Experimental observation of topological transitions in interacting multi-spin systems

5 MB·

by  Zhihuang Luo

Experimental observation of topological transitions in interacting multi-spin systems
Zhihuang Luo 1, Chao Lei 1, Jun Li 2, Xinfang Nie 1, Zhaokai Li 1, Xinhua Peng 1,3,∗ and Jiangfeng Du 1,3†
1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
2Beijing Computational Science Research Center, Beijing, 100094, China and
3Synergetic Innovation Center of Quantum Information & Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Topologicallyorderedphasehasemergedasoneofmostexcitingconceptsthatnotonlybroadens
ourunderstandingofphasesofmatter,butalsohasbeenfoundtohavepotentialapplicationinfault-
tolerantquantumcomputation. Thedirectmeasurementoftopologicalproperties,however,isstilla
challengeespeciallyininteractingquantumsystem. Herewerealizeone-dimensionalHeisenbergspin
chainsusingnuclearmagneticresonancesimulatorsandobservetheinteraction-inducedtopological
transitions, where Berry curvature in the parameter space of Hamiltonian is probed by means of
6
dynamical response and then the first Chern number is extracted by integrating the curvature
1
0 over the closed surface. The utilized experimental method provides a powerful means to explore
2 topological phenomena in quantum systems with many-body interactions.

n
PACSnumbers: 03.67.Ac,03.65.Vf,76.60.-k
a
J

3 Since the first observation of topologically ordered demonstratedinsmall(one-ortwo-qubit)superconduct-
2
phases in quantum Hall effect in 1980s [1, 2], there is ingsystems[25,26],andalsoanexperimentalschemewas

growing interest in studying the topology of quantum proposed to simulate dynamical quantum Hall effect in
]
h systems such as topological insulators [3–7] and spin liq- Heisenberg spin chain with interacting superconducting
p
uids[8–11]. Meanwhile,thegreateffortsinfault-tolerant qubits [27].
-
t quantumcomputationarebeingmadeonthebasisofthe In this Letter, we use several nuclear spins to simulate
n
existence of topologically ordered phases [12–14]. The one-dimensionalHeisenbergspinchainthatwasproposed
a
u different topological phases and their topological transi- in Ref.[24] to have the quantization of first Chern num-
q tions are characterized by robust topological invariants berindynamicalresponse. Theemergentdifferentquan-
[ in physics, most of which arise as integrals of some ge- tized plateaus are related to different topological phases

1 ometric quantity. For example, the first Chern number and the interaction-induced topological transitions are
v [15], the integral of Berry curvature over the closed sur- observed in nuclear magnetic resonance (NMR) systems.
7 face of parameter space of Hamiltonian, is a well-defined In the experiments, we measure the total magnetization
4
topological invariant. It is closely related to Berry phase vectors perpendicular to quench velocity by decoupling.
2
[16, 17]. As emerged in quantum Hall physics, the fill- TheBerrycurvaturesinparameterspaceofHamiltonian
6
0 ing factor known as first Chern number in mathemat- are extracted via the linear response and then the first
. ics is used to distinguish different quantum Hall states Chern numbers are obtained by integrating the closed
1
[18, 19]. When the jumps of their values happen, it un- surface. From the resulting Chern number, we can visu-
0
6 dergos quantum Hall transitions. Topological invariant alizethegeometricstructureofHamiltonian. Theprecise
1 reveals the global properties of topological phases and quantization of first Chern number may be applied to
v: remains unchanged under small perturbations. parameter estimation of Hamiltonian. The full control-
i lability of NMR will make it possible to experimentally
X
However, it is still an experimental challenge to di- investigate many-body phenomena such as the even-odd
r rectly probe the topological properties especially in in- effect of Heisenberg chains [28, 29].
