Signatures of spatial inversion asymmetry of an optical lattice observed in matter-wave diffraction

0.94 MB·

by  C. K. Thomas

Signatures of spatial inversion asymmetry of an optical lattice observed in
matter-wave diffraction
C. K. Thomas1, T. H. Barter1, T.-H. Leung1, S. Daiss2, and D. M. Stamper-Kurn1,3∗
1Department of Physics, University of California, Berkeley, California 94720, USA
2Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
(Dated: 13 June2016)
6
1 Thestructureofatwo-dimensionalhoneycombopticallatticepotentialwithsmallinversionasym-
0 metryischaracterized usingcoherent diffraction of87Rbatoms. Wedemonstratethatevenasmall
2 potential asymmetry, with peak-to-peak amplitude of ≤ 2.3% of the overall lattice potential, can
leadtopronouncedinversionasymmetryinthemomentum-spacediffractionpattern. Theobserved
n
asymmetryisexplainedquantitativelybyconsideringbothKapitza-DiracscatteringintheRaman-
u
Nath regime, and also either perturbative or full-numerical treatment of the band structure of a
J
periodic potential with a weak inversion-symmetry-breaking term. Our results have relevance for
6
both the experimental development of coherent atom optics and the proper interpretation of time-
2
of-flight assays of atomic materials in optical lattices.

DOI:10.1103/PhysRevA.93.063613
]
s
a
g In x-ray crystallography, the diffraction of light is an- inversionasymmetryinthepopulationsofthefirst-order
-
t alyzed to determine the exact crystalline structure of diffraction peaks even while the inversion-asymmetric
n
a material. Similarly, with the availability of ultracold partofthepotentialis≤2.3%oftheoveralllatticepoten-
a
sources of coherent matter waves of atoms, one can use tial. This observation highlights the extreme sensitivity
u
q atomic diffraction to characterize potentials experienced of coherent matter-wave scattering in revealing features
. by the atoms. Of particular interest are the optical lat- of a potential landscape under investigation.
t
a tice potentials produced by periodic patterns of light in- Aside from demonstrating sensitive optical-lattice
m tensity and polarization, formed by the intersection of crystallography,ourobservationalsohasimplicationsfor
- several coherent plane waves of light or by direct imag- thedevelopmentofatomoptics. Matter-waveinterferom-
d ing. Lattice potentials of various geometries and dimen- eters for several applications have employed brief pulses
n
sionalities, some incorporating atomic-spin dependence of light to split and recombine atomic beams coherently
o
c andgaugefields,havebeenproducedorproposedforthe [7, 8]. Kapitza-Dirac diffraction, i.e., the diffraction of
[ purposeofcreatingsyntheticatomicmaterialsbyplacing atomsfromstanding-waveratherthantraveling-waveop-

quantum-degenerateatomswithinthem[1–3]. Justasin tical potentials, has the advantage that it is technically
4
condensed matter, the characteristics of such synthetic simple to implement, requiring only light waves at a sin-
v
7 atomic materials derive from the nature of the optical gle opticalfrequency[9, 10]. However,ascomparedwith
1 crystaluponwhichthey arebased. Matter-wavecrystal- Bragg or Raman diffraction, it has the disadvantage of
1 lography therefore becomes a vital tool in the study of being less efficient and less selective [11]. The tech-
7 such synthetic quantum matter [4]. nical simplicity has inspired modifications of Kapitza-
0 A key first step in determining the structure of a lat- Diracdiffractionemployingseveralpulsesoflightsoasto
.
