CM QUANTIZATION IN CAVITY QED
Chapter 5 Chapter 5 Well-Dressed States for Wavepacket Dynamics in Cavity QED
5.4 Structure
45
fields outside tlie realrrl of quantized opt,ical fields in which hot,li internal arid external (c.m.) atomic degrees of frcedorn rriust be quant,ized in the presence of resorlant or quasi-resonarit excitation. Optical molasses [107, 108, 109, 110, ill], trapped ions j112, 113, 1141 atorn lasers
ills;
1161 and liovel atomic cooling and trapping schenies [lli, 118, 1191 using classical light fields and their associated forces have all been treated in t,his way. Often, either the Ran~an-Nath approximation (for atom?) or the 12arnhDicke lirrlit (for iolls) is invoked, which tends to tieelllpllasize either the extent- of the atomic wavepacket with respect to the cjitantized field or the motion of ille atom over its internal state lifetime. These are things that are irlcorporated here, contingent upon t,hc relative rilagnitudes of the typical energies in the problem, as will be developed further in Sec. 5.4.46
with E,,:* = &,,hi and with the converrtioil that g ( r )
>
0 (g ( r ) < 0) corrcsporlds t,o a decrease (increase) in energy for ID,,..) (ID,,,)). For tire time beingl tiissipation is neglected.Itather thari specify a p;crticuiar form for g ( r ) , ttie generic behavior of g ( r ) is studied around a field ext~rerrnrln zo (in one tlimension for simplicity) where y,,
-
y (zo)allti [ charact,erizes the magnit,utie of the curvature of g ( z o ) , with 7 its sign. In a harmonic approximation, the boirnti st,ates
{ I 9 p ) )
associated witli the dressed st,ate ID,,_) (in the case of a single field rrinximunl with r/ < 0) have an energy spectrum E p = { p + + ) lib',,, ( p = 0 ; 1 , 2 , . . . ), whereIt is mrthwhile to note fronr this eqiration that both the potential well depth and t,he recoil energy set the energy scale for the vibrational spectrum. In general, t,he reyuirerrient for bound st,ates is that
a
< 1, withNote that R1 is analogous t,o tlre paranreter R iri [98].
As the irrtcrition here is to idcritify generic features brought the c.m. s t a t e depemicnce to the usual dressed stat,e structure, tile general solutiori of Eq. (5.1) with arbitrary functioiial forms for y ( r ) and
K:,,
(r) will not be considered. Instc~cd, witli the straightforra:ard cidinitior~s and ol~servatior~ above, the bound-state structure of H in several lirniting cases of experimental relevance car1 be understood. The three energy scales nei:essary for dcterrrrinirrg the appropriate c.m. eigenbisis are the couplirig eriergy fig,& t l ~ e energy split,tirrg for the bountl statcs{I$,))
of I!.,,a i d the eni:rgy splittirig !iVrL tjssocixt,ed with tlie bolmd states
{/@,:!)
of lig (r).47
5.4.2 The Three Different Regimes
Perhaps the simplest case to consider first is that for which the external potcrit,ial is dominar~t, narrlely AE, >> hg,,,v%. In this regime, the well-dressed state stnlct~rre is solvet1 using { / l j q ) ) as a fixed basis for the c.m. ancl fig ( r ) H I P is viewed as a perturbation. The eigensolutioris of H are
{/+,)
@ lL),,,;i), E,
i lifig,,)i where the matrix eleir~ent g,, is defined byThe finite size of the wavepacket inherent in g,, distinguishes this case fro111 much of the previous work in cavity QED. It can cert,airily be the case that (g (r)),
#
g ((r)J for the bourirl states of
i:..,;
which w a tile intended "punch-line1' of Figs.5.l(a,b). One sees that this condition arises when the at,omic c.m. wavefunction has spatial struciure on a scale corrnnensurate with t,hat of the quantum field to which it is couplod. Variations in atomic wavepac:ket probability density strongly influence t,he coupled atonl-field evohltion.
More generally, the c.m. dependent Jaynes-Curnmings lttdder (i.e., the well- dressed states) for tliis regime is illustrated in Fig. 5.2(a). Xot,e that ''Rabi Aop ping" proceeds at the c.m. state dependent rat,e 2&gqqi which cari be conipleceiy suppressetl (y,, -+ 0) due to thc spatial structnre of y [r) with respect to $, (r).