a
teracting quantum systems. Usually, the previous mea- The first Chern number is defined as the integral of
surement of Berry phase relies on the interference ex- Berry curvature over a closed manifold in the pa-
µν
periment [20–23], but this method is limited to systems rameter space R(cid:126)Fof Hamiltonian as [17, 24] S
of weakly interacting quasi-particles. Beyond this limit,
Gritsev et al. [24] proposed an alternative method to di- 1 (cid:73)
h = dS , (1)
rectly measure the Berry curvature via the nonadiabatic C 1 2π µνFµν
S
response on physical observables to the rate of change
of an external parameter. Direct measurement of Berry where =∂ ∂ , and =i ψ ∂ ψ . Here
µν µ ν ν µ µ 0 µ 0
F A − A A (cid:104) | | (cid:105)
curvature provides a powerful and generalizable means weusetheshortennotations,i.e.,∂ ∂ andconsider
µ ≡ Rµ
to explore topological properties in any quantum sys- the ground state ψ . Berry connection and Berry
0 µ
| (cid:105) A
tems where the Hamiltonian can be written in terms of curvature can be viewed as a local gauge potential
µν
F
a set of parameter. Based on this method, some exper- and gauge field, respectively. In analogy to electrody-
imental observations of topological transition have been namics, the local gauge-dependent Berry connection can
hz hz
hx hy hx hy
hz hz hz
hx hy hx hy hx hy 2
never be physically observable. Whereas Berry curva- (a) (b)
8
ture is gauge-invariant and may be related to physical Linearzone Fφθνθ
observable that manifests the local geometric property 1
of the ground state in the parameter space. While the φ
M
firstChernnumbershowstheglobaltopologicalproperty 0.1
of the ground state manifold as a whole. In fact, h
1
C
exactly counts the number of degenerate ground states
enclosed by parameter space . To see this point more 0.1 ν 1 10 100
S θ
intuitively, we substitute into and rewrite the
µ µν
A F
Berry curvature as follows [17], FIG.1. (Coloronline)(a)Thequasiadiabaticevolutionpath
(blue) in the spherical parameter space of external magnetic
(cid:88) ψ0 ∂µ ˆ ψn ψn ∂ν ˆ ψ0 (ν µ) field (cid:126)h (|(cid:126)h| = 1). (b) The magnetization as the function of
=i (cid:104) | H| (cid:105)(cid:104) | H| (cid:105)− ↔ . (2)
Fµν n(cid:54)=0 (εn−ε0)2 vdθasfhoerdtwbloa-cqkubliintec),asMe. I=n tFhevlineaanrdzvomneax(=i.e.1,.5a3t.thTehelefhtigohf
φ φθ θ θ
order terms of v dominate at the right of dashed black line.
θ
Here ε and ψ are the n eigenvalue and its cor-
n n th
responding eig|ens(cid:105)tate of Hamiltonian ˆ, respectively.
H
From Eq. (2), it clearly shows that degeneracies (i.e., velocity is turned on smoothly and the system is not ex-
εn = ε0) are some singular points that will contribute cited at the beginning of the evolution. The generalized
nonzero terms to the integral of , that is, Eq. (1). force at t=π/v is easily derived as
Fµν θ
These degeneracy points act as the sources of h and
1
C N
areanalogoustomagneticmonopolesinparameterspace. (cid:88)
= ∂ ˆ = σˆj . (5)
The first Chern number is essential for understanding of Mφ −(cid:104) φH(cid:105)|φ=0,t=π/vθ (cid:104) y(cid:105)
j=1
the quantized effect. It can be used as the nontrivial or-
derparametertocharacterizedifferenttopologicalphases That is equivalent to total magnetization along y direc-
and their transitions [17]. tion. Figure 1(b) shows the limitation of v calculated
θ
The usual interference experiments for measuring intwo-qubitcase, withsimilarresultsinthree-andfour-
Berry phase [20–23] do not readily generalize to more qubit cases. In the linear zone of v 1.53, M v
θ y θ
≤ ∝
complicated interacting quantum systems. We follow an and the linear response approximation works well. Then
alternative approach as proposed in Ref. [24]. It states Berry curvature can be obtained from Eq. (3). By inte-
that Berry curvature can be extracted from the linear grating over the sphere of (cid:126)h = 1, we get the first
φθ
F | |
response of generalized force µ along the µ-direction, Chern number. Due to the rotational invariance of the
M
i.e., interaction in Eq. (4), the integration will be simplified
into the multiplication of by the spherical area 4π
=const+ v + (v2), (3) Fφθ
µ µν ν [24]. So we have h =2 from Eq. (1).