1 tice is the assignmentofits point-groupandspace-group diffractatoms to selected diffractionorderswith highef-
0 symmetries. The violation of a symmetry is identified ficiency [12], although the diffraction remained inversion
6 in x-ray crystallography by a difference in the intensi- symmetric, with as many atoms diffracted to the wave
1 tiesofdiffractionspots[5]. Followingsuchwork,herewe vector +G as to the wave vector −G. We show that
:
v detect the inversion asymmetry of an optical lattice by this last constraint can be lifted to produce inversion-
i observing significant asymmetries in the diffraction of a asymmetricKapitza-Diracdiffractionofmatter wavesin
X
coherent matter wave from the potential. For this, we two dimensions. Similar to the previous demonstration
ar produce a spin-polarized 87Rb Bose-Einstein condensate in one dimension [13], we explain how this asymmetry
atrest,andthenimposeforavariablepulsedurationthe arises from the interference between different diffraction
two-dimensionalhoneycombopticallatticepotentialpro- pathways to the same final momentum state.
duced by three light beams intersecting at equal angles We begin by describing the optical lattice potential
[6]. The resulting Kapitza-Dirac diffraction is quantified characterized in this work. As in Ref. [6] and illus-
by imaging the gas after it is allowed to expand freely. trated in Fig. 1(a), we form a two-dimensional honey-
By tuning the pulse time and working with a deep opti- comblatticeusingthreebeamsoflightatthewavelength
callattice,weproducehighlyvisible(over50%contrast) λ=1064 nm, with equal intensity, propagating horizon-
tally and intersecting at equal angles, with each beam
linearlypolarizedin the lattice plane. We define a quan-
tization axis orthogonalto the lattice plane and show in
∗ Electronicaddress: [email protected] Fig. 1(b) that the beams produce a periodic pattern of
2
FIG. 1. Three 1064 nm beams interfere at 120◦ with in-planepolarization to create a honeycomb lattice of intensity maxima.
We identify the unit cell of thelattice potential (solid line). A dashed line within theunit cell runsthrough thetwo potential
minima, which are marked with ticks and labeled A and B. One-dimensional profiles of the light intensity (b) and optical
potentials (c) and (d) along this line are shown. The star symbol, located at a minimum-intensity location, serves as a center
forthespatialinversionoperationthatexchangestheAandBsitesofthelattice. (b)Wedefineaquantizationaxisorthogonal
to thelattice plane and show that thelight is predominantly σ+ at site A and σ− at site B. (c) The atoms are polarized by a
uniform magnetic field B at an angle θ from the quantization axis. We show the lattice potential for extreme values of cosθ,
0
where the potential depth at sites A and B maximally differ. (d) The lattice potential is the sum of an inversion-symmetric
potential Vs(r), that arises from the scalar Stark shift and an inversion-antisymmetric potential Va(r), that comes from the
vector Stark shift.
varying intensity and optical polarization. tude and sign of V (r) by tilting the dominant exter-
a
Rubidium atoms exposed to this optical lattice expe- nal magnetic field B0 by an angle θ with respect to the
rience an ac-Stark shift that can be divided into scalar, (vertical) axis defined by the optical helicity. For atoms
vector, and tensor terms acting on the atomic hyperfine spinpolarizedalongB0,theasymmetricpotentialisthen
spin[14]. Thetensorlightshiftisnegligibleinourexper- Va(r)∝cosθ. Figures1(c)and1(d)showthattheresult-
imentowingtothelargedetuningofthelatticelightfrom ing lattice potential has a small, state-dependent offset
the atomic transitions. Figure 1(d) shows the lattice po- in energy between sites A and B.
