Table Of ContentUltracold atoms interacting with a sinusoidal mode of a high Q cavity
J. C. Retamal†, E. Solano††∗ and N. Zagury∗
† Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
††Seccion F´ısica, Departamento de Ciencias, Pontificia Universidad Catolica del Peru, Ap. 1761, Lima, Peru
∗Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945-970 Rio de Janeiro, RJ, Brazil
(February 9, 2008)
We consider the interaction of two level ultracold atoms resonant with a sinusoidal mode of the
electromagnetic field in a high Q cavity. We found that well resolved resonances appear in the
transmission coefficients even for actual interaction and cavity parameters. The probability of
emission of one photon and the probability of transmission of an atom, when number or coherent
states are initially present in the cavity, are discussed. The interplay between the increasing width
of the resonances and multi-peak steady-state photon-statistics are also studied. Furthermore, we
9
compare ourresults with those of a constant field mode.
9
9
PACS numbers: 42.50.-p, 32.80.-t, 42.50.Ct, 42.50.Dv
1
n
a
J I. INTRODUCTION of a sinusoidal electric field mode when ultracold atoms
4 are sent into a cylindrical microwave cavity for actual
1 Recent developments in cooling of neutral atoms [1] experimentalparameters. We willshowthatunderthese
have called the attention on the interaction of ultracold conditionsthebehavioroftherelevantphysicalquantities
1
v atoms with microwave cavities [2–7]. Very cold atoms maybequitedifferentfromthoseofadiscontinuousfield
2 (kinetic energies much smaller than the atom-field inter- mode, although still preserving resonant features if the
3 actionenergy)haveapeculiarbehaviorwheninteracting atoms are cold enough.
0 with a cavity, they are strongly reflected unless the ra-
1
tio of the size of the cavity and the atomic wave length
0
associated to the interaction energy are close to certain
9 II. THE MODEL
resonant values. For example, it has been shown [3,5]
9
/ that ultracold atoms of mass m may be totally reflected
h
by a cavity of size L containing n photons, in a con- Letusassumethatatwolevelatomistravellingalong
p
stantfieldmode,unlesstheRabifrequencyΩ iscloseto thez-directioninthewayofamicrowavehighQcavityof
- n
t 2~(jπ/L)2/(2m), j = 0,1,2..., when only approximately lengthL,containingaquantizedfieldmode inresonance
n
half of the atoms are reflected. with the two level atomic transition e f .
a
| i⇔| i
u Models for the interaction of atoms with a microwave TheHamiltoniandescribingthemotionofthetwolevel
q cavity usually do not take into account the spatial de- atom in the z -direction is given, in the interaction rep-
: pendenceoftheinteractioncomingfromtheelectricfield resentation, by
v
i profile[8]. Thisisjustifiedbecause,ingeneral,wearein-
X terestedintheatom-fieldstateaftertheatomhascrossed H = p2 + ~2κ2u(z)(aσ†+a†σ). (1)
r thecavityandaredealingwithatomicthermal velocities, 2m 2m
a
whichareexceedinglyhighcomparedwiththerecoilones
Here σ f e, a (a†) is the annihilation (creation) op-
experienced by the atom inside the cavity.
≡| ih |
erator of the electromagnetic field mode whose spatial
In orderthat we may observeany effect onthe center-
modulationpatternisu(z)and~κ2/(2m)ishalftheRabi
of-mass motion of the atoms, their velocities have to be
frequency associatedwith the transition f e in the
reduced to very low values in such a way that even the
| i↔| i
vacuum. u(z) is normalized in such a way that the area
microscopicinversionofmomentumofthecenter-of-mass
under the mode is L. In refs. [3,5] u(z) is taken as a
could take place. Actually, the only way for an atom
squarefunctionofheight1andwidthL(mesafunction).
to experience mechanical action on its center-of-mass is
In this case it can be shown that the transmission of ul-
through the mechanism of exchange of momentum with
tracold atoms through a cavity containing n photons is
the whole cavity. At high atomic velocities the energy
very small unless
exchange between the field and the atoms takes place
through the Rabi flopping of the internal atomic levels.