M F O 1 φθ
C F
The Heisenberg model can be effectively simulated by
where = ψ (t )∂ ˆ ψ (t ) ,andv isthequench
Mµ −(cid:104) 0 f | µH| 0 f (cid:105) ν NMRsystem,whosenaturalHamiltonianintherotating
velocity. To neglect the nonlinear term, the chosen v
ν frame is
should be small enough or quasiadiabatic.
The one-dimensional Heisenberg spin chain can be ˆ =(cid:88)N ωiσˆz+ (cid:88)N πJijσˆzσˆz, (6)
taken as the example to demonstrate the above idea HNMR 2 i 2 i j
[24,27],whoseHamiltonianinanexternalmagneticfield i=1 i<j,=1
(cid:126)h is described by where ω represents the chemical shift of spin i and J
i ij
the coupling constant between spin i and spin j. Com-
N N−1
(cid:88) (cid:88) pared to Eq. (4), the Hamiltonian of NMR system has
ˆ = (cid:126)h (cid:126)σ J (cid:126)σ (cid:126)σ , (4)
H − · − j · j+1 the similar form. It is suitable for a NMR system to
j=1 j=1
simulate the Heisenberg model [31]. According to aver-
where (cid:126)σ (σˆ ,σˆ ,σˆ ) stands for Pauli matrices, and J age Hamiltonian theory [32], we can design NMR pulse
x y z
istheisot≡ropiccouplinginteractionstrengthbetweenthe sequences to effectively create the desired Hamiltonian.
nearest-neighbor spins. In order to measure Berry cur- The pulse sequences are shown in Fig. 2(a). Using Trot-
vature, let the system start with the initial ground state ter approximation, we have in a short period of τ,
at the north pole of spherical parameter space of exter-
nal magnetic field (cid:126)h (here we fixed (cid:126)h = 1), and then e−iHˆτ =Rˆtyol(θn)e−i(Hˆz+Hˆzz)τ/2e−i(Hˆxx+Hˆyy)τ (7)
undergo a quasiadiabatic evolution a|lon|g the blue path, e−i(Hˆz+Hˆzz)τ/2Rˆy ( θ )+O(τ3),
· tol − n
as illustrated in Fig. 1(a). The path is determined by
fixing φ = 0 and varing θ(t) = v2t2/2π from t = 0 to where ˆ = (cid:80)N (cid:126)hσˆz and ˆ = J(cid:80)N−1σˆασˆα
θ Hz − j=1| | j Hαα − j=1 j j+1
t = π/v [24]. This choice guarantees that the angular (α = x,y,z). During τ/2, i.e., at the front and back
θ
3
(a) (b)
8
Linearzone M?
1 F?383
M?
