Light scattering detection of quantum phases of ultracold atoms inoptical lattices

Abstract

Ultracold atoms loaded on optical lattices can provide unprecedented
experimental systems for the quantum simulations and manipulations of many
quantum phases. However, so far, how to detect these quantum phases effectively
remains an outstanding challenge. Here, we show that the optical Bragg
scattering of cold atoms loaded on optical lattices can be used to detect many
quantum phases which include not only the conventional superfluid and Mott
insulating phases, but also other important phases such as various kinds of
density waves (CDW), valence bond solids (VBS), CDW supersolids and VBS
supersolids.

Full text

Page 1

RAPID COMMUNICATIONS
PHYSICAL REVIEW A 83, 051604(R) (2011)
Light-scattering detection of quantum phases of ultracold atoms in optical lattices
Jinwu Ye,1,2J. M. Zhang,3W. M. Liu,3Keye Zhang,4Yan Li,4and Weiping Zhang4
1Department of Physics, Capital Normal University, 100048 Beijing, China
2Department of Physics and Astronomy, Mississippi State University, P. O. Box 5167, Mississippi State, MS, 39762, USA
3Institute of Physics, Chinese Academy of Sciences, 100080 Beijing, China
4Department of Physics, East China Normal University, 200062 Shanghai, China
(Received 2 June 2010; revised manuscript received 17 November 2010; published 16 May 2011)
Ultracold atoms loaded on optical lattices can provide unprecedented experimental systems for the quantum
simulations and manipulations of many quantum phases. However, so far, how to detect these quantum phases
effectively remains an outstanding challenge. Here, we show that the optical Bragg scattering of cold atoms
loaded on optical lattices can be used to detect many quantum phases, which include not only the conventional
superfluid and Mott insulating phases, but also other important phases, such as various kinds of charge density
wave (CDW), valence bond solid (VBS), CDW supersolid (CDW-SS) and Valence bond supersolid (VB-SS).
DOI: 10.1103/PhysRevA.83.051604PACS number(s): 67.85.Hj, 03.75.−b, 42.50.−p
Various kinds of strongly correlated quantum phases of
matter may have wide applications in quantum information
processing, storage, and communications [1]. It was widely
believed and also partially established that due to the tremen-
dous tunability of all the parameters in this system, ultracold
atoms loaded on optical lattices (OL) can provide unprece-
dented experimental systems for the quantum simulations and
manipulations of these quantum phases and quantum phase
transitions between these phases. For example, Mott and
superfluid phases [2] may have been successfully simulated
and manipulated by ultracold atoms loaded in a cubic optical
lattice [3]. However, there are still at least two outstanding
problemsremaining.Thefirstishowtorealizemanyimportant
quantum phases [1].The second isthatassuming thefavorable
conditionstorealizethesequantumphasesareindeedachieved
in experiments, how to detect them without ambiguity. In this
Rapid Communication, we will focus on the second question.
So far, the experimental way to detect these quantum phases is
mainly through the time-of-flight (TOF) measurement [1,3],
which simply opens the trap and turns off the optical lattice,
lets the trapped atoms expand and interfere, and then takes the
image. The atomic Bragg spectroscopy is based on stimulated
matter waves scattering by two incident laser pulses [4,5]
through the TOF measurements. The momentum [4] transfer
Bragg spectroscopy was used to detect the Bogoliubov mode
inside a Bose-Einstein condensate (BEC). The energy transfer
[5] Bragg spectroscopy was used to detect the Mott gap in
a Mott state in an optical lattice. Optical Bragg scattering
[Fig. 1] has been used previously to study periodic lattice
structures of cold atoms loaded on optical lattices [6]. It was
also proposed as an effective method for the thermometry
of fermions in an optical lattice [7] and to detect a putative
antiferromagnetic (AF) ground state of fermions in an OL [8].
There are very recent optical Bragg scattering experimental
data from a Mott state, a BEC, and an AF state [9]. The
atomic Bragg spectroscopy and optical Bragg scattering are
two different, but complementary, experimental methods.
In this Rapid Communicaton, we will develop a systematic
theory of using the optical Bragg scattering [Fig. 1] to detect
the nature of quantum phases of interacting bosons loaded
in optical lattices. We show that the optical Bragg scattering
not only couples to the density order parameter but also to the
valencebondorderparameterduetothehoppingofthebosons
on the lattice. At integer fillings, when ? q matches a reciprocal
lattice vector? K of the underlying OL, there is an increase in
the optical scattering cross section as the system evolves from
the Mott to the SF state due to the increase of hopping in the
SF state. At 1/2 filling, in the charge density wave (CDW)
state, when ? q matches the CDW ordering wave vector ? Qn
and ? K, there is a diffraction peak proportional to the CDW
order parameter squared and the density squared, respectively
[Fig. 3(a)]; the ratio of the two peaks is a good measure of the
CDW order parameter. In the valence bond solid (VBS) state,
when ? q matches the VBS ordering wave vector? QK, there is a
much smaller, but detectable diffraction peak proportional to
the VBS order parameter squared; when it matches? K, there
is also a diffraction peak proportional to the uniform density
in the VBS state [Fig. 3(b)]. All the diffraction peaks scale as
the square of the numbers of atoms inside the trap. All these
characteristics can determine uniquely the CDW and VBS
states at 1/2 filling and the corresponding CDW supersolid
(CDW-SS) and Valence bond supersolid (VB-SS) slightly
away from the 1/2 filling. In the following, we just take two-
dimensionalopticallatticesasexamples.Theone-dimensional
and three-dimensional cases can be similarly discussed.
The extended boson Hubbard model (EBHM) with various
kinds of interactions, on all kinds of lattices and at different
filling factors, is described by the following Hamiltonian
[2,10–17]:
?
+V1
?ij?
where ni= b†
hopping which can be tuned by the depth of the optical lattice
potential; the U, V1, and V2are onsite; nearest neighbor (nn)
and next nearest neighbor (nnn) interactions, respectively;
and the ··· may include further neighbor interactions and
possible ring-exchange interactions. The filling factor n =
Na/N, where Nais the number of atoms, and N is the number
HBH= −t
?ij?
(b†
ibj+ H.c.) − µ
?
nink+ ···,
i
ni+U
2
?
i
ni(ni− 1)
?
ninj+ V2
?
?ik?
(1)
ibiis the boson density; t is the nearest neighbor
051604-1
1050-2947/2011/83(5)/051604(4)©2011 American Physical Society

