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Dynamic optical lattices of sub-wavelength spacing for ultracold atoms
Sylvain Nascimbene,1, ⇤ Nathan Goldman,1, 2 Nigel Cooper,3 and Jean Dalibard1
1Laboratoire Kastler Brossel, College de France, ENS-PSL Research University,CNRS, UPMC-Sorbonne Universites, 11 place Marcelin Berthelot, 75005 Paris, France
2CENOLI, Faculte des Sciences, Universite Libre de Bruxelles (U.L.B.), B-1050 Brussels, Belgium3T.C.M. Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
(Dated: June 2, 2015)
We propose a scheme to realize lattice potentials of sub-wavelength spacing for ultracold atoms.It is based on spin-dependent optical lattices with a time-periodic modulation. We show that theatomic motion is well described by the combined action of an e↵ective, time-independent, latticeof small spacing, together with a micro-motion associated with the time-modulation. A numericalsimulation shows that an atomic gas can be adiabatically loaded into the e↵ective lattice groundstate, for timescales comparable to the ones required for adiabatic loading of standard optical lattices.We generalize our scheme to a two-dimensional geometry, leading to Bloch bands with non-zeroChern numbers. The realization of lattices of sub-wavelength spacing allows for the enhancementof energy scales, which could facilitate the achievment of strongly-correlated (topological) states.
Optical lattices have allowed experiments on ultracoldatomic gases to investigate a large range of lattice modelsof quantum many-body physics [1]. Their developmentled to the realization of strongly-correlated states of mat-ter, such as bosonic and fermionic Mott insulators, andlow-dimensional gases [2]. In its simplest form, an opticallattice consists of the optical dipole potential associatedwith a standing wave of retro-reflected laser light. It canbe described as a periodic potential V (x) = U
0
cos2(kx),of spatial period d = �/2, where � is the laser wavelengthand k = 2⇡/�. More complex optical lattices, such as su-perlattices [3, 4] or two-dimensional honeycomb lattices[5, 6], can be generated with suitable laser configurations.The recoil energy E
r
= h2/(8md2), where h is Planck’sconstant and m is the atom mass, sets the natural energyscale for elementary processes, such as atom tunnelingbetween neighboring lattice sites, as well as the temper-ature range T . E
r
/kB
⇠ 100 nK, typically required forquantum degeneracy.
For a large class of models, the physical behavior isdictated by processes associated with even much smallerenergies, such as super-exchange or magnetic dipole in-teractions [1]. The associated temperature scales remainout of reach in current experiments. In order to circum-vent this limitation, it is desirable to find novel schemesfor generating optical lattices with spacing d
e↵
⌧ �,in order to enhance the associated energy scale Ee↵
r
=h2/(8md2
e↵
) [7]. Schemes have been proposed to gener-ate lattices of sub-wavelength spacing, based on multi-photon optical transitions [8] or on adiabatic dressing ofstate-dependent optical lattices [7]; the realization of lat-tices with spacing d
e↵
= �/4 was reported in Ref. [9]. Aninteresting alternative would be to trap atomic gases inthe electromagnetic fields of nano-structured condensed-matter systems [10–12].
In this letter, we propose a novel scheme leading tolattices of spacing d
e↵
= d/N , N being an arbitrary inte-ger, based on spin-dependent lattices with time-periodic
0
1
0 t < T/4
V(x,t)
0
1
T/4 t < T/2
0
1
T/2 t < 3T/4
0
1
3T/4 t < T
0 1 20
0.25
x[d]
Ve↵(x)
FIG. 1. Stroboscopic scheme for engineering short-spacinglattices, illustrated on the case N = 4. We make use of aperiodic potential V (x, t) of spatial period d, that is shifted ofthe distance d/N after every time step of duration T/N (bluecurves). The e↵ective potential Ve↵(x) (red curve), resultingfrom time averaging, exhibits a spatial period de↵ = d/N .
modulation. In the regime of large modulation frequency[13–16], the atom dynamics is governed by an e↵ec-tive static periodic potential of spacing d
e↵
, with anadditional micro-motion. This description is confirmedby a numerical simulation, which shows the possibilityto load adiabatically the ground state of the e↵ectivelattice and to perform Bloch oscillations. We discussthe extension of the scheme to two-dimensional latticeswith non-trivial topology. Lattices with artificial mag-netic fields, generally leading to topological bands, wererecently realized in experiments, with standard latticespacing [17]. For those systems, increasing the energyscale using short-spacing lattices could prove important
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2
for creating strongly-correlated states such as fractionalChern insulators [18, 19].
