MSc in Pho onics PHOTONICSBCN
Uni e si a Poli ècnica de Ca alunya (UPC)
Uni e si a Au ònoma de Ba celona (UAB)
Uni e si a de Ba celona (UB)
Ins i u de Ciències Fo òniques (ICFO)
h p://www.pho onicsbcn.eu
Mas e in Pho onics
MASTER THESIS WORK
TOWARDS INTERFEROMETRY WITH MATTER
WAVE BRIGHT SOLITONS
Juan Polo Gomez
Supe ised by D . Ve ònica Ahu inge B e o (UAB)
P esen ed on da e 4 h o Sep embe , 2012
Regis e ed a
Towa ds in e e ome y wi h ma e wa e b igh
soli ons
Juan Polo Gomez
G up d’ `Op ica, Dep . de F´ısica, Uni e si a Au `onoma de Ba celona, E-08193
Bella e a, Spain
E-mail: zes zes zes gmail.com
Abs ac . We s udy he implemen a ion o a ma e wa e b igh soli on
in e e ome e o med by a Gaussian po en ial ba ie placed a he cen e o a
ha monic ap po en ial. A e nume ically e alua ing he ansmission coe icien
o he po en ial ba ie as a unc ion o he a io be ween he kine ic ene gy o an
inciden ma e wa e b igh soli on and he heigh o he ba ie , we ocus on he
case o which he ansmi ed and he e lec ed spli soli ons ha e he same numbe
o a oms. This beam spli e beha io is ully cha ac e ized in e ms o he phase
and he ene gies o he wo ou come soli ons. The ecombina ion p ocess h ough he
ba ie is also in es iga ed in e ms o he phase di e ence be ween he wo a ms o
he in e e ome e .
Keywo ds: ma e wa e soli ons, in e ac ion wi h po en ial ba ie s, a om
in e e ome e .
1. In oduc ion
In e e ome y has been o many yea s an impo an scien i ic ool wi h a my iad o
applica ions, anging om p ecision measu emen s o undamen al s udies. A omic
in e e ome e s ha exploi he wa e cha ac e o a oms allow o e y high sensi i i ies
due o he associa ed small wa eleng h. Mo eo e , he inhe en a omic p ope ies like
mass o pola izabili y o e new possibili ies o high-p ecision measu emen s o , o
ins ance, undamen al cons an s, in e nal o ces o o a ions [1–4]. Speci ically, he use
o ul acold a oms, ha can be p ecisely manipula ed a he quan um le el and can
be held o long imes in aps [5], p o ides e y high p ecision and sensi i i y [6].
Ne e heless, in linea in e e ome y, he minimum unce ain y in de e mining he
phase di e ence is gi en by he s anda d quan um limi [7], i.e., he expec ed o a
classical measu emen . I has been shown ha his limi can be su passed, app oaching
he Heisenbe g limi , using nonclassical inpu s a es [8–12] o en angled s a es wi hin
he in e e ome e in he egime o la ge a om numbe wi h con olled in e ac ions [7].
One po en ial p oblem ha a omic in e e ome e s ace is based on he ac ha he
wa e unc ion dispe ses du ing i s e olu ion. A possible solu ion is o use ma e wa e
b igh soli ons in Bose-Eins ein condensa es (BEC). Ma e wa e b igh soli ons a e
Towa ds in e e ome y wi h ma e wa e b igh soli ons 3
soli a y wa es wi h a ac i e in e a omic in e ac ions, ha p o e o be a he obus
when colliding wi h each o he o when in e ac ing in an ex e nal po en ial and p esen
a e y impo an cha ac e is ic: hey can p opaga e wi hou dispe sion [13,14]. Wi hin
he mean ield app oxima ion, hei dynamics is well desc ibed by he G oss Pi ae skii
equa ion (GPE), al hough e ec i e pa icle models also exis and ha e been used o
cha ac e ize he cen e o mass ajec o ies o he soli ons [15–17]. No e ha he use
o soli ons o in e e ome y [18–26] opens new pe spec i es and allows o add ess
undamen al ques ions due o he dual na u e (pa icle and wa e) o soli ons in BEC.
