Vib a ional Mechanics in an Op ical La ice: Con olling T anspo
ia Po en ial Reno maliza ion
A. Wickenb ock,
1
P. C. Holz,
1
N. A. Abdul Wahab,
1
P. Phoon hong,
1
D. Cube o,
2
and F. Renzoni
1
1
Depa men o Physics and As onomy, Uni e si y College London, Gowe S ee , London WC1E 6BT, Uni ed Kingdom
2
Depa amen o de Fı
´sica Aplicada I, EUP, Uni e sidad de Se illa, Calle Vi gen de A
´ ica 7, 41011 Se illa, Spain
and Fı
´sica Teo
´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Se illa 41080, Spain
(Recei ed 30 Sep embe 2011; published 13 Janua y 2012)
We demons a e heo e ically and expe imen ally he phenomenon o ib a ional esonance in a
pe iodic po en ial, using cold a oms in an op ical la ice as a model sys em. A high- equency (HF)
d i e, wi h a equency much la ge han any cha ac e is ic equency o he sys em, is applied by
phase modula ing one o he la ice beams. We show ha he HF d i e leads o he eno maliza ion o he
po en ial. We used anspo measu emen s as a p obe o he po en ial eno maliza ion. The e y same
expe imen s also demons a e ha anspo can be con olled by he HF d i e ia po en ial
eno maliza ion.
DOI: 10.1103/PhysRe Le .108.020603 PACS numbe s: 05.60.k, 05.40.a, 05.45.a, 67.85.d
The con ol o anspo is a ecu en opic in physics,
chemis y, and biology. The ypical scena io co esponds
o pa icles di using on a pe iodic subs a e, wi h anspo
con olled by he applica ion o dc and ac ex e nal ields
[1,2]. The ul ima e limi o he con ol o anspo is o en
he impossibili y o uning he pe iodic po en ial, as i is
usually he case in solid s a e.
In his wo k we p o ide a p oo -o -p inciple o how his
limi a ion can be o e come, and demons a e heo e ically
and expe imen ally he con ol o a pe iodic po en ial
ampli ude ia a s ong high- equency (HF) oscilla ing
ield. The po en ial is eno malized, wi h i s ampli ude
con olled by he s eng h and equency o he HF ield.
The mechanism unde lying he po en ial eno maliza ion is
he so-called ib a ional esonance, in ially in oduced [3]
and obse ed [4–7] in bis able sys ems. Ou expe imen
uses cold a oms in a dissipa i e op ical la ice as a model
sys em. Howe e , he phenomenon demons a ed he e is
e y gene al, and is ele an o any classical sys em o
pa icles in a pe iodic po en ial. This may also o e a
possibili y o uning he po en ial in solid s a e sys ems,
whe e his is usually conside ed impossible. Combined
wi h p e ious wo k which showed how ac ields can be
used o con ol anspo ia dynamical symme y b eaking
[1,2] and unnel coupling eno maliza ion [8,9], he p esen
wo k demons a es ha a comple e con ol o anspo can
be achie ed ia ac ields.
Ou expe imen al wo k elies on he s udy o he ans-
po p ope ies o a oms in an op ical la ice o di e en
s eng hs o he applied HF ield. We will demons a e ha
by uning he HF ield i is possible o con ol he ampli ude
o he po en ial, and o make i anish. In his espec ,
he use o cold a oms in dissipa i e op ical la ices is e y
con enien as he anspo p ope ies in hese sys ems ha e
been s udied in de ail [10–12], and his allows us o use he
anspo measu emen s o cha ac e ize he po en ial. We
will p o ide wo di e en se s o measu emen s, as sup-
po ing e idence o he po en ial eno maliza ion. Fi s , we
will demons a e ha he di usion p ope ies, which a e
known o s ongly depend on he po en ial dep h [10,11],
can be con olled by he HF ield in a way which co e-
sponds o he po en ial eno maliza ion. Second, we will
show ha also di ec ed anspo , as induced by ha monic-
mixing (HM) [13] o a biha monic d i e, can be con olled
by he HF ield. In ac , anha monici y, oge he wi h he
b eaking o a dynamical symme y, leads o he c ea ion o
di ec ed cu en s in ha monic-mixing. The e o e, when-
e e he po en ial, eno malized by he HF ield, anishes,
di ec ed anspo should cease [14–16].
