PHYSICAL REVIEW A 75, 053612 共2007兲

Species-specific optical lattices
L. J. LeBlanc and J. H. Thywissen
Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
共Received 2 February 2007; published 25 May 2007兲
We examine single-frequency optical schemes for species-selective trapping of ultracold alkali-metal atoms.
Independently addressing the elements of a binary mixture enables the creation of an optical lattice for one
atomic species with little or no effect on the other. We analyze a “tune-in” scheme, using near-resonant
detuning to create a strong potential for one specific element. A “tune-out” scheme is also developed, in which
the trapping wavelength is chosen to lie between two strong transitions of an alkali-metal atom such that theinduced dipole moment is zero for that species but is nonzero for any other. We compare these schemes by
examining the trap depths and heating rates associated with both. We find that the tune-in scheme is preferable
for Li-Na, Li-K, and K-Na mixtures, while the tune-out scheme is preferable for Li-Cs, K-Rb, Rb-Cs, K-Cs,
and 39K- 40K mixtures. Several applications of species-selective optical lattices are explored, including thecreation of a lattice for a single species in the presence of a phononlike background, the tuning of relative
effective mass, and the isothermal increase of phase-space density.
DOI: 10.1103/PhysRevA.75.053612

PACS number共s兲: 03.75.Nt, 03.75.Ss, 33.80.Ps, 39.25.⫹k

I. INTRODUCTION

As the field of ultracold atom research enters its adolescence, experiments are increasingly including more than one
element or isotope. Dual-species experiments offer possibilities for creating heteronuclear polar molecules 关1兴, sympathetic cooling 关2–5兴, and investigating Bose-Fermi mixtures关3–6兴, which may provide opportunities for studying bosonmediated superfluid states 关7,8兴. More recently, experiments
involving up to three atomic species have been implemented
for sympathetic cooling of two fermionic species 关9,10兴.
Dually degenerate experiments have so far used externaltrapping potentials common to both atomic species. A
species-specific trapping potential would add a degree of
freedom to improve sympathetic cooling 关11兴, tune the effective mass, or create a lattice for one species in the presenceof a background reservoir. Though careful selection of internal atomic states can provide differential magnetic trapping,
optical far-off-resonant traps 共FORTs兲 and magnetostatic
traps are not species specific.
Species-selective adiabatic potentials have been proposed
关12兴 and demonstrated 关13兴 in the case of 87Rb- 40K, where
the Landé factors 兩gF兩 are distinct. A radio-frequency transverse field can be resonant with only one of the two species,
selectively deforming its dressed potential. Onofrio and coworkers 关11,14兴 propose using two overlapping FORTs atfrequency detunings far above and below the dominantground-state transitions of both species in a two-species mixture. The confinement of each species can be chosen independently by individually adjusting the intensity of the two
beams used to create the trap. Unfortunately, neither of theseschemes readily lends itself to a uniform lattice potential
for atoms: the radio-frequency scheme fails because it islimited to one-dimensional periodic potentials and the twofrequency balancing because lattice periodicity depends on
wavelength 关15兴.
In this paper we discuss the generation and application of
species-specific optical lattice potentials. Our motivation is
the strong analogy between atoms in optical lattices and elec1050-2947/2007/75共5兲/053612共9兲

trons in crystalline solids. Cold bosons in lattice potentials
can be used to explore strongly interacting many-body physics, such as the superfluid-insulator transition 关16,17兴. At sufficiently low temperatures, cold fermions in lattices 关18,19兴
might be able to address open questions about the groundstate of the Hubbard model 关20兴. Unlike crystal lattices, optical lattices do not support the lattice vibrations responsible
for many physical phenomena. However, phonon mediation
between neutral atomic fermions could arise in the presenceof a condensed bosonic species capable of sustaining
phononlike excitations 关7,8,21,22兴. Although Bose-Fermi
mixtures were recently loaded into optical lattices 关23,24兴,the lattices confined bosons as well as fermions. A latticeinduced increase in the effective mass of the bosons reduces
the speed of sound in the condensate, and thus the mediating
effects of the phonons. If the fermions were confined in a
lattice without effect on the bosonic cloud, this mediation
would be less inhibited 关25兴.
We discuss two approaches to species-specific optical potentials, both of which involve only a single frequency oflaser light. The first approach is to tune the laser wavelength
close to the atomic resonance of one species, making its
induced dipole moment much stronger than that of any otheratomic species present. We refer to this strategy as the “tunein” 共TI兲 scheme. A second approach exists for elements, suchas alkali metals, with an excited-state fine-structure splitting:between the resonances of the doublet, a wavelength can bechosen such that the induced dipole moment is strictly zero.
We refer to this strategy as the “tune-out” 共TO兲 scheme. Bothapproaches allow for the creation of a species-specific opticallattice with a tunable relative potential strength between species.
In the following sections we consider the relative meritsof the TI and TO schemes. We focus our attention on BoseFermi combinations throughout, paying special attention to
mixtures including 87Rb 关18,26–31兴. Section II presents themethods used to calculate potential energies and heating
rates of atoms in a light field. We present the TI and TO
approaches in Sec. III, and quantify their relative merits. For

053612-1©2007 The American Physical Society

PHYSICAL REVIEW A 75, 053612 共2007兲

L. J. LEBLANC AND J. H. THYWISSEN

several of the stable alkali-metal isotopes, we calculate thetune-out wavelengths, scattering and associated heating
rates, and the trapping potentials imposed upon a second
species of alkali-metal atom. Section IV discusses the role ofinteractions, both in limiting the distinct addressability of
one species and in allowing thermalization between two species. In Sec. V we discuss applications of the species-specificdipole potentials for cooling, engineering mobility, and adding phonons to optical lattices, before concluding in Sec. VI.
II. METHODS

We consider the light-atom interaction in the limit of asmall excited-state fraction. An electromagnetic field induces
an electric dipole potential on a neutral atom, calculated inSec. II A, which can be used as a trapping potential for ultracold atomic samples. We consider the residual effect ofspontaneous emission in Sec. II B.

