cosð2pt=T
pﬃﬃﬃ Þ; cosð2pt=T Þ;sinð2pt=T Þ; 0; sinð2pt=T Þ=p
ﬃﬃﬃ 2 shown in Fig. 4A,
starting from j↑ ðq0 Þi [¼ ð 2j1i  j2i þ j4iÞ=2]
at t = 0, and obtained hGðtÞi. Figure 4B showsthat hGðtÞi nearly followed the adiabatic trajectory (red curves), almost oriented parallel to q,
but slightly deflected owing to the nonadiabaticity [TDSE simulation (15) shown by black curves
in Fig. 4B].
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resulting from parameters qn changing with velocity vn is analogous to the Lorentz force. This
relation gives the driving force behind the topological and geometrical charge pumps recently
realized in ultracold atoms (20–22). In both crystalline and optical lattices, the same relation
underlies the anomalous quantum Hall effect(23–25).
Owing to the phase symmetry of the system for
fA, the generalized geometric force from Eq. 5 isconstant for our trajectory, inconsistent with thesign change present in the observed deflection(Fig. 3C). To account for this discrepancy, Eq. 5
can be extended to accommodate non-Abeliangauge fields, giving the generalized geometricforce (15, 26)^ m i ¼ vn hF^ mn i þ constanthM

Fig. 3. State evolution under a non-Abeliangauge field. (A) Schematic of the control fieldtrajectory. The two phases (fA, fB) were ramped
for T = 2 ms with the laboratory parameters
WA/2p = WB/2p = 1.41 kHz and d = 0. (B) Nearlyadiabatic response of pseudospin magnetization within the ground DS Bloch sphere, showing
the nontrivial acquisition of a Wilczek-Zee phase
after a 2p-rotation. The solid lines simulate theexperiment by numerically solving the TDSE (15).
(C) Deflection during the phase ramp. The^ 4 i, resulting
state was slightly deflected along hGfrom our finite ramp time (black circles),changing from positive to negative. The black
curve shows the theoretically expected linear
response based on Eq. 6 (15).

ð5Þ

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Sugawa et al., Science 360, 1429–1434 (2018)^
0:07ð5Þ; 1:00ð3Þ  s,also in good agreement^z.
with the theoretical value, 2q20 F^f2 q2 ¼ s
Just as in classical electromagnetism, where
the fields from electric or magnetic sources fall
off as 1/q2, the non-Abelian gauge field strength
also follows a 1/q2 scaling law, as required by thegeneralized Gauss’s law (see Eq. 2) that defines
the second Chern number. By repeating the same
Berry curvature measurement (F^f2 q2 ) for different q0, while keeping 2p/qT = 0.25 constant toremain in the linear response regime, the Berrycurvature componentshF^f2 q2 i indeed had the 1/q2
scaling of a monopole source (Fig. 5E); this alsosuggests that hF^f2 q2 i diverges at q→0.

Taken together, these fields provide sufficient
information to extract the second Chern number of a 4-sphere with radius q0. We evaluate thesecond Chern number using the relation
C2 ¼

3q40
∫S tr½Ff1 q1 Fq2 f2 d4 S
4p2 4

ð7Þ

where S4 defines the 4-sphere and d4S = sin3 q1 sin
2q2dq1dq2df1df2. Equation 7 relies on the rotational symmetry of Ĥ(q), which gives the numerically confirmed (15) relations tr½Ff1 q1 Ff2 q2 ¼
tr½ Ff1 q2 Fq1 f2  ¼ tr½Ff1 f2 Fq2 q1  . From the nonAbelian Berry curvature measurements in the

