Atoms in a radiofrequency-dressed optical lattice

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by  N. Lundblad

Atoms in a radiofrequency-dressed optical lattice
N. Lundblad,∗ P. J. Lee, I. B. Spielman, B. L. Brown, W. D. Phillips, and J. V. Porto
Joint Quantum Institute, National Institute of Standards and Technology
and University of Maryland, Gaithersburg, Maryland 20899, USA
(Dated: February 3, 2008)
Weloadcoldatomsintoanopticallatticedramaticallyreshapedbyradiofrequency(rf)coupling
of state-dependent lattice potentials. This rf dressing changes the unit cell of the lattice at a
subwavelength scale, such that its curvature and topology departs strongly from that of a simple
sinusoidallatticepotential. Radiofrequencydressinghaspreviouslybeenperformedatlengthscales
frommmtotensofµm,butnotatthesingle-optical-wavelengthscale. Atthislengthscalesignificant
coupling between adiabatic potentials leads to nonadiabatic transitions, which we measure as a
functionoflatticedepthanddressingfrequencyandamplitude. Wealsoinvestigatethedressingby
measuring changes in the momentum distribution of the dressed states.
8
PACSnumbers: 03.75.Mn,03.75.Lm
0
0
2
For almost twenty years, optical lattices have been Fig. 1(b). After holding the atoms in the dressed lat-

n used to cool and confine neutral atoms, recently lead- tice for a variable time, we abruptly turn off the lattice
a ing to condensed-matter-analog systems, optical clocks and observe the momentum distribution by imaging the
J
2 and possible platforms for quantum computing [1, 2, atoms after time-of-flight (TOF) time tTOF. This gives
3, 4, 5, 6]. The appeal of optical lattices arises from informationaboutthestructureofthedressedlattice, as

] their versatility and tunability, e.g., through control of well as a measure of the lifetime of the dressed adiabatic
h the topology, depth, and spin dependence of the lat- eigenstates.
p
tice [7, 8, 9, 10]. Here we extend this control by rf
- Our apparatus, described elsewhere [21], produces
nt dressing a spin-dependent lattice, a technique which has BECs of ∼105 atoms in a Ioffe-Pritchard magnetic trap.
been recently proposed [11]. The dressed lattice com-
a We load the m =−1 condensate into the ground band
F
u prisesunitcellswhoselocalizedadiabaticeigenstatesare
q spatially-varying superpositions of the bare spin states.
[
The adiabatic potentials have new subwavelength struc-
1 ture,whichincertainlimitscanbering-likewithinaunit D2: 780 nm Energy (ER=3.67 kHz)
5
v cell.
0
5 m = –1
Thecreationofrf-dressedadiabaticpotentialsininho- 790.76 nm F
0
4 mogeneous magnetic fields is well-known experimentally, D1: 795 nm
0 from the use of rf transitions in evaporative cooling [12]
1. to the creation of shell-like quasi-two-dimensional sys- mF = +1 5
0
0 tems in magnetically-trapped condensates [13, 14, 15], ω
lattice
8 and the use of rf dressing to effect condensate splitting
0 and interferometry in micron-scale chip traps [16]. Op- -50
:
v tical lattices with sub-half-wavelength structure in 1D
Xi have been generated by several groups through the use mF = –1 - 05 ∆
ω
of multiphoton Raman processes [17, 18, 19] or static rf -20
r
a magnetic field couplings [20]. The approach we present mF = 0
here, using the rf dressing of a state-dependent lattice, -100
m = +1
F −λ/2 position λ/2 −λ/2 position λ/2
creates 2D subwavelength structure beyond simple lat-
(a) (b) (c)
tice period division. We create such a dressed lattice,
and load ultracold atoms into the ground band of the
FIG.1: (coloronline)(a)Theenergylevelsof87Rbrelevantto
uppermost adiabatic potential.
our lattice; the rf coupling can selectively address either m
F
For these state-dependent lattice experiments we use transition due to the quadratic Zeeman shift. (b) Top two
images: the bare lattice potentials for m = ±1, exhibiting
a spin-polarized 87Rb (52S |F = 1,m = −1(cid:105)) Bose- F
1/2 F approximately opposite light shifts. Bottom two images: the
Einsteincondensate(BEC).Thestatedependenceispro-
uppermost adiabatic potential, weakly dressed and strongly
duced by tuning the lattice laser between the 87Rb D1
dressed. (c) From top: bare lattice potentials for U (cid:39)10E ;
R
and D2 resonances, where vector light shifts are im- dressed adiabatic potentials at ω /2π=35.90 MHz, Ω/2π=
rf
portant [9]. Applying an rf field of appropriate ampli- 20 kHz; dressed adiabatic potentials at ω /2π=35.90 MHz,
rf
tude and frequency creates an adiabatic potential whose Ω/2π = 205 kHz. Note the significantly increased gap for
shape is distorted from the bare potentials, as shown in stronger coupling.
