Table Of ContentEnhancement of thermal Casimir-Polder potentials of ground-state polar molecules in
a planar cavity
Simen ˚A. Ellingsen
Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Stefan Yoshi Buhmann and Stefan Scheel
Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London,
Prince Consort Road, London SW7 2AZ, United Kingdom
(Dated: January 4, 2010)
We analyze the thermal Casimir-Polder potential experienced by a ground-state molecule in a
planar cavity and investigate the prospects for using such a set-up for molecular guiding. The
0 resonant atom–field interaction associated with this non-equilibrium situation manifests itself in
1 oscillating, standing-wave components of the potential. While the respective potential wells are
0 normally too shallow to be useful, they may be amplified by a highly reflecting cavity whose width
2 equals a half-integer multiple of a particular molecular transition frequency. We find that with an
ideal choice of molecule and the use of an ultra-high reflectivity Bragg mirror cavity, it may be
n
possibletoboostthepotentialbyuptotwoordersofmagnitude. Weanalyticallyderivethescaling
a
J of the potential depth as a function of reflectivity and analyze how it varies with temperature and
molecular properties. It is also shown how the potential depth decreases for standing waves with
4
a larger number of nodes. Finally, we investigate the lifetime of the molecular ground state in a
thermal environment and find that it is not greatly influenced by the cavity and remains in the
]
h order of several seconds.
p
- PACSnumbers: 34.35.+a,12.20.–m,42.50.Ct,42.50.Nn
t
n
a
u I. INTRODUCTION recent Letter by two of the present authors [17] (cf. sim-
q ilar findings reported in Ref. [18]), these resonant forces
[ produce a different force from that obtained through the
Casimir–Polder(CP)forcesareaparticularexampleof
standard approach, a perturbative expansion of Lifshitz’
2 dispersion forces, which arise due to the fluctuations of
formulausingthegroundstatepolarizabilityoftheatom.
v the quantized electromagnetic field [1]. These forces oc-
2 Only when the atom is fully thermalized, i.e., when it is
cur between polarizable atoms or molecules and metallic
7 inasuperpositionofenergystatesasgivenbytheBoltz-
or dielectric bodies and can be intuitively understood as
2 manndistribution,dothetwoapproachesyieldthesame
the dipole-dipole force that arises from spontaneous and
2 result, and only when the correct, thermal polarizability
. mutually correlatedpolarization oftheatomormolecule
6 isemployed. Formostatomstheresonantcontributionis
and the matter comprising the body. Casimir–Polder
0 smallbecausetherespectiveexcitationenergiesaremuch
forces at thermal equilibrium have been commonly in-
9 larger than the thermal energies, hence the atom is es-
0 vestigatedinthelinear-responseformalism[2–4]. Studies
sentially always in its ground state.
: of a wide range of geometries such as semi-infinite half
v
spaces[2–5],thinplates[6,7],planarcavities[8],spheres FordiatomicpolarmoleculessuchasLiHorYbF,how-
i
X andcylinders[9]aswellascylindricalshells[5,6,10]have ever, whose lowest rovibrational eigenstates are typically
r revealed that thermal CP forces are typically attractive separated by energies that are small on a thermal scale,
a in the absence of magnetic effects. the situation is changed, and the thermalized CP force
Recent theoretical predictions [11] as well as experi- candifferdrasticallyfromthestandard‘Lifshitz-like’ ex-
mental realizations [12] for CP forces in thermal non- pression. An investigation into these effects was under-
equilibrium situations have pointed towards interesting taken in Ref. [19] where it was found that for YbF out-
effects which arise when an atom at equilibrium with its side a metallic half-space the fully thermalized CP force
local environment interacts with a body held at different is smaller than the non-resonant force alone by a fac-
temperature. In particular, depending on the tempera- tor of 870. These results could be of importance for the
tures of the macroscopic body and the environment, the trapping of Stark-decelerated polar molecules [20] near
force can change its character from being attractive to macroscopic bodies.
repulsive and vice versa. Equally interesting is the observation that even for an
Non-equilibrium between atom and local environment atom or a molecule in its ground state the resonant part
can be investigated by means of normal-mode quantum oftheCasimir-Polderforcehasalong-rangeandspatially
electrodynamics (QED) [13, 14] or macroscopic QED in oscillating contribution, due to propagating modes [14,
absorbing and dispersing media [15, 16]. In this case, 19]. Whilethisoscillatorybehaviordiesoutasthesystem
thermal excitation and de-excitation processes lead to thermalizes, the thermalization time of a ground state
resonant contributions to the force. As discussed in a molecule can be quite long, often several seconds [21].
2
The oscillating propagating potential reported in [19], and the potential components for an isotropic molecule
unfortunately, was found to be too small in amplitude read
to be useful for guiding purposes, but nonetheless points
∞
to interesting applications if a way could be found to U (r)=µ k T ′ξ2α (iξ )ReTrG(1)(r,r,iξ )
n 0 B j n j j
enhance these oscillations.
j=0
X
µ
In the present article we investigate the use of a pla- + 0 ω2 |d |2{Θ(ω )n(ω )
3 nk nk kn kn
nar cavity to enhance the amplitude of the potential os-
k
X
cillations. This geometry has been discussed in detail in −Θ(ω )[n(ω )+1]}ReTrG(1)(r,r,|ω |).
