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DOI: 10.1126/science.1157233

, 1326 (2008); 320Science

et al.P. Neumann,

Diamond

Multipartite Entanglement Among Single Spins in

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Multipartite Entanglement Among

Single Spins in Diamond

P. Neumann,1* N. Mizuochi,2* F. Rempp,1P. Hemmer,3H. Watanabe,4S. Yamasaki,5

V. Jacques,1T. Gaebel,1F. Jelezko,1J. Wrachtrup1†

Robust entanglement at room temperature is a necessary requirement for practical applications in

quantum technology. We demonstrate the creation of bipartite- and tripartite-entangled quantum states

in a small quantum register consisting of individual13C nuclei in a diamond lattice. Individual nuclear

spins are controlled via their hyperfine coupling to a single electron at a nitrogen-vacancy defect

center. Quantum correlations are of high quality and persist on a millisecond time scale even at

room temperature, which is adequate for sophisticated quantum operations.

S

more physical objects can be correlated, even

when separated. Since then, the generation and

retrieval of entanglement among several qubits

have become of fundamental importance in quan-

tumscienceandtechnology.Quantumteleportation

error correction, computation, and communication

all benefit from (or require) entanglement. One

current challenge for the field of quantum in-

formation processing has been to engineer a suf-

ficiently large and complex controllable system

in which questions related to entanglement can

be precisely explored. Hence, proving entangle-

ment among an increasing number of qubits is

typically a benchmark for physical systems, in

demonstrating their relevance to engineer quan-

tum states. On the level of single quantum sys-

tems, entanglement has been proven for photons

(1), ions (2, 3), atoms (4), and superconductors

(5). All solid-state qubit devices require low tem-

perature to achieve sufficiently long entanglement

lifetime. Whereas single electron spins can be

accessed by charge transport (6) or optically (7),

nuclei are more promising for quantum engineer-

ing because of their long coherence times, even

under ambient conditions. Because of their weak

interaction with the environment, nuclear spins

are not directly affected by (for example) lattice

phonons, which are a prominent source for de-

phasing in most solid-state systems. In ensemble

studies, (pseudo)entanglement among nuclei has

been demonstrated (8–10), but this has not been

confirmed for single nuclear spins so far.

For the present experiments,

coupled to a single nitrogen-vacancy (NV) defect

center in diamond (Fig. 1A) were chosen. This

chrödinger coined the term“entanglement”

to mean a peculiar mutual quantum inter-

action in which the properties of two or

13C nuclei

system allows for high-fidelity polarization and

detectionofsingleelectronandnuclearspinstates,

even under ambient conditions (11–16). The NV

center’s electron spin (S = 1) exhibits extraordi-

narily slow relaxation, with a longitudinal re-

laxation time T1(i.e., the time for spontaneous

transition between pure states) on the order of

milliseconds (17). The phase memory time T2

is found to be around 0.6 ms (18). Hence, this

defect has been identified as a prominent can-

didate for engineering quantum states and quan-

tum information processing (19–22), as well

as for high-resolution magnetometry (23). Scal-

ability toward larger-scale quantum registers

was proposed, for example, through optical in-

teractions between NV centers. Alternatively,

probabilistic entanglement based on photonic

channels can provide efficient scaling up (24).

The NV center is a point defect (Fig. 1A):

Thatis,theelectronspinismostlylocalizedatthe

defect site. However, about 11% of its electron

spin density is distributed over the nearest-

neighbor carbon atoms (25), mostly those at the

dangling bonds caused by the vacancy. As a

result, substantial hyperfine and dipolar coupling

are detectable for single nuclei localized close to

the defect (26, 27). Here, we use such coupling

(Fig. 1B) to effectively control two nuclear spins

on an individual basis, and by using this tech-

nique, we are able to demonstrate entanglement

of two13C nuclear spins (N1, N2) in the first

coordination shell of the vacancy. All four max-

imally entangled states, namely the Bell states

FT¼1

YT¼1

ffiffi

2

pðj00〉 T j11〉Þ

.

