Phase collapse and revival of a Bose-Einstein condensate induced by an off-resonant optical probe field

1.1 MB·

by  L. G. E. Arruda

PhasecollapseandrevivalofaBose-Einsteincondensateinducedbyanoff-resonantopticalprobe
field
L. G. E. Arruda,1,∗ G. A. Prataviera,2,† and M. C. de Oliveira3,‡
1DepartamentodeF´ısica, UniversidadeFederaldeSa˜oCarlos, 13565-905, Sa˜oCarlos, SP,Brazil
2DepartamentodeAdministrac¸a˜o,FEA-RP,UniversidadedeSa˜oPaulo,14040-905,Ribeira˜oPreto,SP,Brazil
3InstitutodeF´ısicaGlebWataghin, UniversidadedeCampinas, 13083-970, Campinas, SP,Brazil
(Dated:January20,2017)
WeconsiderasinglemodeatomicBose-Einsteincondensateinteractingwithanoff-resonantopticalprobe
field. We show that the condensate phase revival time is dependent on the atom-light interaction, allowing
optical control on the atomic collapse and revival dynamics through frequencies detuning. Also, incoherent
effectsoverthecondensatephaseareincludedbyconsideringacontinuousphoto-detectionovertheprobefield.
We consider conditioned and unconditioned photo-counting events and verify that no extra control upon the
7 condensate is achieved by the probe photo-detection. Whether or not conditioned to the number of photons
1 detected,thephoto-detectionprocesscontributestosuppressesthetypicalcollapseandrevivaldynamics.
0
2 PACSnumbers:03.75.Kk,42.50.Ct,32.80.-t

n
a
J I. INTRODUCTION of the atomic field statistical properties [9–11]. Previously,
withasetuprelyingontheatomicsystem-probefieldinterac-
9
1 CollapseandrevivalphenomenainBose-Einsteinconden- tionmediatedthroughaclassicalpump[12],weshowedthat
sates (BECs) have been investigated since 1996 [1], shortly under continuous photo-counting, the moments of the probe

] afteritsexperimentalachievementwithultracoldatoms[2,3]. light photon number might carry information about the even
h
They are a consequence of the quantized structure of the moments of the atom number. However, neither the possi-
p
- matter field and the coherent interactions between the atoms bility of controlling of atomic properties with the proposed
nt through atomic collisions (See [4] and references therein). setup, oradetailedanalyzesabouttheeffectsofconditioned
a Dynamically originated due the presence of nonlinearities, singlephoto-countingeventsovertheBECstate,wascarried
u phasecollapseandrevivalissimilarinnaturetothecollapse out. Such analyzes is important, given the interest in phase
q andrevivalappearinginRabioscillationswhenasingletwo- collapse time, since in some situations the detection process
[ level atom interacts with a single mode of a quantized opti- allowsanadditionalcontrolorbetterinferenceofparameters.
1 cal field, or when a coherent light field propagates in a non- Forexample,in[13]anopticalprobecontinuousdetectional-
v linear medium [5]. The possibility of coexistence of non- lows one to create relative phase of two spatially separated
2 linear effects due to external influences opens an interesting atomic BEC. Also, as showed in [14], even dissipative envi-
2 venue for investigating and controlling [6] such phenomena ronmentscanbeusefultotailordynamicsofstatesandphase
5 inBECs. Particularly, thecontrolofphasecollapseinBECs incoldatomicsamples.
5
isextremelyrelevantforatomicinterferometrysinceitmight To address these questions, in this paper we analyze the
0
. preventphasediffusion,whileallowingaperformancebelow situation in which a single mode atomic BEC is coupled to
1 thestandardquantumlimit[7]. Severalattemptsonbothde- a quantized optical probe field, through an undepleted opti-
0
velopingnewtoolsandpushingforwardthepossiblefrontiers calpump. Thenonlinearnatureoftheengineeredinteraction
7
1 for collapse control have been investigated for double-well between the atoms and the probe field, induces a proper dy-
: condensates (See [6] and references therein). A possible ap- namics of collapse and revival of the atomic phase, even in
v proachistoconsidertheactionofanexternallightprobeover theabsenceofatomiccollisionsforaverydilutedatomicgas.
