TRAPPED ATOMS IN CAVITY QED
Chapter 7 Chapter 7 Trapping a Single Atom Inside a High Finesse Cavity
7.2 IntraCavity FORT
7.2.1 Far Off-Resonance Traps (FORTS)
148
149
st,rorigly to the GPIl2 and I ~ P . , , ~ states via the D l arid D2 lines respectively); the shifts as a function of magnetic sublevel r r 2 ~ for ari 668 nm FORT are slro~vn in Fig. 7.14. It, is iritcrestirig that in tlie case of a linear FORT, all sublevels are trapped in both cases.
EIo~vever, because the cal-ity QED fielti drives the ( F , m F ) = (4,4) -, ( F ' , m p ) = (5,5) cyclirig t,ransition using n+ light, a n+ PORT was the one implemented in the lab clue to cxpcrirnerital constrairlts (the FORT used the sarrle polarizing optics as the cavity QED probe field). For this poltlrization, the F = 4, mi. = -3, -4 arid
F
= 3,r n p = 3 states are not trapped. The lat,tcr has serious consequences for those at,oms in our F = 4 implenlentatiori ivhiclr are off-resonantly scattered from F = 4 (t'llrough F' = 4) into the F = 3 state and makes repumping of t,he aton~ic populat,ion out of this state critical on time scales conlparable to the inverse scattering rate.
20
not trapped
i?
-20
5 a 4 0
u:
B 4 0
F = 4, a+ FORT F = 3, o+ FORT
-70
4 - 3 - 2 - 1 0 1 2 3 4 -3 -2 -1 0 1 2 3
magnetic hypsmns subievei r n ~ magnetic hyperfine sublevel m~
L
5 a all trapped all trapped
;: -34 -34
F = 4, linear FORT F = 3, linear FORT
3 5 -35
4 - 3 . 2 - 1 0 1 2 3 4 -3 -2 -1 0 1 2 3
magnetic hypemna sublevel m~ magnetic hyperfine sublevel mp
Figure 7.14: The AC Stark shift of the ground state clue to the FORT trapping laser is sho~vn as i~ function of iiyperfine magnetic sublevel for both linear and circularly polarized FOR?' fields iri botli tile F = 4 and F = 3 grour~ti &ate marrifolds. It is iriterestirig to note that all states ase trapped for linear. FOI?Tsl but riot for circular.
The n ~ ~ r ~ r b e r s liscii in the calculation are typical of those for our experime~lt.
The ~v2ivclengt,h clcperrdcilce of tlie FORT potentiaa is shown in Fig. 7.15 for the
150
relevant
(isll2
F = -1. mi; = 4 state for both cir(;illar (a) arrd linear (b) polarizittiom.In the circular case, this state cannot be connected to tile 6P1p state (Dl transition at 894 nm, see Fig. 6.1) due to selection rules and lience in (a); only the D2 line at 852 rim; with all
F
= 5; 7 i t ~ = 5 state eilters into the caicuiation. Fiirally, Fig.7.16 sho~vs tile ~vavelcngtll dt>pmdeiice of the scattering rate
r',,.i,lt.
All three of these calculatiorls (Figs. 7.13. 7.15 and 7.16) use parameters for I (r) which are very close to tile experimentally irnplerncxtoti valnes. For a standing zmve of power P,where in our casei tile intracavity power a,t the eventual FORT wt~velength was ap- proxiruately 150,000 times the rneasi~otl output power due to t,he high finesse cavity buildup. IIcnce, for a typical nreasured oiitput power of about 8 pW at 868 rim; the experin~elltally iniple~nented FORT deptlr and scattering rate are
U ( T )
1
( 2-
75 sin" ( 2 r z j X F O n ~ ) [MHz; zz A m n (T) j27~ (7.18) I',,,,, ( 2 )-
250 s i 1 1 ~ 2 7 i z / X ~ ~ ~ ~ ~ ) [l/s].The shift, of the 6Q2 excited state dnt: to FORT excitat,iori at 860 and 868 nm is the opposite of the ground statc shift ( A;,,,., = -A;.,,,,) in the circula.rly polarizeci case (see Fig. 8.31, indicating that i:lre excited state is not t,rapped. In fact, tlre finite scattering rate into this state is a major cause of heating in tlre FORTT, because the dipole force secri by the atorii as a wl~ole t,errds to Buctuate. A possible strategy t,o overcorrie this lirrlitatiorr of the FORT' will be discussed in Sec. 8.4.
Thc choice of FORT mavelerrgtlls mas dictated, of course, by t,Iie ava,ilable lorigi- tiidinal modes of the pliysics c;~vity. According to Fig. 6.20, nlodes 7~, - 1 and n - 2 at approximately 860 and 868 rim seerned like good candidates. There were various tratleoffs b e t w c e ~ ~ them; inchiding the FORT depth and scattering rat= which, ac- cording to Eqs. 7.16 (n.b., with detuning A = wp0n.r - w ~ ~ ~ , , , a diflerent "A" from the \veil depth scale wit11 det,urii~rg as l / A and l/A%~s~ectively. Ileilce,
500
6S112 F = 4, mF = 4, n+ FORT
t
6Ski2 F = 4, mF = 4. llnear FORT 300
-
300 2002 5 roo.