κ κ√4n+1 (2)
At very low atomic velocities the spatial variation of the n ≡
electromagneticfieldactingontheatomsprovidesanad-
is a multiple of π/L.In ref. [6] the sech 2(z/L)and sinu-
ditional driving mechanism which lies in the momentum
soidal modes were considered and the authors have con-
exchange of the atom with the cavity.
cluded that, for highvalues of L,the resonancesreferred
Theaimofthe presentworkistoinvestigatethe effect
1
above are smeared out. Here we will discuss in more de- ϕ±n(z) cos(kz δ±n). (9)
e,o ∝ − e,o
tail the sinusoidal mode having just one antinode. We
will show that in this case, resonances still exist for high Hereδ±n(k) (δ±n(k))is the phaseshift associatedto the
e o
values of κ L as long as the atoms are cold enough. even(odd) eigensolutionof the Schr¨odingerequation. In
n
The Hamiltonian given in Eq. 1 can be conveniently termsofthesephaseshiftsthereflectionandtransmission
written in the dressed state basis as: coefficients may be written as
H = [ p2 V (z)] ,n ,n, (3) e2iδe±n +e2iδo±n
2m ± n |± ih± | r±n(k)=
2
X
where V (z) is given by (10)
±n
V±n(z)= ~2κ2nu(z), (4) t±n(k)= e2iδe±n −2 e2iδo±n.
± 2m
The phase shifts are related to the logarithmic deriva-
and
tives at z = L/2 by
−
,n =(e,n f,n+1 )/√2. (5)
|± i | i±| i β±n =k tan(κ L/2+δ±n). (11)
e,o n e,o
In the dressed state basis our problem is equivalent
to that of a particle being scattered by the potentials Notice that βe±,on are real in contrast with the complex
V (z). The cavityactsasabarrierwhenthe atom-field logarithmic derivative of a typical, right or left, propa-
±n
stateis +n andactsasawellwhentheatom-fieldstate gatingsolution. ¿FromEqs. 10 and11 it is easy to show
is n|. i that:
|− i
Beforetheatomreachesthe cavitycontainingthefield
k2+β±nβ±n
c(n)n ,theinitialatomicstateisgivenbytheproduct r (k)= e o e−ikL (12)
ofawa|veipacketinmomentumspace z φ = dkφ(k)eikz ±n (k iβe±n)(k iβo±n)
− −
P h | i
times the internal energy state of the atom that, for
R and
definitiveness,weassumetobetheupperstate e . After
the atom interacts with the cavity the total ato|mi-cavity ik(β±n β±n)
t (k)= e − o e−ikL. (13)
state is (for z >L/2): ±n (k iβ±n)(k iβ±n)
| | − e − o
c(n) dkφ(k)exp(−i~22mk2τ)[(eikzt+n(k)+e−ikzr+n(k)) k In thκe,cathseeotfratnhsempiosstieonntiacolebffiarcrieienrt, tfor κinsLve≫ry1smanadll
n +n
Z ≪
X+n +(eikzt (k)+e−ikzr (k)) n ], (6) and close to zero. In the case of the potential well, and
−n −n
| i |− i for k κ the transmission coefficient t is close to
n −n
where the reflection, r , and transmission, t , coeffi- zero un≪less β−nβ−n = k2, that is for very small β−nor
cients are associated w±inth the scattering eigen±fnunctions very small βeo−n.oFor a−fixed value of k/κn, theseeres-
ϕ±(z). These functions are solutions of the time inde- onances in the transmission coefficient occur at special
k
pendent Schr¨odinger equation : values of κnL, which would correspond to the appear-
ance of a bound state with k2 = 0 ( notice that the
d2 +k2 κ2u(z) ϕ±(z)=0, (7) transmission coefficient has a pole at k =iβe−,on).