0.1
0.1 1 10 100 3
8
3
loopn=1,2,...,300 (a) (b)
τ/2 τ τ τ/2
4 Ground state
1 zH First excited state
6 2
2... yˆR(θ)ntol− yˆR(π/2)tol yˆR(π/2)tol− yˆR(π/2)tol− yˆR(π/2)tol yˆR(θ)ntol .412 ygrenE--420
N -1.5 -1 -0.5 0 0.5
J
ωi=−2|~h| ωi=0 fori=1,2,...,N ωi=0 ωi=−2|~h| (c) (d) 1 Theoretical hz
Experimental
0.8
FIG. 2. (Color online) The pulse sequence for simulating the
one-dimensional Heisenberg model of Eq. (4). The gray re- F?300..46 hz hx hy
gions represent the free evolution of isotropic z-coupling in-
tffereerreqanuctteinorcnoit,easit.ioenf.g,RHfˆFrazzpmue=lssebs−yaJcm(cid:80)tienNjag=n−1so1nσoˆfjzNσˆsjezd+tit1ffi.negreTnthhteenyuocafflr-eerie,siwnonhdaicicfhe- Frequence(Hz) 0.02-1h.5x -1 hy-0.5 0 0.5
J
satisfyω =−2|(cid:126)h|duringtheτ/2andω =0duringtheτ for
i i
i=1,2,··· ,N, respectively. Rˆtαol(θn)=(cid:81)Nj=1e−iθnσˆjα/2(α= FIG.3. (Coloronline)(a)ThemolecularstructureofChloro-
x,y)andθ =v2t2/2πforn=1,2,··· ,300. Theloopisused form. The RF pulses act on 13C and1H nuclei independently
n θ n
to approximate the quasiadiabatic evolution path. to to fulfill the desired control tasks. The coupling constant
between two nuclei is J = 214.6 Hz. (b) The energy-level
CH
diagram of N =2 in Heisenberg spin chain. (c) The sum ex-
perimental13Cspectraobtainedbydecouplingtheotherspin
gray regions in Fig. 2, the off-resonance frequencies of 1Handswapping1Htotheobservednucleus13C. Thelongi-
radio-frequency (RF) pulses acting on N different nuclei tudinalaxisistheintegrationoftheresonantpeakoftheex-
are set to satisfy ω = 2(cid:126)h for i = 1,2, ,N. In this perimental spectra, which stands for the total magnetization
i
rotatingframe, ˆ istu−rne|d|onduringthe··f·reeevolution along y direction. (d) The Berry curvature as a function of
z
of ˆ . When ωH = 0 for i = 1,2, ,N, ˆ is turned interactionstrengthJ. Thebluecirclesandreddiamondsrep-
off.Hzˆz /ˆ cain be readily realiz·e·d· by rHotzating ˆ . resent the theoretical and experimental values, respectively.
To imHpxlxemHenyyt ˆzz, it only requires to refocus someHuznz- eCmh1e(r=gin2gFiφnθ)(cid:126)hcpoaurnatmtehteerdsepgaenceer(abciigesan(sdmyaelllloawndsprheedressp)h.eTrehse)
H
wanted interactions in natural Hamiltonian (6) and tune experimental average values of different quantized plateaus
J s into an isotropic coupling constant. This task can are F =−0.0034±0.019 and F =0.99±0.015.
ij θφ θφ
be implemented only using some refocusing π pulses. As
showninAppendixB,twoexamplesofN =3andN =4
were given. For N =2, ˆzz is the natural interaction of havenothingtodowithN. Themethodsutilizedwillbe
H
NMR system. We will observe topological transitions in still valid when extended to larger quantum systems.
two-, three- and four-spin interacting systems as follows. TheexperimentswerecarriedoutonaBrukerAdvance
In the experiments, we need to measure the total III 400 MHz (9.4 T) spectrometer at temperature 303 K
magnetization. However, it will become a challenge . We first present two-spin experiment using the sample
when considering the multi-spin systems with interac- ofthe13C-labeledchloroform,whosemolecularstructure
tion. Within the linear zone of v 1.53, the generated is illustrated in Fig. 3(a). The coupling interaction be-
θ
M under the quasiadiabatic evolu≤tion is small and will tween 13C and 1H nuclei is ˆ = πJ σˆzσˆz, where
y Hzz 2 CH C H
tend to zero in the adiabatic limit, i.e., v 0. In ad- J =214.6Hz. The Heisenberg spin model of N =2 is
θ CH
→
dition, one can only measure the magnetization of each simulatedverywellintheshortperiodbyusingthepulse
spinanditsNMRsignalwillfurthersplitinto2N−1peaks sequenceinFig. 2. Itssimulatedfidelitycanbeachieved
induced by the interactions of Hamiltonian. Therefore, over0.99evenifthatallpulsesareconsideredtherandom