Wi hin he quasi 1-D limi , collisions o ma e wa e b igh soli ons wi h ba ie s ha e
been la gely s udied in o de o cohe en ly spli he soli on [25] and he ecombina ion
o iden ical soli ons h ough a po en ial ba ie has also been add essed [19,20].
In his wo k, we will in es iga e he implemen a ion o a ull ma e wa e b igh soli on
in e e ome e o med by a ha monic po en ial ap and a Gaussian ba ie a i s cen e
in which he inciden ma e wa e soli on spli s and he wo ou pu spli ones ecombine
a e some ime. This se up is cu en ly unde expe imen al s udy in he g oup o P o .
R. Hule [27]. In sec ion 2, we in oduce he physical sys em ha we a e conside ing.
In sec ion 3, we analyze in de ail he ole o he ba ie as a beam spli e cha ac e izing
he wo ou pu soli ons as a unc ion o he sys em pa ame e s. Sec ion 4 is de o ed o
he ecombina ion p ocess and in sec ion 5 we p esen he conclusions.
2. Physical sys em
We conside a Bose-Eins ein condensa e o 7Li a T= 0 whose dynamics wi hin he
mean ield app oach is desc ibed by he 3D ime dependen GPE [28]:
i¯h∂
∂ ϕ( , ) = −¯h2∇2
2m+V( ) + g3D|ϕ( , )|2!ϕ( , ) ; (1)
whe e he i s e m o he igh hand side o Eq. (1) co esponds o he kine ic
con ibu ion, he second e m is he ex e nal po en ial ha eads V( ) = m[ω2
(x2+
y2) + ω2
zz2]/2, and he las e m is he nonlinea e m, whe e g3D= 4π¯h2asN/m,
and N,asand ma e he a om numbe , s-wa e sca e ing leng h and a omic mass,
espec i ely. The wa e unc ion is no malized o 1, and he sca e ing leng h is nega i e,
as<0, which means ha he in e ac ions a e a ac i e. Fo su icien ly igh adial
con inemen (ω ωz), bu allowing ha in e ac ions emain 3D, as(¯h/mω )1/2,
i is possible o sepa a e he adial and he axial dynamics using he ansa z ϕ( ) =
Ψ(z)(mω /π¯h)1/2exp (−mω [x2+y2]/2¯h), ob aining he educed 1D GPE [26]:
i¯h∂
∂ Ψ(z, ) = −¯h2
2m
∂2
∂z2+V(z) + g1D|Ψ(z, )|2!Ψ(z, ),(2)
whe e g1D= 2N¯hω as[29] and he ex e nal po en ial educes o V(z) = 1
2mω2
zz2. By
using C ank-Nicolson me hod in imagina y ime e olu ion we calcula e he g ound s a e
o he sys em o s a e o he a pa ame e alues [13]: ω = 2π×710 Hz, ωz= 2π×78
Hz, as=−0.21 nm, m= 1.165 ×10−26 kg and N= 15000 a oms, ob aining a ma e
Towa ds in e e ome y wi h ma e wa e b igh soli ons 4
wa e b igh soli on.
Once a s able soli on is c ea ed, we displace he cen e o he ha monic po en ial a
dis ance z0and in oduce a Gaussian po en ial ba ie cen e ed a z= 0 on which he
soli on will collide. The ex e nal po en ial eads now:
V(z) = 1
2mω2
z(z−z0)2+Vbez2
2σ2,(3)
whe e Vband σa e he s eng h and he wid h o he ba ie , espec i ely. Wi h
his con igu a ion we implemen an in e e ome e o ma e wa e b igh soli ons ha
can be sepa a ed in o h ee s eps: a) he soli on is spli in wo by colliding wi h he
ba ie ; b) he wo spli soli ons e ol e in he ex e nal ha monic po en ial educing hei
kine ic ene gy and inc easing he po en ial one un il hey s op and go back in opposi e
di ec ions; and c) a ecollision o he soli ons wi h he ba ie occu s a he cen e o
he ap. Fig. 1 shows he p ocess schema ically.