dz2 ∓ n k Assume that initially there are n photons inside the
(cid:20) (cid:21)
cavityandtheatomisincidentfromtheleftintheupper
where ~k is the momentum of the incident particle. state e . After the atom interacts with the cavity, it
Inthispaperweareinterestedindiscussingasituation can be| itransmitted in the upper state, transmitted in
where the longitudinal z-dependence of the electric field the lower state, reflected in the upper state or reflected
is sinusoidal: in the lower state. The probabilities that these events
πz occur are denoted by Tn, Tn, Rnand Rn, respectively
u(z)=θ(L/2 z )(π/2)cos( ). (8) e f e f
−| | L and given by:
This is the case of a cylindrical cavity tuned either to a 1 1
Tn = t +t 2, Tn = t t 2 (14)
TEmn1 or a TM mn1 mode [9]. e 4| +n −n| f 4| +n− −n|
We may try to obtain the values of r (k) and t (k)
±n ±n
and
by numerically integrating a complex solution of Eq. 7
for runaway asymptotic conditions. An alternative way, 1 1
where we found numerical and analytical advantages, is Ren = 4|r+n+r−n|2, Rfn = 4|r+n−r−n|2. (15)
to consider instead the even (ϕ±n(z)) and odd (ϕ±n(z))
e o
real (up to an overall complex constant) eigensolutions Of course Tn+Tn+Rn+Rn = 1. We will show in
e f e f
of Eq. 7, such that for z < L/2 [10] the next section that for k/κ 1, t = 0. In this
− n ≪ +n ∼
2
rcealsaetiToenna≈ndTftnh≈e u|tn−itna|r2i/t4y.coDnudeittioonthwiselcahsotoasepptroodxiismcuatses ϕ−en(z)≈ q1(z)cos( zq(z′)dz′)
Z0
below only two quantities of physical interest: the prob- 1 z
ability ofemissionofa photon,Pn =Tn+Rn, andthe ϕ−n(z) p sin( q(z′)dz′), (18)
probability of transmission of anematom,fTn =fTn+Tn. o ≈ q(z) Z0
e f
We notice also that when r−n ∼=0, Penm ∼=Tn. where q(z) = k2p+κ2u(z). It is well known that Eqs.
Ifweinitiallyhaveaphotonnumberdistributiongiven n
by c 2, the total probability of photon emission is 18 are valid when the condition
n p
| |
P = c 2Pn and the total probability of atomic
treamnsmissio|nnis| Tem= c 2Tn. These quantities should dq(z) q(z)2 (19)
P | n| dz ≪
beeasilymeasurablebydetectingtheinternalenergylev- (cid:12) (cid:12)
P (cid:12) (cid:12)
els of the reflected and transmitted atoms. As we will (cid:12) (cid:12)
is fulfilled [11]. Th(cid:12)is con(cid:12)dition may be satisfied for the
show, these quantities depend strongly on the shape of
sinusoidal mode and for any z, whenever
the field inside the cavity, as long as k/κ 1.
n
≪
π2(κ /k)3
n
ξ 1. (20)
≡ 4κ L ≪
III. ANALYTICAL RESULTS n
In this case the semiclassical solutions (Eqs. 18 ) are
The constant field mode (u(z) = 1, for z < L/2) valid for all z and the reflection coefficient r−n(k) van-
| |
have been discussed in refs. [2,3,5]. For k/κ 1 we ishesasξgoestozero,aswewillshowbelow. Whenξ &1
n
haveβ+n κ tanh(κ L/2),β+n κ coth(κ≪L/2), the inequality 19 is not valid near the regions z .L/2.
βe−n ≈e κn≈t−an(nκnL/2)nand βo−no ≈≈κ−n cnot(κnL/n2), so In this case the modulus of r−n(k) may increa|se|even to
that the resonances appear when κ L equals to an in- 1. Notice that we may have ξ >1 even for large kL and
n
teger multiple of π [2,3,5]. In this case the resonances κnL. For example, for κnL 105 and kL 103, ξ 24.
∼ ∼ ≈
areverysharp,theirwidthbeingapproximatelyconstant Near z = L/2, u(z) may be approximated by a
−
and equal to 4k/κ even for large κ L. straight line and the solutions of Eq. 7 may be written
n n
In the case of the sinusoidal mode, as long as κ L is as:
n
small, it is easy to obtain the even and odd eigensolu-
tions of Eq. 7 by numerical integration . This can be ϕ−e,on(z)∝w1/3(Ae,oJ1/3(w)+Be,oJ−1/3(w)) (21)
done easily and with confidence for κ L. 100π. If we
n
where
wish to predict values closer to real experimental situa-
tionsweneedtoconsiderhighervaluesofκnL,whichare π 2z 4k2 3/2
typicallyoftheorderofO(105 106)forRydbergatoms. w = κ L +1+ . (22)
In these cases the numerical s−olutions do not converge 6 n (cid:18)L π2κ2n(cid:19)
rapidly for the n channel and we found easier, and
|− i The logarithmicderivatives atz = L/2are givenby:
more instructive, to use a WKB-like approximation for −
calculatingtheδ±n andδ±n phaseshiftsor,equivalently,
e o A J′ (ξ−1/3)+B J′ (ξ−1/3)
the logarithmic derivatives βe±,on. β−n =k ξ+ e,o 1/3 e,o −1/3 .