Be o e discussing he expe imen al esul s, we in oduce
a model use ul o he unde s anding o he po en ial
eno maliza ion o a dissipa i e op ical la ice as p oduced
by a HF oscilla ing ield. We conside he simples model
o a dissipa i e op ical la ice: a Jg¼1=2!Je¼3=2
a om, o mass m, illumina ed by wo coun e -p opaga ing
lase ields wi h o hogonal linea pola iza ions. This con-
igu a ion gene a es a 1D op ical la ice [12]. The a om in
he g ound s a e expe iences he po en ial UðzÞ¼
U0½2cosð2kzÞ=2, whe e zis he lase beam p opaga-
ion axis, k he lase ield wa e ec o and U0 he op ical
la ice dep h [12,17].
We now in oduce a HF oscilla ing o ce wi h equency
!HF and ampli ude AHF:
FHFð Þ¼AHF sinð!HF þ0Þ;(1)
wi h 0a (mainly i ele an ) phase which desc ibes he
s a e o he oscilla ing o ce a ¼0. O in e es he e is he
high- equency case, whe e he equency o he HF d i e
is much la ge han any cha ac e is ic equency o he
sys em, in he p esen case he ib a ional equency !
o he a oms a he bo om o he well. In he asymp o ic
PRL 108, 020603 (2012) PHYSICAL REVIEW LETTERS week ending
13 JANUARY 2012
0031-9007=12=108(2)=020603(5) 020603-1 Ó2012 Ame ican Physical Socie y
limi o in ini e ampli ude and equency o he d i e
(!HF !1,AHF !1), i is possible o show [18] ha ,
consis en ly wi h Re s. [3,14–16], he a omic dynamics
co esponds o he mo ion in a s a ic (i.e., wi hou HF ield)
dissipa i e op ical la ice, wi h eno malized ampli ude ~
U:
~
Uð^zÞ¼U0½2J0ð2k Þcosð2k^zÞ=2;(2)
whe e J0is he Bessel unc ion o he i s kind, and ¼
AHF=ðm!2
HFÞis he pa ame e —he e and he ea e e med
he HF a io—which con ols he eno maliza ion o he
op ical la ice.
The abo e analysis shows ha , in he asymp o ic limi o
in ini e equency and s eng h, a HF ield leads o an
e ec i e eno maliza ion o he po en ial. We now conside
ini e alues o d i ing away om in ini y ha a e expe i-
men ally accessible. The a omic anspo in he op ical
la ice in he p esence o a HF ield is nume ically s udied
o wo di e en se ups, which co espond o he ones used
o p o ide he expe imen al e idence.
In he i s se up, a HF o ce o ini e ampli ude and
equency is applied o a oms in a dissipa i e op ical
la ice. Fo his scheme, he e ec i e eno maliza ion o
he op ical po en ial can be de ec ed by s udying he di -
usion p ope ies o he a oms h ough he la ice. In ac ,
o a dissipa i e op ical la ice o he ype conside ed he e,
i is well es ablished [10,11] ha he e is a c i ical po en ial
dep h loca ed a abou Uc 100E , wi h E ¼@2k2=ð2mÞ
he ecoil ene gy, which sepa a es wo e y di e en e-
gimes. Fo po en ial dep hs la ge han he c i ical one, he
di usion is no mal. Ins ead, o po en ial dep h lowe han
he c i ical one, he di usion becomes anomalous, wi h he
exponen o he di usion dependen on he po en ial am-
pli ude. To be quan i a i e, we de ine he di usion expo-
nen as hx2ð Þi  hxð Þi2 in he limi !1.
Acco ding o his de ini ion, ¼1co esponds o no mal
di usion, while >1cha ac e izes supe di usion. In an
und i en op ical la ice, supe di usion is encoun e ed a
shallow op ical po en ials (below he c i ical dep h), wi h
he exponen inc easing o dec easing po en ial dep h.
