, 1326 (2008); 320Science
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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.
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-
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
system allows for high-fidelity polarization and
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):
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
pðj00〉 T j11〉Þ
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,
Tsukuba Central 2, Tsukuba, 305-8568, Japan.
*These authors contributed equally to this work.
†To whom correspondence should be addressed. E-mail:
2Graduate School of
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 ↓
0.80 ± 0.07
0.81 ± 0.06
0.98 ± 0.05
0.96 ± 0.05
0.87 ± 0.06
0.59 ± 0.11
–0.47 ± 0.46
6 JUNE 2008VOL 320
on November 3, 2008
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-
(GHZ) states, as well as the so-called W state
pðj000〉 þ eiϕj111〉Þ
where ϕ and q are arbitrary phases. Again, “0”
and “1” denote the two possible spin states of
pðj110〉 þ eiϕj101〉 þ eiqj011〉Þð4Þ
all three spins (mS¼ 0↦j0〉, mS¼ −1↦j1〉,
The measured optically detected magnetic
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
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
in principle, apply additional preparation steps
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
frequency-selective p pulse in resonance with
the j01〉 and j11〉 transition, eventually F−=
pðj00〉 þ j01〉Þ. By the application of a
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; (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
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=
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.
VOL 3206 JUNE 2008
on November 3, 2008
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
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.
spins in a certain eigenstate and revealing co-
herent phases among them (e.g., j00〉 þ j11〉),
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
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
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
(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
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
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
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
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
spins (32). Entangled spin states in a diamond-
based magnetometer thus might enhance sensitiv-
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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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
Supporting Online Material
Materials and Methods
Figs. S1 to S4
Tables S1 to S3
3 March 2008; accepted 8 May 2008
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.
conductivity (1), excitations with fractional
statistic (2), topological quantum computation
systems (4). One of the main physical mecha-
nisms that gives rise to strong correlations for
low temperatures and for strong elastic repulsive
interactions, particles tend to stay far away from
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-
evolution of the density distribution, etc. cor-
respond to those of fermionic particles. In two
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-
particles: This may seem surprising because in-
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
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
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.
particles are lost because of inelastic collisions.
Our experiment used molecules confined to
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)
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).
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
*Present address: Institute of Quantum Electronics, Eidgenös-
†Present address: Institut für Photonik, Technische Uni-
versität Wien, Gußhausstraße 25-29, 1040 Wien, Austria.
‡To whom correspondence should be addressed. E-mail:
Fig. 1. Time-resolved
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
sponding to K3D= 2.2 ×
10−10cm3/s and, at t =
0, to g(2)= 0.11. The
loss is much slower than
the dashed line because
of strong correlations.
VOL 3206 JUNE 2008
on November 3, 2008