nk nk nk
conjunctionwithexcitedatomsinazerotemperatureen-
(2)
vironment, where an oscillating, standing-wave potential
is known to occur [22–28]. For ground-state molecules where µ is the free-space permeability, k is Boltz-
0 B
in a cavity at finite temperature, an enhancement of up mann’s constant, ξ = 2πjk T/~ is the jth Matsubara
j B
to two orders of magnitude will indeed be shown to be frequency, and G(1)(r,r′,ω) is the scattering part of the
possible when the cavity width is fixed at the resonant classical Green tensor of the geometry the molecule is
lengtha=π~c/E10whereE10istheenergyseparationof placed in. The prime on the Matsubara sum indicates
the ground state and first excited state of the molecule. that the j =0 term is to be taken with half weight. The
molecular polarizability is given by
Thepaperisorganizedasfollows: Thegeneralformal-
ismoftheCPforceonamoleculeinacavityisdeveloped 1 |d |2 |d |2
nk nk
iunnSdeecrt.aIkI,enanidnnSuemc.eIrIicAa,lwcahlecruelawtieonaslsfoorshaogwolhdocwavtihteyparoe- αn(ω)=ǫl→im03~ k (cid:20)ω+ωkn+iǫ−ω−ωkn+iǫ(cid:21). (3)
X
tentialisenhancedasthecavityapproachestheresonant
The photon number follows the Bose-Einstein distribu-
width. In Secs. IIB and IIC we investigate strategies
tion,
for further enhancing the potential by considering differ-
ent cavity resonancesand molecular species. The scaling ~ω −1
n(ω)= exp −1 . (4)
of the potential with the reflectivity of the cavity is in-
k T
vestigated numerically and analytically in Sec. IID, and (cid:20) (cid:18) B (cid:19) (cid:21)
we thereafter discuss how further enhancement can be The first sum in Eq. (2) is the non-resonant force, rem-
achievedusingacavityofparallelBraggmirrorstunedto iniscent of that obtained by a dilute-gas expansion of
frequency ω =E /~ and normal incidence (Sec. IIE). Lifshitz’ formula [2]. The second sum is the resonant
01 01
Finally, inSec.IIFweinvestigatetheeffectofthecavity contribution to the force. We will see how it splits natu-
on the thermalization time of a molecule initially pre- rally into a propagating plus an evanescent part.
pared in its ground state and find that this remains in Weassumethemoleculetobeplacedwithinanempty
thesameorderofmagnitudeasinfreespace,typicallyin planar cavity bounded by two identical plates of infinite
theorderofseconds. WesummarizeourresultinSec.III lateral extension with plane parallel surfaces, separated
and provide a guide to further investigations. by a distance a. We choose the coordinate system such
thatthecavitywallsarenormaltothezaxisatz =±a/2
(z = 0 being the center of the cavity) and denote direc-
tions in the xy plane by the symbol ⊥. The scatter-
ing Green tensor of the system is well known (cf., e.g.,
Ref. [29]), and the relevant diagonal elements inside the
cavity are given by
II. THERMAL CASIMIR-POLDER POTENTIAL
IN A PLANAR CAVITY G(1)(z,z;ω,k )=− ic2β rp eiβacos2βz, (5a)
xx ⊥ ω2 D
p
i r
We consider a polar molecule with energy eigenstates G(y1y)(z,z;ω,k⊥)=β Ds eiβacos2βz, (5b)
|ni, eigenenergies ~ω , transition frequencies ω = s
ωpamre−dωinnaanndindcoiphoelreenmtnasturpixerepleomsiteinotnsodfmitns,ewnheircghyimesingperne-- G(z1z)(z,z;ω,k⊥)=icω22kβ⊥2 Drpp eiβacos2βz, (5c)
states with probabilities p . As shown in Ref. [17],
n where we have performed a Weyl expansion,
the CP force is conservative in the perturbative limit,
F(r) = −∇U(r), where the associated CP potential is G(1)(r,r′,ω)= d2k⊥ G(1)(z,z′,k ,ω)eik⊥·(r−r′)⊥,
given by (2π)2 ⊥
Z
(6)
taken the coincidence limit r → r′ and dropped all
U(r)=− p U (r), (1) position-independent terms (which give rise to an irrel-
n n
evant constant contribution to the CP potential). Here,
n
X
3
r ,r are the reflection coefficients of the (identical) cav- 2.0
s p
ity walls for s,p polarized waves and we have defined )
J
D =1−r2e2iβa, (7) 35 1.0
σ σ -0
1
β = ω2/c2−k⊥2. (8) l ( 0
q a
i
The square root is to be taken such that Imβ ≥ 0. t
n
When the cavity walls are homogeneous, semi-infinite te-1.0 Total potential
half-spaces of an electric material of permittivity ε(ω), o Non-resonant
P Resonant propagating
the reflection coefficients can be written simply as Resonant evanescent
-2.0
β− β2+(ε−1)ω2/c2 -250 0 250
r = , (9a) Position z (μm)
s
β+ β2+(ε−1)ω2/c2
p
εβ−p β2+(ε−1)ω2/c2 FIG. 1: Casimir-Polder potential of a ground-state LiH
r = , (9b)
p εβ+ β2+(ε−1)ω2/c2 molecule inside a gold cavity of width a = 500µm at room
p
temperature (T = 300K). The non-resonant, propagating,
where again the squareproots are chosen such that their andevanescentcontributionstothetotalpotentialareshown
imaginary part is positive. separately.