2
ofa3Dopticallatticeviaanexponentiallyincreasingin-
tensity ramp of duration 300 µs (τ = 50 µs), a timescale
adiabatic with respect to band (vibrational) excitation.
We then remove the magnetic trapping fields, leaving a
uniform bias field of 5.117(3) mT [27], a Zeeman res-
onance ν = 36.12(2) MHz and a quadratic Zeeman
−1,0
shiftδ(cid:48)/2π =ν −ν =376(1)kHz,wherehν is
−1,0 0,+1 m,m(cid:48)
the positive energy difference between Zeeman sublevels
|F =1,m (cid:105).
F
The lattice in the xˆ-yˆ plane (xy lattice) is similar to
that described in [21]. We tune a Ti:sapphire laser to
λ =790.76 nm, where for σ+ light, atoms in the three
F = 1 Zeeman sublevels m =−1, 0, +1 experience
F
total light shifts in the approximate ratio of −3 : 1
: 5. This can be understood as the sum of a state-
independentscalarlightshiftandastate-dependentvec-
tor light shift—an effective magnetic field related to the
local ellipticity of the optical polarization. For appropri-
ate phase shifts of the lattice beams [9], the combination
√
of the light shifts and the bias field along (xˆ − yˆ)/ 2 FIG. 2: (color online) (a) The width of the momentum dis-
results in the potentials shown in Fig. 1. tribution as a function of final dressing frequency, as mea-
The atoms are confined along zˆ(the direction of grav- sured by absorption imaging. The width is calculated as
ity) using a 1D optical lattice, derived from the same Md/(cid:126)tTOF, where d is the Gaussian 1/e radius along one
direction, corrected for the initial cloud size. The displayed
Ti:sapphire laser [28]. These lattice beams are lin-
uncertainties represent 1σ shot-to-shot scatter of five points.
early polarized, yielding an effectively blue-detuned and
Theshadedarearepresentspredictedwidthsgiventhevalues
spin-independent lattice. As measured by pulsed-lattice
and estimated uncertainties of the magnetic field (5.117(3)
diffraction [22] this vertical lattice is 6ER deep, where mT) and the depth of the bare lattice U = 10.0(5)ER. Also
ER =(cid:126)2k2/2M (cid:39)h×3.68 kHz, k =2π/λ and M is the shown are the bare and dressed single-site potentials (solid
atomic mass. This lattice, together with the xy lattice, line) and ground-state wavefunctions (gray). (b,c) Sample
fully confines the atoms in 3D, and we do not observe images of the off-resonant (effectively bare) and dressed mo-
mentumdistributionsrespectively(bothatΩ/2π=205kHz)
any displacement due to gravity over the duration of the
showingsignificantnarrowingofthelatter. (d,e)Examplesof
experiment, even though the magnetic trap has been re-
dressedmomentumdistributionscreatedthroughfaster (>1
moved.
MHz/ms)rframps,suchthathighervibrationalstatesofthe
After loading, we wait 30 ms for the atoms to dephase dressed lattice are populated.
andfillthelowestBlochbandofthelattice[8],suchthat
asudden-releaseTOFimagewillapproximatethesingle-
lattice-site momentum distribution (with resolution lim-
itedmainlybythespatialextentoftheatomcloud). The model of the lattice [21]. Ω is obtained through obser-
dephasing time is well within the 1/e lifetime of atoms vations of rf-driven oscillations between m = −1 and
F
in the lattice due to spin-flip loss from light scattering, m = 0 (in a spin-independent lattice) at rates up to a
F
which we measured to be 85 ms at an xy lattice depth maximum of ω /2π = 200 kHz. Ω differs slightly from
osc
of U = 55E for m = −1 atoms; this lattice depth is ω due to the multilevel nature of this system. Diago-
R F osc
more than twice the largest used in the experiment. The nalizingonlyH (i.e.,theBorn-Oppenheimerapproxima-
1
depth of the xy lattice was measured with two-photon tion) gives rise to adiabatic potentials; failure of this ap-
Raman vibrational spectroscopy [23]. proximation will manifest as momentum-dependent cou-
We dress the lattice with an rf magnetic field aligned plings between the adiabatic potentials.