Inthecaseofthepotentialbarrier,andfork/κn 1, e,o Ae,oJ1/3(ξ−1/3)+Be,oJ−1/3(ξ−1/3)!
≪
we may use the WKBapproximation[11]for calculating
β+n . We get (23)
o,e
ByconnectingtheevenandoddsolutionsofEq. 21with
β+n = k(1 Θ), (16)
e,o ∼− ∓ those given by Eqs. 18 , we obtain :
where the and + signs inside the parenthesis refer to
− Ae sin(ϕ+π/12)
the even and odd solution and =χ(ϕ)= (24)
B −cos(ϕ π/12)
e
a −
Θ=exp(− −k2+κ2nu(z)dz), (17) and Ao/Bo =χ(ϕ+π/2), where
Z−a
p
−L/2
where a are the turning points. Substituting the val-
ues for±β+n in Eqs. 12 and 13 we get r (k) ie−ikL ϕ≃ k2+κ2nu(z)dz (25)
o,e +n ≃− Z0
and t (k) iΘe−ikL. For k/κ 1, a L/2 and p
+n ≃ n ≪ ≃ For ξ 1, we use the asymptotic expressions for
ΘI≃nesxidpe(−th4e.1c9aκvnitLy)t,hwehsioclhutiisovnesrϕy−es,onm(azl)lffoorrtκhneLpo>te1n.tial βJ±−1n/3(ξ−1k/≪(ξ3/)2to coobttϕa)in. Fβreo−mn E≈qs.−1k2(ξa/n2d+13tawneϕo)btaanidn
well may be written, whenever the semiclassical approx- o ≈− −
thesameresult(uptoaphasecomingfromourdefinition
imation holds, as
of r and t ) of ref. [6]:
−n −n
3
r = i(ξ/4)[cos2ϕ+(ξ/4)sin2ϕ]t constant field mode where the narrow resonances assure
−n −n
− (26) a similar behaviorofPenm andPem, Tn andT,except for
accidental numerical coincidences.
t−n =e−ikL (ξ/4)2e−2iϕ+(1+iξ/4)2e2iϕ −1. In Fig. 1 we show our results for the probability of
emission of a photon when there is zero photons inside
J For(ξξ−1≫/3)1t,o ow(cid:2)betaminayβ−unse=thαeksξe1r/i3eχs(ϕex)paann(cid:3)dsiβon−nfo=r thecavity(vacuumfield),Peom,forafixedvalueofk/κn =
±1/3 e − o 0.01, as a function of κnL in the range 100π < κnL <
αkξ1/3χ(ϕ+π/2) where α = (2/9)1/3Γ(2/3)/Γ(4/3). In 104π. We see that the resonances are shifted and wider
this case we get: in the sinusoidal mode (solid line), in comparison with
the constant mode (dashed line), and that their widths
iαξ1/3(χ(ϕ)+χ(ϕ+π/2))
r = − e−ikL are still much smaller than the resonance separation.
−n (1+iαξ1/3χ(ϕ))(1 iαξ1/3χ(ϕ+π/2)) We now consider that the cavity is in contact with a
−
heat bath at a temperature associated with a thermal
(27)
mean photon number n . Assuming, as usual, that we
b
and maytreatgainandlossindependentlyandthattheinter-
val of the atom separation obeys a Poissonian statistics,
1 α2ξ2/3χ(ϕ)χ(ϕ+π/2) it can be shown that the stationary photon distribution
t = − e−ikL.