Ug ⬇

冊

冉

␲c2⌫ 1 − PgFmF 2 + PgFmF
+
I,
⌬1
⌬2
2␻30

共2兲

where P = 0 , ± 1 for ␲ , ␴± polarization, respectively, gF is the
Landé factor, ⌬1共2兲 is the detuning from the D1共2兲 line, ␻0= 共␻D1 + 2␻D2兲 / 3 is the line center weighted by line strength,and ⌫ = 共⌫D1 + ⌫D2兲 / 2 is the average of the D1 and D2 linewidths. Since ⌫D2 / ⌫D1 ⬇ 1 + 3⌬FS / ␻0, one can expect an accuracy between ±7% for Cs and ±0.003% for Li. At small
detunings, we empirically find that Eq. 共2兲 deviates from Eq.
共1兲 by ⱖ1%for detunings min兵兩⌬1 兩 , 兩⌬2 兩 其 / 2␲ ⱗ 1.5冑M GHz,
where M is the mass in atomic units.
As an approximate form, Eq. 共2兲 neglects the counterrotating terms. For 兩⌬ 兩 ⬍ ⌬FS, the strength of the counterrotating contribution is at most ⌬FS / 2␻L relative to the contribution of one near-resonant dipole transition. Thus, theneglected shift is at most −2% for Cs, and −0.001% for Li.
B. Heating rate

A. Dipole potential

An atom in a ground state 兩g典 will experience a potential
shift due to coupling by the light field to the excited states兩e典. We calculate the sum of these shifts on each state 兩g典,
including the counter-rotating term, using second-order perturbation theory:
Ug =

冋

册

円具e兩d · ⑀ˆ 兩g典円
円具e兩d · ⑀ˆ 兩g典円
1−I,兺2⑀0c e ប共␻L − ␻eg兲 ប共␻L + ␻eg兲2

2

Detuning and intensity of optical traps must be chosenwith consideration of the incoherently scattered trapping
light that heats the atoms. For each state 兩g典, we quantify therate of scattering in the low saturation limit,

␥g = 兺

⌫e円具e兩d · ⑀ˆ 兩g典円2

e共1兲

where ␻L is the laser frequency, ប␻eg is the energy difference
between 兩e典 and 兩g典, d is the dipole operator, ⑀ˆ is the polarization of the light, and I is the light intensity. In the case of
atoms in a weak magnetic field, we use the matrix elements
defined in the 兩F , mF典 basis, where F is the total angularmomentum and mF is the magnetic quantum number.
We will consider only alkali-metal atoms, which have twodominant ns → np transitions due to the fine-structure splitting. Using nomenclature established by Fraunhofer for the
3 2S1/2 → 3 2 P1/2 and 3 2S1/2 → 3 2 P3/2 transitions in sodium,
we label the corresponding lines in each of the alkali metals
D1 and D2, respectively. Spin-orbit coupling splits each excited state by a frequency ⌬FS = ␻D2 − ␻D1, while each groundand excited state is further split by the hyperfine interaction⌬HFS and ⌬HFS
⬘ , respectively. The atomic data used for Eq. 共1兲
are the measured linewidths and line centers of the D1 andD2 lines, and the ground- and excited-state hyperfine splittings 关32兴.
Transitions to higher excited states ns → 共n + 1兲p are neglected by our treatment. When detuned within ⌬FS of thens → np transition, the relative magnitude of the ns → 共n
+ 1兲p shift is less than 2 ⫻ 10−5 for Cs and 7 ⫻ 10−8 for Li.
As Eq. 共1兲 requires a sum over several states and knowledge of individual matrix elements, it is useful to have anapproximate but simpler expression for Ug. If the detunings⌬eg = ␻L − ␻eg are small compared to ⌬FS, but large comparedto the excited-state hyperfine splitting ⌬HFS
⬘ , an approximate
expression for the dipole shift is 关33兴

2
⌬eg

I,

共3兲

where ⌫e is the natural linewidth of the g → e transition, and⌬eg = ␻L − ␻eg. The rate of scattering of photons can be converted to an average heating rate Hg = 32 ER␥g, where ER= ប2k2 / 2m is the recoil energy, k = ␻L / c, and c is the speed oflight.
III. SPECIES-SPECIFIC DIPOLE TRAPPING SCHEMES

It is not surprising that optical traps can be species specific, given that optical resonances are unique to atomic elements and isotopes. However, most species-specific opticaltraps, such as magneto-optical traps, are tuned to within a
few linewidths of resonance, which is incompatible withquantum degenerate ensembles. At low temperatures andhigh density, any gain in trap depth close to resonance must
be balanced against the heating due to unwanted light scattering 共Sec. II B兲.
In the sections below we consider two-species mixtures.
The goal is to apply a dipole force to the “target” species
while inducing as little potential as possible on the second
species, which we will call the “spectator.” We define the“selectivity” as

␳=

冏 冏

Ut
,
Us

共4兲

where Ut共s兲 is the potential induced on the target 共spectator兲.
As discussed in Sec. II, in the low-saturation limit, boththe induced dipole potential and the heating rate are proportional to intensity. We define the intensity-independent ratio

053612-2

PHYSICAL REVIEW A 75, 053612 共2007兲

SPECIES-SPECIFIC OPTICAL LATTICES

energy shift (nK)

0.5

␳TI ⬇

0.25

−0.25
780

790

laser wavelength (nm)800

FIG. 1. Energy shift as a function of wavelength for 87Rb in the兩F , mF典 = 兩2 , 2典 state, under linear polarization, for 1 mW/ cm2. This
general structure will arise for each of the alkali-metal elements,
with the divergences located at the D1 and D2 lines. The tune-in
scheme for a 87Rb target is indicated by the square marker on theblue-detuned branch of the potential energy curve. The roundmarker indicates the position of the tune-out wavelength, where theenergy shift is zero. Here, 87Rb is the spectator.