.

ð1Þ

ffiffi

2

pðj01〉 T j10〉Þð2Þ

are generated, where “0” and “1” denote the

two possible nuclear spin orientations (mI¼

−1=2↦ j0〉, mI¼ þ1=2↦ j1〉, jN1N2〉) (Fig. 1B).

13.Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring

57, D-70550 Stuttgart, Germany.

Library, Information and Media Studies, University of Tsukuba,

1-2 Kasuga, Tsukuba-City, Ibaraki 305-8550, Japan.3Depart-

ment of Electrical and Computer Engineering, Texas A&M

University, College Station, TX 77843, USA.

Research Center, National Institute of Advanced Industrial

Science and Technology (AIST), Tsukuba Central 2, Tsukuba,

305-8568, Japan.

Tsukuba Central 2, Tsukuba, 305-8568, Japan.

*These authors contributed equally to this work.

†To whom correspondence should be addressed. E-mail:

wrachtrup@physik.uni-stuttgart.de

2Graduate School of

4Diamond

5Nanotechnology Research Institute AIST,

Fig. 1. (A) Atomic structure of the NV center. The nitrogen atom, vacancy, and three nearest-

neighbor carbon atoms are shown. Two of the carbon atoms are13C isotopes with nuclear spin 1/2.

(B) Energy-level scheme of electronic ground state of the NV center. Two of three hyperfine split

electron spin sublevels (mS= 0,–1) are depicted. Allowed transitions are shown by solid arrows

(blue, electron spins; orange, nuclear spins). Zero and double quantum transitions are shown as

dashed arrows. For further information, see (15). (C) ODMR spectrum (red curve) showing the mS=

0↔–1 transition. The simulated spectrum (blue curve) accounts for the hyperfine splitting of a single

electron spin with two nearest-neighbor13C atoms. a.u., arbitrary units.

Table 1. Fidelity and entanglement measures for Bell, GHZ, and W states. Dashes indicate that the

respective coherence measures were not calculated.

Coherence measure →

Bell state ↓

Y+

0.80 ± 0.07

Y–

0.81 ± 0.06

F+

0.98 ± 0.05

F–

0.96 ± 0.05

GHZ

0.87 ± 0.06

W

0:85þ0:05

−0:1

FidelityConcurrencePartial transpose

0:65þ0:15

0.59 ± 0.11

0:96þ0:04

0:92þ0:07

—

—

−0:08

−0:31þ0:05

−0:32þ0:04

−0:49þ0:04

–0.47 ± 0.46

—

—

−0:06

−0:05

−0:02

−0:09

−0:08

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These entangled states cannot be achieved by in-

dependently bringing the spins into an individual

superposition state: Instead, they are global states

of the two spins. To demonstrate even tripartite en-

tanglement, we take the NV center’s electron spin

(E) into account and generate maximally entan-

gledstatessuchastheGreenberger-Horne-Zeilinger

(GHZ) states, as well as the so-called W state

GHZ ¼1

.

ffiffi

2

pðj000〉 þ eiϕj111〉Þ

.

ð3Þ

W ¼1

where ϕ and q are arbitrary phases. Again, “0”

and “1” denote the two possible spin states of

ffiffi

3

pðj110〉 þ eiϕj101〉 þ eiqj011〉Þð4Þ

all three spins (mS¼ 0↦j0〉, mS¼ −1↦j1〉,

jE N1N2〉).

The measured optically detected magnetic

resonance(ODMR)spectrumoftheelectronspin

of a single center is shown in Fig. 1C. Four lines

separated by hyperfine splittings are observed in

each mS= 0↔–1 and 0↔1 transition (15). The

Bell states F±and Y±are prepared as follows

(Fig.2A).Afterinitializationbyopticalpumping,

the system is set in the mS= 0 state. The nuclear

spin states, however, are undetermined. To ini-

tialize a specific nuclear spin starting state, we

applied a transition-selective microwave (MW)p

pulseandtransferredjE N1N2〉 ¼ j000〉 intoj100〉.

