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Phase-dependent velocity selective coherent population trapping in a folded three-level ([and ]) system under standing wave excitation
optics Communications 103 ( 1993 ) 453-460 North-Holland
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F u l l length article
Phase-dependent velocity selective coherent population trapping in a folded three-level ( A ) system under standing wave excitation M.S. Shahriar a b c a
a, P.R. H e m m e r b, M . G . Prentiss c, A. C h u c, D . P . K a t z a a n d N.P. B i g e l o w e
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Rome Laboratory/EROP, Hanscom AFB, MA O1731 USA Department of Physics, Harvard University, Cambridge, MA 02138, USA Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA Department of Physics and Astronomy and the Laboratory for Laser Energetic& University of Rochester, Rochester, N Y 14627, USA
Received 23 February 1993; revised manuscript received 17 June 1993
Theoretically identified are the conditions under which velocity selective coherent population trapping (VSCPT) takes place when a three-level A system is excited by a pair of Raman resonant standing waves. The efficiency of the VSCPT depends on the relative phase, O, between the standing waves, and vanishes when ~ = 0. Also reviewed are previous experimental observations by Aspect et al. [Phys. Rev. Lett. 61 ( 1988 ) 826 ] which qualitatively validate the predictions. Finally, the implication of this result to the feasibility of a dense trap of sub-recoil temperature A atoms is discussed.
1. Introduction Recently, there has been a great deal o f interest in the forces experienced by a folded three-level ( A ) atom. Aspect et al. [ 1 ] first d e m o n s t r a t e d that such an a t o m can be cooled below the recoil limit via velocity selective coherent p o p u l a t i o n t r a p p i n g ( V S C P T ) , when excited by a pair o f counter-propagating travelling waves. In this paper, we show that V S C P T occurs in a A a t o m excited by a pair o f R a m a n resonant standing waves, and its efficiency d e p e n d s on the relative phase ~l, ¢, between the standing waves. We will also review previous experimental observations m a d e by Aspect et al. [ 1 ] in R a m a n resonant standing waves which are consistent with our theoretical predictions. Finally, we briefly describe the generalization of V S C P T in standing waves to three d i m e n s i o n s [ 3 ]. The interest in V S C P T in R a m a n resonant standing waves stems from our earlier experimental [4] observation o f deflection and cooling o f A sodium a t o m s in an atomic beam. This experiment suggests that these standing wave forces could be used to design a stimulated force trap [4]. In addition, we have developed a theory [3 ] which predicts that the cooling can be m a d e to have characteristics very similar to those o f conventional polarization gradient ( p o l - g r a d ) cooling, so that such a trap should have a sub-Doppler temperature. The results derived in the present p a p e r suggest that it m a y be possible to reach a sub-recoil t e m p e r a t u r e in a R a m a n force trap. The existence o f V S C P T corresponds to absence o f diffusion for zero velocity a t o m s in the t r a p p e d state. However, Chang et al. [5] have previously c o m p u t e d a non-zero diffusion coefficient in R a m a n resonant standing waves by a p e r t u r b a t i v e numerical solution o f the Wigner density matrix equation o f motion. O u r disagreement with Chang et al. seems to stem from the fact that they assumed, a priori, that the distribution o f a t o m s in m o m e n t u m space is smooth, which is in sharp contrast with the result o f VSCPT. at Mauri et al. [2 ] also found a process of VSCPT with a phase-dependent efficiency. 