i
X the atomic system, although, in general this leads to an even We show that the characteristic revival time, depends on the
r worsesituationwherethelightfielditselfinducesphasecol- commensurability between the parameters such as the cou-
a lapse in an irreversible manner. It is expected that, unless a pling constants and the detuning between the transition fre-
timedependentinteractionisemployed, nofurthercontrolis quencyofthetwolevelatoms(thatrepresentsourBEC)and
achieved. thefrequency ofthe pumpfield, as wellas, the atomiccolli-
Nonetheless, the interaction of the atomic system with a sionparameter. This allowscontrol overthe atomiccollapse
quantized light probe field can be engineered with the assis- andrevivalthroughtheeffectiveatom-lightcouplingparame-
tanceofanadditionalpumpfield(See[8]foranexcellentre- ters. Inaddition, weincludecontinuousphoto-countingover
viewonthistopic).Thisschemeallows,e.g.,thesimultaneous the probe field and analyze its possible control over the col-
amplification of atomic and optical fields, as well as control lapseandrevivaltimes. Weconcludethatnofurthercontrolis
achievedbythesemeasurements.
Thepaperisorganizedasfollows: InSectionIIwededuce
the model for a BEC trapped in a single well potential, in-
∗[email protected] teractingwithaclassicalundepletedopticalfieldandaquan-
†[email protected] tized optical probe mode in the far-off resonant regime. In
‡marcos@ifi.unicamp.br Section III, we analyze the collapse and revival dynamics of
2
theBECphaseusingtheHusimifunctionandthevarianceof n,andtogetherwiththecreationoperator,satisfiestheregular
the phase operator. We show how the coupling parameters comutationrelation: (cid:2)c ,c†(cid:3)=δ . Withtheexpansion(2),
n n nm
between atomsand light determineand enable tocontrol the theHamiltonian(1)takesthefollowingform
collapse and revival dynamics. In Section IV we include an
(cid:32) (cid:33)
incoherent process through a continuous photo-detection on H =(cid:126)(cid:88) ω˜ + |g2α2|2 c†c +(cid:126)(cid:88)κ c†c†c c
thequantizedopticalprobefield. Theeffectofopticalphoto- n ∆ n n ijlm i j l m
counting over the BEC phase dynamics is analyzed. Finally, n ijlm
inSectionVwepresenttheconclusions. +(cid:126)(cid:88)(cid:18)g1∗g2α2χ a†c†c + g2∗g1α2∗χ a c† c (cid:19)
∆ mn 1 n m ∆ nm 1 m n
mn
(cid:32) (cid:33)
II. MODEL +(cid:126) δ+ |g1|2 (cid:88)c†c a†a , (3)
∆ n n 1 1
n
We consider a system of bosonic two-level atoms inter-
acting via two-body collisions and coupled through electric- where
dipoleinteractionwithtwosingle-moderunningwaveoptical
U (cid:90)
fireeslpdescotifvefrleyq.uTenhceiepsroωb1eaonpdtiωca2l, afinedldw(a1v)eivsectrteoartsekd1qaunadntkum2, κijlm = 2(cid:126) d3rϕ∗i(r)ϕ∗j(r)ϕl(r)ϕm(r), (4)
mechanically,whilethepump(2)isundepletedandistreated
istheinter-modescollisionparameterand
classically. Both fields are assumed to be far off-resonance
fromanyelectronictransition,andtheexcitedstatepopulation (cid:90)
issmallsothatspontaneousemissionmaybeneglected. Sim- χ = d3rϕ∗ (r)e−ik·rϕ (r), (5)
nm m n
ilarly, collisionsamongexcitedstateatoms,aswellas, colli-
sionsbetweengroundstateatomswithexcitedstateones,are
is the optical transition matrix elements from the mth state
veryimprobableandcanalsobeneglected. Inthisregimethe
to nth state. Hamiltonian (3), accounts for the process of