tentials that result from the scalar and vector parts of Tocharacterizethislatticeusingmatterwaves,wecre-
the ac-Stark shift. The scalar light shift is proportional ate a nearly pure, optically trapped Bose-Einstein con-
to light intensity and produces a honeycomb lattice po- densate of 3×105 87Rb atoms that is spin polarized in
tential Vs(r) with two sites of equal depth per unit cell, the|F =1,mF =−1istatealongtheaxisdefinedbya∼
labeled A and B in the figure. The vector light shift in 0.5Gappliedmagneticfield. We thenintroduce athree-
the presence of a dominant external magnetic field pro- beam lattice potential with |Vs(r)|max= h×87±4 kHz
duces a potential V (r) that is approximately diagonal for a pulse time τ between 10 and 100 µs. This lattice
a
in the Zeeman basis defined by the field direction. V (r) depth is calibrated with independent measurements of
a
is proportional to both intensity and the dot product of the diffraction produced by the one-dimensional lattices
helicity and atomic spin [14]. The helicity in the lattice formedbypairsofthelatticebeams[15]. Afterthepulse,
is staggered so that V (r) is of opposite sign at each of we simultaneously switch off the optical lattice and op-
a
the two sites in the unit cell. tical trapping potentials and allow the atoms to expand
freelyfora20-mstimeofflight. Wefinallytakeanimage
The scalar and vector light shift potentials differ in
their inversion symmetry, with V (r) being symmetric of the density distribution in which the various diffrac-
s
and V (r) being antisymmetric under spatial inversion. tion orders, generated at the reciprocal lattice vectors
a
by exposure to the lattice potential, are seen as separate
Figure 1(a) shows one of the zero-intensity locations
peaks.
within the optical lattice as an example of the center
The first-order diffraction peaks in Figs. 2(a) and 2(c)
of the inversion operation. The result of this operation
show a pronounced inversion asymmetry. To quantify
is to switch sites A and B.
this asymmetry, we identify three reciprocal lattice vec-
For alkali atoms, V (r) is suppressed with respect to
a tors that describe first-orderdiffraction as G =k −k
V (r)owingtothelargeopticaldetuningfromtheatomic 1 3 2
s and its cyclic permutations, where k are the wave
resonance. For the wavelength of light used in our lat- 1,2,3
vectors of the incident beams that form the lattice. We
tice, the ratio 2|V (r)/V (r)| is at most 2.3%, so that
a s define an asymmetry parameter A as
V (r)addsonlyasmallinversion-symmetry-breakingpo-
a
tential atop a graphene-like, inversion-symmetrichoney- A= Pi(PGi −P−Gi), (1)
comb lattice. Within this limit, we control the magni- Pi(PGi +P−Gi)
3
FIG. 2. An asymmetry parameter A is defined as the first-
order population imbalance and measured for data taken as
a function of θ with a pulse time of 50 µs. (a) Time-of-
flight image for θ = 0.44 shows an asymmetry in the first-
orderdiffractionpeaks. (b)Wehighlightthefirst-orderpeaks
withcircles(atGi)andtriangles(at−Gi). (c)Time-of-flight
FIG. 3. Oscillations in A as a function of the Kapitza-Dirac
image for θ =2.2 shows reversal of the observed asymmetry.
pulsetimeτ,shownforθ=0.38radians(circles)andθ=2.8
(d) A is computed for each of five images and the mean and
radians (triangles). The data represent the mean and stan-
standarderrorofthesedataareplotted. Thesolidlineshows
darderroroffiveexperimentalrunsateachpulsetime. Anu-
theexpected dependenceon θ.
mericalcalculation(solidline)withnofreeparametersclosely
reproducesthetimedependenceofA,whileperturbationthe-
ory (dashed line) captures the short time behavior. Inset
i.e., as the contrast between the diffraction intensities time-of-flight images for τ of 8, 50, 59, and 77 µs show di-
at wave vectors Gi and −Gi, the two sets of wave vec- rectly the evolution of the first-order asymmetry. We note
tors being related by inversion. This measure is robust that discrepancies between theory and experiment, e.g., at
against variations in the total atom number and against times around 30, 60, and 80 µs, appear when the total pop-
residual center-of-mass motion of the condensed atoms ulation in thefirst-orderpeaksis small, causing asystematic
withrespectto the lattice potential. We note thatimag- reduction in themeasured magnitude of A.
ing aberrations introduce a slight offset in A (of about
0.1) in our experiment, seen in Figs. 2 and 3.