In o de o cha ac e ize he dynamics o he spli ing o he ini ial soli on and he
Figu e 1. Schema ics o he in e e ome e sequence: (a) ini ial si ua ion in which
he soli on is displaced wi h espec o he cen e o he ha monic po en ial ap by
a dis ance z0; (b) he wo spli soli ons ob ained a e he collision wi h he ba ie
sepa a e om each o he (c) he wo soli ons e u n o he posi ion o he ba ie and
ecollide.
e olu ion o he sys em a e wa ds, we calcula e he expec ed alue o he posi ion and
momen um o he soli on be o e i s collision wi h he ba ie and o he wo soli ons
ob ained a e he spli ing:
hzii=K−1
iZCi
dz|Ψ(z, )|2z, (4)
hpii=−i¯hK−1
iZCi
dzΨ∗(z, )∂
∂zΨ(z, ),(5)
whe e Ki=ZCi
dz|Ψ(z, )|2and Cideno es he egion con aining he soli ons: being
i=I, R, T o Inciden , Re lec ed and T ansmi ed egions, espec i ely. We also de ine
he posi ion and momen um unce ain y:
∆zi= (hz2
ii−hzii2)1/2,(6)
∆pi= (hp2
ii−hpii2)1/2,(7)
whe e:
hz2
ii=K−1
iZCi
dz|Ψ(z, )|2z2,(8)
Towa ds in e e ome y wi h ma e wa e b igh soli ons 5
hp2
ii=−¯h2K−1
iZCi
dzΨ∗(z, )∂2
∂z2Ψ(z, ).(9)
The numbe o pa icles o each soli on is gi en by Ni=NKiand he ene gy pe pa icle
eads:
ei=Ei
N=ZCi
dz 

¯h2
2m
∂
∂zΨ(z, )
2
+V(z)|Ψ(z, )|2+g1D|Ψ(z, )|4
,(10)
whe e he i s e m o he igh hand side co esponds o he kine ic ene gy Ek, he
second one is he ex e nal po en ial ene gy, Eho, and he las one is he in e nal ene gy
due o he a om-a om in e ac ion, Ein . No e also ha by di ec in eg a ion o he
GPE unde he s a iona y ansa z Ψ(z, ) = Ψ(z) exp (−iµ /¯h), whe e µis he chemical
po en ial, we ob ain µ=Ek+Eho + 2Ein [28].
3. Spli ing p ocess
In his sec ion we analyze he main aspec s ha a ec he dynamics o he spli ing
p ocess o a ma e wa e b igh soli on unde he collision wi h a Gaussian po en ial
ba ie .
3.1. T ansmission
By an ins an aneous displacemen o he apping po en ial, he cen e o he soli on is
placed a −z0and he soli on acqui es a po en ial ene gy ha , as p edic ed by pa icle
models [30,31], is ully con e ed in o kine ic ene gy once he soli on eaches he cen e
o he po en ial. The e o e, he kine ic ene gy o he soli on when in e ac ing wi h he
ba ie is Eho =1
2mω2
z(z−z0)2. We ha e checked ha his esul is in ull ag eemen
wi h Eq. (10).
Fig. 2(a) shows he nume ically e alua ed ansmission coe icien as a unc ion o he
a io be ween he kine ic ene gy o he soli on and he heigh o he po en ial ba ie o
a ixed po en ial ba ie heigh o Vb= 17.14¯hωz(squa es), and o a ixed kine ic ene gy
o Ek= 10.83¯hωz(ci cles); in bo h cases σ= 0.5µm. The di e en beha io s ob ained
close o T= 1 can be unde s ood by aking in o accoun ha he changes in he s eng h
o he po en ial o in he kine ic ene gy o he inciden soli on do no p oduce exac ly
he same e ec due o he di e en a ea ha he soli on has o pene a e in o de o
pass h ough he ba ie . Ne e heless, a common ea u e is he high slope in he egion
whe e he ansmission coe icien is 0.5, i.e. whe e he ini ial soli on spli s in o wo
soli ons wi h he same numbe o a oms, which makes his poin e y sensi i e o he
ini ial condi ions. Fig. 2(b) shows he ansmission as a unc ion o he wid h o he
ba ie (ci cles) o a ixed a io Ek/Vb= 0.63. As expec ed, he inc ease o he wid h
o he ba ie co esponds o a dec ease in ansmission. In Fig. 2, we also plo he
analy ical ansmission coe icien o a ec angula ba ie (do ed-dashed line), which

Towa ds in e e ome y wi h ma e wa e b igh soli ons 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
T ansmission T
Ek/Vb
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2
T ansmission T
σ(µm)
(b)
Figu e 2. T ansmission as a unc ion o (a) he a io be ween Ek/Vband (b) he wid h
o he ba ie ob ained by ixing Ek= 17.14¯hωz(ci cles) o Vb= 10.83¯hωz(squa es).