Thus we will ake he exponen o he di usion as a
measu e o he po en ial dep h. To assess he e ec i e
eno maliza ion o he po en ial by a HF ield, we nume i-
cally simula ed he dynamics o he a oms in a deep op ical
la ice wi h a HF d i e. We de e mined he di usion ex-
ponen as a unc ion o he HF a io which o in ini e
equency and ampli ude o he d i e de e mines he la ice
eno maliza ion. Ou esul s, epo ed in Fig. 1, show ha
he di usion exponen can be con olled by he HF ield,
wi h a dependence consis en wi h he po en ial eno mal-
iza ion de i ed in he in ini e limi , Eq. (2): inc eases
whene e he po en ial dep h is dec eased, wi h he la ges
alues o p oduced by he alues co esponding o he
ze os o he Bessel unc ion, i.e., o anishing po en ials.
The uppe bound o ¼3 o U0¼0co esponds o he
anishing o he ic ion mechanism (Sisyphus cooling
[12]) associa ed wi h he op ical la ice. Figu e 1also
epo s he alue o U0which co esponds o he exponen
 o an und i en la ice, so o make explici he co e-
spondence be ween a d i ing wi h HF a io and he dep h
o he eno malized po en ial. Finally, we no ice ha ou
esul s o he di usion exponen essen ially coincides
wi h he alues de i ed in he in ini e limi . We can hus
conclude ha he po en ial is e ec i ely eno malized
acco ding o he dependence ob ained in he in ini e limi
[see Eq. (2)].
In he second se up, besides he HF d i e, a biha monic
o ce o he o m
Fð Þ¼F0½A1cosð! ÞþA2cosð2! þÞ (3)
is also applied o he a oms in he la ice. He e !is he
equency o he d i e, o he same o de o magni ude o
smalle han he ib a ional equency, and  he ela i e
phase be ween ha monics. The ampli ude o he la ice, and
i s eno maliza ion by he HF ield, can be de e mined by
obse ing he di ec ed mo ion o he a oms h ough he
la ice. In ac , he wo ha monics o he d i e a e mixed by
he nonha monic po en ial, hus p oducing di ec ed mo ion
o he a oms h ough he la ice [19]. The a e age cu en
being p opo ional o he nonha monici y o he po en ial,
di ec ed anspo measu emen s gi e access o he po en-
ial ampli ude. Mo e p ecisely, o weak d i ing he a e -
age a omic eloci y is expec ed o be o he o m
¼ max sinðdÞ, wi h da dissipa ion-induced
phase lag [19]. In ou simula ions we de e mined he
01234
k
0
0.5
1
1.5
2
2.5
3
α
Γ ω
Γω
01234
k
0
20
30
40
50
60
70
90
150
U0 / E :
FIG. 1 (colo online). Le axis: nume ical esul s, as ob ained
by Mon e Ca lo simula ions, o he spa ial di usion exponen as
a unc ion o he HF a io o an op ical la ice wi h a dep h
U0¼200E . Righ axis: alue o U0which co esponds o he
exponen  o an und i en la ice. T iangles co espond o
esul s ob ained in he in ini e limi , o a pho on sca e ing
a e 0¼5! and 0¼10! , we e ! is he ecoil equency.
The illed diamonds e e o simula ions wi h a HF ield o ini e
ampli ude, wi h equency !HF=! ¼20 and 0¼10! . The
solid line is a guide o he eye o he esul s co esponding o
he in ini e limi wi h 0¼10! . The do ed line is jJ0ð2k Þj.
The e o ba s on he nume ical esul s co espond o he ini e
s a is ics o he Mon e Ca lo simula ions.
PRL 108, 020603 (2012) PHYSICAL REVIEW LETTERS week ending
13 JANUARY 2012
020603-2
eloci y o di e en alues o he phase , so o de i e
he maximum eloci y max. Then by a ying he s eng h
o he HF d i e, we we e able o de e mine max as a
unc ion o he pa ame e , wi h esul s as in Fig. 2.
Once again, he esul s p oduced wi h a ield o la ge, bu
ini e, equency and ampli ude essen ially coincide wi h
hose ob ained in he in ini e limi . These esul s also show
ha cu en measu emen s can be used o p obe he po en-
ial eno maliza ion. Whene e he HF ield leads o a
shallowe po en ial, as om Eq. (2), he cu en is educed,
wi h ze o cu en obse ed o hose alues o leading o a
anishing po en ial.