Spectroscopic investigation of tire stn~cture %vonltl, in general, involve c.111. st,at,e dependent transitions within each manifold.
A secontl regilr~e interchanges the roles of (r) and h,q ( r ) ; such t,hat Fig,,fi =->
AE, >> hVT,. Here, J: is assumed suificiently large so that the coupling g (r)
-
goacts siniply as a global shift to {E,). In this rcgirne, the well-dressed states of
H
are 3iU,,:i))
, (?&figi, J- E,)) as illiistrated in Fig. 5.2(b). The large curva1;ure J: in the cavity field allows tlle atomic c.m. to relilairl couplet1 predolilinarlt,ly to tile boiind states of 'l<!,, ( r j . The interaction energy associated wirh the internal degrees of freedom appear nlore or less as in the standard Jaynes-Cummings ladder, modified now by the fine si.ructure associateci with the atomic c.rn. In fact tliere are48
additionally small shifts g,, of each eigeristate due to the spatial dependence of g ( r ) . These shifts are assi~mcd to be of higher order arid are not slio~vn. Note that the separate lirrlitv of Figs. 5.2(a,b) begin to coriverge as the hound state spacings of g (r) and
v?,,
( r ) approach one another, openirig the way for more complicaied structure (as for example with iricreasirig n ) m d dyr~an~ics than will be discussed llei-e.Finally, a third reginie takes fig,,L/;;
>
fiV,>
AE,. In fact, ( r ) is consid- ered only a Incans of providing a ~vell-clefined initial state for. an atomic wavepackei, after which its effect is asslirriecl iicgligible compared t,o the cavity field. For a given n-manifold, the Jaynes-Cummings ladder in t,l~e drt;ssetl-state picture is not split by i h g . Instead, the well-dressed st,at,es ( / # p ) } are associated with the minima of the respr:ctive poteritials V= (r) = itiv'?ig (r) forjL>,,+)
as in Fig. 5.2(c) for the bound states of 1/_, with t,he repi~lsive barrier seen by !I),,+) at that spatial location omitted.This is a consequence of the fact that the dressed state ID,,-) is at,tracted towards the regions of negative curvature in ,9 (r) 111 .c 01 while jD,,+) is repulsed: and conversely for rl
>
0. This structure is heavily dependent on xvell depth and changes wit,h the inanifold level n . Nevertheless; it must be enlphasized that. in the h x ~ n o u i c approx- imation (and indeed for Illore general poter~tials as well) the level spacing within a given :nn-manifold is given by fiV, and scales as ( n ) : . Xote that ttlis same fact,or 1plays a cerrtral role in the work of [99j, for scattering of a cold wavepacket from a potential formed by the c;~vit,y field. In t,hat, casee:
E
is set by the lengti~ L of the well and successive scat,tering resoiiarices in t,he low energy liniit, for increasing yoL are associated wit11 irlcremer~ts of tile round trip phase in units of 271; wlnicli is precisely tlie conditiorl for the addition of another botmd state in the a&soci,?ted square well poteritial of dept,li go. Implicit in that, analysis is she use of an external riiechanism, sucli as the poteritial\i:,t
(r); to produce the initially cold wavepacket.111 t e r n s of a piiysical implemienitatiori, a,n example relevarit to the t,wo regimes of Fig. 5.2(a,b) is a trapped ion interacting with a cavity rrlode. In the rnicro\vavt:
clomaiii j103I1 typical cavities have
-
20 kHz, while tlict vibratiorial frequencies for an RF Paiil trap acting as l<:,t ( r ) are-
1 MHz and5 <
:,,,.,A, so thatthe coniiit~ion~ for Fig. 5.2(aj are satisfied. By cor~trast: in the optical domain [9: 6.11
-
10 MHz : as is appropriat,e to Fig. 5.2(b), wliere for a heavy atom, 492
200 kkHz2r
a6
.
5 -i;'
1" an optical standing wave.For the regirrle of Fig. 5.2(c); orie can consider a light atom such as He*? initially prepred in a c.111. eigcnst,ate of I/;,, (r); wi-1icl1 coilld be switched on in the for111 of a dipoleforce trap as in Ref. [I041 to provide this initial set of ~vell-defined cigenstates with g ( r ) = 0. The atom is thc11 alio\c7ed to int,eract with a single mode of an optical cavity. This requires a t,ransit,ion to ( r ) = 0 with g ( r )