−n (1+iαξ1/3χ(ϕ))(1 iαξ1/3χ(ϕ+π/2)) is given by [3]
−
(28) dp ω
n =r(p Pn−1 p Pn ) (n +1)
dt n−1 em − n em − Q b
Resonances occur when β−n, or β−n, are zero, that is
e o ω
when either χ(ϕ) or χ(ϕ+π/2) are zero, which give us [npn (n+1)pn+1] nb[(n+1)pn npn+1]. (31)
− − Q −
a simple condition for localizing them:
Hereristheatomicinjectionrate,Qisthecavityquality
κ L π/2
n (k/κ )2+(π/2)cos(θ)dθ =mπ/2+π/12, factor, ω is the cavity resonance frequency and nb is the
n
π meannumberofphotons inside a cavityinthermalequi-
Z0
p libriumwithareservoirattemperatureT . FromEq. 31
(29) b
we obtain the stationary population :
with m = 0,1,2,... For (k/κ )2 1 we may evaluate
n ≪ n +N Pj−1/j
the integralapproximatelyandget for the resonancepo- p =p b ex em , (32)
n 0
sitions: (nb+1)
Y
κnL 2.092 (mπ/2+π/12). (30) where Nex = rQ/ω. This result is similar to that ob-
∼ × tained in the conventional micromaser, where the value
Therefore, for the sinusoidal field mode we will also of Pn is given by sin2(τκ2/(2m)), τ = (m/~k)L being
em n
obtain a series of resonances at values of κ L that are the time of flight of the atom through the cavity. For
n
separated by an interval approximately equal to π. In k/κ0 1 the photon distribution obtained from Eq.
≪
additiontheirpositionsareshiftedinrelationtothecon- 32 is completely different from that of the conventional
stant field case,their widths are wider andincrease with micromaser. In this case the predictions are stronglyde-
κnL. We will show that these results give rise to new pendent on the field mode profile as long as nb is small
features in the physical quantities of interest. enough and Nex large enough.
Fig. 2 shows the stationary photon distributions, for
N = 1000, n = 1 and k/κ = 0.01, when the κ L
ex b 0 2
IV. NUMERICAL RESULTS value corresponds to the position of the 100th resonance
in the case of the constant mode (κ L = 99π) and of
2
the sinusoidal mode (κ L 103.7π). Fig 2a (constant
We have checked that the WKB-like eigenfunctions 2
≈
mode) showsthe apparitionof twothermaldistributions
agrees extremely well with the numerical solutions at
with peaks at n = 3,12. For such low values of κL
each point, even for low values of κ L and k/κ . For
n n
this may happen in two special cases [6], when there
highervaluesofκ L(realisticvaluesincluded)theWKB-
n
like solution should be increasingly better. is a resonance as in κL√43 = (99π/31/4)(3)1/4 = 99π
For very small values of κ L the resonances in Pn , (n = 3) or when there is a numerical coincidence as in
n em
although wider than in the constant field case, do not κL√412 = (99π/31/4)(12)1/4 140.007π (n = 12). Oth-
≈
correspond to appreciable values of Pm for m = n. For erwise we would obtain an equilibrium thermal distri-
largevaluesofκ LtheresonancesinPenmarewid6eenough bution at the temperature corresponding to the average
n em
to produce appreciable values of Pemm for m6=n , so one photonnumbernb. Thisisnotthecaseforthesinusoidal
should take them in account when predicting the total mode (see Fig. 2b) since the probability of photon emis-
probabilities P , or T. This is not the case for the sion, Pj , may be important for several different values
em em
4
of j besides the resonant probability for n = 0, even at Forwarmatomsitisthe value ofthe Rabiangleandthe
these low values of κL. This is due to the large width of rateofincidentatomsthatdefinesthegain. Forultracold
the resonances for the sinusoidal mode, in contrast with atoms we should also consider the exchange of momen-
the sharp ones for the constant field case. As we men- tum between the cavity and the atom. Most atoms are
tioned before, these widths increase with κL for the si- reflectedwhentheatom-fieldsystemisinthestate +n ,
| i
nusoidal mode, producing an extremely complex photon independentlyoftheshapeofthemode. Whentheatom-
statistics. field system is in the state n and the shape of the
|− i
RealisticvaluesforthecouplingofRydbergatomswith cavityis smoothenough,sothatthe effectivede Broglie
microwave cavities are much larger than the values con- wavelengthdoesnotvaryappreciablythroughthecavity,
sideredabove. Forexample,considertheRydbergtransi- most atoms would be transmitted. When this is not the
tionbetweencircularstateswithprincipalquantumnum- case the atoms would be reflected, unless κ L is close to
n
bers50and51inRubidiumat51GHz,usedbytheEcole certain resonant values.