Adding Eqs. (5a)–(5c) and partially performing the
Fourier integral by introducing polar coordinates in the
term), propagating (contributions to second term with
xyplane,thetraceoftheGreentensorofthecavityreads
0 ≤ k < ω ), and evanescent components (contribu-
⊥ nk
TrG(1)(r,r,ω) tions to second term with ωnk ≤k⊥) according to
= 1 ∞ k⊥dk⊥ 2c2β2 rp − rσ eiβacos2βz. U(z)=Unr(z)+Upr(z)+Uev(z) (12)
2πi β ω2 D D
Z0 " p σ=s,p σ# To illustrate the behavior of the total potential and
X
(10) its three components, we consider a LiH molecule in its
electronic and rovibrational ground state placed inside a
This result can be substituted into Eq. (2) to obtain the goldcavity. Thepermittivityofthe(semi-infinite)cavity
thermal CP potential of a molecule in an arbitrary in- walls may be computed using the Drude model
coherent internal state. In the following, we will assume
the molecule to be prepared in its ground state, so that ωp2
ε(ω)=1− (13)
the thermal CP potential is given by ω(ω+iγ)
U(r)=µ k T ∞ ′ξ2α(iξ )ReTrG(1)(r,r,iξ ) with ωp = 1.37×1016rad/s and γ = 5.32×1013rad/s
0 B j j j [30]. As shown in Ref. [19], the CP potential of ground-
Xj=0 state LiH is dominated by contributions from the ro-
µ
+ 0 ω2 n(ω )|d |2ReTrG(1)(r,r,ω ), tational transitions to the first excited manifold, with
3 0k k0 0k k0 the respective transition frequency and dipole matrix el-
k6=0
X (11) ements being given by ω0k = 2.78973×1012rad/s and
|d |2 = 3.847×10−58 C2m2, respectively [21]. The
k 0k
[α(ω)≡α (ω),ground-statepolarizability]togetherwith potential (11) and its three components (12) for a cavity
0
P
Eq. (10). The first term is the non-resonant part of the oflengtha=500µmatroomtemperature(T =300K)is
potential,itdependsontheGreentensortakenatpurely shown in Fig. 1 as the result of a numerical integration,
imaginaryfrequencies. Sinceβispurelyimaginaryinthis where Eqs. (7)–(10) have been used. For transparency,
case, the Green tensor (10) and hence the non-resonant wehaveshiftedallthreecomponentssuchthattheyvan-
potentialisnon-oscillatingasafunctionofposition. The ish at the center of the cavity. It is seen that the non-
second term in the CP potential is the resonant contri- resonant potential is attractive and has a maximum at
bution, which depends on the Green tensor taken at real the center of the cavity, while the evanescent potential
frequencies. The integral over k in this case naturally is repulsive and has a minimum at the cavity center. As
⊥
splits into a region 0 ≤ k < ω of propagating waves in the case of a single surface [19] these two contribu-
⊥ nk
in which β is real and positive, and a region ω ≤ k tions partially cancel, where the attractive non-resonant
nk ⊥
of evanescent waves in which β is purely imaginary. The contribution is slightly larger and leads to an attractive
contributionsfrompropagatingwavesareoscillatingasa total potential in the vicinity of the cavity walls. The
function of position due to the term cos2βz, while those propagating part of the potential is spatially oscillating
from evanescent waves are non-oscillating, just like the and finite at the cavity walls, it dominates in the central
non-resonant part of the potential. The total poten- regionofthecavitywhereitgivesrisetowell-pronounced
tial (11) can thus be separated into non-resonant (first maxima and minima.
4
It is natural to wonder whether these potential min- 6
ima might be used for the purpose of guiding of polar ) a = 0.90·λk0
J 5 a = 0.99·λk0
molecules. With this in mind, we will in the following a = 1.00·λk0
discussstrategiesofenhancingthedepthofthepotential −3510 4 aa == 11..0110··λλkk00
well by analyzing the dependence of the potential on the ( single wall
molecular species as well as the geometric and material z) 3
(
parameters of the cavity. U
2
l
a
i
t 1
n
e
A. Cavity-induced enhancement of the potential ot 0
P
−1
Webeginouranalysisbydiscussingthedependenceof −400 −300 −200 −100 0 100 200 300 400
the potential on the cavity width. The one-dimensional
Position z (µm)
confinement of the propagating modes in a cavity of
highly reflecting mirrors leads to the formation of stand-
FIG.2: Cavity-inducedenhancementofthethermalCasimir-
ing waves and associated cavity resonances. When the
Polder potential of a ground-state LiH molecule inside gold
moleculartransitionfrequencycoincideswithoneofthese
cavities of various widths close to the second resonance a =
resonances, the thermal CP potential can be strongly λk0 =673µm. The potential of a single plate at z =−λk0/2
enhanced: When the squared reflection coefficient r2 is
σ is also displayed.
close to unity, the denominator D of Eq. (7), featur-
σ
ingintheGreentensor,becomessmalliftheexponential
10
exp(2iβa)isequaltounity,resultinginastrongenhance-
)
ment of the potential Upr. This happens for normal inci- J
dence (k⊥ = 0) of the propagating waves, when the res- −35 8
onance condition 2ω a/c = 2πν, ν ∈ N is fulfilled. In 10
0k ( 6
other words, the cavity length has to be equal to a half-
)
integer multiple of the molecular transition wavelength (z
U 4
λk0 =2πc/ωk0: l
a
i 2
a=νλ /2, ν ∈N. (14) nt
k0 e
t
o 0
P
We say that the molecular transition coincides with the
νth cavity resonance. -2
−600 −400 −200 0 200 400 600
The cavity-induced enhancement of the thermal CP Position z (µm)
potentialisillustratedinFig.2, whereweshowthetotal
thermal CP potential of a ground-state LiH molecule in FIG.3: PropagatingpartofthethermalCasimir-Polderpo-
goldcavitiesofwidthssuchthatthemoleculartransition tentialsofaground-stateLiHmoleculeinsidegoldcavitiesof
is close to the second cavity resonance λk0. As seen, the widths a = νλk0/2 corresponding to resonances of different
amplitude of the spatial oscillations, associated with the orders (ν =1,2,3).