s=

Ut
Ht + Hs

The tune-out wavelength scheme exploits the characteristic doublet structure of the alkali-metal atoms. By choosing awavelength that lies between the two strongest transitions,the large negative energy shift of the D2 line is balanced
against the large positive energy shift of the D1 line 共see Fig.
1兲. This atom becomes a spectator, while any other species
experiences the shift induced by the laser and becomes atarget. Since the potential shift on the spectator can be zero,
the selectivity of the tune-out approach is infinite.
The laser frequency ␻TO at which Us = 0 is determined
numerically using Eq. 共1兲 with all 兩F , mF典 excited states inthe D1 and D2 manifolds. Table II shows the tune-out wavelength for all 87Rb ground states with ␲, linear 关35兴, ␴+, and␴− polarizations. An approximate expression for this wavelength can be derived from Eq. 共2兲, giving

to be the “sustainability,” where Ht共s兲 is the heating rate of
the target 共spectator兲. The absolute value of s sets the scale
for the possible trapping time 关34兴. The laser frequency will
be chosen to maximize both ␳ and 兩s兩.
A. The tune-in scheme

The simplest selective potential is one in which the laser
is tuned close to a resonance of the target 共see Fig. 1兲. Since
the scattering rate is inversely proportional to the square of
the detuning, we consider only heating of the target such thatsTI → Ut / Ht. Considering Eq. 共2兲 in the limit 兩⌬t 兩  兩⌬s兩, asimple estimate is冊冉␻TO = ␻0 −共6兲

1 + PgFmF⌬FS .
3

TABLE I. Sustainability s of two-species mixtures for tune-out and tune-in schemes, in units of seconds,
using Eq. 共5兲. The second column indicates the selectivity. Rows with ␳ = ⬁ are calculated using the TO
scheme while rows with ␳ = 100 and ␳ = 10 use the TI scheme. For the lighter spectators, the TI scheme hashigher 兩s兩. The heaviest elements and isotope mixtures favor the TO scheme.
Spectator

␳

7

6

⬁
100
10
⬁
100
10

0.00134
2.66⫻ 10−5
3.92⫻ 10−6
4.28⫻ 10−7
3.05
29.2

Li

40

K

Li

共8兲

For 87Rb, Eq. 共8兲 predicts tune-out wavelengths of 785.10,
787.54, 790.01, and 792.49 nm for PgFmF = −1, −0.5, 0, and0.5, respectively. Comparing these values to the results ofTable II, we see that the approximations are accurate to
0.04 nm or better. Equation 共8兲 also gives ␻TO = ␻D1 for the

The choice of detuning ⌬t will depend upon the desired selectivity. Assuming that the spectator is far detuned, ⌬s⬇ 共␻0,t − ␻0,s兲 andTarget共7兲

B. The tune-out scheme

共5兲

3ប
sTI ⬇
⌬t .
2ER,t⌫t

冊

Together, Eqs. 共6兲 and 共7兲 explicitly show the opposing dependence on ⌬t and the necessary trade-off between selectivity and sustainability.
Table I shows the sustainability for bosonic species asspectators and fermionic species as targets, calculated using
Eqs. 共1兲 and 共3兲 for tune-in selectivities of 100 and 10. The
inverse scaling predicted by Eqs. 共6兲 and 共7兲 is observed: sTI
drops by approximately a factor of 10 when ␳TI is increased
by a factor of 10. The product of sTI and ␳TI in Table I varies
between 22 and 36 s for 6Li mixtures and between 140 and
530 s for 40K mixtures, whereas Eqs. 共6兲 and 共7兲 would predict ranges of 8 – 21 s and 64– 132 s, respectively, excluding
isotope mixtures from the comparison. Finally, we note that23
Na- 40K is the optimal tune-in mixture.

0

−0.5

冉

3
兩␻0,t − ␻0,s兩 1
2⌫t ␻0,s
.
3
3⌫s
兩⌬t兩
␻0,t

23

Na

39

7.77⫻ 10−4
0.281
2.54
5.77⫻ 10−4
5.78
53.0

−0.0381
0.220
2.30
0.188
3.64⫻ 10−5
4.65⫻ 10−4053612-3

K

87

133

−1.20
0.239
2.49
−9.03
0.251
3.27

−8.45
0.347
3.57
−25.8
1.39
18.8

Rb

Cs

PHYSICAL REVIEW A 75, 053612 共2007兲

L. J. LEBLANC AND J. H. THYWISSENTABLE II. Tune-out wavelengths and scattering rates in 87Rb,
for select polarizations and all ground states. Two states have no
tune-out wavelength, because the D1 line has no F = 3 excited state.

Polarization

兩F , mF典

␭TO共nm兲

␥sc / I
共cm2 / mJ兲␲兩2 , ± 2典
兩2 , ± 1典
兩2 , 0典
兩1 , ± 1典
兩1 , 0典

790.05790.06790.06790.06790.059.0⫻ 10−6

兩2 , ± 2典
兩2 , ± 1典
兩2 , 0典
兩1 , ± 1典
兩1 , 0典

790.04790.04790.03790.03790.049.1⫻ 10−6␴−兩2 , 2典
兩2 , 1典
兩2 , 0典兩2 , −1典兩2 , −2典兩1 , 1典
兩1 , 0典
兩1 , −1典

785.14787.59790.06792.52
共none兲
792.53790.06787.599.1⫻ 10−6
8.1⫻ 10−69.0⫻ 10−6
14.4⫻ 10−6
14.4⫻ 10−69.0⫻ 10−6
8.1⫻ 10−6␴+兩2 , 2典
兩2 , 1典
兩2 , 0典兩2 , −1典兩2 , −2典兩1 , 1典
兩1 , 0典
兩1 , −1典

共none兲
792.52790.06787.59785.14787.59790.06792.53–14.4⫻ 10−69.0⫻ 10−6
8.1⫻ 10−6
9.1⫻ 10−6
8.1⫻ 10−69.0⫻ 10−6
14.4⫻ 10−6

LinearTABLE III. Tune-out wavelengths and scattering rates for various elements. We have assumed linear polarization and stretchedstates and show a comparison of Eqs. 共1兲 and 共2兲.