excited state can be adiabatically eliminated and the ground
atomic and optical parametric amplification due momentum
stateatomicfieldplustheopticalprobeevolvecoherentlyun-
exchangebetweenatomsandopticalfield[10]. Here,weas-
dertheeffectiveHamiltonian[10,12,15]
sume a sufficiently low temperature so that all atoms form a
(cid:90) (cid:34) U |g α |2(cid:35) pureBECinthetrapgroundstate,andtoavoidscatteringpro-
H = d3rΨ†(r) H0+ 2Ψ†(r)Ψ(r)+(cid:126) 2∆2 Ψ(r) cesswhichtransfersatomstoothertrapmodesviamomentum
exchange we consider that k ≈ k (and ω ≈ ω ). With
1 2 1 2
+(cid:126)(cid:90) d3rΨ†(r)(cid:20)g1∗g2α2a†e−ik·r+H.c.(cid:21)Ψ(r) these assumptions, only the atomic operators c†0 (c0) corre-
∆ 1 spondingtocreation(annihilation)ofatomsatthetrapground
(cid:34) (cid:35) stateremains,andtheHamiltonian(3)simplifiesto
|g |2 (cid:90)
+(cid:126) (ω1−ω2)+ ∆1 d3rΨ†(r)Ψ(r) a†1a1. (1) (cid:32) |g |2(cid:33)
H =(cid:126) ω + (cid:101)2 c†c +(cid:126)κc†c†c c
(cid:101)0 ∆ 0 0 0 0 0 0
Ψ(r)isthegroundstateatomicfieldoperatorwhichsatisfies
theusualbosoniccommutationrelations(cid:2)Ψ(r),Ψ†(r(cid:48))(cid:3) = (cid:18)g g∗ g g∗ (cid:19) |g |2
δ(r−r(cid:48)).H0 =−2(cid:126)m2 ∇2+V(r)isthetrappedatomsHamil- +(cid:126)c†0c0 1∆(cid:101)2a+ (cid:101)2∆1a† +(cid:126) ∆1 c†0c0a†a, (6)
tonian,wheremistheatomicmassandV(r)isthetrappoten-
tial.∆=ω2−νisthedetuningbetweentheatomictransition whereg(cid:101)2 = g2α2 andκ = 2U(cid:126)(cid:82) d3r|ϕ0(r)|4 isthecollision
andtheopticalpumpfrequencies,g1andg2aretheatom-light parameter between the atoms in the trap ground state. The
coupling coefficients, and k = k1 − k2. The operators a1 optical probe mode index was dropped in order to simplify
and a† (given in a rotating frame with frequency ω ) satisfy thenotation.
1 2
(cid:104) (cid:105)
the commutation relation a ,a† = 1, and α is the pump Assuming that initially the atoms plus optical field joint
1 1 2
state is completely disentangled, i.e., |ψ(0)(cid:105) = |α(cid:105) ⊗ |β(cid:105),
amplitude. Two-bodycollisionswereincludedinthes-wave
where both, atomic (|α(cid:105)) and optical field (|β(cid:105)) are suposed
scatteringlimit,whereU = 4π(cid:126)2a andaisthes-wavescatter-
m tobecoherentstates,thetimeevolutionofthecombinedsys-
inglength[16].
temcanbeobtainedexactlyfromtheHamiltonian(6)(Seethe
We expand the atomic field operators in terms of the or-
appendixforadetailedderivation),andisgivenby
thogonalsetoftrapeigenmodes{ϕ (r)},as
n
Ψ(r)=(cid:88)cnϕn(r), (2) |ψ(t)(cid:105)=e−|α2|2 (cid:88)αm√e−Φm(t)|m(cid:105)⊗|βm(t)(cid:105), (7)
m!
n m
where the trap eigen-modes satisfies the ortogonality rela-
where
tion (cid:82) d3rϕ∗ (r)ϕ (r) = δ and the eigenvalue equation
m n mn
fHre0qϕune(nrc)ie=s).(cid:126)cω(cid:101)nnisϕtnh(era)to(ω(cid:101)mnainsntihheilactoiorrnesoppoenradtionrgintrathpeemigoedne- βm(t)=βe−i|g1∆|2mt+ gg(cid:101)21(e−i|g1∆|2mt−1), (8)
3
istheprobefieldcoherentstatetimedependentamplitudeand
1(cid:104) (cid:105) g
Φ (t)= |β (t)|2−|β|2 + (cid:101)2 [β (t)−β]
m 2 m g m
1
−iω mt−iκm(m−1)t, (9)
(cid:101)0
istherelativephaseintroducedbythedynamics. Thedensity
operator for the combined system is ρ(t) = |ψ(t)(cid:105)(cid:104)ψ(t)|
from which we can obtain the BEC state by tracing over the
optical state, ρ (t) = Tr ρ(t), where A and L stand for
A L
atomsandlight,respectivelly.