propagating part of the potential U , sharply increases
pr
as the cavity width approaches λ . For comparison,
k0
we have also displayed the potential of a single plate at B. Different cavity resonances
z = −λ /2, where Eq. (10) for the cavity Green tensor
k0
has been replaced with the single-plate result [19]
In the following, we are interested in the cavity-
enhanced oscillations of the thermal potential. As seen
TrG(1)(r,r′,ω) from Fig. 2, they set in at some distance away from the
i ∞ k dk c2β2 cavity walls where the potential is well approximated by
= 4π ⊥β ⊥ rσ−2 ω2 rp eiβ(a+2z). (15) its propagating-wave contribution Upr. We can therefore
Z0 σ=s,p(cid:20) (cid:21) restrict our attention to this part of the total CP poten-
X
tial. The (propagating-wave) potentials associated with
The comparison shows that the amplitude of the oscilla- different cavity resonances ν are shown in Fig. 3. It is
tions, whilehardlyvisibleforthesingleplate, isstrongly seenthattheorderν oftheresonancecorrespondstothe
enhanced for a cavity. The depth of the potential mini- numberofmaximaofthepotential. Potentialsassociated
mumatthecenterofthecavitywithrespecttotheneigh- with resonances of order ν ≥ 2 have minima. The am-
boring maxima is increased by a factor 6.7 when using a plitudes of the oscillations become generally smaller for
resonant cavity rather than a single plate. higher resonance orders ν. As seen from the case ν =3,
5
−34 LiRb BaF YbF
VIB VIB VIB
−35
)
[∆U] (J −−−333876 22J/Cm) 2224 LiHNRHOTROOTHROT(a) LiOHDVIVBIB
0 (
log1−−4309 d2] 20 NaRLbiCCasFRROOTT OHVIB
−41 U/ 18 RbCs ROT
Δ ROT
−42 LiBaOHYbLiNHODCaLiOHOHKCLiODODOHNaODNaNaLiCaYbLiBaKCNaRbKRODNHRbOHKR g[10 16
ROTHVIBFROT(a)VIBFVIBRbROTROT(a)VIBFVIBHROT(c)ROT(b)ROTsVIBCsROT(c)ROT(b)ROT(d)ROTCsROT(d)VIBCsVIBRbROTCsROTFROTFROTRbROTFVIBsROTRbVIBCsVIBbVIBVIBROTCsVIBROTb lo 14
107 108 109 1010 1011 1012 1013 1014 1015 1016
FIG. 4: Depth ∆U of the ν = 2 potential minimum for ro-
Frequency (rad/s)
tational and vibrational transitions of various ground-state
polar molecules inside gold cavities at T =300K. The differ-
ent non-degenerate transitions of OH and OD are labeled as FIG.5: Frequency-dependenceofthedepth∆U oftheν =2
(a)-(d) in order of ascending frequencies, cf. Ref. [21]. potentialminimumforpolarmoleculesinsidegoldcavitiesat
T =300K.
the minima and maxima are slightly more pronounced
(vibrational), LiRb (vibrational), NH (rotational), OD
towards the cavity walls.
(dominant rotational transition), and CaF (vibrational).
Thescalingofthepotentialminimawiththeresonance
The variation in the depth for different molecules is
orderasobservedinFig.3canbeconfirmedbyananalyt-
partly due to its dependence on the molecular transi-
icalanalysis. Foreachcavityorderν,wedefine∆U tobe
ν
tion frequency. As shown in Ref. [19], the resonant
thedepthofthedeepestpotentialminimumwithrespect
part of the CP potential of a single plate is propor-
to the neighboring maxima. As suggested by Fig. 3, this
tionaltoω2 n(ω )foragoodconductorwithfrequency-
deepest minimum will always be the one closest to the k0 k0
independent reflectivities. This remains true in the case
cavity walls. Cavity QED problems can often be solved
of a cavity. In addition, the amplitude of the oscilla-
analyticallyunderthesimplifyingassumptionthatreflec-
tions is inversely proportional to the molecule-wall sepa-
tion coefficients are independent of the transverse wave
ration. Thelargestpotentialmaximumbeingsituatedat
number k [31, 32], and this method is also successful
⊥
z−a/2=λ /4∝1/ω , itsheight carriesanadditional
here. AsshowninApp.A,intheperfectconductorlimit k0 k0
ω -proportionality. The dependence of the potential-
r =−r ≡r →1, we have the simple scaling law k0
p s
minimum depth on molecular transition frequency can
1 thus be given as
∆U ∝ . (16)
ν
ν
ω2 for ~ω ≪k T,
wFohratimlepsesrsfelocwtlcyowndituhctνo.rs, the ∆Uν will decrease some- ∆Uν ∝ωk30n(ωk0)∝(e−k0~ωk0/kBT k0for ~ωBk0 ≫kBT .
Theanalyticalscalinglawobtainedonthebasisofsim- (17)
plifying assumptions supports the observation from the As shown in Sec. IID, this scaling becomes exact for
numerical results in Fig. 3 that the ν =2 resonance pro- cavities with frequency- and k -independent reflectivi-
⊥
vides the deepest potential minimum. In view of poten- ties. For real conductors, the decrease of ∆U for high
ν
tial guiding, we can therefore restrict our attention to frequencies will be stronger than given in Eq. (17) due
this case, ∆U ≡∆U . to the decrease of the reflection coefficients. Note that
2
Eq. (17) also shows that ∆U becomes larger for higher
ν
temperatures due to the increased thermal-photon num-
C. Different molecular species ber. Again, this only holds when disregarding the tem-
peraturedependenceofthereflectioncoefficients,cf.also
The CP potential depends on the molecular transition Sec. IIE below.
in question via the respective transition frequencies and The frequency-dependence of ∆U is illustrated in
ν
dipole matrix elements. Using the molecular data as Fig. 5 where we have plotted its values normalized by
listed in Ref. [21], we have calculated the depth of the dividing by the transition dipole moments d2 (d2 ≡
ν = 2 potential minimum for both rotational and vibra- |d |2). The transition frequencies of some of the
k 0k
tional transitions of the polar molecules LiH, NH, OH, molecules investigated are indicated in the figure. In
P
OD, CaF, BaF, YbF, LiRb, NaRb, KRb, LiCs, NaCs, particular, the vibrational transitions of BaF and YbF,
KCs, and RbCs; the results are displayed in order of de- which have been seen in Fig. 4 to give rise to large
scending∆U inFig.4. Thefigureshowsthatthedeepest potential-minimum depths, are very close to the peak
potential minima are realized when using the rotational of the function ω3 n(ω ), which is at ω = 1.11 ×
k0 k0 k0
transitionofLiH,thevibrationaltransitionofBaForthe 1014rad/s for room temperature.
