TRAPPED ATOMS IN CAVITY QED
Chapter 7 Chapter 7 Trapping a Single Atom Inside a High Finesse Cavity
7.2 IntraCavity FORT
7.2.7 Limits to Trap Lifetime
I ebackground counts
/
*net events with untriggered FORT'7 0.3 -TI
L i j! ,
here without
fit trap lifetime (Ile) 27 ms FORT
- , j I
1
30 40 50 60 70 80 90
time after downstairs PGC [ms]
Figure 7.30: The FORT lifetitne of Fig. 7.28 was indeperideritly verified in a separate experirr~ent in which 6he FORT was not triggered by tlie presence of a single atom, but was gated on a t the fixed time indicated. The resonant probe beam was also left on for t,his experiment. Here, a frt reveals a lifetime of about 27 ms. The background counts associated witti this measurement are also shown.
MOT, albeit with an extrerni:ly large atom number, iri Fig. 7.3(c)). Finally, care was taken to check t i ~ o xiumber of everits seer1 u41e1r tire atom source was blocked to rule out any other spurious signal source. This measurenicnt always produced zero counts independciit of time.
a- -background (no FORT) --- -net events with FORT
k
0 -e=-c;=:- :=-"- . 4
30 40 50 60 70 80 90 100
time after downstairs PGC [ms]
Figure 7.31: To coinplete the discussion of t;Eie FORT lifetime measuror~ierrt~ the sharp increase of t,hc background coiix~ts for very short tirries associat,eti with residual &oms from t,he falling MOT is shown for the rireasureinent of Fig. 7.30.
kick) during the scattering cycle. The second corilponent is due to fluctuat:ions in the clipole force; whidi can be understood from the fact that for a two-level atorn in a, red-tlctnncd FORT, t,lic excited state sees a repulsive force. Repeated scatter- irig cycles through the excited state ter~d t,o heat the atorn due to the fact that t,he net force it feels fluctuates 011 this scattering tirrie-scale betmen at,tractive and r e piilsive. As indicated in Ey. (7.16), 1~1th of these forces are spatially dependent because 17s,.,t, (r) ciI (r)
,
with the spont,aneous heating maximal at FORT field antill- odes (wllero the light, is nraxin~i~l), and the dipole force flucti~ations maximal a t FORT field nodes. Cotier~-7knr1ontlji Iias sltcwn l i G 1 j that the slim of these t ~ v o heating rates is acti~ally spatially ir~depertder~t in a standing-wave FORT, and that the rr~.cl~-I.rnnltrnvalue is the same for each of t,hese contribirtiorrs.
The co~nbined cffcct of tliese kieatirig rates on the atomic energy c a i be qi~ickly estimated as follo~vs. E d i scattering process will heat the atom by approximately 2TR = 200 riK (renierriber: kuT13 = tL'k2/2rn), so that the rat,e of clitarige of atornic energy slioi~ltl obey
E= 2 i l - ~ T l i l ~ ~ C ~ ~ t t
.
(7.30)177
The atom will leave the Trap ~vllen E
-
in timeA& O I ~ T * tin
Ti,< n t
-
2 i " ~ T ~ r , , , i i 2kHTR17 'where the last cquivalericy comes from the general relatiori
-
Ai;ollT(ria)
fora deturii~ig A of the FORT field frorri the excited state of widt,h
T,
aitl shows the heating time is generally iildepender~t of FORT depth. The linear heatirig rate only xvorks in the limit of hAFolri.>>
k B T ; which is satisfied for the 2.5 mK deep FORT versus t,he initial atorriic tcrriperature of about 30 pK. In t,he case of our 868 nm FORT: t,he result is T , , ~ , ,-
50 S, which is-
200 times the observed ~ ~ 0 n . r and cannot explain tlie observations.r 7
lliere is yet another heating mechanism which arises frorll fiuctnations in the FORT poter~tial. Two possible causes are xrariations in the spatial position of tile pot,ential well or time-dependent fluctnations in t,he depth of the potential. Even t,hough the cavity length is servoed, tliere is no mecharlisrn to deterrriinc (and stabilize) the overall position of the cavity with respect to a fixed point in space. However, the frequency coritent of this rrlot,ion is expected to be mid- to sub-acoustic due to passive vibration isolation, so tile former possible root cause of heatirig is all but ruled out.
I-lowever, according t,o Ecl. (7.16), Ai:oryr cc I (r) so that, laser illtensity noise can drive meal-square force fluctuations because the atom will see a timodependent poteritial.