III. CONDENSATEPHASEDYNAMICS
Analyzing each term from Hamiltonian (6) separately, we
are able to visualize different scenarios. For instance, its FIG.1. Husimifunctiondynamicsofthepre-selectedatomicden-
second term plays a key role on the collapse and revival of sity operator in the limit |g1|2 (cid:29) κ. In (a) t = 0; (b) t =
∆
the atomic phase dynamics, typical of one-mode BECs (See 0.1(2π∆/|g1|2);(c)t=0.4(2π∆/|g1|2);(d)t=0.5(2π∆/|g1|2);
Fig. 1 in Greiner et al. [4] for the atomic Husimi func- (e)t = 0.6(2π∆/|g1|2);(f)t = 0.9(2π∆/|g1|2). Afullrevivalof
tion QA(αA,t) = π1(cid:104)αA|ρA(t)|αA(cid:105) corresponding here to theinitialcoherentstatein(a)isreachedattLrev =2π∆/|g1|2.
|g1|2 (cid:28) κ). After evolving into states of totally uncertain
∆
phasesandalsosomeexactsuperpositionsofcoherentstates, several times in Fig. 2(a) for the atomic mode at the chosen
the atomic state fully recover the initial phase at the revival timescale. Howeverthisbehaviorisveryfragileandchanges
timetCrev =π/κ. completelywiththeinclusionofaverysmallperturbationin
ThethirdandfourthtermsinHamiltonian(6)describethe |g |2/κ∆. The revival time for this scenario can be inves-
1
couplingsbetweenatomsandthequantumopticalprobe. The tigated by expanding the collision and interaction terms in a
thirdtermcorrespondstoatransferofphotonsfromtheclassi- Fock basis, {|n(cid:105),|m(cid:105)} for the atoms and light field, respec-
calundepletedpumpopticalfieldtothequantizedprobemode tively. Therevivaltimeisnotdependentonthethirdtermof
mediatedbytheatoms. Thefourthtermisquiterelevantsince theHamiltonian(6),whoseeffectisonlytodisplaceaninitial
it will also contribute to the collapse and revival of the BEC lightfieldcoherentstatedependingonthenumberofatomsin
phase (and of the optical probe) due to the cross-Kerr type the BEC. Then, considering a rotating frame with frequency
of nonlinearity. The regime |g∆1|2 (cid:29) κ, for a small detuning ω(cid:101)0 + |g(cid:101)∆2|2, the revival time coincides with the recurrence of
(largeenoughthoughtopreventspontaneousemission[8])or theinitialphase,whichwilltakeplacewheneverthefollowing
for a very diluted atomic gas (κ ≈ 0), is depicted in Fig. 1. relationissatisfied
Thecollapseandrevivaldynamicsoccursatacompletelydis-
tinctrevivaltimetLrev = 2π∆/|g1|2,dependingontheatom- e−i(cid:20)κn(n−1)+|g∆1|2nm(cid:21)t =e−2ilπ, (10)
lightinteractionparameter. Similarlytheprobelightfieldwill
also show a collapse and revival dynamics. The only extra wherel,m,andnarepositiveintegers. Thisoccursfortimes
effect is due to the third term in Hamiltonian (6), which dis- suchthat
places an initial light field coherent state depending on the
(cid:34) (cid:35)
numberofatomsintheBEC. 2κ∆ l
t = tC , (11)
A nontrivial dynamics occurs when both the second and rev |g |2 κ∆ n(n−1)+nm rev
fourth therms in (6) are of the same order, which can be 1 |g1|2
reached by varying the detuning ∆. In a general way the since tC = π/κ. The revival depends on the commensura-
rev
revival occurs whenever the terms in a expansion spanned bility between |g |2/∆ and κ. Since n and m are integers,
1
by Fock states of the atomic state are in phase. While it is when|g |2/κ∆isrational(≡ p,withp,qintegers)wehave
1 q
easy to describe this when only the collision term or the in-
(cid:20) (cid:21)
teraction with the optical probe are on, the situation is more 2κ∆ lq
t = tC . (12)
complicated when both terms are relevant. To show that, let rev |g |2 pn(n−1)+qnm rev
1
us analyze the behavior of the variance of the phase opera-
tor for the atomic mode [17]. For an initial coherent state Given that pn(n−1)+qnm is an integer and l is arbitrary,
of a large amplitude α, the phase variance is approximately there always exists a lq = pn(n−1)+qnm in the numer-
V(φ) = 1/4|α|2 andclearlyshowsthatthephaseiswellde- ator of Eq. (12), so that the revival time reduces to trev =
fined for large α. For the chosen amplitude of |α|2 = 3 the 2|gκ1∆|2tCrev. However, for an irrational |g1|2/κ∆, there is no
phase revival, as depicted in Fig. 2, occurs every time V(φ) revivalatall.