Weconfirmthatthemomentum-spaceinversionasym- try. First, we consider how the momentum-space asym-
metryiscausedbythereal-spaceinversionasymmetryof metryoriginatesfromlow-orderdiffractioninthelattice.
the lattice potential by varying the magnitude and sign This description, shown schematically in Figs. 4(a) and
of the inversion-symmetry-breaking potential V (r). We 4(b), is valid in the limit of a shallow optical lattice and
a
tune V (r) by rotating the orientation of the magnetic in the Raman-Nath regime, where we can ignore the ki-
a
field from the vertical axis by the polar angle θ before neticenergyofthediffractingatoms[16]. Boththescalar
exposing the condensate to the lattice potential. andvectorStarkshiftopticallatticepotentials,V (r)and
s
Our data emphasize the fact that even an asymmetry Va(r), canbe characterizedin momentum space by their
in the lattice potential of ≤2.3% can lead to highly vis- Fourier transforms Vs,a(±Gi) at the wave vectors ±Gi,
ible asymmetry in the matter-wave diffraction pattern. wheretherelationVs,a(Gi)=Vs∗,a(−Gi)isvalidbecause
The evolution of the momentum-space asymmetry A vs both potentials are real. Considering the C3 rotational
pulse time is portrayed in Fig. 3. The asymmetry grows symmetry of both lattices and their respective inversion
fromsmallvaluesatearlytimestoover50%atτ ∼50µs, symmetrieswehaveVs(±Gi)=βsandVa(±Gi)=±iβa,
and also displays clear modulation in time reflecting the where βs and βa are both real.
coherent dynamics of matter waves within the imposed We now consider the probability amplitudes p(±G )
i
lattice potential. Throughout these dynamics, reversing for atoms diffracting from their initial zero momentum
the sign of the inversion asymmetry of the lattice re- state to a final wavevector±G within a time τ. Figure
i
versestheobservedinversionasymmetryofthediffracted 4(a) illustrates that such diffraction can be achieved by
atoms. one first-order process, with amplitude −i(β ∓iβ )τ/~,
s a
We present two physical pictures that explain the ori- andbytwosecond-orderprocesses,whichsumto anam-
gin of the observed momentum-space inversionasymme- plitude (−i)2(β ±iβ )2τ2/~2. We ignore higher order
s a
4
lattice asymmetry (|β | ≪ |β |) we obtain A ≃ 6β t/~,
a s a
which is plotted as a gray dotted line in Fig. 3 and de-
scribes the data well for small τ.
While the model above provides a simple analytic ex-
pression for A, its assumptions are violated under the
conditions of our experiment. For one, our experiments
are performed with a deep lattice that leads to diffrac-
tiontohighorder,asexemplifiedbythemanydiffraction
peaks in our images. Second, the measurements are per-
formed with pulse times that are long enough to be out-
sidetheRaman-Nathregime,whichisshownbythehigh
kineticenergyofthelargemomentumstatesproducedin
our experiment. Therefore, the diffraction pattern pro-
ducedinourmeasurementisbetterdescribedasresulting
from coherent dynamics governed by the band structure
of the optical lattice.
We performed numerical calculations that trace the
evolution of a noninteracting gas, produced initially at
zero momentum, within the lattice band structure. The
numericalresultsshowninFig.3areforθ =0.44radians
and a lattice depth of 87 kHz with no free parameters.
The calculationmatches well with the observedtime de-
pendence of the diffraction asymmetry.
To provideanintuitive descriptionofthe coherentdy-
namicsin Athat weboth observeandcalculate,we con-
sider the effect of a small inversion-symmetry-breaking
perturbation to the band structure of an inversion-
symmetric lattice potential. The unperturbed Hamil-
tonian H , which includes the kinetic energy and the
0
inversion-symmetric lattice potential V (r), has eigen-
s
states |ψ(0)i that are either even (labeled by +) or odd
i,±
(labeled by −) under the action of the spatial inversion.