Element兩F , mF典

6兩2 , 2典
兩2 , 2典
兩2 , 2典
兩2 , 2典
9 9
兩2 , 2典
兩2 , 2典
兩4 , 4典

Li
Li
23
Na
39
K
40
K
87
Rb
133
Cs
7

3

3

␭TO 共nm兲
Eq. 共1兲
Eq. 共2兲
670.99670.97589.56768.95768.80790.04880.29670.99670.97589.56768.95768.80790.01880.06

␥sc / I
共cm2 / mJ兲
2.8
2.4
2.0⫻ 10−3
1.4⫻ 10−4
1.7⫻ 10−4
9.1⫻ 10−6
1.5⫻ 10−6

Spectators with larger scattering cross sections at ␭TO havelower sustainability. In addition to these data, we note thatthe 133Cs- 87Rb spectator-target combination gives the highest possible sustainability among alkalis metals in thetune-out scheme: s = 34 s at a tune-out wavelength of ␭TO= 880.29 nm 共not shown in tables兲.
As an example, consider making a lattice for 40K only,
leaving 87Rb unaffected by the lattice and confined only by abackground magnetic trap or FORT. If we consider the mixture in a three-dimensional lattice of arbitrary depth, the 40K
potential shift is 1.54⫻ 10−5 ␮K ⫻ 关I 共mW/ cm2兲兴; withbeams of 100 ␮m waist, the potential shift is 98 nK/ mW.
For a lattice that is 8ER deep, we find a 87Rb heating rateof 98 nK/ s.
C. Discussion: Tune-in vs tune-out

case of PgFmF = 1, which is inconsistent with the assumptionthat 兩⌬1 兩  ⌬HFS. In fact, since PgFmF = 1 corresponds to adark state with respect to the D1 excitation, there is no tuneout wavelength for this case.
We note that in Table II the tune-out wavelength for linear
polarization is nearly independent of the choice of groundstate, in contrast with the ␴+ or ␴− polarizations. Given thisindependence, we calculate the tune-out wavelengths using
Eq. 共1兲 for several of the common alkali-metal isotopes intheir stretched ground states under linear polarization 共Table
III兲 关36兴.
Tables II and III also give the scattering rates per unit
intensity 共scattering cross sections兲 for the spectator species
at the tune-out wavelength. Due to the large dispersion offine splitting among the alkali metals, scattering cross sections vary by over six orders of magnitude. To understandthe implications for trapping, we need to consider the sustainability s of various pairs of atomic species. Table I
shows sTO in the same mixtures for which sTI is calculated.

With the data of Table I, we can evaluate the practicalityof both the tune-in and tune-out schemes for Bose-Fermi
mixtures of neutral alkali-metal atoms. Since the scattering
rate in the tune-out scheme can be smaller for elements withlarger fine-structure splittings, this approach is better suited
to more massive elements 关37兴. In particular, the tune-in
scheme is preferable for Li-Na, Li-K, and K-Na mixtures,and for applications requiring selectivity of less than 10:1.
The tune-out scheme is preferable for Li-Cs, K-Rb, andK-Cs mixtures when the selectivity required is greater than
10:1, and for Li-Rb mixtures at selectivity of greater than
20:1.
An isotope mixture of potassium could be compatible
with the tune-out scheme, where s ⬎ 100 ms. Isotopespecific manipulation within a lithium mixture is less practical due to sustainabilities of 1 ms or less.
Other factors may also influence whether the tune-in or
tune-out approach is preferred. For experiments with timescales that are slow compared to the thermalization rate 共discussed further in Sec. IV B兲, it may be preferable to heat theminority species and allow for sympathetic cooling. If thereservoir of spectator atoms is large, the tune-in scheme
might be preferred since extra energy due to near-resonant
heating would be transferred to the reservoir. For experiments on time scales fast compared to the thermalization

053612-4

PHYSICAL REVIEW A 75, 053612 共2007兲

SPECIES-SPECIFIC OPTICAL LATTICES

rate, the tune-out scheme might be preferred even if s issmaller, since spectator heating will not affect the target.
Finally, we note that if a third species is included, it willplay a spectator role in the tune-in scheme and a target rolein the tune-out scheme.
D. Lattice confinement effects

In Secs. III A and III C, we have discussed the ratio of
induced potential to heating rate as a local quantity. However, the spatial dependence of intensity in an optical latticemeans that moving atoms may not sample all laser intensities
uniformly, and the heating rate should be modified accordingly. A more relevant characterization of trapping and dissipation is

sglobal =Upkt,
具Ht典 + 具Hs典

共9兲

where the angular brackets indicate expectation values andis the maximum potential depth of the target. In theUmaxtcase where only one heating rate dominates, we define aquantity ␤ to represent the enhancement of sustainability due
to the lattice:

␤TO,TI ⬅

I0
sglobal,→
具I典s,ts

共10兲where the expectation value is taken for the spectator species
in the tune-out case and for the target in the tune-in case. In
a one-dimensional lattice, the intensity is I共x兲 = I0sin2共kx兲,where k = 2␲ / ␭L.
For the tune-out scheme, neither the position nor the motion of the spectator is coupled to the standing wave intensity. We can assume that all intensities are equally sampled
and use the average intensity 具I典s,t = I0 / 2 to give ␤TO = 2.
For the tune-in scheme, the target is the species for which
the heating rate is dominant and for which the lattice enhancement of the sustainability must be taken into account.
We first consider ␤ in the case of a localized atom. Classically, we expect deeply bound states of a blue-detuned lattice
to avoid the intensity maxima where incoherent scattering ishigh, such that the intensity experienced by the atom is lessthan the average. Heating is therefore reduced and ␤TI is
increased. For an atom localized to a single site by interactions, we can use Wannier states to calculate the expectation
value of the intensity 共see Fig. 2兲. For lattice depths ␩= Upkt / ER  1, these states are approximately harmonic oscillator states with oscillator frequencies ␻ho = 2冑ERUt / ប. In
this case, ␤TI ⬇ 2冑␩; in an arbitrary number of dimensions d
the advantage is increased as ␤TI ⬇ 2d␩d/2 for ␩  1.
Considering that classical oscillators spend more time atturning points than at potential minima, blue tune-in lattices
may not always have ␤TO ⬎ ␤TI. For a delocalized q = 1.5បkBloch function 共in the first excited band兲, where q is thequasimomentum, we show ␤ versus ␩ in Fig. 2. For ␩ ⬍ 10,
confinement effects give ␤TI ⬍ 2.