approaches zero. Note that the collapse and revival dynam- Letusexemplifywiththeinclusionofperturbationsinthe
ics similar to the one in Fig. 1 in Greiner et al. [4] repeats interactionwiththelightfield(bydecreasingthedetuning).In
4
Fig. 2(b), for|g |2/κ∆ = 1/50, weseeaninhibitionofthe where|α (cid:105)areoddandevencoherentstates. Suchstatesare
1 ±
number of revivals at this time scale, even though an actual useful,e.g.,forteleportation[18].
revivaloccursatadifferenttimescaleat100tC .Inasimilar
rev
wayFig.2(c)for|g |2/κ∆=1/5,showsthattherevivaltime
1
is10tC ,andinFigs.2(d)and2(f),for|g |2/κ∆=1/2,and IV. PROBEFIELDPHOTO-DETECTION
rev 1
|g |2/κ∆ = 1, where the revival time is 4 tC and 2 tC ,
1 rev rev
respectively. However, for g2 = 2, as in Fig. 2(e), there Sofarwehaveassumedacoherentunitaryevolution. Now,
κ∆ π
is no phase revival, as we expected. The relevant aspect on we turn to the situation where the probe light field is being
detected. In several situations detection allows an additional
control [13, 14]. Here we will see that the continuous pho-
todetection over the probe field does not give any additional
controloverthesinglemodeBECphase.
We employ a continuous photodetection model [12, 17]
characterizedbyasetofoperationsN (k),suchthat,
t
N (k)ρ(0)
ρ (t)= t , (14)
k P (k,t)
where P (k,t) = Tr[N (k)ρ(0)] is the probability that k
t
photocountsareobservedduringthetimeintervalt. Theop-
eration
(cid:90) t (cid:90) tk (cid:90) tk−1
N (k)ρ= dt dt dt ...
t k k−1 k−2
0 0 0
(cid:90) t1
... dt S JS ...JS ρ (15)
1 t−tk tk−tk−1 t1
0
accounts for all possible one-count process, with Jρ =
γaρa† (where γ is the detector counting rate) followed by
thenon-unitaryevolutionbetweenconsecutivecountsS ρ =
t
eYtρeY†t, with Y = −iH − γa†a. After k-counts on the
(cid:126) 2
probefield,theconditionedjointstatebecomes
1 (cid:88)
ρ (t)= C(k) (t)|m(cid:105)(cid:104)n|⊗|β (t)(cid:105)(cid:104)β (t)|,
k P (k,t)k! m,n m n
m,n
(16)
whereCm(k,)n(t)=e−|α|2α√mα∗n [Fm,n]keΦm(t)+Φ∗n(t),with
m!n!
(cid:26) Λ Λ∗ (cid:104) (cid:105) (cid:27)
Fm,n(t)=γ −Γ m+Γn∗ e−(Γm+Γ∗n)t−1 +GmG∗nt
m n
(cid:26)G Λ∗ G∗Λ (cid:27)
+iγ m nf∗(t)− n mf (t) , (17)
Γ∗ n Γ m
n m
FIG.2. PhasevariancefortheBECowingtobothatomcollisions
andinteractionwiththeopticalprobe. Thelatteronedisturbsenor- wherefm(t)≡(cid:0)e−Γmt−1(cid:1)and
mouslytherevivaltime,whentheatomicstatereturnsapproximately 1(cid:16) (cid:17)
to a coherent state. From top to bottom, (a) |g |2/κ∆ = 0, (b) Φ (t)=− |β|2−|β (t)|2 +iG Λ f (t)
|g |2/κ∆ = 1/50,(c)|g |2/κ∆ = 1/5,(d)|g |12/κ∆ = 1/2,(e) m 2 m m m m
1 1 1
|g1|2/κ∆=2/π,and(f)|g1|2/κ∆=1. −|G |2Γ∗ t−i[(ω + |g2|2)m+κm(m−1)]t.