A basic scheme of our method is pictured in Fig. 1.Consider a periodic potential V (x) of period d, whichis abruptly shifted by the distance d/N at stroboscopictimes tn = (n/N)T , n 2 Z, leading to a time-periodicpotential V (x, t) of period T . Provided that T is muchsmaller than typical timescales of atomic motion, theatoms experience an e↵ective time-averaged potentialVe↵
(x) =RV (x, t)dt/T . A simple calculation shows that
Ve↵
(x) is given by the sum of all harmonics of the po-tential V (x), whose orders are multiples of N [20]. Thee↵ective potential V
e↵
(x) is thus spatially periodic, ofspatial period d
e↵
= d/N .Conventional optical lattices present a spatial modu-
lation proportional to the intensity pattern of interferinglight waves, which exhibit spatial frequencies of at mosttwice the light momentum k. Thus, applying the stro-boscopic scheme in Fig. 1 to these potentials could notlead to e↵ective lattices of period d
e↵
< �/2. This re-striction does not apply to spinful particles subjectedto spin-dependent optical lattices. As an illustration,consider a spin-1/2 particle evolving in the potentialV (x) = V
L
cos(2kx)�z + VB
�x, where �u (u = x, y, z)are the Pauli matrices. In a dressed state picture, theatom may follow adiabatically the state of lowest energyV�(x) = �
pV 2
L
cos2(2kx) + V 2
B
. As this potential ex-hibits harmonics of the spatial frequency 2k of all orders,the lattice spacings achievable by applying the strobo-scopic scheme to V�(x) can be made arbitrarily small.
We describe in the following a modified, more practical,version of this scheme, which consists of a spin-dependentoptical lattice with smooth temporal variations, given by
V (x, t) = VL
cos(2kx� ⌦t)�z + VB
cos(N⌦t)�x. (1)
This potential satisfies V (x + d/N, t + T/N) = V (x, t),with d = ⇡/k, thus, it can be viewed as a continuousversion of the stroboscopic scheme. Understanding thephysical e↵ects of the potential (1) falls within the de-scription of time-periodic Hamiltonian systems [13–16].Following Ref. [15], we describe the dynamics of an atombetween the times t
i
and tf
as
U(ti
! tf
) = e�iK(tf )e�i~ (tf�ti)Heff eiK(ti), (2)
where we introduce a time-independent, e↵ective Hamil-tonian H
e↵
and a time-periodic kick operator K(t). Thethree operators in (2) describe, from right to left, therole of the initial phase of the Hamiltonian at time t
i
, theevolution from t
i
to tf
according to a stationary Hamil-tonian, and the micro-motion related to the final phaseof the Hamiltonian at time t
f
.The expressions for the e↵ective Hamiltonian
He↵
and kick operator K(t) can be calculatedthrough a perturbative expansion in powers of1/⌦, see Refs. [14, 15]. To lowest-order, this yields
0
5
10(a)
�1 0 1
0
50
100
150
q [k]
![E
r/~]
0
5
10
![E
e↵
r/~]
(b)
�4 �2 0 2 4
0
50
100
150
q [k]
FIG. 2. Band structure of a dynamic optical lattice of spac-ing de↵ = d/4, corresponding to the parameters N = 4, andVL = VB = ~⌦ = 200Ee↵
r . In (a), we make use of the spa-tial and temporal translational symmetries T
x
, Tt
and labelthe eigenstates by their quasi-momentum �k q < k andquasi-energy �~⌦/2 ~! < ~⌦/2. The Bloch-Floquet bandscan be unfolded using the additional symmetry T ⇤, leadingto the band structure in (b), indexed by the modified quasi-momentum �4k q < 4k. The unfolding of the band struc-ture can be followed from the di↵erent coloring of successivebands.
He↵
=p2
2m+ V
e↵
(x), (3)
Ve↵
(x) =Ue↵
2cos(2Nkx)�x, U
e↵
=2V
B
N !