FIG. 2. A standing wave of laser light increases the ratio ofconfinement depth to scattering rate, modifying the sustainability s
calculated for uniform intensity. The enhancement ␤TI is shown as afunction of ␩ 共trap depth兲 for a localized state 共solid line兲, a q = 0
Bloch state in the lowest band, and a q = 1.5បk Bloch state in thefirst excited band 共both dashed兲. At high lattice depths, both wave
functions in the lowest band approach the tight-binding limit 共dotted line兲. For comparison, ␤TO = 2 for all lattice depths.
IV. INTERACTIONS BETWEEN SPECIES
A. Mean-field interactions

Interactions between elements may couple the environment experienced by the target to the spectator, spoiling thespecies-specific potential shaping of both schemes. If target
atoms are trapped in a lattice in the presence of a background
spectator species, a periodic interaction potential could arise
for the spectator due to its interaction with the periodicallymodulated density of the target. A mean-field approach isused to make a simple estimate of the magnitude of thiseffect. The interaction potential of the spectator due to thetarget is given by Uint,s = 共4␲ប2ast / ␮st兲nt, where ast is thescattering length between species, nt is the density of thetarget, and ␮st = 2msmt / 共ms + mt兲 is the reduced mass. As
Ut / Uint,s → 1, interaction effects begin to degrade selectivity.
Estimating the three-dimensional density of the target at thecenter of a single lattice site using the harmonic oscillator
ground state with frequency ␻ho and assuming one atom persite, we find an interaction-limited selectivity

␳max ⬅

冏 冏

Utប␮st␩1/4=
1/2 .
Uint,s32␲兩ast兩m3/2
t ER

共11兲

In the case of the 6Li- 87Rb target-spectator mixture, theinteraction-limited selectivity is ␳max = 3.1␩1/4 关31兴, while forthe 40K- 87Rb target-spectator mixture, it is ␳max = 0.22␩1/4. To
be in a regime where interaction effects might be ignored, we
require ␳max  1, which gives ␩  0.01 for 6Li- 87Rb and␩  400 for 40K- 87Rb. While for the lithium target this condition is quite reasonable, the large lattice depths required forpotassium would completely localize the atoms to individualsites and prevent the exploration of interesting tunnelingdriven physics. Overcoming this interaction limitation andachieving the high selectivity discussed in Sec. III may bepossible by tuning a magnetic field to a value where aRbK
⬇ 0 near a Feshbach resonance 关38兴. For instance, with depth␩ = 8 and ␳max ⱖ 10, the scattering length between species islimited to 兩ast 兩 ⱕ 7.5a0, where a0 is the Bohr radius.

053612-5

PHYSICAL REVIEW A 75, 053612 共2007兲

L. J. LEBLANC AND J. H. THYWISSEN

Where species selectivity is not the goal, this interactioninduced periodic potential for a second species could be used
as an alternative lattice potential. Such potentials are nonsinusoidal, do not involve a Stark shift, and may have adynamic structure if target atoms are mobile. The strength of
this induced potential could be controlled through the interaction strength, such as at a Feshbach resonance, as described above.
B. Inter-species thermalization

An understanding of the thermalization between species isimportant when considering heating or cooling in the trap.
The rate at which energy is transferred between the species
will be relevant in setting the time scales on which adiabatic
experiments can take place. Thermometry is also possible if
there is good thermalization between species; a high-density
target could be confined within a species-specific latticewhile the spectator remains extremely dilute and thus atlower quantum degeneracy, where temperature is more easily
measured. Conversely, thermal isolation could be useful inshielding one species from the spontaneous heating in theother.
In the classical limit, the thermalization rate is proportional to the collision rate of the atoms in the trap, given by␥coll = n␴v, where n is the overlap density, ␴ the scatteringcross section, and v the relative velocity between species.
Random collisions act to equilibrate the system and the rateof rethermalization is ␥therm ⬇ ␥coll / 2.7 关39兴.
For a degenerate mixture of bosons and fermions, theclassical picture of scattering breaks down, and the rate ofthermalization decreases as the fermionic system becomes
more degenerate. An estimate of the sympathetic cooling of a
uniform system in the degenerate regime using the quantum
Boltzmann equations gives the rate of change in the degeneracy 关40兴:

冉 冊

冉 冊

6␨共3兲
T 2
d T
⬘
=−
,
2 ␥coll
dt TF
␲
TF

共12兲

where ␥coll⬘ = 共3 / 8兲nB␴vF is the collision rate between spe2
cies, with the scattering cross section ␴ = 4␲aBF, Fermi ve1/3
locity vF = ប共6␲nF兲 / mF, ␨共3兲 ⬇ 1.202 06, where ␨ is the
Riemann zeta function, and TF is the Fermi temperature.
Though other assumptions 关41兴 of this treatment are not valid
for the systems we consider in this paper, we use this expression to determine an order of magnitude for the thermalization rate.
Taking the boson density nB = 1 ⫻ 1014 cm−3, fermion density nF = 1 ⫻ 1013 cm−3, and using the mass for 40K, we calculate that d / dt共T / TF兲 ⬇ −350 s−1 ⫻ 共T / TF兲2, which gives atemperature relaxation time of ⬇30 ms at T = 0.1TF. This isan order of magnitude larger than the the classical expectation for the rethermalization time of ⬇2 ms for particles
moving at the Fermi velocity. Thus, for rapid experiments inthe deeply degenerate regime, thermal contact is essentiallybroken, allowing, for instance, the target to be unaffected by
the heating of the reservoir in the tune-out scheme.