Page 2

RAPID COMMUNICATIONS
YE, ZHANG, LIU, ZHANG, LI, AND ZHANGPHYSICAL REVIEW A 83, 051604(R) (2011)
q
ω
k ,
1ω1
k ,
2ω2
A
PM
ω
∆
Ω
Ω
a
∆
ΩΩ
ωa
(a)
i
j
(b)
i
FIG. 1. (Color online) Optical Bragg scattering of cold atoms
movingintwo-dimensionalopticallattices.The ? q =?k1−?k2andω =
ω1− ω2are momentum and energy transfers from the laser beams to
the cold atoms, respectively. The A stands for an aperture, and the
PMstandsforaPhotomultiplier.Theoffresonantscatteringprocesses
lead to (a) the onsite term and (b) the offsite term in Eq. (3).
of lattice sites. The onsite interaction U can be tuned by the
Feshbach resonance [2]. Various kinds of optical lattices, such
as honeycomb, triangular [18], body-centered-cubic [18], and
Kagome lattices [19], can be realized by suitably choosing the
geometryofthelaserbeamsformingtheopticallattices.There
are many possible ways to generate longer range interactions
V1,V2,... of ultracold atoms loaded in optical lattices. Being
magnetically or electrically polarized, the52Cr atoms [20] or
polar molecules [21]40K +87Rb (or39K +87Rb) interact
with each other via long-range anisotropic dipole-dipole
interactions. Loading the
a two-dimensional optical lattice with the dipole moments
perpendicular to the trapping plane can be mapped to Eq. (1)
with long-range repulsive interactions ∼p2/r3, where p is
the dipole moment. The CDW supersolid phases studied by
Quantum Monte Carlo (QMC) simulations [11] and described
in [15] by the dual vortex method was numerically found
to be stable in large parameter regimes in this system [22].
The generation of the ring exchange interaction has been
discussed in [23]. Some of the important phases with long
range interactions are listed in Fig. 2. Recently, the quantum
entanglement properties of the VBS state were addressed
in [24].
The interaction between the two laser beams in Fig. 1 with
the two level bosonic atoms is:
?
+?
2
l
where ?(? r) = (ψe,ψg) is the two component boson annihi-
lation operator, the incident and scattered lights in Fig. 1
have frequencies ωl, and mode functions ul(? r) = ei?kl·? r+iφl.
The Rabi frequencies ? are much weaker than the laser
beams(not shown in Fig. 1), which form the optical lattices.
When it is far off the resonance, the laser light-atom detunings
?l= ωl− ωa, where ωais the two level energy difference,
are much larger than the Rabi frequency ? and the energy
transfer ω = ω1− ω2[Figs. 1(a) and 1(b)], so ?1∼ ?2= ?.
Afteradiabaticallyeliminatingtheupperleveleofthetwolevel
atoms,expandingtheground-stateatomfieldoperatorψg(? r) =
?
52Cr or the polar molecules on
Hint=
d2? r?†(? r)
?
? p2
2ma
+ VOL(? r) +¯ hωa
2
σz
?
[e−iωltσ+ul(? r) + H.c.]
?
?(? r),
(2)
ibiw(? r − ? ri) in Eq. (2), where w(? r − ? ri) is the localized