✓VL
~⌦
◆N
, (4)
K(t) =�V
L
~⌦ sin(2kx� ⌦t)�z +VB
N~⌦ sin(N⌦t)�x. (5)
The expression (4) is derived under the assumption thatN is an even integer. In the Supplementary material weshow that the perturbative expansion can be resumed,with respect to either the variable V
L
/(~⌦) or VB
/(~⌦)[20]. The e↵ective potential (4) describes a periodicpotential of depth U
e↵
and spatial period de↵
= d/N .In order to test the validity of the e↵ective Hamil-
tonian, we performed a numerical study of the time-periodic Hamiltonian using the Floquet formalism. Sincethe Hamiltonian H is invariant under the symmetriesTx : x ! x+d and Tt : t ! t+T , we look for eigenstateswritten as Bloch-Floquet wave functions q,!(x, t) =ei(qx�!t)uq,!(x, t), where uq,!(x, t) is d-periodic in x andT -periodic in t [21, 22]. Eigenstates are labelled bytheir quasi-momentum �k < q k and quasi-energy0 ~! < ~⌦. An example of the band structure calcu-lated numerically for N = 4 is plotted in Fig. 2a. Theband structure exhibits gap openings once every fourbands, at the momenta Nkp, where p 2 Z⇤, as expectedfor a lattice of spacing d/N .The band structure can be unfolded, making use of the
additional symmetry T ⇤ : x ! x + d/N, t ! t + T/N .
3
�4 �3 �2 �1 0 1 2 3 4
x [d]
n(x)[a.u.]
(a)
05
10
x [d]0
1
2
t [⌧B
]
n(x,t)
(b)
0 0.5 1
0
0.5
1
t [⌧B
]
xcm(t)[W
e↵/F]
(c)
FIG. 3. (a) Atomic density of a wave-packet loaded into adynamic lattice of spacing de↵ = d/4. We start from a gaus-sian wave-packet, spin-polarized along x, of wave function (x, t = 0) = exp[�x2/(2�2)], with � ' 1.4 d (red dashedline). The lattice depth VL is slowly ramped up for a du-ration tramp = 20~/Ee↵
r from VL = 0 to VL = V 0L , and lat-
tice parameters N = 4, V 0L = VB = ~⌦ = 200Ee↵
r . Theatom density after loading is spatially modulated, with a pe-riod d/4 (blue line). (b) Evolution of the density distributionduring Bloch oscillations, calculated for the dynamic latticeparameters of (a), and for a force F = We↵/(8 de↵), whereWe↵ ' 0.06Ee↵
r is the expected bandwidth of the lowest bandfor Ue↵ = 10.9Ee↵
r . (c) Evolution of the center-of-mass posi-tion during Bloch oscillations, calculated for a standard op-tical lattice of depth Ue↵ = 10.9Ee↵
r (red dashed line), andfor the dynamic optical lattice (blue line). The time and spa-tial coordinates are plotted in units of the ideal Bloch period⌧B
= 2N~k/F and amplitude We↵/F .
As explained in the Supplementary Material, eigenstatesassociated with the symmetries Tx, Tt and T ⇤ can bewritten as q,!(x, t) = ei(qx�!t)vq,!(x, t), where vq,!(x, t)is d/N -periodic in x and 2⇡-periodic in (kx�⌦t) [20]. Weshow the band structure calculated within this formalismin Fig. 2b, which is very close to that expected for alattice of spacing d/N and depth U
e↵
' 10.9Ee↵
r
[23] .
The practical relevance of the short-spacing lattice de-scribed above is based on the ability to load atoms intothe ground state of the e↵ective potential (4). The anal-ysis of this loading protocol requires special care, as thee↵ective-Hamiltonian approach inherent to Eq. (2) as-sumes a constant lattice depth [15]. In fact, we find thatthe concept of the e↵ective Hamiltonian can be modifiedso as to describe the time-evolution under a ramp of themoving-lattice depth V
L
, see Ref. [20]. We simulate thelattice loading from a numerical calculation of the fulldynamics of an atomic wave packet under the action of
�0.5 0 0.5
0
5
10
15
q [2Nk]
!�
qv/~[E
e↵
r/~]
v = 0
�0.5 0 0.5
�5
0
5
10
q [2Nk]
v = vlatt
FIG. 4. Band structure corresponding to the dynamic latticefor the parameters N = 2, VL = VB = ~⌦ = 10Ee↵
r . Thepanels correspond to di↵erent frames of reference, of veloc-ity v = 0 (left) and v = vlatt = ⌦/(2k) (right). The bluepoints correspond to the band structure of an optical latticeof spacing de↵ = d/N and depth Ue↵ ' 2Ee↵
r , at rest in thelaboratory frame. The red dots correspond to the band struc-ture of an optical lattice of spacing d and depth U ' 74Er
(' 9Ue↵), at rest in the frame of velocity v = vlatt [24].
the potential (1). Starting from a gaussian wave packet,spin-polarized along x, we solve the Schrodinger equa-tion, discretized in space and time, with a lattice depthVL
slowly ramped up for a duration tramp
. As shownin Fig. 3a, a ramp duration t
ramp
= 20~/Ee↵
r
leads to astate with strong spatial modulations of spacing d/N , asexpected for a wavepacket prepared in the lowest bandof the e↵ective lattice (4). The calculated populationin the e↵ective lowest band is 93%, close to the valueexpected with standard optical lattices for such a rampduration. In the Supplementary Information we analyzethe momentum distribution, which corresponds to theone expected for the ground state of the e↵ective lattice,slightly modified by the micro-motion [20].