The analy ic esul o a ec angula ba ie (do ed-dashed line) and o he WKB
app oxima ion (dashed line) in he linea case a e also plo ed. The pa ame e s o he
sys em a e N= 15000, ωz= 2π×78 Hz, ω = 2π×710 Hz, as=−0.21 nm. In (a),
σ= 0.5µm and in (b) Ek/Vb= 0.63.
eads [32]:
TuE<Vb=1
1 + V2
bsinh2(k1(2σ))
4E(Vb−E)
TuE>Vb=1
1 + V2
bsin2(k1(2σ))
4E(E−Vb)
; (11)
whe e k1=q2m|Vb−Ek|/¯h2; and he Wen zel−K ame s−B illouin (WKB)
app oxima ion (dashed line). The WKB app oxima ion es ima es he ansmission
coe icien , o small alues o Tand o po en ials ha do no change ab up ly, aking
in o accoun he shape o he po en ial and conside ing he ac ha he ampli ude and
he phase o he wa e unc ion a y slowly. The de ailed analy ical de i a ion o he
ansmission coe icien can be ound in [32]:
TWKB ≃e1
¯hRb
adz√2m(V(z)−Ek)+1
4e
−1
¯hRb
adz√2m(V(z)−Ek)−2
; (12)
whe e aand ba e he z-axis poin s whe e he kine ic ene gy and he ba ie coincide,
which o he Gaussian shape ba ie a e b=−a=q2σ2ln (Vb/Ek).
No e ha hese analy ical ansmission unc ions (Eq. (11) and (12)) a e only alid in
he linea egime. The e o e, in o de o be alid o ou nume ical simula ion Ekhas
o be la ge enough o he wid h o he ba ie small enough o minimize he in e ac ion
ime o he soli on wi h he ba ie . Ne e heless, ou speci ic se up limi s hese wo
pa ame e s: he minimum wid h o he ba ie is ixed by he di ac ion limi , since
he Gaussian po en ial is expe imen ally c ea ed by using a ocalized lase beam wi h
a beam wais equal o σand he kine ic ene gy is limi ed by he ini e po en ial ap
displacemen , z0.
Mo eo e , he e is an addi ional aspec ha makes he compa ison wi h he analy ical
unc ions di icul : he soli on is a wa e packe desc ibed by a mac oscopic wa e unc ion,
whe eas he analy ical cases conside plane wa es. The e o e, while plane wa es ha e a
Towa ds in e e ome y wi h ma e wa e b igh soli ons 7
well de ined momen um, he soli on has a e y well de ined posi ion and he momen um
unce ain y is e y la ge, ul illing he unce ain y ela ion ∆xI∆pI≥¯h/2 close o
he sa u a ion alue. Fo ins ance, o an inciden soli on wi h N= 15000 he
unce ain ies using Eq. (6) and (7) a e ∆xI= 0.583 nm and ∆pI/¯h= 0.898 1/nm,
gi ing ∆xI∆pI/¯h= 0.524.
In spi e o he discussed limi a ions, he ec angula po en ial ba ie i s a he well wi h
he nume ical esul s (see Fig. 2). Howe e he slope o he ansmission unc ion o
he ec angula ba ie nea T= 0.5, egion o in e es o in e e ome ic applica ions,
is lowe han he one ob ained nume ically wi h he Gaussian ba ie . The WKB
app oxima ion ag ees wi h he nume ical ansmission coe icien when T→0. Howe e
his egion is ou o he egime o in e e ome ic applica ions whe e he aim is o spli
he soli on in wo iden ical soli ons (T= 0.5).