Ou expe imen al demons a ion o po en ial eno mal-
iza ion ia HF ield elies on he wo de ec ion schemes
ou lined abo e. In bo h se ups, 87Rb a oms a e cooled and
apped in a magne o-op ical ap (MOT). A e a comp es-
sion phase o 50 ms, and 8 ms o op ical molasses, he
a oms a e loaded in o a 1D dissipa i e op ical la ice. The
la ice is c ea ed by he in e e ence o wo linea ly pola -
ized and coun e p opaga ing lase beams, ed de uned om
esonance wi h he D2-line Fg¼2!Fg¼3a omic an-
si ion. One o he la ice beams is sen h ough a double
pass elec o-op ical modula o (EOM), so o be able o
apply a HF phase modula ion. In he e e ence ame o
he la ice, such a phase modula ion ansla es in o a ock-
ing o ce o he o m o Eq. (1). Quan i a i ely, a phase
modula ion ð Þleads o a o ce Fð Þ¼m€ð Þ=ð2kÞin he
e e ence ame o he la ice [19]. In he expe imen s, he
modula ion is p og essi ely u ned on s a ing a e 1 ms
equilib a ion ime in he op ical la ice, wi h a u n-on amp
o 1 ms. The ea e he p ocedu e di e ed o he wo
se ups.
In he i s expe imen , we s udy he di usion o he
a oms in he op ical la ice in he p esence o he HF d i e.
The wid h o he a omic cloud is measu ed by luo escence
imaging a e di usi e expansion inside he d i en la ice.
The wid h is measu ed a a ixed se o expansion imes
wi hin he in e al be ween 1 and 16 ms om he la ice
u n-on. The ime ange o e which images a e aken is
limi ed by he a om loss, pa icula ly impo an o he
alues o he HF a io leading o a anishing eno mal-
ized po en ial. Measu emen s o he spa ial wid h o he
a omic cloud on such a sho empo al ange do no allow
us o de i e an accu a e alue o he exponen o he
di usion [20]. Ins ead, we cha ac e ize he di usion
by an e ec i e di usion coe icien D, as ob ained by
i ing he da a wi h hx2ð Þi  hxð Þi2¼2D . Clea ly,
supe di usion leads o a la ge enhancemen o he de i ed
e ec i e di usion coe icien . Thus a la ge inc ease in he
e ec i e di usion coe icien can be aken as signa u e o
he educ ion o he po en ial, as p oduced by he eno -
maliza ion by he HF ield.
Ou expe imen al esul s o he e ec i e di usion co-
e icien as a unc ion o he HF a io a e epo ed in
Fig. 3. The da a clea ly show ha he a omic di usion is
signi ican ly modi ied by he HF d i e, wi h a dependence
o he e ec i e di usion coe icien on he HF a io con-
sis en wi h he po en ial eno maliza ion [see, e.g., Eq. (2)
o he analy ic exp ession in he limi o in ini e equency
and ampli ude]. Indeed, he e ec i e di usion coe icien
inc eases whene e he HF a io co esponds o dec eas-
ing dep h o he op ical la ice, wi h he la ges alues o
he di usion cons an obse ed in co espondence o he
01234
k
0
0.5
1
1.5
2
max /
FIG. 2 (colo online). Nume ical esul s, as ob ained by Mon e
Ca lo simula ions, o he ampli ude o he cu en max, escaled
by he ecoil eloci y , as a unc ion o he HF a io unde a
biha monic d i ing o ce o he o m o Eq. (3) wi h A1¼A2¼
1,F0¼140@k! , and !¼! . The solid line co esponds o
he esul s ob ained in he in ini e limi and he diamonds o he
simula ion esul s wi h !HF=! ¼20, bo h cases wi h 0¼
10! . The do ed line is jJ0ð2k Þj. The e o ba s on he
nume ical esul s co espond o he ini e s a is ics o he
Mon e Ca lo simula ions.