FIG. 4. (a) Atoms at zero momentum are coupled to wave TheperturbationHa resultsfromthesmallantisymmet-
vectors ±Gi by the asymmetric Fourier components of the ric lattice potential Va(r) and mixes the even and odd
potential. (b) Interference between first- and second-order eigenstates. To first order in H , the zero quasimomen-
a
processes creates a population imbalance at ±Gi. (c) We tum eigenstates become
treattheinversion-asymmetricpotentialasaperturbationHa
ontheinversion-symmetriclatticeHamiltonian H andshow
0
momentum-space amplitudes (spot size) and phases (color) |ψ(1)i≈|ψ(0)i+ α |ψ(0)i (2)
of the two lowest energy eigenstates. In our experiment α2,1 i,+ i,+ X j,i j,−
is large and Ha strongly mixes the symmetric ground state, j
|ψ1(0,+)i,andtheantisymmetricexcitedstate,|ψ2(0,−)i. Bothper- |ψj(,1−)i≈|ψj(,0−)i+X−α∗j,i|ψi(,0+)i (3)
turbed states are asymmetric and overlap with a stationary
i
condensate. (d)Muchof theoscillatory behaviorobservedin
A can be attributed to the beating of three states, identified
sinhotwhesetiagteen|sψp3(e01c),t+riu,mwhoifcHh0isatsh|eψ1e(0,x+)cii,te|dψ2(s0,t−)ait,eatnhdat|ψb3(e01)s,+tis.atWise- whTehreeαinj,iiti=alhsψtEja(,0jt−(),0e−|)H−isaE|aψi(,0i(+),0z+)eir.o-momentum condensate that
fiesthecriteriadescribedinthefirstscenarioofthetext. The can be written in the basis of inversion-even eigenstates
energydifferencesamongthesestatesdefinethreefrequencies as |ψ(0)i = c |ψ(0)i. During the lattice pulse time τ
(2,65and67kHz)thatdominatethesignalofA. Ournumer- Pi i i,+
this initial state evolves in time as
icalcalculationsshowthatthisthree-statedescription(dotted
line)capturesmostofthephysicsinthefullsignalofA(solid
line). |ψ(t)i=Xcie−iωi,+t|ψi(,0+)i+Xαj,i|ψj(,0−)i (4)
i  j 
terms. Interference between the first- and second-order +X−αj,kcke−iωj,-t|ψj(,0−)i
scatteringamplitudesresultsinanimbalanceofprobabil-
j,k
ityfordiffractionintooppositewavevectors. Calculating
theasymmetryparameterAatshorttimesandforsmall where ωi,+ =Ei,+/~ and ωj,- =Ej,−/~.
5
The first term of Eq. (4) represents the incorporation these two states are dominant contributors to oscillation
of inversion antisymmetry into the initially even eigen- in A as in the second scenario described, and also cou-
states,andthe secondtermrepresentsfully antisymmet- ple with a third state |ψ(0)i as in the first scenario. In
k,+
ric states for which the perturbation introduces popu-
Fig.4(d)weisolatethe symmetric excitedstate withthe
lation at zero momentum. Figure 4(c) illustrates each
largest population in the zero and first-order diffracted
of these effects on two states at zero quasimomentum momenta,|ψ(0) i. Theenergyofthesethreestatesdefine
that are heavily influenced by the perturbation H : the 31,+
a three frequencies that dominate the time evolutionof A.
initially symmetric ground state and antisymmetric first
Thelargemomentum-spaceasymmetryisobservedwhen
excited state.
the Kapitza-Diracpulse time is tuned so that these tem-
The numerator of the inversion-asymmetryparameter
poral oscillations interfere constructively. We note that
A is the expectation value of an inversion-odd opera-
there are several other symmetric excited states besides
tor M that is diagonal in the basis of reciprocal lattice |ψ(0) i that also play the role of |ψ(0)i in the scenario
momenta, with matrix element ±1 for the wave vectors 31,+ k,+
we have outlined, and provide somewhat smaller contri-
±G . Using the first-order expression above for |ψ(t)i,
i butions to the overalldynamics.