In a harrrlonic trap; this mmm sq~iarc force is proportiollal to tlle energy. which means the heating rate will be proportiorla1 to the energy. Furthermore, x2 perturbations will drive transitions hetween vibrational levels In,) and In i 2 j , which makes tlie spectral density of Auctuatior~s at twice the trap vihrat,ional frequency 21). the irrlportant, quantity. These ideas are discilssecl irl Ref. [I651 arid can be surnrnarized by ttie followir~g t%vo ecjuations for the llciitir~g rate E and tirlic coristar~t T , respectively;
178
Here v, is the trap oscillation frequerlcy (in Hz) and S,(2v::i"') is one sided power spectral density (PSD) of fmctionoi intensity noise evaluated at frequency 2vtr (in i~nits of 1 i f P ~ ) .
It sllonlil be noted tliat Eqs. (7.32) (lo rlot lleccssariiy faithfully describe the details of the heating process [IGGj. The fluctuations of an initial dist~ribiitiori of t,rap levels {in)) cail grow very rapidly due t,o the in,) -i in 1 2 ) healing process, mlch that the upper part of the distribution (and riot the mean) is responsible for heat,ing out of the trap. Nevertheless, T, gives a reasonable estimate for the timescale of this process.
For example. using Eq. 7.21, a rnodest FORT well depth of 1 rrlK ( S F o ~ < ~ =
25 XHz) will correspond to trap frequencies (v:?', vr1"') = (5,450) kHz. This makes it clear that intensit,y noise on the intracavity light out a t 900 kHz and beyond can have a devastating effect on the FORT. Direct measurements of the spectral density of photocurrent fluctuations for the FORT bearn emerging from the cavity (calibrated by coherent, AM at, tile requisite frequcr~cy 2v;P1 = 900 kHz) lead to (S,z(2vr""'): Se(2v;='"')) .=: (5x 10-" 22.3 10-'l)/IIz, so that (T:;"""',
rF1)
.=: (830,23)Ins. An exarn~tle of sl~cll a rne>rsnrenlent at 600 kHz (for a FORT depth of S F o n ~ = 20 MBL) is shourn in Fig. 7.32. Fig. 7.32(a) shou~s tr;ulsrnussion of resonant FORT light through the cavity, with n~:,l,,,,,, = 1.9 % coherent AM purposely put on the TUI diode laser using an AOM a t 600 kHz to calibrate the rloise power spectrum. In (b), it is clear this peak shovils up far above the light noise floor by a factor G = 45 dB in a B = 300 Hz rf haidwidtii, with the electronic noise floor visible to prove that the me;rsurernent was light-noise limit,ed. Using this, t,he PSD at, 2v;P' can be calcnlat,ed
which rrcults in T?"" = 224 rrLs for this ineasurement. This nuriiber niight be ex- pocteil to be sorncwhat intlt?penderrt of freqiency if a presunied singlctime-comtant 20 dBjdecai1e roll-off in S,(2vt,) cancels the u;v dependence in (7.32).
The heating rate for l / ~ p ' in bot,h of these rneasuresnents is in reasonable agree-
menr with tile observed ( l / e ) trap decay rate l / N 1/28 ~ Ins: leading to the ~ ~ ~ ~ conclusion that fiuctuations in iritracavity intensity drive heating along the cavit,y axis and are tlse limiting factor in t,llis .cvorlc. Slicli flilctuatio~ls are exacerbated by tlle corlversioll of FA1 to A% noise of the FORT lascr due to the high cavity finesse a t the ~va~relength of the FORT (here. Fl:oiyr = 3.5 x lo5). This conversion process, along with recerit attenipts t,o eliminate it: are the snbject of cllrrellt intensive work and will be covered in Sec. 8.1
1.9 % A M @ 600 kHzfor nominal 25 MHz FORT depth
'- 0.15
g
0.40time Ips]
expected FORT decay time = 24 ms
,f:
FORT on with 1.9% AM @SO0 kHz,
3
/ '!
peak is approximate 44 dB above-80
,
\, TUI noise @ 600 kHz for a 300 Hzti 1 \ if bandwidth
"
-120--k 1 I I I
590 595 600 605 610
frequency [kHz]
Figure 7.32: The po~x~cr spectral deiisit,y (PSD) of fractioxsal irltensity fluctnatior~s
('v;:xi"')
e ,- was met~ureti by correlt~tirlg the time doirlain out,put (a) of t,lie cavity witti its freclilency spectrnni (b) after erisuring the measurement was lir~litecl by ligllt, noise. The 1.9% coiitaent a~nplitude rriodulat~ion was nseci to calibrate the noise level.
180