dominant rotational transition of OH, followed by YbF The other main dependence of ∆U on the molecular
ν
6
species and transition is the proportionality to the mod- thecavitywallsz =±a/2isproportionaltothedifference
ulus squared of the transition-dipole moments, I(0)−I(1). We have
2
∆Uν ∝ |d0k|2 =d2. (18) I(1/2)= r Im 2πdxx2 eix+1
16πa3 1−r2eix
Xk Z0
r 2π
The transition-dipole moments are typically larger for = Im dxx2 1+(1+r−2)Li (r2eix)
16πa3 0
rotational transitions than for vibrational ones. For this Z0
reason, the rotational transition of LiH gives rise to the (cid:2) (24(cid:3))
largest minimum depth although the vibrational transi- where the polylogarithmic function is defined as
tion frequencies of BaF and YbF are much closer to the
peak frequency 1.11×1014rad/s. ∞ zk
Li (z)= . (25)
s ks
k=1
X
D. Scaling with reflectivity ThefirstterminEq.(24)isrealanddoesnotcontribute.
The second one is easily calculated using the relation
The cavity-induced enhancement of the thermal CP 1
dzLi (Aebz)= Li (Aebz)+C (26)
force strongly depends on the reflectivity of the cavity s b s+1
walls. To understand this dependence in more detail, let Z
validforarbitraryconstantsA,bwhere|A|<1. Partially
us for simplicity investigate how the height of the single
integratingthisrelationtwiceandsubstitutingtheresult
maximum for a ν = 1 resonance depends on reflectivity.
for A=r2, b=i into Eq. (24), one finds
The scaling of the potential extrema with reflectivity is
the same for all ν as is shown in App. A, so considering r+r−1 4π2
I(1/2)= Im Li (r2)+4πLi (r2)
the simplest case will suffice. 16πa3 i 1 2
We begin by writing the propagating part of the reso- (cid:26) (cid:27)
π(r+r−1)
nant CP potential associated with a single transition in = ln(1−r2), (27)
the form 4a3
where we have noted that Li (z) = −ln(1−z). In the
1 1
Upr(z)= n(ωk0)|d01|2I(φ) (19) limitofhighreflectivity, δ ≡1−r →0+ thisexactresult
3ε
0 shows the asymptotic behavior
where we have introduced the dimensionless position π
I(1/2)∼ (lnδ+ln2) for δ →0 , (28)
2a3 +
z
φ= (20) with the first correction term being of order δ.
a
The calculation of I(0) is only slightly more involved.
and the integral We have
r 2π x2eix/2
ωk0/c dk k r ω2 r I(0)= Im dx
I(φ)=ImZ0 2⊥πβ⊥"2β2Dpp − ck20 σX=s,pDσσ# =8πra3 ∞Zr02kIm 1−2πrd2xexix2eix(l+21). (29)
×eiβacos2βaφ. (21) 8πa3
l=0 Z0
X
As in Sec. IIB, we consider the simple model case By partial integration we obtain
of frequency- and k -independent reflection coefficients
rp =−rs ≡r. With⊥this assumption, Im 2πdxx2eix(l+12) = (l4+π21) − (l+41)3. (30)
Z0 2 2
r
σ
=0. (22) After substitution of this result, the sum over l can be
D
σ=s,p σ performed by using the relations (cf. §1.513 in Ref. [33])
X
After introducing the dimensionless integration variable ∞ r2l 1 1+r
= ln ∼−lnδ+ln2 for δ →0 (31)
x = 2βa with k⊥dk⊥ = −4a2xdx, the integral above l+ 1 r 1−r +
takes the form Xl=0 2
(leading corrections being of the order δlnδ) and
r x0 x2eix/2cosφx
I(φ)= Im dx (23)
8πa3 1−r2eix ∞ r2l
Z0
(l+ 1)3
where x0 = 2ωk0a/c. For the ν = 1 resonance, we have Xl=0 2
a=λ /2=πc/ω , so x =2π. ∞ ∞
0k k0 0 1 1
The required height of the potential maximum at the ∼8 l3 − (2l)3 =7ζ(3) for δ →0+,
" #
cavity center (z =0) with respect to the value of Upr at Xl=1 Xl=1
7
where ζ(z) is the Riemann zeta function. We thus find 14
l l
l l
) a a
I(0)∼−2πa3[lnδ−ln2+ π72ζ(3)] for δ →0+, (32) −3510 J 1120 avity w avity w
( C C
with the first correction again being of order δlnδ. z) 8
Substitutingtheresults(28)and(32)intoEq.(19),the (
U
differencebetweenthemaximumandminimumvaluesof l 6
the ν =1 propagating potential reads tia 4 rr == 11--1100--76
n r = 1-10-5
Upr(0)−Upr(a/2)∼−πPk|3dε00ka|23n(ωk0)(cid:20)lnδ+72ζπ(32)(cid:21) Pote 02 rrrG o===l d111 ---(111A000u---)432
−400 −200 0 200 400
for δ →0 . (33)
+ Position z (µm)
This result being representative of the case of arbitrary
ν, we can conclude that FIG. 6: Propagating part of the thermal CP potential for a
ground-state LiH molecule inside different cavities with con-
∆Uν ∝ln(1−r) (34) stant reflection coefficients. The rotational transition of LiH
is assumed to coincide with the ν =2 resonance of the cavi-
in the limit r →1. In the case where the reflection coef-
ties. Forcomparison,theexactresultforagoldcavityisalso
ficients are not the same for both polarizations but still
shown.
