TRAPPED ATOMS IN CAVITY QED
Chapter 7 Chapter 7 Trapping a Single Atom Inside a High Finesse Cavity
7.2 IntraCavity FORT
7.2.2 The Hamiltonian and Eigenvalue Spectrum
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FORT as a positive AC Stark shift Arorrr ( r ) = 2n(SFORT (I-) 011 tile excited state / e ) and a negative AC Stark shift -Arorrr ( r ) on the ground state lg). Again, siriiilar to Eq. (7.31, the eigellvalues of thc rnatrix
for the grou~id state ig, 0) :
A<:>, x,
A* =
7
k/- +
,q2 ( r )+
A F ~ I Y ~ (I-) (AFORT (I-) - At!tL)4
for the first two excited states - 1 (/y, 1) & ie.0)) .
4
These eigc:nst,atcs are very sirnilar to well-dressed eigenstat,es of Ecl. (5.1), with t,he subtle diEereilce that !&+ ( r ) tllcre does not have an irit,ernal st,ate dependence. In Eq. (7.23), Aroltr (r) does not have arly bound eigenstates in tile excit,ed state. An experirrierst,al situation mucii closer to that erivisioned in Part I1 of this thesis will be discussed shortly in Sec. 8.4.
The grourid state shift -AFoiiT ( r ) rnust be taken into accourlt, when considering the shift, of the resonarice frequencies 6w+. The probe laser, at, a aletuning A,,, from the bare atorriic rcsonancc:, mill now see an "extran shift due t o the FORT of
To complete the st,ory, the arlalogoi~s cavity transrnissioa spectra to Eq. (7.7) can be
i" (-1- t i (2Ai:orr.r - A , , ) ) t (a,,,,) =
(A+
-+
iA,,;,) (A-+
Z A ~ , ~ , ) '137 with
(A
+
K+
%A(.,) (A+
71 2 A ~ 0 1 1 ~ )+
g: = 0 (7.28)Tliese equations are tieriueil from Eq. 7.7; Ijy noting that the bare atornic resonance frequency (wlricli was chosen as the 'keference frequency" in Eq. (7.7)) is now rmodi- fied by ,yw,,
,,,,,
-i wiriliiZi + 2AFoIrr (r) .'Ib rnake all of this very concrete; below in Figs. 7.18, 7.19, 7.20 and 7.21 are presented three specific examples using the actual experimental sit,uat,ion of an 1 = 44 j ~ ~ n long cavity and 868 nm FORT as in Fig. 7.17(b). In Fig. 7.18(a), the registration of g(z) with a 20 ~ I W A tlecp FORT Al,~olvr ( z ) is shown for ref(:rence near the ceriter of the cavity. In 7.18(b), the eiger~value structure 6wi is shown, along with A i . 0 ~ ~ (2)
,
which represents the shift of t,he gronritl stat,e. Note that A,,, = 0 for tlus example.
Corlcent,rating on the line AA', along which g = g,,,,, = 32 :\/fHz and Al.-om = 20 MHz (perfect registration at the center of the cavity), Fig. 7.18(b) is the evaluation of Eq.
(7.27) for the cavity transmission for an atom located at this position. If the probe was chosen to be on resonance A,,, = then an atom wllich entered this pari~icular potential well woi~liltl he seen as a ciown-going transit,. Tile effect of t l ~ e FORT in this case is to reduce the splitting of tlle cavity-like eigenvalue from y,,,,, = 32 MHz in the "nrrFORTn case t,o approximately 18 MHz here.
In Fig. 7.19: the eigeilvalue spect,ra near the point. z = 114 is shown for tlle salne parameters as in Fig. 7.18. Alorig the line AA' is tlie point where rnaxilnal cavity QED couplirig coincides wit,h a zero of the FORTi giving rise to the synlmetric spet:tr~ini in (c). which h s heen seer1 before in Fig. 7.2. A shift to the left a fraction of a wavelength to the line
RB'
ciianges this situatioi~ drastically, as is shown in (d), where there is a, cleas identification of the cavity-like eigenvalue similar to Fig. 7.18, because the atorn has niovetl away horn the FOEIT node.Tlle third Fig. 7.20 illustrates tile situation of a rather deep FORT of AFORT = 80 MHz. In this case, all four quaritities (gjz), A l - o R T ( ~ ) and (2)) are showri on the sarrie graph so that (a) shows the situation asourid z = l j 2 at tlre center of the cavity and (b) at 2 = 1j4. There is very little relative effect along the line AA' at the cavity
distance alon cavity axis
a
bm] distance alo g cavlty axls1
. . bm]-100 -80 -60 4 0 -20 0 20 40 60 80 100 probe detuning Spa [MHz]
Figure 7.18: Tlle situation for an atom trapped in a FORT poterltial well near tlie center of the cavity is clepicted here. 111 (a), the FORTjcavii-y QED fielcl registration is stlrrirnarized again. Fig. (h) gives the spatial dependence of the upper aud lower eigenv,thles (both solid) along wit'li the FORT potential (daslied). Finally for a11 aton1 t,rappcct in the potential ~vell along the lirle AA', the cavity transmission it ( A , , ~ ~ ) / ~ of Eg. 7.27 sliows the lower eigcri\.alue as "ca\-ity-like" and the upper as "atorr-like."