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Non-Abelian Berry curvatures andChern numbers
With the ultimate goal of evaluating Chern numbers in mind, we characterized the non-Abelian
Berry curvatures on spherical manifolds in parameter space. Accordingly, we adopt sphericalcoordinates described by a radius q and fourangles q1 ∈ [0, p], q2 ∈ [0, p/2], f1, and f2 that are
related to our experimental control parameter
space via WA = q sin q1 cos q2, WB = q sin q1 sin q2,
d = q cos q1, f1 = (fA + fB)/2, and f2 = (fA − fB)/2.
After preparing the system in its ground state
at q0, we measured the deflection along the q1 direction, while rotating the control field alongqT ðtÞ ¼ q0 ½cosð2pt=T
pﬃﬃÞ;ﬃ cosð2pt=T Þ; ∓sinð2pt=
T Þ; 0; ∓sinð2pt=T Þ= 2 by ramping f1 from 0 to±p (half-circles in Fig. 5A). The geometric force
Mq1 is directly obtained from the deflection of^ 4 i. Figure 5B plots the deflection during thishGramp for four different initial states (marked by
jAi, jBi, jCi, and jDi in Fig. 5D) within the DS,
manifesting the state dependence of the geometric force in the non-Abelian gauge field incontrast to Abelian cases. The net deflection
during any given ramp gives the integrated geometric force.
To confirm that our drive was in the linear response regime, we measured the geometric force
as a function of ramp time T (Fig. 5C). From both
the data and TDSE simulations (dashed curves),
the geometric force (solid curves) is almost linearwith respect to velocity for T ≥ 12p/q.
The components of the Berry curvatures canbe reconstructed from the integrated geometricforce. Owing to the symmetry of our experimental Hamiltonian, the geometric force components
must be almost constant in spherical coordinates
during the ramp in the linear response regime.
By measuring the geometric force experienced
by four independent initial states all within theDS, we determined the four independent parameters present in the 2-by-2 matrices describing each element (labeled by b and g) of therepresentation of the non-Abelian Berry curvabg. Following this procedure for T ≥ 12p/q,
ture Fml
we obtained 2q20 F^q1 f1 ¼ 0:01ð3Þ^I 0 þ ½0:06ð5Þ;^
0:08ð5Þ; 0:98ð3Þ  s,in agreement with the the2^^ z (here, Î0 is the idenoretical value, 2q0 Fq1 f1 ¼ stity operator).
We thoroughly investigated the state dependence of the geometric force by studying theevolution of 225 initial states covering the Bloch
sphere of the initial DS (Fig. 5D). For each initialstate, we recorded the deflection after a 250-ms
ramp to obtain the Berry curvature component
hF^q1 f1 i. Figure 5D shows the initial-state Bloch
sphere colored according to the curvature; thetheoretically computed result (top) is in good
agreement with experimental result (bottom).
By changing the path and the direction along
which we measure the deflection, other components of the curvatures can be measured. For
example, by rotating the control field alongqT ðtÞ ¼ q0 ½cosð2pt=T
pﬃﬃﬃ Þ; cosð2pt=T Þ; ∓sinð2pt=
T Þ; 0; Tsinð2pt=T Þ= 2 by ramping f2 and evaluating the deflection along the q2 direction at q0,
we obtained 2q20 F^q2 f2 ¼ 0:08ð3Þ^I 0 þ ½0:12ð5Þ;Fig. 5. Deflection of states within the ground-state manifold owing to non-Abelian Berrycurvatures. (A) Schematic of the control field trajectory. (B) Deflections along q1 were measured^ 4 i was measured for four independent initial states (jAi, jBi, jCi, andjDi)
during the T = 6 ms ramp. hG
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
j1iﬃﬃﬃ  j2i þ j4iÞ=2 and jBi ¼
within the DS at q0. Here jAi ¼ ð 2 p
pﬃﬃﬃðj2i þ 2 j3i  j4iÞ=2 are the basis
states for the DS, jCi ¼ ðjAi þ jBiÞ= 2, andjDi ¼ ðjAi þ ijBiÞ= 2. (C) Geometric force as a function of
1/T measured for the four initial states (jAi, jBi, jCi, and jDi) at q0. The dashed lines assume linearity,and the solid curves are the outcomes of our TDSE simulations (15). (D) Expectation values of the^ q f i in the ground state manifold at q0 are mapped onto Bloch spheres
non-Abelian Berry curvature hF
1 1
associated with the state within the DS at q0. The four initial states used in (A) to (C) are also shownin the theoretical (top sphere) and the experimental (bottom sphere) plots. (E) 1/q2 scaling in the^ f q i are evaluated for various q0.
strength of the curvature. The matrix components of the curvature hF
2 2
The data show excellent agreement with the theory that exhibits 1/q2 dependence (solid lines).

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previous section, we directly obtained C2 ¼
2q40 tr½Fq1 f1 ðq0ÞFf2 q2 ðq0Þ ¼ 0:97ð6Þ for the groundstate, consistent with the theoretical value C2 = 1.
We repeated the measurements for the excitedstate and found C2 ¼ 2q40 tr½Ff1 q2 ðq 0 ÞFf2 q1 ðq0 Þ ¼
0:93ð6Þ, also in agreement with the theoretical value C2 = –1. These nonzero Chern numbers inform us that the manifold is topologically
nontrivial.
Because the system is time-reversal symmetric, the first Chern form is zero, and therefore
Eq. 3 for the first Chern number should be zerofor both degenerate manifolds. Indeed, all themeasured non-Abelian Berry curvatures weretraceless (q20 tr½F^f2 q2 ¼ 0:08ð3Þ and q20 tr½F^q1 f1 ¼
0:01ð3Þ for the ground state, and q20 tr½F^q1 f2  ¼
0:02ð3Þ and q20 tr½F^f1 q2  ¼ 0:00ð3Þ for the excited state), so that the first Chern number,which is the surface integral of the trace of theindividual curvatures, was also zero. Thus, thenontrivial topology of the monopole field is not
expressed by a first Chern number.
Topological transition
We concluded our measurements by inducing
a topologically nontrivial-to-trivial transitionof the manifold by displacing the 4-sphere inparameter space from the origin by an amountqoffset = qoffset(q0/q0) (Fig. 6A). The topologicaltransition occurs at the critical displacement
qcrit = q0 when the Yang monopole departs the
manifold. Figure 6B shows our observed transition of the second Chern number from ±1, forthe ground and excited states, to zero as the offset coupling qoffset was increased. This transitionis associated with the topology of the manifold
changing from topologically nontrivial to trivial.
The smoothness of the observed transition wascaused by the breakdown of the linear response
near the transition point. Our theory [continuous
curves in Fig. 6B, and see (15)] shows that slower
ramps enlarge the region in which linear response is valid and make the transition sharper(Fig. 6B). Topological transitions have been observed in a range of experiments (6, 7, 25); however, in all of these cases, the observed topologicalphases were only identified by a Dirac monopoles’
first Chern number and enclosing 2D manifolds.
By contrast, in our system, the first Chern number
is zero everywhere and the second Chern number
characterizes the topological phase, arising froma Yang monopole at the origin of parameter space.
The opposite topological charges observed in the
ground or excited manifolds result from a monopole in one manifold acting as an antimonopolein the other. With these Chern number measurements, we confirmed that the engineered topological singularity in our system indeed simulated a
Yang monopole.