assumed constant, the coefficient of the term ∝ lnδ in
Eq. (33) will change, leading to a slight quantitative but
no qualitative difference to the scaling of the potential- part of the ground state force on a two-level molecule
minimumdepth. NotethatEq.(33)immediatelyimplies depends on the reflection properties of the cavity at a
thescalinglaw(17)forthefrequencydependenceof∆U . single frequency, ω = ω . In addition, the resonance of
ν k0
The fact that the potential depth diverges only log- the cavity is also associated with a single value of the
arithmically as reflectivity tends to unity poses severe wave vector k , namely normal incidence. An enhance-
⊥
restrictions on the potential which is obtainable using mentofthepropagatingpotentialhencedoesnotrequire
a planar cavity. The mathematical reason for the rela- a good conductor like gold which is a good reflector for
tive weakness of the resonance is that the integrand of a broad range of frequencies and all angles of incidence;
the k -integral only becomes large at a single point, at instead, cavity walls whose reflectivity has a sharp peak
⊥
k =0. The physical reason is that the photonic modes atnormalincidenceandthesinglefrequencyω aresuf-
⊥ k0
in the cavity are only confined in one out of three spa- ficient. TheobviouscandidateistousemultilayerBragg
tial dimensions. We conjecture that the potential due to mirrors, which consist of alternating layers of two differ-
theresonantCPforceonagroundstatemoleculecanbe ent materials, each layer of thickness being equal to one
muchincreasedbyaresonantcavityifconfinementisim- quarter of the wavelength λ = 2π/nω in that layer
10 k0
posedintwooreventhreedimensions,i.e.inacylindrical where n is the respective refractive index.
or spherical cavity. The reflection coefficient of a stack of layers with per-
The logarithmic scaling law of the potential-minimum mittivitiesε andthicknessesd isfoundbyrecursiveuse
j j
depth ∆U for the case ν = 2 is confirmed by a nu- of the formula
ν
mericalcalculationinwhichreflectioncoefficientsareset
cfoornstthaentr,otrapti=on−alrstr≡ansritaionndocfloLseiHtoisunshitoyw.nTihneFreigs.ul6t rijk··· = 1ri+j +rijrrjjkk((ll······))ee22iiββjjddjj (35)
where the exact result for a gold cavity is also included.
By comparing the latter curve to the potentials for con- (βj = n2jω2/c2−k⊥2), which relates the reflection co-
stantreflectioncoefficients,onecanreadofftherelatively efficienqt of a set of three adjacent layers ijk··· (and all
small ‘effective’ reflectivity of gold between 1−10−2 and thelayersbehind)totherespectiveresultforthenextset
1−10−3 at the respective transition frequency of LiH. of adjacent layers jkl···. If the kth layer is the last one
For a molecule with a smaller eigenfrequency ωk0 the of the stack, the coefficients rjk(l···) reduce to the two-
gold cavity does slightly better because the permittivity layer coefficients r . In straightforward generalization
jk
islarger. ConsiderthevibrationaltransitionofYbFwith of Eqs. (9a) and (9b), the two-layer coefficients read
ω ≈ 9·1010rad/s as an example, for which the ‘effec-
k0
tive’ reflectivity of the gold cavity (in the sense of figure rs =βi−βj ; (36a)
6) increases to about 1−10−3.5. ij β +β
i j
ε β −ε β
rp = j i i j , (36b)
ij ε β +ε β
E. Enhanced reflectivity using Bragg mirrors j i i j
for s- and p-polarized waves, respectively. The Casimir
In contrast with the non-resonant CP force which de- effect for such multilayer stacks has been extensively
pends on a very broad band of frequencies, the resonant studied in the past [34–37].
8
100 3.5 l l
l l
) a a
r}10-1 −35(10 J 32..05 Cavity w Cavity w
Re{10-2 z) 2.0
- (
1 U
1.5
10-3 al
Dissipation as indicated i
No dissipation t 1.0
ℑm{ε} reduced by factor 10 en
10−40 10 20 N 30 40 50 Pot 0.5 GSoalpdp hciarvei/tvyacuum
Bragg mirror
100 0
−400 −200 0 200 400
Position z (µm)
10−2
14
} l l
r1-Re{1100−−64 T=300K −35(10 J) 1102 Cavity wal Cavity wal
10−8 T= 77K ) 8
z
(
0 5 N 10 15 l U 6
a
i
t 4
FIG. 7: Reflection coefficients of Bragg mirrors for normal en
t Gold cavity
incidenceattherotationaltransitionfrequencyofLiHvs the Po 2 Sapphire/vacuum
numberofdoublelayersN. Above: GaAs/AlAsmirror,where Bragg mirror
0
theresultsforreducedandvanishingabsorptionarealsodis- −400 −200 0 200 400
played for comparison. Below: Vacuum/sapphire mirror at Position z (µm)
two different temperatures T =77K and 300K.