The system description as an e↵ective d/N lattice isalso supported by a numerical simulation of Bloch os-cillations. We calculate the action of a linear potential�Fx applied to the state obtained after the lattice load-ing. As shown in Fig. 3b,c, the wave packet undergoesBloch oscillations, revealed as real-space oscillations ofits center of mass. Both the amplitude and period of thisoscillation agree well with those expected for an e↵ectivelattice of period d/N and depth U
e↵
inferred from bandstructure calculations.
The potential V (x, t) written in (1) corresponds tothe sum of a time-modulated magnetic field and a spin-dependent optical lattice moving at the velocity v
latt
=⌦/(2k). In the above discussion we considered the ef-fect of this potential as an e↵ective static optical lattice.
4
�x
�x
�y
�y
�z
�z
qx
qy
a
e1
e2
e3
b
�x
1
�y
i�z
1
⇡/2
FIG. 5. (a) Momentum-space representation of the e↵ectivecouplings in Eq. 6, illustrated as arrows of length 2Nk, ori-ented along the unit vectors±e
i
(i = 1, 2, 3), and proportionalto Pauli matrices. Quantum states are represented in the ba-sis {|+
z
i (filled dots), |�z
i (circles)}. (b) Phase accumulatedaround a triangular subcell of the k-space lattice. Due tothe internal-state degree of freedom, the unit cell of the lat-tice is formed by four triangular subcells. The same phaseof � = ⇡/2 is found to be accumulated around all subcells,indicating that the lowest energy band is associated with anon-trivial Chern number ⌫Ch = 1 [25].
An alternative view is obtained in the frame of referencemoving at the velocity v = v
latt
, where the potentialV (x0 = x � vt, t) consists of the sum of a modulatedmagnetic field and a static lattice V
L
cos(2kx0)�z, withVL
⇠ ~⌦ � Ue↵
. Both points of view can be reconciledby a proper interpretation of the band structure, as illus-trated for the case N = 2 in Fig. 4. Among the eigenener-gies (q,!) calculated numerically in the laboratory framev = 0, we identify the Bloch bands corresponding to astatic e↵ective lattice of spacing d
e↵
. The eigenenergies(q,!0) corresponding to a frame of reference moving at avelocity v can be deduced from those in the laboratoryframe using the relation !0 = !�qv/~. In the frame mov-ing at v = v
latt
, we observe Bloch bands correspondingto a very deep static optical lattice of period d.
We now consider a 2D extension of our scheme. Thetime-dependent part of the Hamiltonian is taken as
V (r, t) = VL
cos(2ke1
· r� ⌦1
t)�x + VB
cos(N⌦1
t)�y
+ VL
cos(2ke2
· r� ⌦2
t)�y + VB
cos(N⌦2
t)�z
+ VL
cos(2ke3
· r� ⌦3
t)�z + VB
cos(N⌦3
t)�x,
where the unit vectors e1,2,3 have directions as repre-
sented in Fig. 5. For a suitable choice of the frequencies⌦
1,2,3 [26], each line of the equation above can be treatedindividually, which results in an e↵ective potential of theform [27]
Ve↵
(r) ' Ue↵
2[ cos(2Nke
1
· r)�x + cos(2Nke2
· r)�y
+ cos(2Nke3
· r)�z], (6)
where N is taken to be an even integer. These couplings
are illustrated in quasi-momentum space in Fig. 5a. Fol-lowing Ref. [25], the topological Chern number associ-ated with the lowest energy band can be readily obtainedfrom these couplings. Indeed, the Chern number mea-sures the flux of the Berry curvature ⌦(q) over the entire(momentum-space) unit cell:
⌫Ch
=1
2⇡
Z
unit cell
⌦(q)d2q, (7)
which can be directly evaluated by calculating the phasesaccumulated by a state as it performs a loop aroundthe triangular subcells [25]. For the e↵ective lattice de-scribed by eq. (6), each unit cell is constituted of fourtriangular subcells, and we find an accumulated phase of⇡/2 within each of them (see Fig. 5b). In this configu-ration, the Chern number of the lowest band is given by⌫Ch
= (1/2⇡) ⇥ 4 ⇥ (⇡/2) = 1. Generally the reasoningabove is valid only in the weak-binding regime; however,for the coupling (6), ⌫