0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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2. Theory of standing wave V S C P T In general, it is rather difficult to find an ideal /k system. One example is when a j = l~-,j' = 1 transition is excited by a pair of left- and right-circularly polarized waves, which turns into a A system after a few optical p u m p i n g cycles [ 1 ]. This is illustrated in fig. la, for the case where each circular polarization forms a standing wave. The lines j o i n i n g the magnetic sublevels indicate the allowed electric-dipole transitions. As can be seen, the m=O~m' = 0 transition is not allowed. In addition, the fields do not excite any z transitions (the dashed lines). As a result, after a few spontaneous lifetimes, the a t o m s originally in the m = O sublevel get completely depleted, and we get a perfect A system, as indicated by the thick lines. In the coming analysis, we will use this j = l ~ j ' = 1 system. We emphasize, however, that the analysis is valid for any ideal A system, independent o f the polarizations o f the excitation fields. The three atomic states ]a>, ]b>, and ]e> in fig. l b correspond respectively to the i n = - 1, m = + 1, and m' = 0 states. The electric field is expressed as E = ½El (a) [exp( - i ~ o , t) +c.c. ] +
½E2(z) [exp( - i ( 0 2 t ) +c.c. ] ,
( 1)
where
El(z)=6-+Emcos(kiz),
E2(z)=6-_E2ocos(k2z+O),
(2)
z is the position o f the center o f mass o f the atom, ~ is the phase difference between the standing waves, and 6-+ and 6 - are the polarization vectors, corresponding respectively to right- and left-circular polarizations. We define the Rabi frequencies as gl (Z)=--,Uae " E l / f l ~ g , o
COS(k, Z) ,
g 2 ( Z ) =,/Abe
"Ez/h=-g2o C O S ( k 2 2 + q ) )
(3)
.
Ignoring the anti-resonant terms, and making the rotating-wave transformation, we get the h a m i l t o n i a n for the system:
-~ H=7
-g,(z)
-g~(=)/
-g2(z)
-2~
]
=(h/2)[A(ta>
.
(4)
Here, the R a b i frequencies are real, and the c o m m o n detuning c~and the difference detuning A are as defined in fig. lb. In what follows, we will consider p r i m a r i l y the case where A = 0 . We also assume that states la> and ]b > are degenerate, so that 0), = o)2 = o), 6, = ~ 2 - 6, and k, = k 2 - k. In addition, for simplicity we consider only the case ofglo=g2o-4g. So far, we have assumed that z, the position of the center o f mass of the atom, can be treated as a classical
ZZ ZZZ~A
J/= i m'=-i
@
I
I
o-'! / \
/\
I I
m=-I
@ d=l
(a) 454
81 . . . .
l
I I
1
la>
(b)
Ib>
Fig. 1. (a) A J = l ~ J ' = 1 transition, excited by a pair of circularly polarized standing waves. (b) The resulting ideal A system: is the average detuning, A is the differential detuning, and F is the excited state decay rate.
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variable. However, we want to allow for the possibility that the atoms become cold enough that the de Broglie wavelengths become comparable to or bigger than the optical wavelength [6]. In this regime, it is necessary to treat z as a quantum mechanical operator, ~, with the atom's linear momentum in the z-direction,/~, being the canonical conjugate. Denoting by IP) the eigenstate of/~ (i.e.,/~IP) = P I P ) ), the basis states can now be written as la, p a ) = - l a ) ® l P a ) ,
Ib, P b ) - l b ) ® l P b ) ,
le, p c ) - - l e ) ® l p ~ ) ,
(5)
where the linear m o m e n t u m is a continuous variable: - o e < p ~ < o e , ( i = a , b, e). In order to find the hamiltonian in this basis, we note first that +oo
exp(+ik:?)=
j dplp)(p~hk[,
(6)
--oo
which simply means that a travelling wave optical field with a wavenumber k couples a state with a momentum
p-hk to a state with a momentum p, conserving linear momentum. Given eq. (6), and noting that the kinetic energy of a state IP) is p2/2M, where M is the mass of the atom, we get the following form for the hamiltonian: H= p a , P~b , p e L2Mla, Pa)(a, Pal+-~-~lb, pb)(b, Pbl+ \2M -hg j dp ( I P ) (p+hkl + IP) (p-hkl) ( [ a ) (el +h.c.) --oo
-Foo
-hg J dp [ I P ) (p+hklexp( - i 0 ) + IP) (p-hklexp( + i ~ ) ] ( I b ) (el + h . c . ) .