Wannier function of the lowest Bloch band corresponding to
VOL(? r),andbiistheannihilationoperatorofanatomatthesite
i in the Eq. (1), then we get the effective interaction between
the offresonant laser beams and the ground level g:
⎛
i
Hint= ¯ h?2
?e−iωt
⎝
N
?
Ji,ini+
N
?
?ij?
Ji,jb†
ibj
⎞
⎠,
(3)
where the interacting matrix element is Ji,j=?d? rw(? r −
onsite termˆD =?N
w(? r) can be taken as real in the lowest Bloch band, the offsite
termcanbewrittenasˆ K =?N
neighbor kinetic energy of the bosons Kij= b†
It is easy to show that
? ri)u∗
1(? r)u2(? r)w(? r − ? rj) = Jj,i. The first term in Eq. (3) is the
iJi,ini[Fig. 1(a)]. The second term is the
offsite term [Fig. 1(b)]. Because the Wannier wave function
?ij?Ji,jb†
ibj=?N
?ij?Ji,j(b†
ibj+
H.c.), which is nothing but the offsite coupling to the nearest
ibj+ H.c.
ˆD(? q) = f0(? q)
N
?
i=1
e−i? q·? rini= Nf0(? q)n(? q),
(4)
where ? q =?k1−?k2, f0(? q) =?d? re−i? q·? rw2(? r), and n(? q) =
N
the density operator at the momentum ? q. Note that n(? q) =
n(? q +? K) where the ? K is a reciprocal lattice vector. The
wavevectorisconfinedtoL−1< q < a−1,wherethetrapsize
L ∼ 100 µm, and the lattice constant a ∼ 0.5 µm in Fig. 1.
In fact, more information is encoded in the offsite kinetic
coupling in Eq. (3). In a square lattice, since the bonds are
either oriented along the ˆ x axis, ? rj− ? ri= ˆ x, or along the ˆ y
axis, ? rj− ? ri= ˆ y, we have
ˆ K?= N[fx(? q)Kx(? q) + fy(? q)Ky(? q)],
1
?N
i=1e−i? q·? rini=?
?kb†
?kb?k+? qis the Fourier transform of
(5)
where
eiqα/2?
α = x,y bonds at the momentum ? q and the “form” factors
fα(? q) = f(? q,? ri− ? rj= α) =?d? re−i? q·? rw(? r)w(? r + ? ri− ? rj).
approximation used in [2], we can estimate that f0(π,0) ∼
exp[−1
π2
4(V0/Er)1/2], so |fx(π,0)/f0(π,0)| ∼ exp(−π2
where V0and Er= ¯ h2k2/2m are the strength of the optical
lattice potential and the recoil energy, respectively [2]. The
f0(π,0) is close to 1, when V0/Er> 4. It is instructive to
relate this ratio to that of the hopping t over the onsite
interaction U in the Eq. (1): |fx(π,0)/f0(π,0)| ∼
as is the zero field scattering length and a = λ/2 = π/k is
the OL constant; using the typical values t/U ∼ 10−1and
as/a ∼ 10−2, one can estimate |fα/f0| ∼ 10−3. Note that
the harmonic approximation works well only in a very deep
optical lattice V0? Er, so the above value underestimates
the ratio, so we expect |fα/f0| ? 10−3.
The differential scattering cross section of the light from
the cold atom systems in the Fig. 1 can be calculated by using
Kα(? q) =
1
N
?N
i=1e−i? q·? riKi,i+α=
?kcoskαb†
?kb?k+? q,α = x,y are the Fourier transforms
of the kinetic energy operator Kij= b†
ibj+ H.c. along
Note that Kα(? q) = Kα(? q +? K). Following the harmonic
fx(π,0) ∼ iexp[−1
4(V0/Er)−1/2],
4(V0/Er)−1/2−
4
√V0/Er),
t
U
as
a, where
051604-2