we obtain hMi=M (t)+M (t) with
1 2 The observations and theoretical descriptions offered
in this work illustrate how matter-wave diffraction can
M1(t)= X(cid:16)c∗icke−i(ωk,+−ωi,+)tα∗j,iMj,k+c.c.(cid:17) (5) be made highly sensitive to, and strongly manipulated
by, fine features of anoptical lattice. Our workalso sug-
i,j,k
gests an explanation for the momentum-space asymme-
M2(t)= X(cid:16)c∗kcie−i(ωj,−−ωk,+)t(−αj,i)Mj∗,k+c.c.(cid:17) tryobservedinthediffractionofaBose-Einsteinconden-
i,j,k sate of two spin states of 87Rb and released from a spin-
dependent optical lattice reported in Ref. [17] (see also
and M =hψ(0)|M|ψ(0)i. Ref. [18]). The asymmetry was interpreted as evidence
j,i j,− i,+
of a ground-state superfluid that forms with a spatially
These expressions identify two generic scenarios that
dependent phase in the superfluid order parameter. A
lead to a large momentum-space asymmetry. The
later theoretical study [19] found no evidence for such a
first results in oscillations described by both M (t)
1
“twistedsuperfluid”state,whichisconsistentwithnaive
and M (t) and involves a trio of eigenstates of the
2
expectationsgiventhattheopticallatticeandmean-field
unperturbed Hamiltonian H at zero quasimomentum:
0
interactionpotentials experiencedbythe atomsareboth
two inversion-symmetric, |ψ(0)i and |ψ(0)i, and one
i,+ k,+ real valued.
(0)
inversion-antisymmetric, |ψ i. These states can be We suggest that the inversion-asymmetric diffraction
j,−
identified by three key features. First, the symmetric patterns observed in the experiment [17] may have re-
states have significant population at zero momentum so sulted from matter-wave diffraction from the inversion-
astooverlapwiththestationarycondensate,givinglarge asymmetric transient honeycomb lattice that repulsion
ci and ck. Second, the inversion-antisymmetric state is from one atomic spin state creates for the second spin
close in energy to one of the inversion-symmetric states, state. Such a transient lattice potential would have an
say |ψ(0)i, so that α is large and they are strongly interaction-energyasymmetrybetweentheAandBsites
i,+ j,i
mixed by the perturbation H . Finally, the inversion- of the honeycomb lattice that is on the order of the su-
a
antisymmetric state and at least one of the inversion- perfluid chemical potential (around h × 1 kHz). This
symmetric states, say |ψ(0)i, have large population in potential would persist for a time somewhat less than
k,+
thefirst-orderdiffractionmomenta,sothatM islarge. the recoil time (i.e., around 100 µs). The strength and
j,k
When these criteria are satisfied, we expect modulations duration of this asymmetric potential are comparable to
of equal strength in M (and thus in A) at frequencies thosestudiedinthepresentwork. Theinteraction-driven
ω −ω and ω −ω . The second scenario is de- diffractionofonematterwaveoffanothercancanbe de-
k,+ i,+ j,− k,+
scribedbyM (t)whenk =iandinvolvesjusttwostates scribed equivalently as nonlinear coherent wave mixing
2
– |ψ(0)i and |ψ(0)i. These states are againcharacterized inducedbyinteratomicinteractions[20]. Theobservation
i,+ j,− in Ref. [17] that the sign of the asymmetry parameter A
by large c and α , and must both have large popula-
i j,i wasconsistentbetweenexperimentalrepetitionssupports
tion in the first-order diffraction momenta so that M
j,i ourviewthattheasymmetryresultedfromdeterministic
is large. This scenario results in a modulation of A at
matter-wave dynamics rather than by spontaneous sym-
frequency ω −ω .