If the system was in j000〉 after the laser irra-

diation, a pure j00〉 (¼j100〉) state is created. If

the system was in another state after laser irra-

diation, a subsequent MWexcitation toj00〉 does

not occur, and no observable signal is visible. To

increasethesuccessrateofinitialization,onecan,

in principle, apply additional preparation steps

(27).

Generation of Bell states proceeds in two

steps. At first, a coherent superposition of states

j01〉 and j00〉 is generated with a p/2 pulse on

one radio frequency (rf) transition, which yields

1=

ffiffiffi

frequency-selective p pulse in resonance with

the j01〉 and j11〉 transition, eventually F−=

1=

ffiffiffi

2

pðj00〉 þ j01〉Þ. By the application of a

2

pðj00〉 − j11〉Þisformed.TheremainingBell

Fig. 2. (A) Pulse sequence for Bell state generation

(F–) among two nuclear spins. Spin-selective pulses

are represented by squares, operating on a target

qubit. Vertical lines represent logical connections.

The control qubit state j1〉 and the state j0〉 are

displayed as closed and open circles, respectively.

For example, an open circle indicates that the pulse

is applied to the target qubit if the quantum state of

the controlling qubit isj0〉. Bell states evolve during

time t followed by state tomography. During six

different transfer steps, the six possible coherences

among the four nuclear spin states are unitarily

mapped onto nuclear spin N1. Each coherence is

completely analyzed by performing two nutations

(nut.) with two 90° phase-shifted rf fields. The

results are the density matrix elements. (B) Ramsey

fringes of Bell states: (i) F+, (ii) F+and F–, (iii)

Y+, and (iv) Y+and Y–. The frequency offsets

of rf1 (Dw1), rf2 (Dw2), and fitted curve (wfit) are:

(i) Dw1= 0.5, Dw2= 0.5, and wfit= 1.0 MHz ≈

Dw1+Dw2;(ii)Dw1=0,Dw2=0.5,andwfit=0.5MHz≈

Dw1+Dw2; (iii) Dw1= –0.3, Dw2= 1.0, and wfit=

1.3 MHz ≈ Dw2−Dw1; and (iv)Dw1= 0, Dw2= 0.4,

and wfit= 0.4 MHz ≈ Dw2−Dw1. (C) Density matrix

reconstruction (real part) of states F–and Y+.

Imaginary parts are found in (15). The high abso-

lute values of the main off-diagonal elements

(F–:j00〉〈11j,j11〉〈00j;Y+:j10〉〈01j,j01〉〈10j)provethattheseare,infact,entangledstatesandnotjustincoherentmixtures(otherwise,alloff-diagonalelements

would be zero). (D) Coherence time of Bell states as compared with the relaxation time of the electronT1.

Fig. 3. (A) Pulse sequence for generation and detection of the GHZ state. The

preparation sequence is similar to that of Bell states, followed by a selective MW

pulse. For readout, a tomography is performed for the ideally nonzerodensitymatrix

elements. (B) Ramsey fringes of the GHZ coherence quantified by showing the

sums of the respective off-diagonal elements of the density matrix (r1,8+r8,1=

|000〉〈111|+|111〉〈000|).Thedecayisfittedtoexp[–(t/T2*)2],withT2*=1.3±0.2ms.

Error bars indicate the accuracy of the measured data points. (C) Tomogram of

GHZ state’s main density matrix entries. Unmeasured elements are discarded.

Thetop graph shows the densitymatrixforevolutiontimet =0ms,whereas the

bottom graph shows the decay of off-diagonal elements for t = 2.4 ms.

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states are prepared in a similar fashion (28).

Having arranged the Bell states, we proved their

generation via Ramsey fringes: that is, letting the

Bell state evolve freely for a certain time t and

analyzing its phase afterwards (15). Phases of

different Bell states evolve differently under

rotation around the quantization axis (28). F

and Y are characterized by an angular frequency

DwF,Y= Dw1+Dw2and Dw2–Dw1, respectively

(Fig.2B),whereDw1andDw2aredetuningsfrom

resonance of the two radio frequency fields rf1

and rf2 involved in generation and detection of

the respective Bell states. The “+” and “–” states

are distinguishable by their phase with respect to

the phase of the driving rf field (Fig. 2B).