FIG. 3. A cooling procedure using species-specific trapping;
one-dimensional trap shape represents three-dimensional trapaveraged shape. The solid line represents a FORT, the dashed line
the species-specific trap; open circles are the spectator species andclosed circles are the target. 共a兲 Two species are trapped in a FORT;
共b兲 the species-selective beam is turned on, compressing and heating the target species; 共c兲 the target species rethermalizes with thespectator; 共d兲 the spectator is removed; 共e兲 the target is adiabaticallydecompressed to a lower temperature and transferred to a FORT.
V. APPLICATIONS OF SPECIES-SPECIFIC TRAPPING
A. Isothermal phase-space increaseTwo-species mixtures can be used to realize various cooling schemes. For example, dark-state cooling by superfluid
immersion is discussed in Ref. 关42兴. We present two simple
cooling scenarios in which the presence of an uncompressed
spectator allows the target species to be compressed withnegligible temperature increase but significant improvementin phase-space density. In both cases, target atoms are firstcompressed isothermally, spectator atoms are then removed
from the trap, and finally, the target atoms are decompressed
adiabatically.
We consider a species-selective single-well dipole trap
and a species-selective lattice. A closed cycle is described forboth scenarios with atoms beginning and ending in a FORT.
We assume that the heat capacity ratio between spectator andtarget is infinitely large, and that thermalization in the spectator and between the spectator and target species is fasterthan any other time scale considered. The latter assumption
may restrict fermion cooling to the nondegenerate regime
共see Sec. IV B兲.
In the first scenario, schematically represented in Fig. 3, asingle species-specific beam crosses a FORT. Adiabatic cooling reduces the temperature in proportion to the ratio of average trapping frequencies, that is, Ti / Tf = ␻i / ␻f, where i andf indicate “initial” and “final.” As an example, consider40
K- 87Rb and 6Li- 87Rb mixtures, confined by a 1064 nm,
500 mW single-beam FORT with a 1 / e2 radius of 20 ␮m
and a corresponding trap-averaged harmonic oscillator frequency ␻i = 2␲ ⫻ 540 Hz for 40K and ␻i = 2␲ ⫻ 950 Hz for6
Li. A 500 mW, 50 ␮m waist beam at ␭TO 共see Table III兲 isturned on perpendicular to the FORT to compress the fermi-

053612-6

PHYSICAL REVIEW A 75, 053612 共2007兲

SPECIES-SPECIFIC OPTICAL LATTICES

ons in a trap with frequency ␻f = 2␲ ⫻ 3040 Hz for 40K and␻f = 2␲ ⫻ 3810 Hz for lithium. After providing sympathetic
cooling during the compression of the target atoms, the rubidium is ejected from the trap by temporarily removing the
FORT or by using a resonant pulse of light. The speciesspecific trap is then adiabatically ramped down and turnedoff, leaving the fermions in the FORT approximately 5.7
共4.0兲 times colder than the initial temperature for potassium
共lithium兲. Though this is a modest change in temperature, thephase-space density in a harmonic trap is proportional to theinverse cube of the temperature, and increasing by a factor of
180 for potassium and a factor of 60 for lithium.
In the second scenario, we consider a three-dimensionallattice created by the tune-out wavelength 关43兴. The targetspecific lattice is ramped up until peak lattice intensity isreached. The spectator is evaporatively cooled and ejected by
reducing the spectator trap depth. The lattice is then ramped
back down isentropically, leaving the target in the initial trapwith an entropy and temperature limited by the “plateau entropy” discussed in 关44兴. As shown there, a target of fermions
with unity filling has an entropy plateau of zero, which
would suggest no lower limit to the achievable temperature.
An important limitation of these schemes will be the competition between adiabaticity and heating. For an ideal gas,
the condition of adiabaticity requires any changes to takeplace in times longer than the inverse of the smallest trap
frequency. Using the numbers given in the 40K- 87Rb exampleof the crossed dipole trap in the tune-out scheme, we findthat adiabaticity requires a relaxation time of 35 ms, at an
intensity of 3.5⫻ 107 mW/ cm2, yielding a heating of 1.3 ␮Kduring this time, which sets a lower bound for the temperature attainable in this scheme. Another possible limitation of
the cooling schemes is the interspecies thermalization, which
limits the speed of the isothermal step 共see discussion in Sec.
IV B兲.

FIG. 4. 共a兲 The ratio of the effective mass to bare mass of thetarget species is shown as a function of trap depth ␩t, for both q= 0 and qc = 0.5បk, as labeled. The tight-binding approximation
共dashed line兲 approaches the exact calculation for ␩t  1. 共b兲 The
effective-mass ratio is shown for the case of a 6Li target and a 133Cs
spectator. Whether the ratio increases or decreases depends on theselectivity ␳ of the lattice. The critical point is when ␳ = ms / mt
共⫽22.2 in this case兲, as explained in the text. The tight-binding
approximation of the effective-mass ratio is also shown 共dashedline兲.

interaction. A finite selectivity does not prevent mediation ofinteractions, since the bosons remain superfluid at depths lessthan the Mott-insulator superfluid transition 关17兴, permitting
both tune-in and tune-out schemes to be used for this application.
C. Effective-mass tuning