perpendicular to the bias field. The single-particle We apply rf beginning well below resonance((cid:39) −1
rotating-frame Hamiltonian, taking the rotating-wave MHz), sweeping at constant rate (300 kHz/ms) over (cid:39)
approximation (RWA), is H=p2/2M +H (r) , where 3 ms to a final near-resonant frequency, and hold the
1
newly created dressed state for a variable time. The lat-
  ticebeams,rffield,andbiasfieldarethenturnedoffin(cid:46)
V (r)−δ Ω/2 0
−1
1 µs, (cid:39) 10 µs, and (cid:39) 300 µs, respectively. After TOF of
H1(r)= Ω/2 V0(r) Ω/2  (1) 12.2 ms we observe the atoms with resonant absorption
0 Ω/2 V (r)+δ+δ(cid:48)
+1
imaging along the zˆdirection.
and δ = ω − 2πν . The state-dependent lattice Fig.2(a)depictstheobservedwidthofthemomentum
rf −1,0
potentials V (r) (see Fig. 1) are calculated using our distribution(fromaGaussianfittotheTOFdensitydis-
mF
3
tribution) as a function of final frequency at maximum
rf power. The optical power in the xy lattice was fixed
such that the bare depth U was (cid:39) 10E . Also shown 103
R
is the expected width from the calculated ground state
momentumwavefunctionintheadiabaticpotential. The )-1s
data show narrowing of the momentum distribution as ( γ
the lattice is dressed, implying significant alteration of e
at
the wavefunction and hence the lattice structure itself. s r
s
Atthesedresseddepths,whilethepotentialexhibitsring- o
d l
like character, the ground state is still simply connected: cte 102
seeFig. 2(a). Theobserveddressedstate(corresponding e
r
r
to mF = −1 without rf) is predominantly a superposi- co lattice depth U
tion of m = ±1 at these rf detunings [29]. Unlike in a
F 8 E
two-levelsystem,theavoided-crossingenergygap(cid:126)∆be- R
10 E
R
tweenadiabaticpotentialsisnotsimplyrelated√toΩ. For 101 12 ER
Ω (cid:28) δ(cid:48), ∆ (cid:39) Ω2/δ(cid:48), and for Ω (cid:29) δ(cid:48), ∆ (cid:39) Ω/ 2−δ(cid:48)/4. 16 ER
ForΩ/2π =205kHz,thelargestusedinthisexperiment,
and ω = 35.90 MHz, we calculate ∆/2π (cid:39) 75 kHz, a
rf
40 50 60 70
valueonlyweaklydependentonU. Tofurthercharacter-
avoided-crossing gap ∆/2" (kHz)
ize our dressed lattice, we performed Raman vibrational
spectroscopy, but observed that the Raman signal did
FIG. 3: (color online) Dressed-state losses as a function of
not persist beyond weak dressing. This may be due to
calculated avoided-crossing gap ∆/2π, for four bare lattice
to shrinking matrix elements between the combined spin
depths. Vertical bars represent uncertainties-of-fit; multiple
and spatial eigenfunctions of the dressed bands, or in-
points near a given ∆ indicate scatter in multiple runs. The
creasing width of the excited bands of the (shallower) data are well-represented by γ(∆) = Ae−B∆/2π, with B =
dressed lattice. 83(6)µs,83(11)µs,82(13)µs,and100(30)µs,forU =8E ,
R
Three adiabaticity criteria are relevant to loading and 10ER, 12ER, and 16ER respectively. The scaling of the data
holding atoms in the ground band of the dressed lattice: does not change if plotted versus Ω.
(1) Adiabatic following of the local spin eigenstates of
H (r) during the upward rf sweep δ(t) is well-satisfied.