(7)
-oo
To illustrate this, consider some representative matrix elements. For example, the energy of the state l a, Pa) is given by (a, pa}Hla, Pa) =PZ/2M, while the coupling between states la, Pa) and le, Pe) is given by
(a,p, JHle, pc)=-hg[~.po_hk+~p~,po+~k], ~ , j = 0 i f i ~ j a n d l i f i = j .
(8)
This hamiltonian is illustrated in fig. 2 for a few of the infinite number of basis states. As can be seen, each of the ground states is connected to two excited states. Therefore, it is not possible to find closed families of states, as is possible for two travelling waves. Nevertheless, we will be able to find a velocity selective dark state and calculate the VSCPT rate into this state (as a function of ~) both perturbatively and by numerically integrating the optical Bloch equations [6]. In analogy to the travelling wave VSCPT, we are interested in finding a state (the dark state) which satisfies the following conditions: (i) it does not contain any excited state, so that it is completely decoupled from the vacuum fields, and (ii) the net amplitude for coupling this state to any of the excited states must vanish. It can be shown that if p = 0 a n d / o r ~ = 0 then the state [NC(p))
=½[ la, p - h k ) e x p ( - i ~ ) + la, p+hk) e x p ( i 0 ) - I b , p - h k ) - Ib, p+hk) ]
(9)
does not couple to any excited state. This is illustrated in fig. 3 for two cases: p = 0 (levels denoted by solid lines) and p ~ 0 (levels denoted by dashed lines). In either case, m o m e n t u m conservation allows for only three excited states to be coupled by the hamiltonian directly to INC ( p ) ) . We will show that the coupling to each of these states vanishes (for p = 0 a n d / o r 0 = 0 ) by showing that if the system is in the state I N C ) at t = 0 , then it remains in INC ) after an infinitesimal amount of time, dt. We introduce the following notation for the seven levels of fig. 3: 455
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le,-aT~k >
(~_2nv%l
le, O>
le,ahk >
le,p+ahk>
le,p>
(3
e
I
e,p-~'hk >
//I
J aav 1 ~b,p-hk >
Ia,p-hk >
• "q*la,-hk>
p-hk)
,
15)-Ie,
p-2hk)
13)=-Ib, p-hk)
12)-=la, p+hk),
,
[6)-Ie.,p),
17)-Ie,
÷~la,hk >
-Ib,]hk >
Fig. 3. Illustration of the dark state, INC(0) ) (levels denoted by solid lines) and the state INC(p)) (levels denoted by dashed lines). Here, q_=exp(i¢), 0=-exp(i4g2d), and ~=-exp(i-Qvt). The energy zero is chosen to be (p2)/(2M). Note that this causes overlap of the states le, p> and le, 0). The symbols in the transition lines represent the effects of the standing wave phase shift, whereas those next to the solid ( p = 0 ) levels are associated with the phases produced by the motional energy shifts. The symbols next to the dashed (p # 0) levels are the additional phases associated with the center-of-mass momentum, p. Finally, the weights for the state INC ) are shown to the ground state labels.
Fig. 2. Illustration of the energy levels of and couplings between a typical set of momentum states in a A system excited by standing wave fields. Here, we have chosen ~= -t2~.
II)-Ia,
-Ib,-Xk>
p+2hk)
,
14)=-Ib, p + h k )
, (10)
.