Fig. 3 shows he po en ial s eng h Vbas a unc ion o Ek o which T= 0.5 and o
wo di e en alues o he numbe o pa icles N. We can obse e ha he dependence
is app oxima ely linea and independen o N o la ge kine ic ene gies. Ne e heless,
o small kine ic ene gies he nonlinea i y s a s o play an impo an ole and b eaks
he linea dependence (see inse o Fig. 3). In he ollowing we conside a soli on wi h
N= 15000. This alue has been chosen in o de o ob ain wo soli ons in he spli ing
p ocess i.e, he nonlinea i y is su icien o compensa e he dispe sion a e he spli ing,
whe e he numbe o a oms o each o he ou come soli ons is N/2.
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60 70
Vb/¯hωz
Ek/¯hωz
0.3
0.45
0.6
0.75
0.9
0 20 40 60
Ek/Vb
Ek/¯hωz
Figu e 3. Po en ial ba ie heigh as a unc ion o he kine ic ene gy o he inciden
soli on o a ixed ansmission coe icien T= 0.5, o an inciden soli on o N= 15000
(squa es) and N= 7500 (ci cles). The inse shows he a io Ek/Vbas a unc ion o
he kine ic ene gy. The es o he pa ame e s o he sys em a e ωz= 2π×78 Hz,
ω = 2π×710 Hz, as=−0.21 nm and σ= 0.5µm.
3.2. Ene gy and Phase
The sensi i i y o an in e e ome e has a s ong dependence on how he phase di e -
ence is measu ed and o how long he spli pa hs can accumula e phase wi hou losing
cohe ence [6]. The e o e, in sec ion 4, we will analyze he phase e olu ion o he inciden
Towa ds in e e ome y wi h ma e wa e b igh soli ons 8
soli on and o he wo spli soli ons p oduced by he in e ac ion wi h he ba ie .
Ne e heless, in his sec ion we s udy i s he phase di e ence be ween he ansmi ed
and e lec ed soli ons, in oduced by he in e ac ion wi h he ba ie du ing he spli -
ing p ocess, by se ing he ha monic ex e nal po en ial equal o ze o. Fig. 4(a) shows
0.5
0.75
1
1.25
1.5
1.75
2
2.25
20 30 40 50 60 70 80
∆φ( ad)
Ek/¯hωz
(a)
58
60
62
Ek/¯hωz
(b)
-17
-15
-13
3 3.5 4 4.5 5 5.5 6
Ein /¯hωz
(ms)
Figu e 4. (a) Phase di e ence be ween he ansmi ed and he e lec ed soli on as
a unc ion o he kine ic ene gy o he inciden soli on o T= 0.5. (b) Kine ic and
in e nal ene gy o a o al e lec ed soli on wi h ini ial kine ic ene gy Ek= 58.85¯hωz.
The solid (dashed) line shows he ime a e aged ene gy be o e (a e ) he collision wi h
he ba ie o bo h Ekand Ein . The in e al o ime in which he collision wi h he
ba ie occu s is ma ked wi h he black egion. The pa ame e s used a e: N= 15000,
ω = 2π×710 Hz, ωz= 0 Hz, σ= 0.5µm and as=−0.21 nm.