FIG.8: ResonantpartofthethermalCPpotentialassociated
withtherotationaltransitionsofaground-stateLiHmolecule
AverycommonpairofmaterialstouseforBraggmir- at ν = 2 resonance with a gold cavity and a cavity bounded
byvacuum/sapphireBraggmirrorsattwodifferenttempera-
rors is GaAs and AlAs. At the rotational transition fre-
tures: 77Kabove,300Kbelow. Thesolidblacklinesrepresent
quency of LiH, the permittivity of the two materials can
calculationsatconstantreflectioncoefficientsasinFig.6;the
be roughly given as εGaAs = 12.96+0.02i [38, 39] and corresponding values of 1−r decrease in powers of 10 from
εAlAs = 10.96+0.02i [40]. The reflection coefficient of 10−2 (lowest curve) to 10−7 (highest curve). The same per-
a GaAs/AlAs Bragg mirror is plotted as a function of mittivity is used for gold for both temperatures.
the number of (double) layers N in the upper panel of
Fig.7. ForagivenN,theBraggmirrorconsistsof2N+1
layers in total, i.e. N pairs of GaAs and AlAs layers of the reflection coefficients of the vacuum/sapphire mirror
thickness λ10/4 (beginning with GaAs) and a terminat- asdisplayedinthelowerpanelofFig.7. Atroomtemper-
ing GaAs layer of infinite thickness. As Fig. 7 shows, ature,thecoefficientsaturatesatN &6toδ =5.5·10−6.
the reflectivity initially increases for increasing N and AtT =77K,thereflectioncoefficientsaturatesatN &8
theneventuallysaturatesforN &30tosomefinitevalue to δ = 5.5·10−8, the the increase in reflectivity is ob-
where 1−Rer ≃ 10−2. This saturation is due to ab- viously due to the reduction of material absorption for
sorption, as is illustrated by the other two curves, where the lower temperature. Note that in comparison to the
we have given the results that would be obtained for a GaAs/AlAsmirror,thenumberoflayersrequiredforsat-
reduced or vanishing imaginary part of the permittivi- uration is significantly lower because of the larger dielec-
ties. For a reduced imaginary part, the saturation sets tric contrast; and the room-temperature reflectivity at
in for higher N, and consequently to a lower δ. In the saturation is increased by about four orders of magni-
absence of absorption, the reflectivity could be brought tude.
arbitrarilyclosetounitybyaddingmoreandmorelayers. TheresultingpropagatingpartoftheresonantCPpo-
A higher reflectivity could hence be obtained by using tential at resonant cavity width using the sapphire/vac-
materials with very small dielectric loss. One example of uum Bragg mirror at T = 77K and 300K are shown in
suchaBraggmirrorcouldbealternatinglayersofvacuum Fig. 8, where the corresponding graphs at various con-
and sapphire, which can have an extremely low loss tan- stant reflection coefficients have also been displayed for
gent (Imε/Reε ≃ 10−5 and 10−7 at room temperature reference. The ‘effective’ reflection coefficients achieved
and 77K, respectively [41]) combined with a refractive at the two temperatures are around δ = 10−4.8 and
index considerably larger than unity (Reε ≃ 10 [42]). δ = 10−6.7 respectively, and the potential depths ap-
Using the approximative values ε = 10+10−4i at proximately a factor 2.45 and 1.77 greater than that of
sapph
300Kandε =10+10−6iat77K,wehavecomputed the gold cavity at the same temperatures. Note, how-
sapph
9
ever, that the effect of the enhanced reflectivity at 77K 1.0
c
is counteracted by the overall decrease of the potential ) a
due to the lower photon number. 1-s( 0.8 ytiv
Γ law
l
e
F. Lifetime of the ground state in the cavity ta 0.6
r
g
Resonant CP potentials are only present for molecules nit 0.4
which are not at equilibrium with their thermal environ- a
e
ment, i.e., on a time scale given by the inverse heating H
rate [19]. When enhancing the thermal CP potential via 0.2
0 100 200 300 400
a resonant cavity, it is necessary to ascertain that the si-
Distance from leftmost interface (µm)
multaneouscavity-enhancementofheatingratesdoesnot
reduce the lifetime of the resonant potential by so much
FIG. 9: Ground-state rotational heating rate of a LiH
as to render it experimentally inaccessible. We show in
molecule in a gold cavity at ν = 1 resonance a = πc/ωk0
the following that the lifetime of the molecular ground
(solidline)andtotherightofagoldhalf-space(dashedline)
state is not radically changed even by the presence of a
at room temperature (T =300K).
resonant planar cavity.
The total heating rate of an isotropic molecule out of
its ground state may be written as [17] Γ = Γ0 +Γcav these oscillations is enhanced when placing the molecule
where inside a suitable cavity such that a molecular transition
|d |2ω3 n(ω ) frequency coincides with a cavity resonance. We have
Γ0 = k 30kπ~c3kε0 k0 (37) analyzed the dependence of this oscillating potential on
P 0 the parameters of the molecule and the cavity by both
is the heating rate in free space and analyticalandnumericalmeansandfoundthatthedepth
of potential minima ...
2µ
Γ = 0 |d |2ω2 n(ω )ImTrG(1)(r,r,ω )
cav 3~ 0k k0 k0 k0 • Cavity resonance: ... decreases with increasing or-
k
X der of the cavity resonance approximately as 1/ν;
(38)
is its change due to the presence of the cavity. Apart
• Molecular eigenfrequency: ... is proportional to
from the prefactor, this additional term has the same
ω3 n(ω )forgoodconductors,wheren(ω )isthe
form as the expression for the potential, except that the k0 k0 k0
thermal photon number;
imaginary part of the Green tensor is taken rather than
the real part. • Molecular dipole moments: ... is proportional to
In Sec. IID, we had shown that for real and constant the modulus squared |d |2 = d2 of the total
k 0k
reflection coefficients, the Green tensor exhibits a loga- transition dipole moment;
rithmicdivergenceasr →1withapurelyrealcoefficient, P
whereasallothercontributionsremainfinite. Thisshows • Temperature: ...increaseswithtemperaturedueto
that the imaginary part of the Green tensor responsible an increase of the thermal photon number n(ω );
k0
for the decay rate can be expected to remain finite even
forstronglyincreasedreflectivity. Itfollowsthatthepres- • Reflectivity of cavity walls: ... scales as ln(1−r)
enceofthecavitydoesnotdrasticallychangethelifetime for high reflectivity r.
of the ground state of the molecule, which will typically
In view of observing this potential and possibly uti-
be in the order of seconds. This is confirmed in Fig. 9
lizing it for the guiding of cold polar molecules, these
where we display the ground-state heating rate of a LiH
observationsimplythefollowingstrategiesforenhancing
molecule inside a ν =1 gold cavity and near a gold half-
the depth of the potential minima:
space. The lifetime is reduced by only a factor 2 at the
center of the cavity, remaining in the order of seconds.