Page 3

RAPID COMMUNICATIONS
LIGHT-SCATTERING DETECTION OF QUANTUM PHASES ...
PHYSICAL REVIEW A 83, 051604(R) (2011)
the standard linear response theory:
dσ
d?dE= S(? q,ω) ∼
??2
?
?2
N2
×
?
|f0(? q)|2Sn(? q,ω) +
?
α=ˆ x,ˆ y
|fα(? q)|2SKα(? q,ω)
?
,
(6)
where
?n(−? q, − ω)n(? q,ω)? is the dynamic density-density response
function, whose Lehmann representation was listed in [4].
The SKα(? q,ω) = ?Kα(−? q, − ω)Kα(? q,ω)? is the bond-bond
response function, whose Lehmann representation can be ob-
tainedfromthatoftheSn(? q,ω)simplybyreplacingthedensity
operator n(? q) by the bond operator Kα(? q). The integrated
scattering cross section over the final energydσ
is proportional to the equal-time response function
S(? q) ∼ (?2
We first look at the superfluid to Mott transition at the
integer filling factor n. When ? q is equal to the shortest
reciprocal lattice vector
? K = (2π,0); in the Mott state,
dσM
d?∼ |fM
|fSF
erage kinetic energy on a bond in the superfluid side. Because
|fSF
in the superfluid side, we expect a dramatic increase of the
scattering cross section
? q =?k1−?k0,
ω = ω1− ω2, andthe
Sn(? q,ω) =
d?=?dE
dσ
d?dE
dσ
d?=
?)2N2[|f0(? q)|2Sn(? q) +?
α=ˆ x,ˆ y|fα(? q)|2SKα(? q)].
0(2π,0)|2N2n2; in the superfluid state,
0(2π,0)|2N2n2+ 2|fSF
dσSF
d?∼
x(2π,0)|2N2B2, where B is the av-
0(2π,0)|2∼ |fM
0(2π,0)|2∼ 1, and B is appreciable only
dσSF
d?
−dσM
d?
= 2??fSF
x(2π,0)??2N2B2
(7)
across the Mott to the SF transition due to the prefactor N2.
This prediction could be tested immediately. Surprisingly,
there is no such optical Bragg scattering experiment in the
superfluid yet.
IntheCDWwith? Qn= (π,π)inFig.2(a),duetothelackof
VBS order, the second term in Eq. (6) can be neglected, so that
????
amplitude scales as the square of the number of atoms
inside the trap ∼ |f0(π,π)|2N2m2, where m = nA− nB
is the CDW order parameter [15]. When ? q =? K, then
dσ
d?dE
CDW
∼
??2
?
?2
N2|f0(? q)|2SN(? q,ω),
(8)
which should show a peak at ? q =? Qn [Fig. 3(a)], whose
(a)
(b)
(c)
(d)
FIG. 2. (Color online) (a) CDW phase in a square lattice at n0=
1/2 with ordering wave vector? Qn= (π,π). (b) Valence bond solid
(VBS) phases with ordering wave vector ? QK= (π,0), where the
kinetic energy ?Kij? = ?b†
the two sites connected with a dimer, but takes 0 in the two sites
withoutadimer.(c)StripeCDWorderat? Qn= (π,0)and(d)plaquette
VBS order at? QK= (π,0),(0,π) [2,10–17].
ibj+ H.c.? takes a nonzero constant K in
FIG. 3. (Color online) Optical scattering cross section in (a)
CDW,wheretheratioofthepeakat? Qnoverthatat? K is∼ m2/n2∼ 1,
and (b) VBS state, where the ratio of the peak at? QKover that at? K is
∼ |fx/f0|2K2/n2? 10−5, which still should be visible in the current
optical Bragg scattering experiments.
SCDW(? K) ∼ |f0(2π,0)|2N2n2, where f0(2π,0) ∼ f2
[Fig.3(a)].SotheratioofthetwopeaksinFig.3(a)is∼ m2/n2
if one neglects the very small difference of the two form
factors. Slightly away from 1/2 filling, the CDW in Fig. 2(a)
may turn into a CDW-SS [25] phase through a second order
phase transition [15]. Then, we have ?n(? q)? = mδ? q,? Qn+ nδ? q,0
where n = nA+ nB= 1/2 + δn. The superfluid density
ρs∼ δn = n − 1/2. The scattering cross section inside
theCDW-SS:
SCDW-SS(? Qn) ∼ |f0(π,π)|2N2m2
more or less the same as that inside the CDW, but
SCDW-SS(? K) ∼ |f0(2π,0)|2N2n2+ 2|fx(2π,0)|2N2(δn)2B2
will increase. The B is the average bond strength due to
very small superfluid component ρs∼ δn = n − 1/2 flowing
through the whole lattice. So the right peak in Fig. 3(a)
will increase due to the increase of the total density and the
superfluid component inside the CDW-SS phase.
Now we discuss the VBS state with
Fig. 2(b). Due to the uniform distribution of the density in
the VBS, when ? q =? K, the second term in Eq. (6) can be
neglected, so there is a diffraction peak [Fig. 3(b)] whose
amplitude scales as the square of the number of atoms inside
the trap ∼ |f0(2π,0)|2N2n2, where f0(2π,0) ∼ f4
n = 1/2 is the uniform density in the VBS state. However,
when one tunes ? q near ? QK, the first term in Eq. (6) can be
neglected; then,
????
which should show a peak at ? q =? QK signifying the VBS
ordering at? QK, whose amplitude scales also as the square of
thenumberofatomsinsidethetrap∼ |fx(π,0)|2N2K2,where
K = Kx− KyistheVBSorderparameter[15].Sotheratioof
theVBSpeakat ? q =? QKovertheuniformdensitypeakat ? q =
? K is∼ [K2/n2]|fx(π,0)/f0(2π,0)|2? 10−5inFig.3(b).Note
that the smallness of |fx|2is compensated by the large number
of atoms N ∼ 106, |fx|2N2= (|fx|2N) × N ∼ N ∼ 106.
Therefore, the Bragg scattering cross section from the VBS
order is?10−5smaller than that at ? q =? K atthe same incident
energy Iin[Fig. 3(b)], but still ∼ 106above the background,
so very much visible in the current optical Bragg scattering
experiments. Slightly away from 1/2 filling, the VBS may
0(π,π)
stays
? QK= (π,0) in
0(π,0) and
dσ
d?dE
VBS
∼
??2
?
?2
N2?
α=ˆ x,ˆ y
|fα(? q)|2SKα(? q,ω),
(9)
051604-3