he linea dependence o he phase di e ence be ween he ansmi ed and he e lec ed
soli on, in oduced by he e lec ion p ocess, wi h he kine ic ene gy o he inciden
soli on. We conside N= 15000, po en ial wid h σ= 0.5µm and a heigh such ha
he T= 0.5 condi ion is ul illed, i.e., he wo esul ing soli ons ha e he same numbe
o a oms. By using exp ession (10), one can also calcula e he e olu ion o he di e en
con ibu ions o he ene gy du ing he spli ing p ocess. We ob ain ha he e lec ed
soli on has less kine ic ene gy han he ansmi ed one, and mo eo e , in he e lec ion
p ocess some kine ic ene gy is ans e ed o in e nal ene gy. The e o e, he ansmi ed
and e lec ed soli ons a e no iden ical ene ge ically. As an example, an inciden soli on
wi h an ini ial ene gy Ek= 20.08¯hωzand Ein =−14.84¯hωzspli s in o wo soli ons wi h
he same numbe o pa icles when in e ac ing wi h a ba ie wi h Vb= 20.57¯hω and
σ= 0.5µm. The kine ic (in e nal) ene gy o he e lec ed and he ansmi ed soli on
is EkT= 5.20¯hωz(Ein T=−1.73¯hωz) and EkR= 3.57¯hωz(Ein R=−1.80¯hωz), espec-
i ely. Al hough he o al ene gy is conse ed, EkR+Ein R+EkT+Ein T=Ek+Ein , i
is clea ha he e lec ed soli on has less ene gy han he ansmi ed one. An ex eme
case occu s o he o al e lec ion o a soli on o N= 15000 whe e he ans e o kine ic
o in e nal ene gy is clea ly obse ed (Fig. 4(b)): he solid (dashed) line shows he ime
a e aged ene gy be o e (a e ) he collision wi h he ba ie o bo h Ekand Ein . I is
wo h no icing ha he e lec ion p ocess can also induce b ea hing exci a ions in which
some ac ion o he ene gy is exchanged pe iodically be ween Ekand Ein as shown in
Towa ds in e e ome y wi h ma e wa e b igh soli ons 9
Fig. 4(b), o he o al e lec ion case (T= 0).
4. Recombina ion p ocess
In his sec ion we add ess he las s ep o he p oposed implemen a ion o a ma e
wa e b igh soli on in e e ome e , i.e., he ee e olu ion o he spli soli ons and hei
ecombina ion a he cen e o he ha monic po en ial (Fig. 1(c)). We conside he
case desc ibed in sec ion 3.2 in which one soli on collides wi h a Gaussian ba ie and
spli s in o wo soli ons wi h he same Nand a ela i e phase induced by he po en ial
ba ie . Du ing hei e olu ion in he ha monic po en ial, he phases o he spli soli ons
e ol e di e en ly due o he di e en eloci y acqui ed du ing he spli ing p ocess (see
sec ion 3.2) as shown in Fig. 5. The inciden soli on inc eases i s phase as i p opaga es
owa ds he ba ie . A e he collision wi h he ba ie a = 4.34 ms, wo soli ons
appea wi h a phase di e ence (see inse Fig.5) as desc ibed in sec ion 3.2. A e hei
e olu ion in he ha monic po en ial pe o ming a dipole oscilla ion, he wo spli soli ons
collide a he posi ion o he ba ie a = 10.40 ms. The ou come o his collision
depends on he ela i e phase be ween he wo soli ons ha ecombine. In gene al,
one ob ains wo soli ons wi h di e en numbe o pa icles a e he ecombina ion.
The e o e, he accumula ed ela i e phase in he wo a ms o he in e e ome e can be
de e mined by measu ing he ela i e pa icle numbe o he wo ou going soli ons [25].
Fig. 6 shows di e en ou pu s o he s udied a omic in e e ome e co esponding o
42
44
46
48
50
52
54
56
58
60
62
4 5 6 7 8 9 10
φ( ad)
(ms)
45
46
47
48
4.2 4.4 4.6
φ( ad)
(ms)
Figu e 5. Time e olu ion o he phase o he inciden soli on (black line) be o e he
in e ac ion wi h he ba ie and o he e lec ed (blue line) and ansmi ed ( ed line)
soli on c ea ed in he spli ing p ocess du ing i s dipole oscilla ion in he ha monic
ap. The pa ame e s used a e N= 15000, ωz= 2π×78 Hz, ω = 2π×710 Hz,
as=−0.21 nm, Ek= 10.74¯hωz,σ= 0.5µm, he po en ial used o spli ing in o wo
soli ons wi h he same numbe o pa icles is Vb= 17.70¯hωz. The ime in e al in
which he collision wi h he ba ie occu s is ma ked wi h he black egion. The inse
shows an enla ged iew o he spli ing p ocess.
di e en phase di e ences be ween he ecombined soli ons. The phase di e ence in his