Ch
is unchanged for all values ofUe↵
. Note that the size of the unit cell in real spacescales as 1/N2; we thus expect the flux density to be in-creased by a factor of N2 compared to standard opticallattices.In conclusion, we introduced a novel scheme to en-
gineer spatially periodic atom traps of sub-wavelengthspacing, based on the application of spin-dependent op-tical lattices. The scheme could be implemented withalkali atoms, and more favorably with Lanthanide atomsfor which one expects lower Rayleigh scattering e↵ects[28, 29]. We mention that, since short-spacing latticesare associated with typically higher energy scales com-pared to usual optical lattices, their realization requireslarge laser intensities. A natural extension of this workwould be to include interactions between atoms in thee↵ective lattice description, and to understand whethermicro-motion plays a significant role in scattering proper-ties [30–32]. This aspect will play a central role for inves-tigating quantum many-body physics with short-spacinglattices.The authors are pleased to acknowledge Fabrice Ger-
bier and Jerome Beugnon for valuable discussions.This work is supported by IFRAF, ANR (ANR-12-BLANAGAFON), ERC (Synergy UQUAM), the RoyalSociety of London and the EPSRC. N.G. is financed bythe FRS-FNRS Belgium and by the BSPO under the PAIproject P7/18 DYGEST.
⇤ sylvain.nascimbene@lkb.ens.fr[1] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski,
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80, 885 (2008).[3] J. Sebby-Strabley, M. Anderlini, P. Jessen, and J. Porto,
Phys. Rev. A 73, 033605 (2006).
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[4] S. Folling, S. Trotzky, P. Cheinet, M. Feld, R. Saers,A. Widera, T. Muller, and I. Bloch, Nature 448, 1029(2007).
[5] P. Soltan-Panahi, J. Struck, P. Hauke, A. Bick,W. Plenkers, G. Meineke, C. Becker, P. Windpassinger,M. Lewenstein, and K. Sengstock, Nature Phys. 7, 434(2011).
[6] L. Tarruell, D. Greif, T. Uehlinger, G. Jotzu, andT. Esslinger, Nature 483, 302 (2012).
[7] W. Yi, A. Daley, G. Pupillo, and P. Zoller, New J. Phys.10, 073015 (2008).
[8] B. Dubetsky and P. Berman, Phys. Rev. A 66, 045402(2002).
[9] G. Ritt, C. Geckeler, T. Salger, G. Cennini, andM. Weitz, Phys. Rev. A 74, 063622 (2006).
[10] M. Gullans, T. Tiecke, D. Chang, J. Feist, J. Thompson,J. I. Cirac, P. Zoller, and M. D. Lukin, Phys. Rev. Lett.109, 235309 (2012).
[11] J. Thompson, T. Tiecke, N. de Leon, J. Feist, A. Akimov,M. Gullans, A. Zibrov, V. Vuletic, and M. Lukin, Science340, 1202 (2013).
[12] O. Romero-Isart, C. Navau, A. Sanchez, P. Zoller, andJ. I. Cirac, Phys. Rev. Lett. 111, 145304 (2013).
[13] P. Avan, C. Cohen-Tannoudji, J. Dupont-Roc, andC. Fabre, J. Phys. 37, 993 (1976).
[14] S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. A 68,013820 (2003).
[15] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027(2014).
[16] M. Bukov, L. D’Alessio, and A. Polkovnikov,arXiv:1407.4803 (2014).
[17] N. Goldman, G. Juzeliunas, P. Ohberg, and I. B. Spiel-man, Rep. Prog. Phys 77, 126401 (2014).
[18] E. J. Bergholtz and Z. Liu, Int. J. Mod. Phys. B 27(2013).
[19] S. A. Parameswaran, R. Roy, and S. L. Sondhi, C. R.Phys. 14, 816 (2013).
[20] See supplementary material for a discussion on the stro-boscopic method, the Bloch-Floquet calculation, the re-summation of the perturbative expansion of He↵ , micro-motion e↵ects in momentum space, and the descriptionof the lattice loading.
[21] M. Holthaus, Z. Phys. B Cond. Mat. 89, 251 (1992).[22] M. Grifoni and P. Hanggi, Phys. Rep. 304, 229 (1998).[23] This depth value slightly di↵ers from the perturbative
result Ue↵ ' 16.7Ee↵r , from eq. (4), since V
L,B
6⌧ ~⌦. Wechecked numerically that the di↵erence can be accountedfor by higher-order terms.