FIG. 3 (colo online). Expe imen al esul s o he e ec i e
di usion coe icien Das a unc ion o he HF a io o
di e en alues o !HF, as indica ed in he igu e. The da a a e
escaled by he alue o he di usion cons an o an und i en
la ice. The ib a ional equency o he a oms a he bo om o
he well, as de e mined by measu ing he la ice beam powe and
wais , is ! ¼ð91Þ105 ad=s. The solid line, wi h alues
on he igh axis, is jJ0ð2k Þj.
PRL 108, 020603 (2012) PHYSICAL REVIEW LETTERS week ending
13 JANUARY 2012
020603-3
alues o leading o a anishing (in he in ini e limi )
op ical la ice. This shows ha he HF d i e eno malizes
he op ical po en ial, in ag eemen wi h he gene al heo y
[3] and wi h ou nume ical analysis o he speci ic sys em.
In he second expe imen , we p obe he ampli ude o he
eno malized po en ial by s udying di ec ed anspo ol-
lowing ha monic mixing o wo ha monics, as ou lined in
he nume ical analysis. Wi h espec o he p e ious ex-
pe imen de o ed o he s udy o he a omic di usion, an
addi ional biha monic d i e, wi h equencies !,2!and
phase di e ence is in oduced. This is done using addi-
ional acous o-op ical modula o s (AOMs). In he e e -
ence ame o he la ice, he biha monic phase
modula ion co esponds o a d i ing o ce o he o m o
Eq. (3). In he expe imen , he HF d i ing is i s amped
up, as in he p e ious expe imen . Then he biha monic
d i e is p og essi ely u ned on wi h a amp-up ime o
4 ms. The eloci y o he cen e -o -mass o he a omic
cloud is de i ed by posi ion measu emen s ob ained ia
luo escence imaging. The measu emen s a e epea ed o
10 di e en alues o he phase di e ence be ween
ha monics. The da a a e hen i ed by he expec ed depen-
dence ¼ max sinðdÞ, hus de i ing a alue o
max which can be aken as a measu e o he eno malized
po en ial dep h. In ac , he mixing o ha monics equi es
an anha monic po en ial, wi h he cu en gene a ed p o-
po ional o he anha monici y. Ou esul s o max as a
unc ion o he HF a io a e p esen ed in Fig. 4. These
da a o he di ec ed anspo ampli ude a e consis en
wi h he eno maliza ion o he po en ial dep h by he HF
d i e. In ac , whene e he alue o he HF a io co e-
sponds o a educed po en ial dep h, he cu en dec eases,
wi h ze o cu en obse ed o he alues co esponding
o he ze os o he Bessel unc ion, a signa u e o he
anishing op ical la ice. These esul s also demons a e a
new scheme o he con ol o he anspo ia ac ields:
he ampli ude o he cu en can be con olled by a a ia-
ion in he HF ield and he di ec ion e e sed ia a shi
in he ela i e phase be ween ha monics. Finally, we no ice
ha he e is a small de ia ion, bo h in he expe imen and
in he nume ical simula ions (see Fig. 2) om he beha io
expec ed om he Bessel unc ion a small alues o , wi h
he da a showing an ex a peak a k 0:75. This peak
could be explained by a supe imposed esonance co e-
sponding o he ma ching o he equency o he biha -
monic o ce wi h he oscilla ion equency o he a oms a
he bo om o he eno malized well.
In conclusion, in his wo k we demons a ed expe imen-
ally he phenomenon o ib a ional esonance in a dissi-
pa i e op ical la ice. The applica ion o a HF d i e, wi h
equency much la ge han any cha ac e is ic equency o
he sys em, leads o he eno maliza ion o he po en ial.
The eno malized ampli ude can be con olled by he HF
d i e pa ame e s. We used anspo measu emen s as a
p obe o he po en ial eno maliza ion. The e y same
expe imen s also demons a ed ha anspo can be con-
olled by he HF d i e ia po en ial eno maliza ion.
The possibili y o eno malize a po en ial ia ac ields, as
demons a ed he e, is e y gene al, and i is applicable o
any sys em o pa icles in a pe iodic po en ial. As such, i
pa es he way o he con ol o po en ials in sys ems in
which hey a e no di ec ly accessible, and i may also be
applicable o solid s a e sys ems whe e ac d i es can be
in oduced by he applica ion o elec ic ields.