m m (cid:101)0 ∆
the atomic revival time change is the possibility to control it (18)
throughthevariationof∆. Whenevertherevivaloccurs,and
only then, both optical pump and the BEC are left disentan- In Eqs. (17) and (18), Gm = ∆g1∗Γg(cid:101)m2 m, Λm = β + iGm,
gled. Besidesthecontrolovertherevival,onecouldbeinter- Γm =i|g∆1|2m+ γ2,andβm(t)=Λme−Γmt−iGm. Besides
estedinthisavailableentanglement. Severalentangledcoher- P (k,t)isgivenby
entstatesoccur,beingatypicalexampleatexactlyhalfofthe
rleefvtivinalatsimtaete- the optical probe and BEC are approximately P (k,t)= e−k|α!2|2 (cid:88)(cid:0)|αm|2!(cid:1)m [Fm,m(t)]ke−Fm,m(t).
m
|ψ(cid:105)≈|β(cid:105)|α (cid:105)+|β(cid:105)|α (cid:105), (13) (19)
+ −
5
In Fig. 3 we plot the phase variance and the correspond-
ing probability of occurrence for (a) k = 0 and (b) k = 1,
respectively, forthe same parameters foundin Fig. 2(f), and
withγ =2×10−2κ. WeseeinFig. 3(a)thattheno-counting
doesnotaffecttherevivalofthestate,atthetypicaltimescale
for a no-count event (Fig. 3(d)). We only see some change
after a few collapse and revivals. At the revival time scale
theprobabilitythatk = 1countingoccursislargeenoughso
that the chance to get a revival of the BEC phase is severely
compromisedasweseeinFig. 3(b). Infactthesameoccurs
for any k (cid:54)= 0. Therefore the phase evolution given by the
postselected state is very sensitive, and whenever a photode-
tection event occurs the whole atom-light system state is so
affectedthatthereisnochanceforarevivaloftheinitialBEC
phase. This contrasts to other situations where the detection
processhelpstodefineaphase. Thereforenofurthercontrol
isachieved, otherthantheonebythedetuningoftheoptical
pump frequency. For completeness, in Fig. 3(c) we plot the
phase variance for unconditioned (pre-selected) state of the
jointBEC-lightsystemundertheeffectofcounting,
(cid:88)
ρ(t)= P (k,t)ρ (t)
k
k
(cid:88)
= C (t)|β (t)(cid:105)(cid:104)β (t)|⊗|m(cid:105)(cid:104)n|, (20)
m,n m n
FIG.3. Therevivalofthephaseisseverelyaffectedbythecounting
m,n
processasdepictedbythephasevariancefortheBECwiththephoto-
with Cm,n(t) = e−|α|2α√mα∗neΦm(t)+Φ∗n(t)+Fm,n(t). This is countingprocessataγ =2×10−2κcountingrate.Otherparameters
m!n! aresimilartotheonesinFig.2(f).(a)Varianceforano-countevent,
thesituationwhenoneisabsolutelyignorantaboutthecount-
and(b)thevarianceforasinglecountevent.(c)Varianceforthepre-
ing events, and therefore the best one can do is to assume selectedstate(20). (d)P(0,t)(purpleline),P(1,t)(redline),and
a convex sum of all conditioned states ρk(t) occurring with probabilityofanarbitrarycountingevent,[1−P(0,t)](blueline).
probabilityP(k,t).Effectivelytheevolvedstate(20)isequiv-
alent to the situation where the light field is under the action
ofanamplitudedampingchannel,duetocontactwithazero-
temperaturereservoir.Thereforethepre-selectedstatereflects
thejointsystemincoherentevolution.
WeseeinFig3(c)thattheremightbeachanceofapartial
orcompleterevivalunderdampingdependingonγ. Thede-
pendence on the counting rate is better seen in Fig. 4 where
thephasevariancevalueattherevivaltimesisplottedagainst
the variation of γ for the pre-selected state. We can see that
above γ/2κ = 6×10−4 (γ/2κ = 1.2×10−3) there is not
a single phase revival, inside a tolerance of 10% (20%). All
thefollowingrevivalscanbetrackedinasimilarmannerand
obviouslywillbemoresensitivetothevariationofγ. Were-
markthattheobservedeffectforonecountingevent(k = 1) FIG. 4. Phase variance at the revival time calculated with the pre-
wouldbesimilarlyobservedhadwetakenanarbitrarycount- selected state (20) as a function of the detector rate γ for the case
ingevent,givenbythestate showninFig. 2(f). (a)-(e)correspondtothefirst-fifthrevivals,re-
spectively. Thedashedanddottedlinesrepresenttoleranceof10%
1 and20%intherevival,respectively.