[24] The modulated magnetic field VB
cos(N⌦t)�x
renormal-izes the depth U of the optical lattice according toU = 2J0[VB
/(N⌦)]VL
, as observed in the calculatedband structures.
[25] N. R. Cooper and R. Moessner, Phys. Rev. Lett. 109,215302 (2012).
[26] The frequencies ⌦1,2,3 should not be chosen too close toeach other, and their ratios should not approach simplefractions.
[27] N. R. Cooper, Phys. Rev. Lett. 106, 175301 (2011).[28] S. Nascimbene, J. Phys. B: At. Mol. Opt. Phys. 46,
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Supplementary Material for
Dynamic optical lattices of sub-wavelength spacing for ultracold atoms
Sylvain Nascimbene,1, ⇤ Nathan Goldman,1, 2 Nigel Cooper,3 and Jean Dalibard1
1Laboratoire Kastler Brossel, College de France, ENS-PSL Research University,CNRS, UPMC-Sorbonne Universites, 11 place Marcelin Berthelot, 75005 Paris, France
2CENOLI, Faculte des Sciences, Universite Libre de Bruxelles (U.L.B.), B-1050 Brussels, Belgium3T.C.M. Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
(Dated: June 1, 2015)
S.I. EFFECTIVE POTENTIAL CREATED BYTHE STROBOSCOPIC SCHEME
In this Section, we present the calculation of the ef-fective lattice potential produced via the stroboscopicscheme illustrated in Fig. 1 (main text). We start froma periodic potential V (x) of period d, which can be de-composed in Fourier series as
V (x) =X
p2ZVp e
i2⇡px/d.
The method consists in shifting the potential V (x) ofthe distance d/N , after each time interval T/N . For Ninteger, this leads to a time-periodic potential V (x, t)of time period T . For a su�ciently short period T ,the atomic motion is governed by the e↵ective potentialVe↵
(x), equal to the time average of V (x, t):
Ve↵
(x) =1
T
Z T
0
V (x, t)dt
=1
N
NX
j=1
V (x+ jd/N)
=X
p2ZVp e
i2⇡px/d 1
N
NX
j=1
ei2⇡pj/N
=X
p multiple of N
Vp ei2⇡px/d.
It is then apparent that the e↵ective potential Ve↵
(x) isperiodic, of period d
e↵
= d/N .
S.II. EXPRESSION FORTHE BLOCH-FLOQUET HAMILTONIAN
The modulated potential (1) is invariant under thespace and time translational symmetries Tx, Tt and T ⇤,which all commute with each other. The eigenstates of
⇤ sylvain.nascimbene@lkb.ens.fr
the Hamiltonian can thus be written as eigenstates ofthose symmetries, which can be expressed as
q,!(x, t) = ei(qx�!t)X
j,l2Zcj,l e
il(kx�⌦t)eijNkx,
where �Nk < q Nk and 0 ! < ⌦. The spinorcoe�cients cj,l are determined by the equations
~(! + l⌦)cj,l =~2[q + (l +Nj)k]2
2mcj,l
+VL
2�x(cj,l+1
+ cj,l�1
)
+VB
2�z(cj+1,l�N + cj�1,l+N ).
The numerical data represented in Fig. 2 (main text) iscalculated using the above equations, in a truncated basis�10 j, l 10.
S.III. RESUMMATION OFTHE PERTURBATIVE EXPANSION OF He↵
The e↵ective potential Ve↵
can be calculated as a se-ries expansion in powers of the (potentially small) dimen-sionless parameters V
L
/(~⌦) and VB
/(~⌦), in the high-frequency limit ⌦ ! 1. In the main text, we provide itsexpression in Eq. (4), which corresponds to the lowest-order term. We note that this derivation, which is basedon the general formula of Ref. [S1], was obtained by ne-glecting the non-commutativity of the position and mo-mentum operators; indeed, we verified that the momen-tum operator is irrelevant in the derivation of the e↵ectivepotential, which essentially relies on the spin-dependenttime-modulated components of the Hamiltonian. Thus,in the following of this Section, which aims to derive thee↵ective potential in the strong-modulation regime, weexplicitly neglect any e↵ects associated with the kineticenergy term of the full Hamiltonian.In this Section, we first derive the expression for the
e↵ective potential Ve↵
, in the case where VL
/(~⌦) is al-lowed to take arbitrary large values (still assuming thatVB
⌧ VL
, ~⌦). Following Refs. [S2, S3], we perform aunitary transformation
| 0i = R(t) | i , R(t) = exp
✓�i
VL
~⌦ sin(kx� ⌦t)�z
◆,
2
0 1 2 3 4 50
0.5
1
1.5
VL [~⌦]
Ue↵[V
B]
(a)
0 2 4 60
0.05
0.1
VB [~⌦]
Ue↵[V
N L/(~⌦)N
�1]
(b)
FIG. S1. Depth Ue↵ of the e↵ective lattice, calculated forarbitrary values of VL/(~⌦) (a) or VB/(~⌦) (b) in the caseN = 4. The dashed lines correspond to the lowest-order per-turbation result (4), and the solid lines to the resummationresults (S.1) and (S.2).