The errrpty cavity transr~iission is shoxi~n as a dashed line here.
distance along c v ty axls [um]
I
distance along cgv/ty axis [pm]I
probe detuning Sprobe [MHz] probe detuning tiprobe [MHz]
Figure 7.19: liloving to the situation close to 1/4 of tlie way alorig the cavity axis, there is riow perfect anti-registration of FORT (dashed in (a)) and cavity QED field (solid in (a)). Tlie spatial eiger~valrics are shown in (h) as solid lines for t,he upper atoxl-like 6w+ ( 2 ) arid lower cavity-like 6s.- (2) eigerivalues. An atorri situat,ed at the posit,ioll AA' will see no FORT potential, so it,s cavity trarisrrlission in (c) is the same as in Fig. 7.2 with g = g,,,;,, = 32 MHz. Tliere is no atorn/cavicy-like distinction here.
.41ong the lint: BB', hoivever, the situatiorl ciiarigt?~ dri~stically sricll that the FORT depth is approximately 16 XIHz tint1 tlie cavity QED coupli~lg is g = 13 lIHz to give the transrnissiorl spectra in (cj
.
160
center clue to an atom trappctl there, as the large FORT depth overwliel~x~s the cavit,y couq~ling g xr~d therefore tile ability to see thc atom ~v11en A,:, = 0. Using Eq. 7.27:
the fractioilal c;liange in the i~ircr~sity of the cavity trar~srnissiori when probing on resorral~ce (A,,, = 0) is
which is just detectable only for the very best ator~ls with g (z) go; assuming A I > o n T ( ~ )
-
AFolrr (niax) for trapped ato~lis. This ex<ample illustrates that it is probably more ~ o r t h w h i l e to turn ojJ large FORTS (Sec. 7.2.5) or use A,,,#
0 (See.7.2.4) in order to see atonis trappod in them.
In the final Fig. 7.21: a neat proposal higl~ligl~ts the role of A,:,> which tias been absent until now. Examining Eq. (7.25) shows that making A,, = 2max (AFon-r (z)) sliould "remove" the presence of tlie FORT from the eigenvalue spect,runi such that
positioris where AFonT ( z ) is a n~axirni~m, &w+ =
a,:,
& g (z) . Furthermore, if g ( 2 )happens t,o be a rrmximum at this same poinit,, as along the line AA' in Fig. 7.21, then clet,uning tho probe to 6,,, = 6,,;, will allow one to "see" t,he at0111 as if the FORT were not there at all (i.e., without sacrificing any contrast in g). Unfortiu~ately; for reasoriably sized FORTS with AFoRT
2
20 - 30 MHz: the required S,:, t,o implement, this scherrie quickly moves out of the experirnentally accessible dynamic range in frequer~cies due to limitations on modi~lator and detect,or bandwidths. F~irtherrrore, one rrust always keep in rr~ir~d that ariy population in the atomic excited state will tend to cause heiitir~g, as sliowri 11y the iianti-r(?gistration' of the spatial ciependence of the upper and lo~vcr eige~lrralues (soliti) witli respect to the FORT (dashed) in Fig.7.21. Here, these eiger~values are once itga,iri equal con~binat,iolls of atorrt and ca,vity (with the "atorr-like" anii "cavity-like" distirictior~s cleverly rernoved) so that any population in eitlier of these states c ~ t positions of (I trapped a t o m (FORT mirumum) will terid to calm sorrle lioating.
distance along cavity axis Lum] distance along cavity axis bm]
1 h :;
I\::
1 1 , ; .O 0.7
.B U) 0.6
$d
0.5c 2 0.4 I :j
* I : \ :
,? 0.3
.-
> ;;I",;
t ,
m 0.2
0 I ' \, ',
0.1 0
..
\ .&.
-100 -80 -60 -40 -20 0 20 40 60 80 100 probe detuning a,, [MHz]
..
-Figure 7.20: Tlie limit of a very big FORT wit,h Al,.oir.r = SO MHz is s2iown near the center of the cavity (a) and 114 of the way alorrg tile cavity axis in (b). The cavit,y QED field arid upper a.xid lower cigerivalues are sliown as solid lines, and the FORT potcritial is tiasl~ed. Note that the eige~istatcs "rneet" in (a) at positions where g (r) = Ai:onr (r) = 0. Tlie prohl(?~ii of detecting ari aturn trapped in tiit: \veil along the line AA' is erriphi~3i~t:tl in t:kie spectri~rn of ( c ) ? w21c:re the cavity-like eigc?rivaluc is only shifted t>y 6,,;, N -g'/2AFolir frorri the err1pt;y cavity case: versus the
4,.
N -y shift orie 'r'rpi>uld get in the case of no FORT field. As a result, the contrast for detecting single atorris on resoriaricc (with A,, = 0) is poor for big FOlXTs.
distance alopg cavity axis [jirn]
A'
Figure 7.21: I11 an irlterestirig proposi~l; a r ~ attenipt is made lo use tlie atorrl-cavity det,uning A,, to eliniinate tlie effects of t,he FORT potential on tlie cavity t,rans- nlissior~ spectra. In (a); upper arid lom-(-1:r eigenvalucs ( ~ o l i d ) ~ cavity QED couplirig (dot-dashed) and FOIIT potential (dasheil) arc; all s2iourn. Tlie t,ra~~smission spectra in (b) for the axon1 trapped at the position correspo~lcling to the line AA' is once again tkie syr~lrnetric eigerivalue strni:ture. Filrtllerrnore, the eigerlvalue splitting is the rnaxirrlal value of 2g3 allo~virig for rn:~xi~ln~rn "'visibilityn of a. trapped atom in the cavity Irarisrnissiori.
163