1
(2) Deformation of the dressed lattice with respect to
vibrational excitation is nearly adiabatic, yielding pre- counted as loss. Fig. 3 shows loss rates as a function
dominantly ground-band occupation. Faster sweep rates of calculated avoided-crossing gap ∆ as a function of U,
yielded interesting deviations from this condition, shown at dressing frequencies near 35.90 MHz. The rates are
inFig.2(d,e), inwhichweobservelobedstructureinthe firstobtainedthroughtwo-parameterfitstoanexponen-
momentum distributions indicating higher vibrational tial decay and then slightly corrected via subtraction of
states. (3) The Born-Oppenheimer approximation, i.e., theappropriatelyscaledspin-fliplossrate(1/85ms−1 for
thedegreetowhichzero-pointmotioninthelatticedoes 55ER). The data are well-described by γ(∆) = Ae−B∆.
not induce transitions between adiabatic potentials, is Figure 4 shows loss rate at constant Ω as a function of
more difficult to satisfy. In previous experiments involv- U, well described by γ(U)=CeDU.
ingrf-dressedadiabaticpotentials,thelengthscaleswere We consider our observations in the context of semi-
large enough such that this condition was easily met for classical theory for the losses. The traversal of a two-
cold atoms [13, 14, 15, 16]. For our experiment, the bare level avoided crossing is often described by Landau-
lattice confinement is such that, even for our largest rf Zener (L-Z) theory, which for a particle traveling along
coupling,nonadiabaticlossisafactor; inourexperiment x at velocity v through an avoided crossing character-
these loss rates are 101–103 s−1. These losses do play a ized by a gap ∆ and bare energy levels whose differ-
role in the loading process itself, but this is irrelevant as ence E(x) = Ex, the adiabatic following probability is
long as enough atoms are loaded such that subsequent PA = 1−e−21π(cid:126)∆2/(vE(cid:48)). A semiclassical scaling argu-
loss is measurable. ment for the motion of an atom in the ground state of a
We load atoms into the uppermost adiabatic dressed lattice site of depth U yields v ∝ U1/4 and vE(cid:48) ∝ U5/4,
potential using the procedure described above, and mea- and a loss rate scaling as γ ∝ ω (1−P ), where ω is
l A l
sure the number of atoms remaining in the central mo- a trapping frequency and (1−P ) (cid:28) 1. For a parti-
A
mentum feature, as in Fig. 2(b-e), as a function of hold cle of insufficiently large energy with respect to ∆ (i.e.,
time. Atoms transitioning to lower adiabatic potentials atomstrappedinthegroundstateofanadiabaticpoten-
appear in absorption images as high-momentum rings of tial), the simple L-Z approach may not apply. In a more
∼ 10(cid:126)k (well outside the range of Fig. 2(b-e)), and are sophisticated semiclassical approximation [24, 25], more
4
ficient rf coupling; based on the data presented here, we
-1s) 103 extrapolate that ringlike wavefunctions of lifetime (cid:39)100
(γ
e ms should appear for a coupling Ω/2π (cid:39) 400 kHz and
at
r U (cid:39)55E .
s R
s
d lo 102 This paper presents observations of a novel lattice po-
e tential consisting of rf-coupled components of a state-
ct
re dependent optical lattice. The lifetime of this dressed
r
o
c lattice depends exponentially on both coupling strength
andopticallatticedepth. Weobservelifetimesupto100
101
msforstrongly-dressedshallowlattices,atimescalesuffi-
8 12 16 20 24
cient for the realization of many-body condensed-matter
bare lattice depth U (E )
R systems. Extensions of the band-structure engineering
demonstrated by this experiment lead to lattice geome-
FIG. 4: (color online) Dressed-state losses as a function of
tries with structures much more complex than possible
barelatticedepthU,atconstantΩ/2π=205kHzanddress-
with conventional optical lattices, leading to intriguing
ingfrequency35.900(35.875)MHzforthefirstfive(lastfour)
depths, resulting in approximately constant gap ∆/2π (cid:39) 75 single-site wavefunctions and the next-nearest-neighbor
kHz. Vertical bars represent uncertainties-of-fit; multiple interactions required for extended Bose-Hubbard mod-
pointsatagivendepthrepresentthescatterinmultipleruns. els [26].