Page 4

RAPID COMMUNICATIONS
YE, ZHANG, LIU, ZHANG, LI, AND ZHANG PHYSICAL REVIEW A 83, 051604(R) (2011)
turn into a VB Supersolid (VB-SS) through a second-order
transition [15]. We have ?Kx(? q)? = Bδ? q,0+ Kδ? q,? QKand
?n(? q)? = (δn + 1/2)δ? q,0.
δn = n − 1/2. The scattering cross section inside VB-SS:
SVB-SS(? QK) ∼ |fx(π,0)|2N2K2stays more or less the same
as that inside the VBS, but SVB-SS(? K) ∼ |f0(2π,0)|2N2n2+
|fx(2π,0)|2N2(δn)2B2
1/2 + δn and the Bx,By are the average bond strengths
along x and y, respectively, due to a very small superfluid
component ρs∼ δn = n − 1/2 flowing through the whole
lattice. So the right peak in Fig. 3(b) will increase due to the
increase of the total density and the superfluid component
inside the VB-SS phase. Very similarly, one can discuss
the VBS order at ? q =? QK= (0,π). For the plaquette VBS
order in Fig. 2(d), then one should be able to see the SK(? q)
peaks at both (π,0) and (0,π). So the dimer VBS and the
plaquette VBS can also be distinguished by the optical Bragg
scattering.
Thesuperfluiddensity
ρs∼
x+ |fy(2π,0)|2N2(δn)2B2
y, where n =
In this Rapid Communication, we only focused on the
optical Bragg scattering detections of the various ground
states in a square lattice. The detections of the excitation
spectra, the generalization to frustrated lattices, the effects of
finite temperature and a harmonic trap will be discussed in a
future publication [26].
We thank A. V. Balatsky, G. G. Batrouni, Jason Ho,
R. Hulet, S. V. Isakov, Juan Pino, and Han Pu for helpful
discussions. J. Ye’s research is supported by NSF-DMR-
0966413, NSFC-11074173 is supported at KITP in part by the
NSFunderGrantNo.PHY-0551164,andissupportedatKITP-
C by the Project of Knowledge Innovation Program (PKIP) of
the Chinese Academy of Sciences. W. M. Liu’s research was
supported by NSFC-10874235. W. P. Zhang’s research was
supported by the National Basic Research Program of China
(973 Program) under Grant No. 2011CB921604, and NSFC
under Grants No. 10588402 and 10474055.
[1] For a review, see M. Lewenstein et al., Adv. Phys. 56, 243
(2007); I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.
80, 885 (2008).
[2] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller,
Phys. Rev. Lett. 81, 3108 (1998).
[3] M. Greiner et al., Nature (London) 415, 39 (2002).
[4] M. Kozuma et al., Phys. Rev. Lett. 82, 871 (1999); J. Stenger
et al., ibid. 82, 4569 (1999); D. M. Stamper-Kurn et al., ibid. 83,