Normale Sup´erieure Group in recent experiments [12]. We have calculated, in the case of a sinusoidal mode,
For a cylindrical cavity, with a length L 0.40cm and a the probability of transmissionof an ultracold atom and
≈
radius R 0.68cm, this transition is resonant with the the probability of emission of a photon comparing them
TE mod≈e. For a dipole moment d 10−26Cm, we get withthecorrespondingmesafunctionresults. Inthecase
121
κ 2.1 105cm−1and κ L 27000≈π. ofa mesa function sharpresonancesappearin the trans-
0 0
≈ × ≈
InFig. 3weplottheprobabilityofdetectingonetrans- missioncoefficientasafunctionofκ L. Inthecaseofthe
n
mitted atom, Tn = Tn +Tn, and the probability that sinusoidalmoderesonancesarepresentwhenthevalueof
e f
toinaellyphnot=on0b,1e,e2m,3ittinedt,hPeenmca=vitRy,fna+s Tafnf,unifctwioenhaovfeκi0nLi-, gπi2v(4eκκnnu/Lks)3a&cha1n,ceevetnowtehsetnthκensLe iesffevcetrsyilnarager.eaTlihsitsicmeaxy-
for 30000π < κ0L < 30005π and k/κ0 = 0.01. This ra- periment (when κ0L 105) using a cylindrical cavity.
tio corresponds to Rubidium atom velocities of approxi- We also found, in the∼case of the sinusoidal mode, that
mately0.2mm/s. Atomicvelocitieslowerthanthathave the width of the resonances are larger, but still well re-
been already obtained by evaporative cooling [13]. For solved for realistic parameters, and increases with κ L,
0
these velocity values the injection rate should be very as long as k/κ is very small. As a consequence, the
0
small, if we wish to have only one atom at a time in transmission of atoms through a cavity containing a co-
tbheesmcaavlilteyr t(hra≃n 10.e0v6esn−f1o)r, Qand t3he10v1a2lu!eInotfhNisexcawseotuhlde haecroemntpsletatetleywdiitffherfeewntpshhoatpoensfo,rasthaeftuwnoctmionodoefsκw0eL,hhavaes
≈ ×
stationary photon distribution would be very close to a studied. The broadeningofthe resonancesproducesalso
thermal one, inhibiting the new features shown in Fig. amorecomplexphotonstatisticsforthestationarystate,
2. In Fig. 3, after a careful inspection, we note that atleastfor theoreticalparameters,to ourknowledge,far
Tn and Penm show a similar, but not identical, behavior. from realistic ones.
Penm presentsbetterresolvedresonancesthanTnandthis These effects show the importance of considering the
could be useful for determining the more convenient ex- details of the cavitymode when analyzing the scattering
perimental setup. of ultracold atoms by a resonant cavity.
In Fig. 4 we plot the total probability of detecting We should remark that a serious difficulty in realizing
thetransmittedatom,T,forthesinusoidalmode(dotted theseexperimentsisthelongtimeofflightthatultracold
line)andtheconstantmode(solidline),whenacoherent atoms need to reach the cavity after their production.
state, either with n = 0.25 or n = 2, is initially present
insidethecavity. Thefigureshowsverysharpresonances
in κ L for the constant mode. They are a consequence VI. ACKNOWLEDGMENTS
0
of the fact that only one resonance is contributing for a
given n, with a statistical weight c(n)2. For the sinu- ThisworkissupportedinpartbyCentroLatinoAmer-
| |
soidalmodetheresonancesinTnarebroaderandseveral icanodeF´ısica(CLAF),TheBrazilianConselhoNacional
resonances may contribute for a fixed κ0L. The case in de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq),
which n = 2 is much more sensible to these features, Programade Apoio a Nu´cleosde Excelˆencia(PRONEX)
showing, for the constant mode, many little resonance and Fundac¸a˜o Universit´aria Jos´e Bonif´acio (FUJB).
peaks and, for the sinusoidal mode, very bad resolved
resonances.