B. Phonons

Unlike crystal lattices of solids, optical lattices do notsupport phonons. Since these quasiparticles play a leadingrole in the physics of condensed matter, it is of interest to
introduce phononlike excitations into a system of ultracold
atoms in an optical lattice. The boson-mediated interaction
between fermions has been studied in both uniform
关7,8,21,22兴 and lattice systems 关25,45,46兴. Little theoreticalwork has been done on a system in which only the fermionsare confined to the lattice 关47兴. Here, the background bosons
are free to interact with the fermions and with one another. If
the bosons are degenerate, the condensate can sustain
phononlike excitations and allow for boson-mediated interactions between fermions on spatially separated lattice sites.
For the phonons in the condensate to play a role in mediating interactions, their spatial extent must exceed the latticespacing. The healing length of a uniform Bose condensate,␰ = 共8na兲−1/2 where n is the condensate density and a is thescattering length between species, sets the relevant lengthscale. Species-specific trapping allows the superfluid bosons
to remain at low density while fermions are tightly bound in
the lattice, thereby maximizing the range of the mediatedAn optical lattice can be used to change the effective massof the atoms in it, allowing for the tuning of experimental
parameters including interaction strength 关48兴 and tunnelingrate 关49兴, which can be used, for example, to explore different regimes of collective dynamics 关50兴. The effective massof a wave packet centered at quasimomentum qc is
m*共qc兲 =冋 册

d2E −1,
dq2 qc共13兲

where E共q兲 is the band energy. Figure 4共a兲 shows the effective mass for qc = 0 and 0.5បk in a one-dimensional opticallattice potential U共x兲 = U pksin2共kx兲.
For deep lattices, an approximate form for the qc = 0 caseis m* = ប2k2 / 2␲2J, where J is the tunneling energy 关51兴, giving an effective-mass enhancement
冑

m*
e2 ␩
⬇
.
m
4␲3/2␩3/4

共14兲

The ratio of effective masses for the target and spectator canbe estimated from Eq. 共14兲:

053612-7

PHYSICAL REVIEW A 75, 053612 共2007兲

L. J. LEBLANC AND J. H. THYWISSEN

m*t

* ⬇

ms

冉 冊

exp关2冑␩t共1 − 冑ms/␳mt兲兴 mt 1/4
.
␳3/4
ms

共15兲

Figure 4共b兲 shows the effective-mass ratio for the case of a6
Li target and a 133Cs spectator. In particular, it is striking
that for ␳ = 10 the ratio decreases with lattice depth, while for␳ = 100 the ratio increases with lattice depth. The critical selectivity is well predicted by Eq. 共15兲; at ␳ = ms / mt, theeffective-mass ratio is independent of lattice depth.
Both the tune-out and tune-in schemes provide the meansof choosing selectivity. Several tune-in selectivities areshown in Table I; the tune-out selectivity can be chosen simply by choosing a wavelength slightly different from ␭TO. In
the example used here, the sustainability for ␳ = 10 is better
*
*
⬎ mCs
at moderate
for the tune-in scheme, but tuning to mLi
lattice depths requires a ␳ = 100, for which the tune-out
scheme has higher sustainability.
VI. CONCLUSIONS

spectator elements, and is most successful for the 40K- 133Cs
fermion-boson mixture. The alternative tune-in scheme uses
a near-detuned optical potential, creating a much stronger
potential for one element than the others, without the ability
to strictly cancel the potential for one element. Mixtures involving Li, Na, and K as spectators are most compatible withthis approach, where the 40K- 23Na mixture is the most promising fermion-boson mixture.
The power of species selection lies in its use to engineer
specific lattice environments for the atoms, including adding
a bosonic background to mimic the phonons present in solids
and tuning the relative effective mass of two species. Applications for which experimental timescales are rapid compared to the sustainabilities calculated in Sec. III are especially promising. Species selection enables cooling in a twospecies mixture, and in the case of fermions trapped in a
lattice, reduces the temperature of fermions as they areloaded into a lattice, in contrast with current experimental
realizations 关44,52兴.
ACKNOWLEDGMENTS

We have discussed how the choice of wavelength used tocreate an optical lattice can tune the selectivity between elements or isotopes. This control should increase the range ofparameters that can be explored in multispecies ultracold
atom experiments. The tune-out wavelength scheme allows
for the complete cancellation of the trapping potential for
one species while providing a confining or lattice potential
for any other species in the system. This scheme will work
best using the heavier alkali-metal atoms, Rb and Cs, as the

We would like to thank E. Arimondo, S. Aubin, M. Extavour, D. Jervis, A. Mazouchi, J. Sipe, and E. Taylor for
stimulating conversations, D. James for help with atomic calculations, A. Paramekanti for help with thermalization calculations, and D. Weiss for pointing out the localization advantage of near-detuned blue lattices. This work is supported by
NSERC and the Canada Research Chairs program. L.J.L.
acknowledges support from NSERC.

关1兴 For a review of ultracold polar molecules, see J. Doyle, B.
Friedrich, R. V. Krems, and F. Masnou-Seews, Eur. Phys. J. D
31, 149 共2004兲, and references therein, in a special issue, ibid.
31 共2004兲.
关2兴 C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E.
Wieman, Phys. Rev. Lett. 78, 586 共1997兲.
关3兴 F. Schreck, G. Ferrari, K. L. Corwin, J. Cubizolles, L. Khaykovich, M.-O. Mewes, and C. Salomon, Phys. Rev. A 64,
011402共R兲 共2001兲.
关4兴 A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, Science 291, 2570 共2001兲.
关5兴 G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni,and M. Inguscio, Science 294, 1320 共2001兲.
关6兴 K. Mølmer, Phys. Rev. Lett. 80, 1804 共1998兲.
关7兴 M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, Phys. Rev. A
61, 053601 共2000兲.
关8兴 H. Heiselberg, C. J. Pethick, H. Smith, and L. Viverit, Phys.
Rev. Lett. 85, 2418 共2000兲.
关9兴 M. Taglieber, A.-C. Voigt, F. Henkel, S. Fray, T. W. Hänsch,
and K. Dieckmann, Phys. Rev. A 73, 011402共R兲 共2006兲.
关10兴 E. Wille, F. Spiegelhalder, G. Kerner, A. Trenkwalder, C.
Aiello, G. Hendl, F. Schreck, and R. Grimm 共unpublished兲.
关11兴 C. Presilla and R. Onofrio, Phys. Rev. Lett. 90, 030404
共2003兲.
关12兴 Ph. W. Courteille, B. Deh, J. Fortágh, A. Günther, S. Kraft, C.