2876 (1999); J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson,
ibid. 88, 120407 (2002); S. B. Papp et al., ibid. 101, 135301
(2008); P. T. Ernst et al., Nature Phys. 6, 56 (2010).
[5] T. Stoferle, H. Moritz, C. Schori, M. Kohl, and T. Esslinger,
Phys. Rev. Lett. 92, 130403 (2004).
[6] G. Birkl, M. Gatzke, I. H. Deutsch, S. I. Rolston, and W. D.
Phillips, Phys. Rev. Lett. 75, 2823 (1995); M. Weidemuller,
A. Hemmerich, A. Gorlitz, T. Esslinger, and T. W. Hansch,
ibid. 75, 4583 (1995); M. Weidemuller, A. Gorlitz, T. W.
Hansch, and A. Hemmerich, Phys. Rev. A 58, 4647
(1998).
[7] J. Ruostekoski, C. J. Foot, and A. B. Deb, Phys. Rev. Lett. 103,
170404 (2009).
[8] T. A. Corcovilos, S. K. Baur, J. M. Hitchcock, E. J. Mueller, and
R. G. Hulet, Phys. Rev. A 81, 013415 (2010).
[9] I. Bloch (private communication).
[10] G. Murthy, D. Arovas, and A. Auerbach, Phys. Rev. B 55, 3104
(1997).
[11] F. Hebert et al., Phys. Rev. B 65, 014513 (2001); P. Sengupta,
L. P. Pryadko, F. Alet, M. Troyer, and G. Schmid, Phys. Rev.
Lett. 94, 207202 (2005).
[12] S.V.Isakov,S.Wessel,R.G.Melko,K.Sengupta,andY.B.Kim,
Phys. Rev. Lett. 97, 147202 (2006); K. Damle and T. Senthil,
ibid. 97, 067202 (2006).
[13] L.Balents,L.Bartosch,A.Burkov,S.Sachdev,andK.Sengupta,
Phy. Rev. B 71, 144508 (2005).
[14] L. Jiang and J. Ye, J. Phys. Condens. Matter. 18, 6907
(2006).
[15] J. Ye, Nucl. Phys. B 805, 418 (2008).
[16] Y. Chen and J. Ye, e-print arXiv:0804.3429v3 [cond-mat],
e-print arXiv:cond-mat/0612009v4.
[17] J. Y. Gan et al., Phys. Rev. B 75, 214509 (2007).
[18] G. Grynberg, B. Lounis, P. Verkerk, J. Y. Courtois, and
C. Salomon, Phys. Rev. Lett. 70, 2249 (1993).
[19] L. Santos et al., Phys. Rev. Lett. 93, 030601 (2004); B. Damski
et al., Phys. Rev. A 72, 053612 (2005).
[20] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau,
Phys. Rev. Lett. 94, 160401 (2005).
[21] K.-K. Ni et al., Science 322, 231 (2008).
[22] B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller,
and G. Pupillo, Phys. Rev. Lett. 104, 125301 (2010).
[23] H. P. B¨ uchler, M. Hermele, S. D. Huber, M. P. A. Fisher, and
P. Zoller, Phys. Rev. Lett. 95, 040402 (2005).
[24] A.Chandran,D.Kaszlikowski,A.SenDe,U.Sen,andV.Vedral,
Phys. Rev. Lett. 99, 170502 (2007).
[25] For a supersolid in a continuous system (namely without an
underlying OL), see J. Ye, Phys. Rev. Lett. 97, 125302 (2006);
Europhys. Lett. 82, 16001 (2008); J. Low Temp. Phys. 160, 71
(2010).
[26] J. Ye et al., e-print arXiv:0812.4077v4 [cond-mat].
051604-4