the direct observable or the value of each peak will be errors in the range of 5 degree. The quasiadiabatic evo-
v /N2N−1. For example, the direct observation for lution path was approximated by n=300 discrete steps
θ
∝
N =4willbe32timesaslittleasthatforN =1without with reliable accuracy. However, after the whole loop
interaction. Asthesizeofinteractingsystemincreases,it the experimental error will accumulate a lot even if the
requireshighermeasurementaccuracy. Toenhanceit,we pulse errors are very small. To overcome this, we packed
employedthedecouplingdetectionthatcancelsthefactor theloopsequenceintooneshapedpulsecalculatedbythe
of 2N−1 induced by interactions. Moreover, we swaped gradientascentpulseengineering(GRAPE)method[33],
all other nuclei to an observable nucleus, sum all exper- withthepulselengthof8ms. Theinitialgroundstateat
imental decoupling spectra, and measured the combined thenorthpolealsowaspreparedbyaGRAPEpulsewith
signal once, which avoids the error caused by multiple pulse length of 5 ms, from the pseudo-pure state (PPS):
readout. Therefore, the final measurement values ( v ) ρˆ = 1−(cid:15)I+(cid:15)00 00,withIrepresentingthe4 4iden-
∝ θ 00 4 | (cid:105)(cid:104) | ×
4
(a) (b)
2
la1
rgetn0
I-1
-1.5 -1 -0.J5 0 0.5 Frequence(Hz) Frequence(Hz)
(c) (d)
1.5 2
Theoretical Theoretical
Experimental hz Experimental hz
1.5
hx hy hx hy
F?3 1 hz F?3 1 hz hz
0.5
hx hy hx hyhx hy
0.5 0
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
J J 4
tityoperatorand(cid:15) 10−5 thepolarization. PPSρˆ was (a) (b)
00
≈ 10
prepared using line-selective approach [34], by which the 6 Ground state Ground state
signal strength is larger than that by the spatial average 4 First excited state 5 First excited state
mtieesthoovder[3959]%. ,TahnedGwReArePEdepsiuglnseesdhtaodbteheroorbeutsictaalgfiadinelsit- ygrenE-202 ygrenE 0
the inhomogeneity of RF pulses. The sum experimental -4 -5
-6
decouplingspectraformeasuringtotalmagnetizationare -10
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
illustrated in Fig. 3(c). According to the linear response J J
(c) (d)
theory, wecanobtaintheresultsof andfurther h ,
as shown in Fig. 3(d). Figure 3(b)Fdeφpθicted the eneCrgy1- 1.5 TEhxpeoerreimticeanltal hhzzhhzz hz 2 TEhxpeoerreimticeanltal hhzzhhzz hz
1.5
level diagram of N = 2 in Heisenberg spin chain. The
hhxxhhxx hhyhyhhxyy hy hhxxhhxx hhyhyhhxyy hy
energy-level crossing between the ground state and first F?3 1 hhzzhhzz hz Fh?3hzzhhzz1 hz hhzzhhzz hz
excited state exactly corresponds to the jumping point
of . It can be seen that the quantized plateaus char- hhxxhhxx hhyhyhxhyy hy hhxxhhxx 0.5hhyhyhhxyy hhyhxxhhxx hhyhyhhxyy hy
φθ
actFerized the interaction-induced topological transition. 0.5 0
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
ThefirstChernnumberrevealsthenumberofenergyde- J J
generacies emerging in closed manifold of (cid:126)h parameter
FIG. 4. (Color online) (a)(b) The energy-level diagrams
space.
of Heisenberg spin model for N = 3 and N = 4, respec-
We now turn to three- and four-spin experiments per- tively. (c)(d) The Berry curvatures as a function of inter-
formed on the samples of diethyl-fluoromalonate and action strength J in the three- and four-qubit experiments
iodotrifluoroethylene (see Appendix A). Using the same respectively. The blue circles and red diamonds represent
the theoretical and experimental values, respectively. The
methods in 2-spin experiment, we measured the total
average values of different plateaus are F = 0.48±0.029
magnetization obtained by integrating the sum exper- θφ
and F = 1.49±0.027 in (c), and F = −0.011±0.034,
imental decoupling spectra (see Appendix B) and ex- θφ θφ
F = 1.033±0.032, and F = 1.99±0.024 in (d), respec-
θφ θφ
tracted the Berry curvatures shown in Figs. 4(c) and tively.