Marzok, S. Slama, and C. Zimmerman, J. Phys. B 39, 1055
共2006兲.
关13兴 M. H. T. Extavour, L. J. LeBlanc, T. Schumm, B. Cieslak, S.
Myrskog, A. Stummer, S. Aubin, and J. H. Thywissen, At.
Phys. 20, 241 共2006兲.
关14兴 R. Onofrio and C. Presilla, Phys. Rev. Lett. 89, 100401
共2002兲.
关15兴 In principle one could use intersection angle to match the periodicity of two-color standing waves. In practice, the angle
and relative phase would have to be exquisitely controlled forcancellation to occur.
关16兴 D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller,
Phys. Rev. Lett. 81, 3108 共1998兲.
关17兴 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I.
Bloch, Nature 共London兲 415, 39 共2002兲.
关18兴 M. Köhl, H. Moritz, T. Stöferle, K. Günter, and T. Esslinger,
Phys. Rev. Lett. 94, 080403 共2005兲.
关19兴 J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C.
Sanner, K. Xu, and W. Ketterle, Nature 共London兲 443, 961
共2006兲.
关20兴 W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D.
Lukin, Phys. Rev. Lett. 89, 220407 共2002兲.
关21兴 L. Viverit, Phys. Rev. A 66, 023605 共2002兲.
关22兴 D. H. Santamore, S. Gaudio, and E. Timmermans, Phys. Rev.
Lett. 93, 250402 共2004兲.

053612-8

PHYSICAL REVIEW A 75, 053612 共2007兲

SPECIES-SPECIFIC OPTICAL LATTICES
关23兴 K. Günter, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger,
Phys. Rev. Lett. 96, 180402 共2006兲.
关24兴 S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst, K.
Sengstock, and K. Bongs, Phys. Rev. Lett. 96, 180403 共2006兲.
关25兴 D.-W. Wang, M. D. Lukin, and E. Demler, Phys. Rev. A 72,
051604共R兲 共2005兲.
关26兴 G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys. Rev.
Lett. 89, 150403 共2002兲.
关27兴 J. Goldwin, S. Inouye, M. L. Olsen, B. Newman, B. D. DePaola, and D. S. Jin, Phys. Rev. A 70, 021601共R兲 共2004兲.
关28兴 C. Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs,
Phys. Rev. Lett. 96, 020401 共2006兲.
关29兴 S. Aubin, S. Myrskog, M. H. T. Extavour, L. J. LeBlanc, D.
McKay, A. Stummer, and J. H. Thywissen, Nat. Phys. 2, 384
共2006兲.
关30兴 M. Anderlini, E. Courtade, M. Cristiani, D. Cossart, D. Ciampini, C. Sias, O. Morsch, and E. Arimondo, Phys. Rev. A 71,
061401共R兲 共2005兲.
关31兴 C. Silber, S. Günther, C. Marzok, B. Deh, P. W. Courteille, andC. Zimmermann, Phys. Rev. Lett. 95, 170408 共2005兲.
关32兴 D. Steck, Los Alamos National Laboratories Report No. LAUR-03-7943, 2003 共unpublished兲; Report No. LA-UR-038939, 2003 共unpublished兲; Report No. LA-UR-03-8638 2003
共unpublished兲; M. Gehm, 共unpublished兲; M. A. Zucker, A. R.
Kishore, R. Sukumar, and R. A. Dragoset, http://physics.nist.gov/EDI; E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 共1977兲, and references therein.
关33兴 R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Adv. At.,
Mol., Opt. Phys. 42, 95 共2000兲.
关34兴 The sustainability s has units of seconds and gives the approximate time it would take to heat the atoms out of the trap.
关35兴 By “linear” polarization, we refer to an equal superposition of
␴+ and ␴− polarizations.
关36兴 Unlike for ␴± or ␲ polarization, geometric considerations allow that linear polarization can be chosen for all directions of

a 3D optical lattice when a weak quantizing magnetic field ispresent along one of the coordinate axes.
关37兴 The fine-structure splitting increases approximately as the third
power of mass.
关38兴 S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn,
and D. S. Jin, Phys. Rev. Lett. 93, 183201 共2004兲.
关39兴 H. Wu and C. Foot, J. Phys. B 29, L321 共1996兲.
关40兴 L. D. Carr, T. Bourdel, and Y. Castin, Phys. Rev. A 69, 033603
共2004兲.
关41兴 We assume that the species are of equal mass, that the system
is highly degenerate T / TF  1兲, and that the bosons are either
in the condensate or removed immediately from the system
when excited.
关42兴 A. Griessner, A. J. Daley, S. R. Clark, D. Jaksch, and P. Zoller,
Phys. Rev. Lett. 97, 220403 共2006兲.
关43兴 1D and 2D lattices may be a way to increase the confinement
strength of the first scheme. Assuming high per site occupancy,
the cooling attained will be governed by single-site physics.
关44兴 P. B. Blakie and A. Bezett, Phys. Rev. A 71, 033616 共2005兲.
关45兴 M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann,
Phys. Rev. Lett. 92, 050401 共2004兲.
关46兴 R. Roth and K. Burnett, Phys. Rev. A 69, 021601共R兲 共2004兲.
关47兴 A. Klein, M. Bruderer, S. R. Clark, and D. Jaksch 共unpublished兲.
关48兴 For example, see B. Paredes, A. Widera, V. Murg, O. Mandel,
S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I.
Bloch, Nature 共London兲 429, 277 共2004兲.
关49兴 H. Pu, L. O. Baksmaty, W. Zhang, N. P. Bigelow, and P. Meystre, Phys. Rev. A 67, 043605 共2003兲.
关50兴 M. Krämer, L. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 88,
180404 共2002兲.
关51兴 W. Zwerger, J. Opt. B: Quantum Semiclassical Opt. 5, S9
共2003兲.
关52兴 M. Köhl, Phys. Rev. A 73, 031601共R兲 共2006兲.

053612-9