ρ(t)= [ρ(t)−P(0,t)ρ (t)], (21)
(cid:101) 1−P(0,t) 0
occurring with probability 1 − P(0,t), with no further ob- one-mode quantized optical probe, is thoroughly dependent
servedadvantagethantheoneconsideredinFig. 3. ontheatom-lightinteractionparameters. Thisdependenceal-
lowssomedegreeofopticalcontrolontheatomiccollapseand
revivaltimesbyadjustingthecouplingbetweentheatomsand
V. CONCLUSION the optical field, through the variation of frequency detuning
∆. Inaddition,weanalyzetheeffectsofdissipationandpho-
In conclusion, the observed collapse and revival of the todetectionovertheopticalfieldupontheBECphasedynam-
macroscopic matter wave field of a BEC interacting with a icstocheckwhetheranadditionalcontrolwouldbepossible.
6
Weshowthatincontrastwithothersituations,suchastheone c-number expansion coefficients. The propagator for this
presentindouble-wellcondensates[13],thephaserevivalfor Hamiltonianwillsatisfythefollowingequation
a single mode BEC is very sensitive to detection and never
occurs for k (cid:54)= 0 counting events on the optical probe. The ∂U (cid:88)
i(cid:126) = h (t)a†lamU. (A.5)
overalleffectofthedetectionistoinduceaphasedampingin ∂t l,m
l,m
thecondensate. ThereforeforthesinglemodeBECinteract-
ingwithaquantumopticalprobe,theonlypossiblecontrolof
Now, consider the theorem [? ] that says if m
phasecollapseandrevivalisthroughthefrequencydetuning. is an integer and f(cid:0)a,a†(cid:1) = f(n)(cid:0)a,a†(cid:1) (where the
superscript denotes normal order), then amf(cid:0)a,a†(cid:1) =
N (cid:110)(cid:16)β+ ∂ (cid:17)mf¯(n)(β,β∗)(cid:111) = N (cid:8)(cid:104)β|amf(cid:0)a,a†(cid:1)|β(cid:105)(cid:9),
ACKNOWLEDGMENTS ∂β∗
wheref¯(n)(β,β∗)isaordinaryfunctionofthecomplexvari-
This work is partially supported by the Brazilian Na- ableβ,andN isanoperatorthattransformsanordinaryfunc-
tionalCouncilforScientificandTechnologicalDevelopment tion f¯(n)(β,β∗) to an operator function f(n)(cid:0)a,a†(cid:1) by re-
(CNPq),FAPESPthroughtheResearchCenterinOpticsand placing β by a and β∗ by a†. With the help of this theorem
Photonics(CePOF)andbyPNPD/CAPES. wecanrewrite(A.5)as
i(cid:126)∂U =(cid:88)h (t)a†lN (cid:26)(cid:18)β+ ∂ (cid:19)mU¯(n)(β,β∗,t)(cid:27),
Appendix:DerivationofEquations(7),(8)and(9) ∂t l,m ∂β∗
l,m
(A.6)
To obtain the state given by Eq. (7), we apply the prop- whereU¯(n)(β,β∗,t) = (cid:104)β|U(β,β∗,t)|β(cid:105). Ifwetakediag-
agator U(t) = exp(cid:0)−iHt(cid:1) into the initial state given by onalcoherentstatematrixelementsofbothsidesof(A.6)we
(cid:126)
Eq. |ψ(0)(cid:105) = |α(cid:105)⊗|β(cid:105)withH givenby(6). Sincethefirst obtainthefollowingc-numberequation
twotermsintheHamiltonian(6)commutewiththeremaining