4(d), respectively for N = 3 and N = 4. Note that
the plateaus in 3-qubit experiment start with nonzero
Berry curvature, which is different from the even-spin
in the total simulation error. Therefore, it is necessary
results. It reflects that there are different degeneracies
to prepare high fidelity PPS ρˆ in our experiments.
0
ofgroundstatesinodd-spinandeven-spinantiferromag-
In conclusion, we realized one-dimensional Heisenberg
neticHeisenbergchains[28,29]. Moreover,thegeometric
spin model using the interacting nuclear spins and ob-
structure of Hamiltonian can be visualized from the ex-
served the interaction-induced topological transitions in
perimental results. That means, without the calculation
NMR systems. The topological properties of the ground
of Hamiltonian, one can foresee where the level crossings
states were analyzed by measuring the Berry curvature
betweenthegroundstateandfirstexcitedstatewillhap-
and hence the first Chern number. The experimental
pen, as illustrated in Figs. 4(a) and 4(b), and how many
method utilized for measuring Berry curvature can be
degeneraciesthereareinsidetheclosedmanifoldin(cid:126)hpa-
used in a variety of generic quantum systems. From the
rameter space. The first Chern number can be used as
resultingBerrycurvatureorfirstChernnumber,onecan
nontrivialorderparametertocharacterizedifferenttopo-
get the geometric information of Hamiltonian about the
logical phases and their topological transitions.
degeneracies. For instance, the different degeneracies of
Theseexperimentalresultsareingoodagreementwith ground states in odd-spin and even-spin antiferromag-
theoretical expectations. The relatively minor devia- netic Heisenberg chains were observed in Figs. 4 (c) and
tions can be attributed mostly to the imperfections of (d). The quantized plateaus can be applied for precise
the preparation of PPS ρˆ00 and the spectral integrals. measurement of the parameter of Hamiltonian. Com-
Wecalculatedthestandarddeviationsofexperimentand pared to other platforms such as superconducting sys-
(cid:113)
simulation via σ = (cid:80)M (xi xi )2/M. The
i=1 Exp/Sim− Th
results are listed in Tab. I. The readout error can be
estimated by σRead = σExp −σSTioml, which mainly came TABLEI.Thestandarddeviationsofexperiment,simulation
from spectral integrals. From the two columns of σ and readout. The total simulation includes PPS ρˆ , ground
Exp 0
and σTol, it shows that the controllability became worse state (GS) preparations and quasiadiabatic evolution.
Sim
as the number of qubits increases. We numerically simu-
Qubit σ σTol σPPS σGS σEvol σ
latedtheerrorscausedseparatelybythePPSρˆ ,ground Exp Sim Sim Sim Sim Read
0
state preparations and quasiadiabatic evolution, respec- 2 0.0171 0.0052 0.0049 0.0046 0.0007 0.0119
tively. FromthemiddleresultsinTab. I,wefindthatthe 3 0.0283 0.0173 0.0159 0.0049 0.0020 0.0110
4 0.0368 0.0243 0.0235 0.0098 0.0070 0.0125
imperfectionofPPSρˆ preparationplaystheleadingrole
0
5
tems, NMR systems have the notably advantage in con- Rev. A. 87, 060303 (2013).
trollability and measurement accuracy, which will pro- [24] V. Gritsev and A. Polkovnikov, Proc. Natl. Acad. Sci.
videabettertestingplatformtoexploretopologicalphe- USA 109, 6457 (2012).
[25] M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M.
nomena in more complex quantum systems with inter-
Sandberg, J. Gao, M. R. Vissers, D. P. Pappas, A.
actions. Actually, it is possible for NMR systems with
Polkovnikov, and K. W. Lehnert, Phys. Rev. Lett. 113,
strong coupling interactions, e.g., in low magnetic field
050402 (2014).