5, 0 -
& -roo
0
!J, ZOO -300 -400 -500
~ .-
800 820 840 860 880 BOO 920 840 980 880 1000
- i I
I : , ,
; :
-
--
/./' ~I~ 1_____
~ - - - . - -
t ,-
, , 868 nm FORT -
- - (a)
1
1; 1 ; ! i ; 4:
, m FORTwavelength *<OR? Lnml
800 820 840 880 880 OD0 920 B40 960 980 1000 wavelength LFORT [nml
Figure 7.15: For realistic experirrental parilrneters, the FORT poteiitial depth AF.oirr ( z , , , , ~ )
/
( 2 r l i ) of Eq. (7.18) for tlre 6SiP2 F = 4 > m ~ = 4 ground state as a fi~rrctiori of wavelerrgth is siiowrr for both circularly (a) ant1 liriearly (b) polarized FORTS? ivit,h the two ivavelcnglrts rt:levarit to tlle experirnerits hiere shown i>xplicitly.In the circvtlar case of (a), t,lie ground state only couples to the L32 line at 852 nnr, wliereas tlre effects of tile D l line at, 894 nm rnrist also be incliltied in the linear case.
1 o+ polarlratlon
I
', I.
I $60 nm FORT
/ 4 '
I>,' ;
;>--
,-._.--_- ~~*__i
0
800 820 840 860 880 900 920 940 960 990 1000 wavelength 'FORT fnml
Figure 7.16: The scaticrirlg rates
r,,.,,,,
at the position z,,,;,, associated wit11 the FORT depths of Fig. 7.15 are slioiin for lirletir and circularly polmized FORTS. This scat- tering causes ilcating of the atorn in the FORT as explained in Section 7.2.7.153
going from 860 nrn to 868 nm would only decrease the FORT tlepth by a factor of 2 (for the same iritracavity power) but \\~ould reduce the scattering rate four-fold. Figs.
7.15 and 7.16 have these two ~~~avclerrgths higllligilted on them for convenience.
Fig. 7.17 investigates tire registration of the FORT pot,entitil si11"27rz/X~~~~) [Eq. (7.18)j and tlie cavity QED field
-
gosin (27r~/X,.;,,.~~,) : with X,.,,itY = 852 nm.When the FORT is chose11 to he oliiy one mode order below the atornic line, at 860 nm, the FORT potential rriinirna and cavity QEI) rrraxima coincide exactly only at the tivo mirror surfaces, ancl the FORT minima coiricide wit,h nodes of the cavity QED field at tlie cavity ccrrt,c!r, as sl~oivn in Fig. 7.17(a). This sit,uation seenls not to rnaximizc the probability of "seoing" an at,om using the cavity QED coupling (as in See. 7.1.1) once it is loaded into the FORT. On the other hand, n~oving one mode ortier fiuther along to 868 nm, the standing-wave patterns of the two rrrodffi at (vF(jnT, v,:,~.~,,) are such that, ttlrere are approximately coincident antirkodes near the cent,er and ends of the cavity. Hence, the trapping potential of the FORT has lriaxirnri~rli depths at t,lle positions of rnaxirna (go) for cavit,y QED coupling in these regions. Experiment,ally, work first started at 8'0 nm; but rrroved quickly to 868 nrn in the hopes of taking atfvailtage of this fortuitous overlap of fields at. the ceriter of tlie cavity.
This scction colicludes with a very cjuick estimate of the trap vibrationd frequen- cies. The spatial dcpendence of the intensity t<&s the form
For the t,ypical sized FORT of AF~j1vr/2r = 6FOliT
-
75 ?\IHzj wc can use either Eq. (5.3) or a simple c;xparisiori of (7.20) to eva,h~;lte the radial (z, 9) arid axial (z)Cavity QED field mode n. FORT mode n .1 z = O r = L
around r = U2 around z
-
0 , L(a)
Cavity QED field mode n, FORT made n - 2
around = U4.3U4 (b)
Figire 7.17: TIic i~lignrncnt or registratiori of tile (dashed) PORT poteritial propor.
ljort(11 to the scjual.e of the FOXTJirld with respect to the cavity QED coupling probe field is skiown for two casils. In (a); they ;ire offset by only 1 lorlgitudirlal mode of the cavity, so t i n t :in atorrl trapped in a FORT poterit,ial b i d 1 at the cexlt,er of tlie cavity will see a ~iull of the cavity QED field. In (b), the perfect registration of rrlaxi~rlurn of tlie FORT with rnaxinii~m of tile probe field now occurs at the cavity center for a two lorigitutlinal mode diffcrencc between tlie wavelengtlls.