Horizontal bars represent known drifts in laser power; not
We thank P. Zoller for inspiring this particular exper-
shownisaconstantsystematicuncertaintyindepthdetermi-
iment and A. J. Daley, G. Pupillo, and P. Z. for ongoing
nationontheorderof5%. Thedataarewell-representedbya
simple exponential fit γ(U)=CeDU, yielding C =1.2(4)s−1 helpful discussions. This work was partially supported
and D=0.27(3)E−1. by DTO and ONR. N.L., P.J.L., and B.L.B. acknowl-
R
edge support from NRC fellowships.
appropriate to ground-state trapped atoms, the result-
ing transition rates can be estimated as γ ∝ ωle−α∆/ωl,
(cid:112)
where ω ∝ U/∆ is the trapping frequency (ω (cid:28) ∆)
l l ∗ Electronic address: [email protected]
of the uppermost adiabatic potential, and α ∼ 1 is a
[1] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998).
constant. The shape of our adiabatic potentials differs [2] M. Greiner et al., Nature 415, 39 (2002).
significantly from this simple case, but the scaling sug- [3] D. Jaksch et al., Phys. Rev. Lett. 82, 1975 (1999).
gests transition rates that change exponentially with the [4] G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H.
coupling gap and bare lattice depth. Deutsch, Phys. Rev. Lett. 82, 1060 (1999).
[5] M. Anderlini et al., Nature 448, 452 (2007).
Unsurprisingly, the L-Z model does not describe the
[6] M. Takamoto et al., Nature 435, 321 (2005).
datainFigs.3and4;theexponentialcoefficientscalesas
[7] O. Mandel et al., Phys. Rev. Lett. 91, 010407 (2003).
−∆ratherthan−∆2 andalsoscalesasU, not−1/U5/4. [8] M. Greiner et al., Phys. Rev. Lett. 87, 160405 (2001).
Fits to the L-Z model are significantly worse than those [9] P. J. Lee et al., Phys. Rev. Lett. 99, 020402 (2007).
in Figs. 3 and 4, although they agree with the general [10] J. Sebby-Strabley et al., Phys. Rev. Lett. 98, 200405
trends. Similar results were found upon calculating a (2007).
[11] A. J. Daley, G. Pupillo, Y. Wei, P. Zoller et al., to be
Fourier-weightedsumoftheL-Zformulaoverthedressed
2D ground-state wavefunction, (cid:82) |ψ |2P (k)dk. The published (2007).
k NA [12] W. Ketterle and N. J. van Druten, Adv. At. Mol. Opt.
moresophisticatedsemiclassicalapproximationperforms
Phys. 37, 40 (1996).
better than the L-Z approach, but still fails to satisfac- [13] O.ZobayandB.M.Garraway,Phys.Rev.Lett.86,1195
torily predict the observed scaling. It is likely that 1D (2001).
semiclassical descriptions are simply inadequate for this [14] M. White et al., Phys. Rev. A 74, 023616 (2006).
2D quantum problem; perhaps a more complete solution [15] Y. Colombe et al., Europhys. Lett. 67, 593 (2004).
[16] T. Schumm et al., Nat. Phys. 1, 57 (2005).
will yield the very simple scaling we observe. Further-
[17] G. Ritt et al., Phys. Rev. A 74, 063622 (2006).
more, heating to higher vibrational levels of the upper-
[18] R. Gupta et al., Phys. Rev. Lett. 76, 4689 (1996).
most adiabatic potential, likely caused by instability in
[19] R. Zhang et al., Phys. Rev. A 72, 043409 (2005).
the rf dressing frequency (measured to be on the order [20] D. L. Haycock et al., Phys. Rev. Lett. 85, 3365 (2000).
ofseveralkHz),islikelytoincreasethenonadiabaticloss [21] J.Sebby-Strableyetal.,Phys.Rev.A73,033605(2006).
rate due to increased mean velocity in higher bands. [22] Phys. Rev. Lett. 83, 284 (1999).
[23] H. Perrin et al., Europhys. Lett. 42, 395 (1998).
While the dressed potentials shown in Fig. 1(b) are
[24] A.Kazantsev,G.Surdutovich,andV.Jakovlev,Mechan-
ringlike, the corresponding ground-state wavefunctions
ical Action of Light on Atoms, §1.3 (World Scientific,
are generally not. Nevertheless, ringlike wavefunctions 1990).
with reasonable lifetimes would be possible had we suf- [25] A. J. Daley, private communication.
5
[26] V. W. Scarola, E. Demler, and S. D. Sarma, Phys. Rev.
A 73, 051601 (2006).
[27] Unless otherwise stated, all uncertainties herein reflect
the uncorrelated combination of 1σ statistical and sys-
tematic uncertainties.
[28] Theverticallatticebeamsdeviatefromcounterpropaga-
tion by 18◦, and also differ in frequency from the 2D
lattice beams by (cid:39) 160 MHz.
[29] Separation of the dressed clouds using a magnetic field
gradient during TOF revealed the expected spin charac-
ter.