(<10−3 T)[36], to observe natural topological phases. [26] P. Roushan et al., Nature (London) 515, 241 (2014).
This work is supported by NKBRP(2013CB921800 [27] X. C. Yang, D. W. Zhang, P. Xu, Y. Yu, and S. L. Zhu,
and 2014CB848700), the National Science Fund for Dis- Phys. Rev. A. 91, 022303 (2015).
[28] P. Politi, and M. G. Pini, Phys. Rev. B. 79, 012405
tinguished Young Scholars (11425523), NSFC(11375167,
(2009).
11227901and91021005),theStrategicPriorityResearch
[29] S. Oh, M. Friesen, and X. D. Hu, Phys. Rev. B. 82,
Program (B) of the CAS(XDB01030400), and RFDP
140403 (2010).
(20113402110044). [30] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
[31] X. H. Peng, J. F. Du, and D. Suter, Phys. Rev. A. 71,
012307 (2005)
[32] U.HaeberlenandJ.Waugh,Phys.Rev.175,453(1968).
[33] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrggen,
∗ [email protected] and S. J. Glaser, J. Magn. Reson. 172, 296 (2005).
† [email protected] [34] X. Peng, X. Zhu, X. Fang, M. Feng, K. Gao, X. Yang,
[1] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. and M. Liu, Chem. Phys. Lett. 340, 509 (2001).
Lett. 45, 494 (1980). [35] D. G. Cory, A. F. Fahmy, and T. F. Havel, Proc. Natl.
[2] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Acad. Sci. 94, 1634 (1997).
Rev. Lett. 48, 1559 (1982); R. B. Laughlin, ibid. 50, [36] S. Appelt, F. W. Hasing, U. Sieling, A. Gordji-Nejad,
1395 (1983). S. Gloggler, and B. Blumich, Phys. Rev. A. 81, 023420
[3] B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, (2010).
106802 (2006).
[4] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98,
106803 (2007).
APPENDIX
[5] M.Knig,S.Wiedmann,C.Brne,A.Roth,H.Buhmann,
L.W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science
318, 766 (2007). A. Three- and four-qubit quantum registers
[6] D.Hsieh,D.Qian,L.Wray,Y.Xia,Y.S.Hor,R.J.Cava,
and M. Z. Hasan, Nature (London) 452, 970 (2008). We selected the 1H,13C, and 19F nuclear spins of
[7] M. Levin and A. Stern, Phys. Rev. Lett. 103, 196803
Diethy-fluoromalonate and one 13C and three19F of
(2009).
[8] V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, Iodotrifluoroethylene as the three-qubit and four-qubit
2095 (1987). registers, whose molecular structures and relevant pa-
[9] X.-G. Wen, Phys. Rev. B 44, 2664 (1991). rameters are listed in Fig. 5(a) and (c), respectively.
[10] R.MoessnerandS.L.Sondhi,Phys.Rev.Lett.86,1881 Because both above samples we used are unlabeled, the
(2001). molecules with a 13C nucleus (as the quantum registers)
[11] L. Balents, Nature (London) 464, 199 (2010).
werepresentataconcentrationofabout1%. The1Hand
[12] A. Kitaev, Ann. Phys. (N.Y.) 303, 2 (2003).
19Fspectraweredominatedbysignalsfromthemolecules
[13] C. Nayak, A. Stern, M. Freedman, and S. Das Sarma,
Rev. Mod. Phys. 80, 1083 (2008). containing the 12C isotope. To effectively separate this
[14] A. Stern and N. H. Lindner, Science 339, 1179 (2013). signal from that of the dominant background, it requires
[15] S.-s. Chern, Ann. Math. 47, 85 (1946). to transfer the state of the 1H and 19F qubits to the 13C
[16] M.V. Berry, Proc. R. Soc. A 329, 45 (1984). qubit by a SWAP gate and read the state through the
[17] D.Xiao,M.C.Chang,andQ.Niu,Rev.Mod.Phys.82, 13C spectrum. Figure 5(b) and (d) are their correspond-
1959 (2010).
ing 13C equilibrium spectra.