which removes the diverging term ⇠ VL ⇠ ~⌦ from thetime-dependent potential V (x, t) in Eq. 1 (main text).This leads to a novel time-dependent potential
V 0(x, t) = R(t)V (x, t)(t)R†(t) + i~@tR(t)R†(t)
= R(t) [VB
cos(N⌦t)�x]R†(t).
Making use of the identity e�i��z�xei��z = cos(2�)�x +
i sin(2�)�y, we obtain the expression
V 0(x, t) = VB
cos(N⌦t)
cos
✓VL
~⌦ sin(kx� ⌦t)
◆�x
+ sin
✓VL
~⌦ sin(kx� ⌦t)
◆�y
�.
In the large-frequency limit ⌦ ! 1, the atom dynam-ics can be described by an e↵ective stationary potential,given by [S2, S3]
Ve↵
(x) =1
T
Z T
0
V 0(x, t)dt
= JN
✓2V
L
~⌦
◆VB
cos(2Nkx)�x, (S.1)
assuming N even, and where JN is a Bessel function ofthe first kind. This e↵ective potential corresponds to a
spin-dependent optical lattice of spacing d/N and depthUe↵
= 2JN�2VL~⌦
�VB
.A similar resummation with respect to V
B
/(~⌦) canalso be derived. Here, we make use of the Floquet repre-sentation of time-periodic Hamiltonians. We first writethe exact eigenstates of the coupling VB
2
cos(N⌦t)�x,which read
||n, sxi =X
p2ZJp
✓2sxVB
~⌦
◆|n+ pN, sxi,
where n denotes the Floquet quantum number, and sxis the spin projection along x. The energy of the state||n, sxi is equal to n~⌦. The e↵ect of the couplingVL
cos(kx�⌦t)�x can be understood using perturbationtheory in the degenerate subspace ||n,±i, which must beperformed at order N . We obtain the expression
Ve↵
(x) =Ue↵
2cos(2Nkx)�x,
Ue↵
= 4~⌦✓
VL
2~⌦
◆N������
X
PNi=1 pi=�1
QNi=1
Jpi [(�1)i 2VBN~⌦ ]QN�1
i=1
Pij=1
(1 +Npj)
������.
(S.2)
We plot in Fig. S1 the lattice depth Ue↵
given by the re-summation formulas in Eqs. (S.1)-(S.2) discussed above.We checked that the formulas (S.1) and (S.2) accountwell for the numerical results obtained via direct di-agonalization of the Bloch-Floquet equations (see Sec-tion S.II).
S.IV. MICRO-MOTION EFFECTSIN THE MOMENTUM DISTRIBUTION
In this section, we analyze how the micro-motion as-sociated with the time-modulation in Eq. (1) a↵ects themomentum distribution of atoms prepared in the e↵ec-tive potential V
e↵
of spatial period d/N [Eq. (4)]. Specif-ically, we consider an atom prepared in the ground stateof the e↵ective potential. This state can be expanded onthe family of states of momentum multiple of 2Nk (seeFig. S2a).The actual state created using time-modulated lattices
is expected to be modified by the micro-motion, as
| (t)i = e�iK(t) | 0
i , (S.3)
where the expression for the kick operator K(t) is givenin the main text [Eq. (5)]. The latter leads to additionaldi↵raction peaks at all momenta multiple of 2k, whoseamplitude vary periodically in time, with a period T/N(see Fig. S2b). This shows that Bragg di↵raction doesnot give a direct information on the ground state of thee↵ective lattice.