A standard procedure to quantify the amount

of entanglement is density matrix tomography.

Bymeasuringtheprobabilitiestofindthenuclear

spins in a certain eigenstate and revealing co-

herent phases among them (e.g., j00〉 þ j11〉),

researchershaveperformeddensitymatrixtomog-

raphy for all Bell states (26) (Fig. 2A). Tomog-

raphy results are plotted in Fig. 2C. For brevity,

only the real part of F–and Y+is shown.

Concurrence and the negative eigenvalue of the

partial transpose were calculated (Table 1). To

compare the measured density matrix r with the

ideally expected one s, we estimated the fidelity

F={tr[(s1/2rs1/2)1/2]}2[F=tr(sr)forpurestates

s], where tr is the trace of a matrix. As demon-

strated in Table 1, especially for F±, the respec-

tive values get close to ideal. It has been shown

that a state r of two qubits is entangled if and

only if its partial transpose rpthas a negative

eigenvalue (29, 30). As shown in Table 1, this is

indeed the case for all four states. We also have

measured the coherence decay time T2of F and

Y (Fig. 2D). Both types of Bell states decay on a

time scale of 3 to 5 ms, which is similar to the

relaxation time of the electron spin (T1= 6 ms).

After entangling two nuclear spins, the next

step is to generate a tripartite entangled state via

the electron spin of the NV center itself as the

third qubit. The easiest tripartite entangled state

to generate is the GHZ state. For its generation,

we start off with a nuclear F±state and apply a

nuclear spin state–selective p pulse on the elec-

tron transition (Fig. 3A). For W state generation

and further details of state preparation and read-

out, see (15). Tomography of these tripartite

entangled states can be performed in a fashion

similar to that of the Bell states. Note that these

three spin coherences are directly observable in

our spin resonance experiment. In the present

work, we restricted our measurements to the

main (ideally nonzero) density matrix entries

of the respective entangled states (Figs. 3B and

4B). It can be shown that these elements are

sufficient to calculate the fidelity F = tr(sr)

(15) (Table 1).

Because, for both states, an electron spin is

now involved in the entangled states, one would

expect the coherences to decay much faster than

those where only nuclear spins were involved.

Indeed, by measuring Ramsey fringes of GHZ

coherence after a waiting time of 2 ms, we find

that only an incoherent mixture of j111〉 and

j000〉 isleft(Fig.3,BandC).TheRamseyfringe

decay time is determined by inhomogeneities

caused by slow spectral diffusion. Refocusing

such inhomogeneities or decoupling these inter-

actions will greatly lengthen the respective

coherence time. Interaction with other spin

impurities limits T1and T2of the NV center’s

electronspin[highestreportedvalues:T1=380s,

(31); T2= 350 ms, (18)]. Thus, the GHZ state

prepared here would be an even more robust

resource for quantum state engineering in a purer

lattice environment.

The aforementioned GHZ decoherence is

only expected to be due to the electron spin. This

is reflected in the dephasing behavior of the W

state. After the electron spin coherence has de-

cayed, the remaining W state is written as an in-

coherent mixture of states |0〉|11〉 and |1〉(|10〉 +

|01〉). Thus,for the electron spin measuredtobe

instatej1〉,nuclearspinentanglementisexpected

to be found, namely the Y+nuclear spin Bell

state. Observation of Ramsey fringes of all W

state coherences indeed shows the expected

behavior. The two coherences involving elec-

tron spin states decay within 2 ms, whereas the

coherence with mainly nuclear spin character

(j1〉jYþ〉) persists on this time scale (Fig. 4, A

and B). Nuclear spin coherence outlasts electron

spin decoherence.