[18] Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
[19] Avron, Joseph E.; Osadchy, Daniel, Seiler, Ruedi,
Physics Today 56 (8): 38. (2003).
[20] X. Peng, S. Wu, J. Li, D. Suter, and J. Du, Phys. Rev. B. The pulse sequences of generating isotropic
Lett. 105, 240405 (2010). z-coupling interaction and the sum experimental
[21] P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. decoupling spectra for N =3 and N =4
Gppl, J. M. Gambetta, D. I. Schuster, L. Frunzio, R.
J.Schoelkopf,andA.Wallraff,Science318,1889(2007).
In a short period of τ, we can effectively cre-
[22] S. Berger, M. Pechal, S. Pugnetti, A. A. Abdumalikov,
ate the isotropic z-coupling interactions, i.e., ˆ =
L. Steffen, A. Fedorov, A.Wallraff, and S. Filipp, Phys. J(cid:80)N−1σˆzσˆz only using some refocusing πHpuzzlses.
Rev. B. 85, 220502 (2012). − j=1 j j+1
[23] S. Berger, M. Pechal, A. A. Abdumalikov, C. Eichler, Figures 6(a) and 6(b) show two examples of pulse se-
L. Steffen, A. Fedorov, A. Wallraff, and S. Filipp, Phys. quences for generating ˆ of N = 3 and N = 4, re-
zz
H
6
(a) 13C (c) 19F312C 19F1 (a) τ (b) τ
13C 15479.2  13C 7.9 13C
1H 2032.0 19F 1H 1.2 19F1 -297.6 -33125.0 4.4 19F
1139CF 14617H0.9.4 8-151953C46..49 -73198F53 T112..(13s) 1199FF23 -231793C5.1.7 651914F..461 -4-121926F982.01.0 -5611949F7F332.2 T642..(88s) 13C 19F2
(b) (d) 19F
3
1H
τ1 τ1+τ2 τ2 19F1
τ τ τ+ττ+τ τ τ
1 2 2 31 3 3 3
FIG. 5. (Color online) The molecular structures and pa-
rameters of (a) diethyl-fuoromalonate, where three qubits FIG. 6. (Color online) (a)(b) are the 3-qubit and 4-qubit
are labeled as 1H,13C and 19F and (c) iodotrifluoroethylene, pulsesequencesforeffectivelycreatingtheisotropicz-coupling
where four qubits are labeled as 13C,19F1,19F2 and 19F3. interactons, i.e., Hˆzz = −J(cid:80)jN=−11σˆjzσˆjz+1, respectively. The
Thechemicalshiftsandscalarcouplingconstants(inHz)are red rectangles represent π pulses. In (a), τ =J τ/[2(J −
1 12 12
givenasthediagonalandoff-diagonalelementsintwotables, J )] and τ = −J τ/[2(J −J )]. In (b), τ = J (J +
23 2 23 12 23 1 23 12
respectively. Thelastcolumnshowsthetransversalrelaxation J )τ/[4J (J −J )],τ =J (J −J )τ/[4J (J −J )],
34 12 23 34 2 23 12 34 12 23 34
time T of each nucleus. Due to the interactions, their corre- and τ =−J τ/[4(J −J )]
2 3 34 23 34
sponding13Cequilibriumspectraof(b)and(d)weresplitted
into 2N−1 (i.e.,4 and 8) peaks, respectively.
FIG.7. (Coloronline)(a)(b)Thesumexperimental13Cspec-
tra obtained by decoupling and swapping all other nuclei to
the observable nucleus 13C, respectively for three- and four-
qubit cases.
spectively. Thesumexperimentaldecouplingspectraare
illustratedinFig. 7. Theintegrationoftheresonantpeak
oftheexperimentalspectrastandsforthetotalmagneti-
zation along y direction. The experimental results show
that there are well precise plateaus that reflect the hap-
pening of interaction-induced topological transitions.