3
0
1
2
3n(k)[a.u
.](a)
0
1
2t = 0
(b)
0
1
2t = 0.25T/4
0
1
2t = 0.5T/4
�8 �4 0 4 80
1
2t = 0.75T/4
q [2k]
FIG. S2. (a) Momentum distribution associated with theground state of the e↵ective lattice with spacing d/4 anddepth Ue↵ = 10.9Ee↵
r . (b) Momentum distribution of thestate in (a), taking into account the micro-motion expectedfor the dynamic lattice parameters, according to Eq. S.3. Themicro-motion leads to a more complex structure comparedto (a), periodically evolving in time. The lattice parameterscorrespond to the ones of Fig. 2 in the main text.
S.V. EFFECTIVE HAMILTONIANDURING LATTICE LOADING
In this Section, we analyze the adiabatic preparation ofthe ground state associated with the e↵ective potentialVe↵
(x) of depth U0
e↵
. We consider a slow ramp of themoving-lattice depth V
L
(t) = 0 ! V 0
L
during the timeinterval 0 t t
ramp
, such that the e↵ective potential’sdepth U0
e↵
corresponds to the final value VL
(tramp
) = V 0
L
.As the definition (2) of the e↵ective Hamiltonian andkick operators assumes a constant lattice depth [S1], weexpect these notions to be modified during the ramp. Itis the aim of this Section to show how the adiabatic ramp
can still be captured by an e↵ective-Hamiltonian picture.To analyze this situation, we decompose the ramp into
N steps, and we assume that the time interval �t =tramp
/N is short enough, such that VL
can be consideredto remain constant within each step. More precisely, weassume that the lattice depth is equal to V
L
(j�t) duringthe step j�t t<(j+1)�t. We then apply the e↵ective-Hamiltonian formalism of Ref. [S1] within each time-step,and write the full time-evolution operator as
Uramp
=0Y
j=N�1
Uj ,
Uj = e�iK0[VL(j�t)]e�iHeff [VL(j�t)]�teiK0[VL(j�t)].
In the latter expression, and for the sake of simplicity,we assumed that �t was a multiple of the modulationperiod, so that the kick operators at the beginning andat the end of each step only depend on the value of V
L
(in fact, they correspond to the kick operator at the timet = 0, hence the notation K
0
).Assuming �t short enough, we write
e�iK0[VL((j+1)�t)]e�iK0[VL(j�t)] ' e�i�t(dVL/dt)dK0/dVL ,
leading to
Uramp
= e�iK(tramp)T⇢exp
✓�i
ZHramp
e↵
(t)dt/~◆�
,
where T denotes time-ordering, and where one intro-duced the slowly varying Hamiltonian
Hramp
e↵
(t) = He↵
|VL(t)+ ~dVL
dt
dK0
(tramp
)
dVL
����VL(t)
=Ue↵
(t)
2cos(2Nkx)�x � 1
⌦
dVL
dtsin(2kx)�z.
(S.4)
We now estimate a criterion for identifying the adia-batic regime of the lattice loading. Describing the latticeloading solely with the first term of eq. (S.4) would lead tothe standard adiabaticity criterion t
ramp
� ~U0
e↵
/(Ee↵
r
)2
[S4]. We expect the second term in (S.4) to drive non-adiabatic transitions for V
L
& ⌦Er
. Adiabatic latticeloading thus requires the additional constraint t
ramp
�(V 0
L
/Er
)/⌦. As we choose VL
. ~⌦, this constraintshould not be the most restrictive.
[S1] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027(2014).
[S2] P. Hauke, O. Tieleman, A. Celi, C. Olschlager, J. Si-monet, J. Struck, M. Weinberg, P. Windpassinger,
K. Sengstock, M. Lewenstein, et al., Phys. Rev. Lett.109, 145301 (2012).
[S3] N. Goldman, J. Dalibard, M. Aidelsburger, and N. R.Cooper, Phys. Rev. A 91, 033632 (2015).
4
0
0.5
1
VL(t)cos(⌦t)
(a)
0 0.5 1 1.5
0
0.5
1
U0 U1U2
�t
UN�2UN�1
t [tramp]
VL(t)cos(⌦t)
(b)
FIG. S3. (a) Evolution of the amplitude of the moving optical lattice at position x = 0, during and after the lattice ramp ofduration tramp. (b) Scheme of the ramp discretization: the time interval 0 t tramp is decomposed into N steps of duration�t. Within each step the depth VL is constant, leading to a time-periodic potential.
[S4] J. H. Denschlag, J. Simsarian, H. Ha↵ner, C. McKenzie,A. Browaeys, D. Cho, K. Helmerson, S. Rolston, and
W. Phillips, J. Phys. B: At. Mol. Opt. Phys. 35, 3095(2002).
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