ones,wecanwritethepropagatorasfollows i(cid:126)∂U¯(n) =(cid:88)h (t)β∗l(cid:18)β+ ∂ (cid:19)mU¯(n), (A.7)
∂t l,m ∂β∗
U(t)=e−i[ωAn0+κn0(n0−1)]te−i(Fa+F∗a†+ξa†a)n0t, l,m
(A.1) sincetherighthandsideisinnormalorder. Solving(A.7),we
whereωA =ω(cid:101)0+ |g(cid:101)∆2|2,n0 =c†0c0,F = g1∆g(cid:101)2∗ andξ = |g∆1|2. obtain |β(t)(cid:105) by |β(t)(cid:105) = N (cid:8)U(n)(β,β∗,t)(cid:9)|β(t0)(cid:105). In
By expanding the initial atomic state in the Fock basis, the ourparticularcase,withthegeneratorH ,wehave
DHO
application of the propagator over the global initial state is
givenby ∂U¯(n) (cid:18) ∂ (cid:19)
i(cid:126) =(cid:126)ξ β∗ β+ U¯(n)
∂t m ∂β∗
U(t)|ψ(0)(cid:105)=e−|α2|2 (cid:88)m √αm2!e−i[ωAm+κm(m−1)]t|m(cid:105) +(cid:126)(cid:20)Fm(cid:18)β+ ∂∂β∗(cid:19)+Fm∗β∗(cid:21)U¯(n).(A.8)
⊗e−i(Fa+F∗a†+ξa†a)mt|β(cid:105). (A.2)
If U¯(n) = eG(β,β∗,t) where G(β,β∗,t) = A(t) +
Tosolvee−i[Fma+Fm∗a†+ξma†a]t|β(cid:105),whereFm = g1∆g(cid:101)2∗mand Bcom(te)sβ i+(cid:2)dAC+(td)Bββ∗++dCβD∗(+t)dβD∗ββ∗,β(cid:3)the=n (ξA.8β)∗βb+e-
ξm = |g∆1|2m, we employ the normal ordering method for ξmβ∗(C+dDt β) +dt Fmβd+t Fm∗β∗dt+ Fm(C+Dβm), which
solvingSchro¨dingerequation[19]. Thegeneratoroftheevo- canbeseparatedinthefollowingsetofequations
lution is the Hamiltonian of a Driven Harmonic Oscillator:
H =(cid:126)ξ a†a+(cid:126)F a+(cid:126)F∗a†. TheSchro¨dingerequa- dD
DHO m m m i =ξ (D+1), (A.9)
tion dt m
∂|β(t)(cid:105)
i(cid:126) =H |β(t)(cid:105), (A.3)
∂t DHO
dB
i =F (D+1), (A.10)
hasasolutiongivenby|β(t)(cid:105)=U (t,t )|β(t )(cid:105)where dt m
DHO 0 0
|β(t )(cid:105)=|β(cid:105)andU alsosatisfies
0 DHO
∂U dC
i(cid:126) DHO =H U , (A.4) i =ξ C+F∗, (A.11)
∂t DHO DHO dt m m
subject to the initial condition U (t ,t ) = 1. In gen-
DHO 0 0
eral, a Hamiltonian H(cid:0)a,a†,t(cid:1) in the normal order is given
dA
by H(cid:0)a,a†,t(cid:1) = (cid:80)l,mhl,m(t)a†lam, where hl,m(t) are i dt =FmC, (A.12)
7
and whose solutions are given by D(t) = e−iξmt − 1, andf(a)|β(cid:105)=f(β)|β(cid:105),then
B(t) = Fm (cid:0)e−iξmt−1(cid:1), C(t) = Fm∗ (cid:0)e−iξmt−1(cid:1) and
A(t)= |Fξmm|2 (cid:0)e−iξmt−1(cid:1)+i|Fm|2t.ξSmince |β(t)(cid:105)=eA(t)+B(t)βeC(t)a†eD(t)βa†|β(cid:105). (A.14)
ξm2 ξm
By identifying |β(cid:105) = e−|β2|2eβa†|0(cid:105), then we are able to
recognisethat
(cid:110) (cid:111) |β(t)(cid:105)=eA(t)+B(t)β−|β2|2e{[1+D(t)]β+C(t)}a†|0(cid:105). (A.15)
|β(t)(cid:105)=U(t)|β(cid:105)=N eA+Bβ+Cβ∗+Dβ∗β |β(cid:105)
IfwesubstitutetheA(t),B(t),C(t)andD(t),ξ ,andF
(cid:110) (cid:111) m m
=eA(t)eC(t)a†N eD(t)β∗β eB(t)β|β(cid:105), (A.13) in(A.15)werecovertheEquations(7),(8)and(9).
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