• Cavity resonance: The ν = 2 resonance is most
suitable, since it gives the deepest minimum.
III. CONCLUSIONS AND OUTLOOK • Molecular species: Atroomtemperature,thedeep-
est minima are realized for molecules whose tran-
We have studied the thermal Casimir-Polder potential sitions are not too far from the peak frequency
of ground-state polar molecules placed within a planar 1.11×1014rad/sandwhichatthesametimefeature
cavity at room temperature. As was previously found suitably large transition dipole moments. Good
in Ref. [19], the resonant absorption of thermal photons candidates are, e.g., LiH (rotational transitions),
by a molecule gives rise to spatial oscillations of that BaF (vibrational transitions) or OH (rotational
potential. Ourresultsdemonstratethattheamplitudeof transitions).
10
• Cavitywalls: Highlyreflectingcavitiesarerequired theoneimmediatelytotherightatz =(ν−3)λ /4. The
k0
in order to enhance the potential. Bragg mirrors requireddepthofthedeepestminimumishencegivenby
consisting of materials with small absorption such
as sapphire are favorable to single layers of good ∆Uν =U[(ν−3)λk0/4]−U[(ν−2)λk0/4]. (A1)
conductors like gold.
We calculate this depth for a cavity whose reflection co-
• Temperature: Temperaturesshouldbeintherange efficients are independent of the transverse wave number
of room temperature or even higher in order to k⊥,rp =−rs ≡r,andclosetounity,δ =1−r ≪1. Not-
achieve large photon numbers. This should be bal- ingthattheoscillatingpartofthepotentialisdetermined
anced, however, against the adverse reduction of by Upr and introducing the definition (21), cf. Sec. IID,
reflectivity of most materials with increasing tem- we thus have
perature.
∆U ∝I 1 − 3 −I 1 − 1 . (A2)
ν 2 2ν 2 ν
With an optimum choice of all these parameters, the
We consider in the (cid:0)followin(cid:1)g onl(cid:0)y the t(cid:1)erms which do
planar cavity can be used to enhance the resonant po-
not vanish as δ →0 . For arbitrary φ, we have
tential by one or at most two orders of magnitude with +
respect to the single-plate case. However, the thermal
r 2πν x2eix/2cosφx
potentials achievable with planar cavities are in all like- I(φ)= Im dx (A3)
2πν3λ3 1−r2eix
lihood still too small to facilitate the guiding of polar 0k Z0
molecules.
which, afterexpandingthefractioninpowersofr2, solv-
The limitations of the enhancement of the potential
ing the integral over x and taking the imaginary part,
in a planar cavity are ultimately due to the weak (loga-
can be written as
rithmic)scalingwithreflectivity. Astrongerscalingmay
be expected in geometries providing mode confinement ∞
r
in more than just one dimension such as cylindrical or I(φ)= y(j+ 1 +φ)+y(j+ 1 −φ)
2πν3λ3 2 2
spherical cavities. This will be investigated in a future 0k j=0
X(cid:2) (cid:3)
publication. Note that apart from the different expected (A4)
scaling with reflectivity, all other conclusions regarding with
thedependenceofthepotentialontherelevantmolecular
2 2 4ν2π2
andmaterialparametersasgivenaboveholdirrespective y(p)=− + − cos(2πνp)
p3 p3 p
of the geometry under consideration. The strategies for (cid:18) (cid:19)
the enhancement of thermal CP potentials developed in 4νπ
+ sin(2πνp). (A5)
this work will thus present a valuable basis when consid- p2
ering more complicated cavity geometries.
In particular, this implies
2r ∞ ν2π2 ν2π2
Acknowledgments I(1 − 3 )= r2j +
2 2ν πν3λ3 j+1− 3 j+ 3
0k j=0 (cid:20) 2ν 2ν
X
This work was supported by the Alexander von Hum- 1 1
− − , (A6a)
boldt Foundation, theUKEngineering andPhysical Sci- (j+1− 3 )3 (j+ 3 )3
ences Research Council, and the SCALA programme of 2ν 2ν (cid:21)
the European Commission. S.˚A.E. acknowledges finan- I(1 − 1)=− 2rπ ∞ r2j + r2j (A6b)
cial support from the European Science Foundation un- 2 ν νλ3 j+1− 1 j+ 1
0k j=0(cid:20) ν ν(cid:21)
der the programme ‘New Trends and Applications of the X
Casimir Effect’. The evaluation of sums with simple denominators can
be performed by using the relation (formula 9.559 in
Ref. [33])
Appendix A: Scaling of potential depth with
resonance order ∞ r2j 1
= F(1,b;1+b;r2), (A7)
j+b b
j=0
For a cavity of width a = νλ /2 the potential has ν X
k0
peaks, roughly located at z =−νλk0/4+(µ−1/2)λk0/2 valid for any b 6= 0,−1,−2,... Here, F(a,b;c;z) ≡
(µ=1...ν) and ν−1 minima at z =−νλk0/4+µλk0/2 2F1(a,b;c;z) is a hypergeometric function which in turn
(µ = 1...ν − 1). For a given resonance order ν, the has the following expansion in powers of δ = 1−r (for-
deepest minima are the ones closest to the cavity walls mula 15.3.10 in Ref. [43])
(andwhichhaveamaximumonbothsides), forexample
the rightmost one at z = (ν − 2)λk0/4. It has to be 1F(1,b;1+b;r2)∼−lnδ−ln2−γ−ψ(b) (A8)
comparedwiththelowerofthetwoadjacentmaxima,i.e., b