V. CONCLUSIONS
Thebehaviorofultracoldatomsinteractingwithafield [1] See,forexample,LaserManipulationofAtomsandIons,
inside a high quality microwave cavity depends in an es- edited by E. Arimondo, W. D. Phillips and F. Strumia,
sential way on the profile of the mode inside the cavity. VarennaonLakeComoSummerSchool(NorthHolland,
Amsterdan, 1992).
5
[2] B. G. Englert, J. Schwinger, A. O. Barut and M. O.
Scully,Europhysics Lett.14, 25 (1991).
[3] M. O. Scully, G. M. Meyer, and H. Walther, Phys. Rev.
Lett.76, 4144 (1996).
[4] S. Haroche, M. Brune and J. M. Raimond Europhysics
Lett.14, 19 (1991).
[5] G. M. Meyer, M. O. Scully, H. Walther, Phys. Rev. A,
56, 4142 (1997).
[6] M. L¨offler, G. M. Meyer, M. Schr¨oder, M. O. Scully, H.
Walther, Phys.Rev.A, 56, 4153 (1997).
[7] M. Schr¨oder, K. Vogel, W. P. Schleich, M. O. Scully, H.
Walther, Phys.Rev.A, 56, 4164 (1997).
[8] See,forexample,P.Meystre,M.SargentIIIinElements
of Quantum Optics, p. 446, Springer-Verlag, Berlin Hei-
delberg, 1990.
[9] See, for example, D. Jackson in Classical Eletromag-
netism, p.353, Addison-Wesley,New York (1975).
[10] J.H.Eberly,Am.Jour.Phys.33,771(1965).Ourphase
shifts where defined with a different convention.
[11] See,forexample,MorseandFeshbachinMethodsofThe-
oreticalPhysics,ps.1096-1106, McGraw-Hill,NewYork
(1953).
[12] M. Brune, E. Hagley, J. Dreyer,X.Maˆitre, A. Maali, C.
Wunderlich,J. M. Raimond and S. Haroche, Phys. Rev.
Lett.77, 4887 (1996).
[13] M. O. Mewes, M. R. Andrews, N. J. van Druten, D. M.
Kurn, D. S. Durfee and W. Ketterle, Phys. Rev. Lett.
77, 416 (1996).
FIG.1. Probabilityofemissionofaphotonwhenthecavity
is initially empty as a function of κL when the mode is: the
sinusoidal mode (solid line) and the constant mode (dotted
line)
FIG. 2. Stationary photon distribution for Nex = 1000,
th
k/κ = 0.01 and nb = 1 when κ2L corresponds to the 100
resonance:(a)constant mode; (b) sinusoidal mode.
FIG. 3. (a)Probability that an atom being transmitted
throughthecavityand(b)Probabilityofemissionofaphoton
when the cavity has initially n photons, as a function of κL:
n = 0 (solid line), n = 1 (dashed line), n = 2 (dotted line).
k/κ=0.01. Sinusoidal mode
FIG.4. Probability that an atom being transmitted when
initially there is a coherent state in the cavity as a func-
tion of κL for the constant mode (full line) and the cosine
mode (dashed line ) (a) : n¯ = 0.25 (solid line); (b): n¯ = 2.
k/κ=0.01. Sinusoidal mode
6
1.00
o
P
em
0.75
0.50
0.25
100p 101p 102p 103p 104p
k
L
Figure 1: Retamal
0,5
(a)
0,4
0,3
0,2
0,1
0,0
0 2 4 6 8 10 12 14 16 18
n
0,15
p
(b)
0,10
0,05
0,00
0 5 10 15 20 25 30 35
n
Figure 2: Retamal
1.0
(a)
0.8
0.6
n
T
0.4
0.2
30000p 30001p 30002p
1.0
(b)
0.8
0.6
m
e 0.4
n
P
0.2
30000p 30001p 30002p
k L
Figure 3: Retamal
1.0
(a)
0.8
0.6
0.4
0.2
n
o
i
s0.0
s
i
m1.0
s
n
(b)
0.8
a
r
T
0.6
0.4
0.2
0.0
30000p 30002p
k
L
Figure 4: Retamal
