Model for the Interaction

MICROSPHERES

Chapter 4 Chapter 4 Cavity QED with High Q Whispering Gallery Modes

4.3 Model for the Interaction

33

plificd into the follo\virg foriri:

Using the mode function $ ( r ) , the atomic dipole couplir~g rate (vacuum Rabi fre- cluerlcy) for a single atom is rewrit,ten as y ( r ) = y,n,,$ (r) with g,:,, = b/3~~2r.i/4xi/, so that go = y (a) is the value at rhe surface of the sphere (though the LVGhI field m;rximum is ac:tually inside the sphere close to t,he surface ;is shown in Fig. 2.2, t,he rrraximurn vahie accessible to the atorriic vapor, yo/27i

-

²⁰ !vl\/IIiz, is right at the surface). The qilaritity

plays the role of an effectjive susceptibility for the atornic sarnple in its interaction with the WGkL The Fomier transiorrn of the niode function,

4

(k) = ( 2 ~ ) - " i 2 d3r$ ( r ) exp (-ik. r ) (4.5)

is normalizeti such that V, ⁼ j"d% j jd (k)j2 =

j"

d"r

14

(r)j2. Because it is extremely diRcult t,o evaluate ~ ( w , , ) exactly for the actual furlctions

( 4

^{( r ) . p} ( v ) ) , calculations have been performed numerically for sirr~pliticd approxiniat,iot~ to $ ( r ) external to the rriicrosphere (e.g., $ (r)

-

exp (-ZT(T - a ) /A) exp (-82:18i) exp

(im.4)

as a11 ap- proximation to J!, (rj ^.Y hjl' ( k r )

Y;,,,

(H, 4;)). Via nurnc:rical ir~tcgratiori~ transmission functions T ( w L )

=

it ^(wI,)

l2

are folrrld that are in quar~titat~ive accord with t,lic rilea- sured spectra for low Q

5

^{Qo = 5} x 10') but whic11 deviat.e frorn t,he observations for Q

2

QQo due to a near abserlce of nai.iout feutures of iciidth

-.

^7.

35

4.3.1 Ansatz for the Three Contributions to the Susceptibil- ity

Nonetheless, tliese calci~latioiis rrrotivate an ansats which takes

with ^{m d}

+ ivt + ^N,

⁼

ivI.

The first, cornpor~ent in Eq. (4.6) is physically n~otivat.ed by noting that there must be a Doppler-broadened response

due to velocity conlpor~ents tangential to the sphere in the direction of circulatiorl of the mode. In additioil, due to the geometry of the 'VZTG&i, there is also signific,mt transit broadening due to resicience tirnes of only 10-'r for lllotion along the radial coordi~iate to 10-'7- aloi~g tiicr Go direction (for which there is no Doppler broadelring).

TfiougI~ any giver1 at,ornic trajectory will yield a colnplicated funct,iorr of both of tliese mecharlisms (as in Eq. (4.4)), in Eq. ( l . G ) , a transit broacleneti component of H'AXM Sut/27r -- 25 hh1Hz is simply added on; correspoi~ding to a lir~ear traject,ory of lengtti I , -- through the mode.

The solid lines through the dat,a of Fig. 4.4 are based on Eq. (4.3a) with the nnsatz of Etl. (4.6). For Q

5

Qo, orily the first t,wo coniponerrts with fld = 0.75 i

0.05 arrd

,Vt

= 0.25

+

0.03 are needed in order to explain all t~aces. From t,his, it is inferred that tlie Doppler-broadened set of atoms act only as a broad ahsorber (since Awl < ^ti

<

&,I) and that the absorptive dip at line center is accounted for t ~ y tlre width A w t ? implying tluit cavity geometry is a doniinarlt factor below Qo.

That is, the geonretry of the cavity correctly accour~ts for t,he coexistence of both transit ant1 Doppler broaciening, where of cowse the sirnplc siurl of c o n t r i b u t i o ~ ~ stlggested phcrtor~~enologicdly in Eri. (4.6) is more properly ir1tt:rpreted as an iriterplay of frequency scales as in Eq. (4.4).

By contrast, for Q

>

Qn it is essential to include a small componmrt

r\i,

0.015 of

atorris which respond with their natural linewidtl~ 7 (the inclusion of which does not, change the quality of the fits for Q

5

Qo). In fact, this componerit now completely determines the properties of the narrow absorptive feature since n

<

(Alit, A&),).

hlthougll the need for this small s ~ ~ h s e t of atoms moving sloxvly er~ougii and iri direc- tions s~~cicil that t,liey are ~ieitl~ev itppreciably Doppler- nor transit-broadeneci is thus operationally motivatedl their existeui:e is also supported within the context of other measurements near dielectric surfaces, as, for example; in the work on Doppler-free evanescent-wave spectroscopy [69].