Finally, ou se up can also be aken as he demons a ion
o a senso able o de ec signals wi h equency exceeding
any in e nal equency o he senso [14–16]. He e, he
signal de ec ed is he HF d i e whose p esence, al hough
no coupling o any in e nal mode o he sys em, can be
p ecisely de ec ed due o i s e ec ia he po en ial
eno maliza ion.
We acknowledge inancial suppo om he Le e hulme
T us , he Minis e io de Ciencia e Inno acio
´n o Spain
FIS2008-02873 (D. C.), and he DAAD (P. C. H).
[1] P. Reimann, Phys. Rep. 361, 57 (2002).
[2] P. Ha
¨nggi and F. Ma chesoni, Re . Mod. Phys. 81, 387
(2009).
[3] P. Landa and P. V. E. McClin ock, J. Phys. A 33, L433
(2000).
[4] V. N. Chizhe sky, E. Smeu, and G. Giacomelli, Phys. Re .
Le . 91, 220602 (2003).
[5] J. Casado-Pascual and J. P. Bal ana
´s, Phys. Re . E 69,
046108 (2004).
FIG. 4 (colo online). Expe imen al esul s o he ampli ude
o he cu en max, escaled by he ecoil eloci y , o di ec ed
anspo h ough he op ical la ice as a unc ion o he HF a io
. In addi ion o he HF d i e, a biha monic o ce o he o m o
Eq. (3) is applied o he a oms, wi h pa ame e s A1¼1,A2¼2,
!¼9:42 104 ad=s,F0¼112@k! . The ib a ional e-
quency o he a oms a he bo om o he well is ! ¼ð91Þ
105 ad=s. The solid line, wi h alues on he igh axis, is
jJ0ð2k Þj.
PRL 108, 020603 (2012) PHYSICAL REVIEW LETTERS week ending
13 JANUARY 2012
020603-4
[6] J. Casado-Pascual, D. Cube o, and J. P. Bal ana
´s,
Eu ophys. Le . 77, 50 004 (2007).
[7] D. Cube o, J. P. Bal ana
´s, and J. Casado-Pascual, Phys.
Re . E 73, 061102 (2006).
[8] A. Ecka d , C. Weiss, and M. Hol haus, Phys. Re . Le .
95, 260404 (2005).
[9] H. Lignie , C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O.
Mo sch, and E. A imondo, Phys. Re . Le . 99, 220403
(2007).
[10] S. Ma ks eine , K. Ellinge , and P. Zolle , Phys. Re . A 53,
3409 (1996).
[11] H. Ka o i, S. Schlip , and H. Wal he , Phys. Re . Le . 79,
2221 (1997).
[12] G. G ynbe g and C. Menne a -Robillia d, Phys. Rep. 355,
335 (2001).
[13] K. Seege and W. Mau e , Solid S a e Commun. 27,603
(1978); W. Wonnebe ge and H. J. B eymaye , Z. Phys. B 43,
329 (1981);F.Ma chesoni,Phys. Le . A 119, 221 (1986).
[14] M. Bo omeo and F. Ma chesoni, Eu ophys. Le . 72, 362
(2005).
[15] M. Bo omeo and F. Ma chesoni, Phys. Re . E 73, 016142
(2006).
[16] M. Bo omeo and F. Ma chesoni, Phys. Re . Le . 99,
150605 (2007).
[17] See Supplemen al Ma e ial a h p://link.aps.o g/
supplemen al/10.1103/PhysRe Le .108.020603 o a
comple e de ini ion o he model.
[18] See Supplemen al Ma e ial a h p://link.aps.o g/
supplemen al/10.1103/PhysRe Le .108.020603 o he
comple e de i a ion o he sys em eno maliza ion by
he HF ield.
[19] F. Renzoni, Ad . A . Mol. Op . Phys. 57,1
(2009).
[20] A om losses lead o an unde es ima e o he di usion
exponen , as many supe di using a oms a e los .
PRL 108, 020603 (2012) PHYSICAL REVIEW LETTERS week ending
13 JANUARY 2012
020603-5