Given the long decoherence times found for

nuclear spin entangled states, there is ample op-

portunity to create even higher entangled states:

for example, by including the nitrogen nuclear

spinthatispresentinthedefectbutwhichwasnot

usedhere.Large-scalenetworksmaybeachieved

by entangling distant defects through emitted

photons. Proposals similar to this have been pub-

lished (22). Finally, we note that GHZ-type spin

states are ideal candidates for quantum-improved

measurements.Inmagnetometryapplications,they

outperformmixedstatesinsignal-to-noiseratioby

afactorof

ffiffiffiffi

spins (32). Entangled spin states in a diamond-

based magnetometer thus might enhance sensitiv-

ityconsiderably.

N

p

,whereNisthenumberofentangled

References and Notes

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Fig. 4. (A) Ramsey fringes of all three coherences

[sum of the corresponding off-diagonal elements: for

example, r5,2+r2,5= j011〉〈110j þj 110〉〈011j]. The

respective off-diagonal elements are marked with a

correspondingly colored arrow in (B). Only coherence

between the two energy eigenstates containing the

same electron spin orientation (r3,2= j101〉〈110j,

r2,3 = j110〉〈101j) shows no decay during the

observation time. The others decay within 2 ms. The

decay is fitted to exp[–(t/T2*)2], with T2* = 1.2 ±

0.2 ms for the circles and T2* = 1.6 ± 0.3 ms for the

triangles. The straight line for the squares is a guide

to the eye. Error bars indicate the accuracy of the

measured data points. (B) Tomograms of the main

entries of the W state. The top and bottom graphs

display the density matrix right after preparation and

after 4.4 ms, respectively. Gray density matrix ele-

ments have not been measured.

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24. L. M. Duan, R. Raussendorf, Phys. Rev. Lett. 95, 080503

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33. We acknowledge financial support by the European

Union (QAP, EQUIND, and NEDQIT) and Deutsche

Forschungsgemeinschaft (SFB/TR21 and the JPN-GER

cooperative program). N.M. is supported by the

program KAKENHI (grant no. 18740175), and V.J. is

supported by the Humboldt Foundation. We thank

D. Twitchen from Element Six (UK) Ltd. for providing us

with ultrapure chemical vapor deposition diamond

samples.

Supporting Online Material

www.sciencemag.org/cgi/content/full/320/5881/1326/DC1

Materials and Methods

Figs. S1 to S4

Tables S1 to S3

References

3 March 2008; accepted 8 May 2008

10.1126/science.1157233

Strong Dissipation Inhibits Losses

and Induces Correlations in

Cold Molecular Gases

N. Syassen,1D. M. Bauer,1M. Lettner,1T. Volz,1* D. Dietze,1† J. J. García-Ripoll,1,2

J. I. Cirac,1G. Rempe,1S. Dürr1‡

Atomic quantum gases in the strong-correlation regime offer unique possibilities to explore a variety

of many-body quantum phenomena. Reaching this regime has usually required both strong elastic and

weak inelastic interactions because the latter produce losses. We show that strong inelastic collisions

can actually inhibit particle losses and drive a system into a strongly correlated regime. Studying

the dynamics of ultracold molecules in an optical lattice confined to one dimension, we show that

the particle loss rate is reduced by a factor of 10. Adding a lattice along the one dimension increases

the reduction to a factor of 2000. Our results open the possibility to observe exotic quantum

many-body phenomena with systems that suffer from strong inelastic collisions.