Our work can be extended to other quantum systems, including ions, thermal atoms, and superconducting qubits. The Chern number characterizes
a source of gauge field with high symmetry, asymmetry that naturally arises in particle physics
in contexts such as quantum chromodynamics.
The monopole field and the second Chern
number have been discussed theoretically in thecontext of 4D quantum Hall effect (4DQH) (27, 28),
spin-Hall effect (29), exotic charge pumping (30),and fermionic pairing (31) in condensed mattersystems. The model we explored experimentallyis equivalent to the (4 + 1)-D lattice Dirac Hamiltonian relevant to 4DQH. The 4DQH is a generalized quantum Hall effect and is the root state ofa family of topological insulators, which are obtained by a dimensional reduction procedure(32). The observed transition in Fig. 6B can be
regarded as the type of phase transition present
in the band topology of 4DQH systems. A conformal mapping from a 4D spherical manifoldin parameter space to a 4D crystal momentum
space, 4-torus, directly recasts our Hamiltonianas the Dirac Hamiltonian.
Our observation lays the groundwork for simulating objects in high-energy physics with atomic
quantum systems. Lattice extensions of our work,where lattice sites or bands play the role of spinstates, may allow quantum simulation of emergent
many-body dynamics with non-Abelian gauge
fields with highly controllable ultracold quantum
gases systems (33–36).
REFERENCES AND NOTES

Fig. 6. Topological transition from a Yang monopole. (A) Schematic illustration of the topologicaltransition. Suitable spherical manifolds with radius q0 were offset from the origin by qoffset toevaluate both C1 and C2. At the critical value (qcrit = q0), the monopole exits the manifolds. (B) Chern
numbers extracted from experiment. When the manifold crossed qcrit = q0, jC2 j decreased fromunity to zero (top panel), whereas C1 was constantly zero for both the ground (red) and the excited(blue) states (bottom panel). Numerical simulations (solid lines; T = 3, 10, and 50 ms) andanalytic theory (dash-dot lines) are also shown. The data were taken for T = 3 ms.
Sugawa et al., Science 360, 1429–1434 (2018)29 June 2018

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ACKN OWLED GMEN TS

The authors would like to thank A. Polkovnikov and M. Kolodrubetz
for discussion. Funding: This work was partially supported by
the U.S. Army Research Office’s Atomtronics MURI, and by the
U.S. Air Force Office of Scientific Research’s Quantum Matter
MURI, National Institute of Standards and Technology, and
NSF through the Physics Frontier Center at the Joint Quantum

Institute. S.S. acknowledges the Japan Society for the
Promotion of Science (fellowship for research abroad). Author
contributions: S.S. conducted the experiment, performed thetheoretical work, and analyzed the data. S.S., A.R.P., F.S.-C.,
and I.B.S. contributed to the rubidium Bose-Einstein condensate
apparatus. All authors substantially participated in thediscussion and the writing of the manuscript. S.S. and I.B.S.
conceived of the project. Competing interests: The authors
declare no competing interests. Data and materials
availability: All data are available in the manuscript or the
supplementary materials.
SUPPLEMENTARY MATERIALS

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Materials and Methods
Supplementary Text
Figs. S1 to S3
References (37, 38)
Database S1
15 February 2017; accepted 27 April 201810.1126/science.aam9031

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Sugawa et al., Science 360, 1429–1434 (2018)29 June 2018

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Second Chern number of a quantum-simulated non-Abelian Yang monopole
Seiji Sugawa, Francisco Salces-Carcoba, Abigail R. Perry, Yuchen Yue and I. B. Spielman

Science 360 (6396), 1429-1434.
DOI: 10.1126/science.aam9031

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MATERIALS

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REFERENCES

This article cites 35 articles, 5 of which you can access for free
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Going beyond the first Chern number
Topological properties of physical systems are reflected in so-called Chern numbers: A nonzero Chern number
typically means that a system is topologically nontrivial. Sugawa et al. engineered a cold atom system with a nonzero
second Chern number, in contrast to condensed matter physics, where only the first Chern number is usually invoked.
The exotic topology relates to the emergence of a type of magnetic monopole called the Yang monopole (known fromtheoretical high-energy physics) in a five-dimensional space of internal degrees of freedom in a rubidium Bose-Einsteincondensate. The results illustrate the potential of cold atoms physics to simulate high-energy phenomena.
Science, this issue p. 1429