j,− i,+ metry breaking at a phase transition. Moreover,in a re-
In Fig. 4(d) we show that just one trio of states in
centexperimentwiththesamesystemasinRef.[17],the
this perturbationpictureexplainsmostofthe dynamical
diffraction was modified by eliminating one spin popula-
variation in A. Figure 4(c) shows that the state |ψ1(0,+)i tion from the lattice just before the atoms were released
haslargepopulationinthezeroandfirst-orderdiffracted [21]. Theconsequenteliminationoftheasymmetrysignal
momenta, that |ψ(0)i has large population in the first- is consistent with our suggested explanation.
2,−
order momenta, and that these states are heavily mixed This workwassupportedbythe NSF andthe AFOSR
by the perturbation, i.e., that α2,1 is large. As a result, through the MURI program.
6
[1] K.I.Petsas, A.B.Coates, andG.Grynberg,Phys.Rev. Pritchard, Phys.Rev.Lett. 60, 515 (1988).
A 50, 5173 (1994). [12] S.Wu,Y.-J.Wang,Q.Diot,andM.Prentiss,Phys.Rev.
[2] I.Bloch, Nat. Phys.1, 23 (2005). A 71, 043602 (2005).
[3] I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 8, [13] G.-B. Jo, J. Guzman,C. K.Thomas, P.Hosur, A.Vish-
267 (2012). wanath,andD.M.Stamper-Kurn,Phys.Rev.Lett.108,
[4] J. Sebby-Strabley, M. Anderlini, P. S. Jessen, and J. V. 045305 (2012).
Porto, Phys. Rev.A 73, 033605 (2006). [14] C. Cohen-Tannoudji and J. Dupont-Roc, Phys. Rev. A
[5] M. Ladd and R. Palmer, Structure Determination by X- 5, 968 (1972).
ray Crystallography: Analysis by X-rays and Neutrons [15] O. Morsch and M. Oberthaler, Rev.Mod. Phys.78, 179
(Springer,New York,2013), 5th ed. (2006).
[6] P. Soltan-Panahi, J. Struck, P. Hauke, A. Bick, [16] B. Gadway, D. Pertot, R. Reimann, M. G. Cohen, and
W. Plenkers, G. Meineke, C. Becker, P. Windpassinger, D. Schneble,Opt.Express 17, 19173 (2009).
M. Lewenstein, and K. Sengstock, Nat. Phys. 7, 434 [17] P. Soltan-Panahi, D. S. Luhmann, J. Struck, P. Wind-
(2011). passinger, and K. Sengstock, Nat.Phys. 8, 71 (2012).
[7] A. Wicht, J. M. Hensley, E. Sarajlic, and S. Chu, Phys. [18] O. Ju¨rgensen, K. Sengstock, and D.-S. Lu¨hmann, Sci.
Scr.2002, 82 (2002). Rep. 5, 12912 (2015).
[8] M. Kasevich and S. Chu,Appl.Phys. B 54, 321 (1992). [19] S.ChoudhuryandE.J.Mueller,Phys.Rev.A87,033621
[9] P. E. Moskowitz, P. L. Gould, S. R. Atlas, and D. E. (2013).
Pritchard, Phys.Rev. Lett.51, 370 (1983). [20] D.Pertot,B.Gadway,andD.Schneble,Phys.Rev.Lett.
[10] Y. Ovchinnikov, J. Muller, M. Doery, E. Vredenbregt, 104, 200402 (2010).
K. Helmerson, S. Rolston, and W. Phillips, Phys. Rev. [21] M. Weinberg, O. Ju¨rgensen, C. O¨lschl¨ager, D.-S.
Lett.83, 284 (1999). Lu¨hmann, K. Sengstock, and J. Simonet, Phys. Rev. A
[11] P. J. Martin, B. G. Oldaker, A. H. Miklich, and D. E. 93, 033625 (2016).