This sirriple model also allows the issue of the relationship of the qxantity q of Fig. 4.5 to the actual ernpty cavit:y Q to be addressed. For Q

5

Qo, t,he relationship q N Q holds; with the inferred erripty cavity transmission showri as the dashed trace in Fig. 4.4(a). I-Iowever, for

(2

> Qo, the t,wo broadly absorbing components in (*GdTd,

a)

(which account for niost of the atoms) significantly alter the lineshape relative to that inferred for the empty cavity with q ^1.Q/2.5 for Q = Q2 and tile peak t,ransmission of the cavit,y reduced by a factor w 4.5. I11 fact, within the cont,ext, of the ar~satz (Eq.

(4.6)) for

xA;

tlie si~b-nat,ural widths in Fig. 4.5 are an artifact of ho\v tliese differerit contributions (each of width

2

y) corribine t o produce T(ur,j, as shown by the solid curve in Fig. 4.4(b).

Finally, the result,s of this plienornenological rriodel are summarized in Fig. 4.6(b) where the inference of the effective at,onlic susceptibility X , in Eq. (4.6) is shown. The pwanleters

(;qcl, %; !XTG)

⁼ (0.75: 0.25,0.015) wl.iicli best fit tlle data across the whole range in Q are ilsccl. Note that

&Gc,

^{t 1%} - ^,-i

N,

^N 1, which agrees rather nicely wit11 the

~~revious estimate bitsetl upon V: a ~ i d p. 11ite.restirigly) tlle profile of Fig. 4.6(b) bears a striking rescrnblallce t o those seeri in nltral~igh resolution trilrlsit broadening-lirniterl molecular saturation spectroscopy 1701.

4.3.2 What About the Extremely Narrow Features?

2 . .

.

Because

iV,j'nTr

^N 10- , it is perhaps riot surprising that aiternpts t o sirnplify the full irit,egral of Eq. (4.4) failed to proviiie an accurat,e accourltirig of tbe narrow cornponenit,

intracavity photon number

I .

-100 -50 0 50 100 detuning [MHz]

Figure 4.6: (a) Dependerice of the size of the narrow absorption dip on intracavit.j- plrotorr nurrrber in the low Q regime of Fig. 4.4(a). (b) The at,orriic response

/X

^(~j,,);'

(norinalized to inrity) inferred frorri the phenomenological model discussed in the text.

[see Eq. (4.4);.

of ~ ( w , , ) . Nonetheless, even assumirlg a11 exact evaluation, there are several niecha- nisms wluch could protluce narrow features and are not accoimnted for in Eq. ( 4 . 3 ~ ~ ) . First, as Q increases, a greater percentage of tlre counter-propagating ( q I 1 ) -m) mode is exciteti? with a corresporlding increase in the possibilit,~ for iritracavity standing wave structure [7l! along the direct,ioil of nrode propagation. Sirch structure is capa- ble of prodiicing narrow feat,urcs by isolating thc slow atom cornponerts in a thermal

2 ' 2

gas. In addition, tire exp (-6 /0,) dependence of tlie mode frinction in the transverse direction is strictly only valici for the rn = 1 mode with 0: ^N 2/1, and, as one rnovcs amray fro111 m ⁼ 1, the WGX dependence on B develops ailxiliary rnaxirna 1721. Sec- ond, p (v) may depart fmrri a 1\4axuwll distribution, especially for t,hose a t o m ~vit,h

> 0 wkiich are leaving tlre s~ufacc. 023vio11sly, a distributit~n ~a4iicl1 was peaked at, lower velocities or which favored directions orthogonal to tho direction of propagation of tire node ~vould lead to ~iari-ow featnres. Firtally, possible atomic level sllifts [73]

(6, * 6,

+ ^A)

and rriodificatiorls to the xvitltli (y -. 7') due to elrhal-rcettient or iniri- bit,iori i74, 75j of radiative decay iri tl-re vicinity of the sphere's dielectric boundai-y liax.1: not beerr taken iriio accoinit,. Tlris urouid he extremely difficult to tio correctly

(e.g.. oire nrust eor~sider both norr-idealities sucll as asphericity which splits the tie-

38

generacy in mode number nt and Q's which are typically non-radiatively linrited and the spatial dependcrrce of

r'

and A).