S

conductivity (1), excitations with fractional

statistic (2), topological quantum computation

(3),andaplethoraofexoticbehaviorsinmagnetic

systems (4). One of the main physical mecha-

nisms that gives rise to strong correlations for

bosonicparticlescanbeunderstoodasfollows.At

low temperatures and for strong elastic repulsive

interactions, particles tend to stay far away from

eachotherinordertokeeptheenergylow.Thatis,

thewavefunctiondescribingtheparticlestendsto

vanish when two of them coincide at the same

position. In order to fulfill these constraints, this

wave function has to be highly entangled at all

times, which may give rise to counterintuitive

effects both in the equilibrium properties as well

as in the dynamics. In one dimension, for exam-

ple, this occurs in the so-called Tonks-Girardeau

gas (TGG) (5, 6), where the set of allowed wave

functions for bosonic particles coincide (up to

some transformation) with those of free fermions.

trong interactions are responsible for many

interesting quantum phenomena in many-

body systems: high-temperature super-

Despitebeingbosons,theexcitationspectrum,the

evolution of the density distribution, etc. cor-

respond to those of fermionic particles. In two

dimensions,thesamemechanismleadstothefrac-

tional quantum Hall effect (7), where the ground

state as well as the low energy excitations fulfill

the above-mentioned constraint, giving rise to

the existence of anyons, which behave neither

like bosons nor fermions but have fractional sta-

tistics (2).

Weshowthatinelasticinteractionscanbeused

toreachthestrongcorrelationregimewithbosonic

particles: This may seem surprising because in-

elasticcollisionsaregenerallyassociatedwithpar-

ticle losses. This behavior can be understood by

using an analogy in classical optics, where light

absorption is expressed by an imaginary part of

the refractive index. If an electromagnetic wave

impinges perpendicularly on a surface between

twomediawithcomplexrefractiveindicesn1and

n2, then a fraction |(n1− n2)/(n1+ n2)|2of the

intensity will be reflected. In the limit |n2| → ∞,

the light is perfectly reflected off the surface, ir-

respective of whether n2is real or complex. In

our case, bosons interacting with large imaginary

(8,9)scatteringlengthalmostperfectlyreflectoff

each other for an analogous reason, thereby giving

rise to the same constraints in the particles' wave

function as the ones corresponding to elastic col-

lisions and thus to the same physical phenomena.

Inourexperiment,thecorrelationsmanifestthem-

selvesinastrongsuppressionoftheratesatwhich

particles are lost because of inelastic collisions.

Our experiment used molecules confined to

onedimensionbyanopticallattice,bothwithand

withouta periodic potentialalongtheonedimen-

sion. We started with the transfer of a Bose-

Einstein condensate (BEC) of87Rb atoms into a

three-dimensional (3D) optical lattice in such a

way that the central region of the resulting Mott

insulator contains exactly two atoms at each lat-

tice site. A Feshbach resonance at 1007.4 G (10)

wasusedtoassociatetheatompairsto molecules

(11).Subsequently,themagneticfieldwasheldat

1005.5 G. Atoms remaining after the association

were removed with blast light. This procedure

prepared a quantum state that contains one mol-

ecule at each site of a 3D optical lattice (12, 13).

Theoptical-latticepotentialseenbyamoleculeis

−V⊥cos2(kx)−V⊥cos2(ky)−V||cos2(kz),wherel =

2p/k=830.440nmisthelightwavelength.Atthe

end of the state preparation, V||is equal to V⊥,

1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-

Straße 1, 85748 Garching, Germany.2Universidad Complu-

tense, Facultad de Físicas, Ciudad Universitaria s/n, 28040

Madrid, Spain.

*Present address: Institute of Quantum Electronics, Eidgenös-

sischeTechnischeHochschule(ETH)–Hönggerberg,8093Zürich,

Switzerland.

†Present address: Institut für Photonik, Technische Uni-

versität Wien, Gußhausstraße 25-29, 1040 Wien, Austria.

‡To whom correspondence should be addressed. E-mail:

stephan.duerr@mpq.mpg.de

Fig. 1. Time-resolved

lossofmoleculesatV||=

0. The loss begins at t =

0. The solid line shows a

fit of Eq. 2 to the exper-

imental data (•) with t ≤

1 ms. The best-fit value

iscn3(0)=4.3/ms,corre-

sponding to K3D= 2.2 ×

10−10cm3/s and, at t =

0, to g(2)= 0.11. The

dashedlineshowstheex-

pectationforanuncorre-

latedsystem.Theobserved

loss is much slower than

the dashed line because

of strong correlations.

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