T h e w a v e f u n c t i o n ~2 c a n b e w r i t t e n as Iq/)=
~Ai[i),
(11)
i = 1 , 2 ..... 7 .
i
T h e r e d u c e d h a m i l t o n i a n f o r t h i s s u b s e t o f s t a t e s is g i v e n , f r o m eq. ( 7 ) b y 0
H=h
_
0
0
g
g
0
0
12v
0
0
0
g
g
0
0 g g
0 0 0 g
- g2v 0 grl grl*
0 .(2, 0 g~l
grl* 0 - 2Y2,, + 4Y2r 0
gq grl* 0 0
0 grl 0 0
0
g
0
gr/*
0
0
-
(12)
2Y2~+ 41"2r _
H e r e , d e n o t i n g b y v t h e v e l o c i t y c o r r e s p o n d i n g to t h e m o m e n t u m p, we d e f i n e t h e D o p p l e r shift, Y 2 v - P k / m = kv, t h e recoil energy, £2r = h k 2 / 2 A L a n d q-= e x p ( i 0 ) . N o t e t h a t f o r s i m p l i c i t y , we h a v e c h o s e n f i = -Y2r, a n d h a v e shifted the zero of energy to coincide with the energy of state 16). ~2 The use of wave function is exactly valid for the case where [NC) is a dark state, which implies that As=A6=A7=O for t~>0. When INC } is not a dark state, the wave function approach is still valid as long as dt << 1/1-'. 456
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Note now that the dark state, INC ( p ) ) , is a linear combination of states I 1 ) through 14). We define IC~ ( p ) ) , [C2(p)) and IC3(p)) as three other mutually orthogonal states such that IC, ) [ = 1 r/* IC2)/ 2 / q* IC3)d L-,•*
q r/
1 1 1
12) 13) 145
1
(13)
The wavefunction can now be written as
I~')=ANcINC)+Ac, ICI)+Ac2ICz)+Ac3 C3) +As 15) +A616) +A7 [7) .
(14)
Using eqs. (12) and (13), we get the hamiltonian for this basis:
-0 0 0 t2v 0 0
H=h
_ 0
0 0 --Qv 0 grl 0
0 -g2v 0 0 gq 2Cg
Ov 0 0 0 0 -i2Sg
-gr/*
g~/*
0
0 gq* gq* 0
0 0 2Cg i2Sg
0 -grl g~l 0
2(2~r -£2v)
0
0
-
0
0
0
0
0
2(2£2r+f2v)_
(15)
Here, we have defined S -= sin ~ and C - c o s ~ so that q= C + iS. For p = 0 , so that .Qv=0, it is clear from eq. (15) that the state ]NC(0) ) is completely decoupled from any state. Therefore, INC ( 0 ) ) is a pure dark state. For p :~ 0, consider first the case of ~ # m z, where n is an integer. Note that INC ( p ) ) is still not coupled directly to any of the excited states. However, it is coupled to IC3 ( p ) ) , which in turn is coupled to one of the excited states, 16 ( p ) ) . Thus, for ~ ¢ mr, atoms with non-zero velocity will be optically pumped into INC ( 0 ) ) , which is the zero velocity dark state. This is of course velocity selective coherent population trapping (VSCPT). Consider next the case of ~=nn, with p:~0. In this case, S = s i n ~ = 0 , so that I N C ( p ) ) is coupled only to IC3 ( p ) ) , which in turn is coupled only to INC ( p ) ) . Thus, both ]NC ( p ) ) and IC3 ( p ) ) are pure trapped states, independent of the value of p, as long as
A6(t)= i dt' i dt"A'6(t"). 0
(16)
o
In the perturbative limit, where at t = 0 all the atoms are in the state I N C ) , the integrand can be found from applying the Schroedinger equation twice:
]t'6( t" ) = ( - i / h )2 ( NC IHIC3 ) (C3 IHI6 ) ANc( t" ) .
(17)
Since the hamiltonian is independent of time, and ANc(t" ) ~ 1 is a constant in the perturbative limit, we find that the amplitude of the state 16) after a differential amount of time, dt, is given by rim6(P)
=
(dt)Z ( - i / h ) 2 ( N C IHI C3 ) (C3 IHI 6) = - 2i(dt)2g2vg sin 0 = - 2 i ( d t ) 2 ( p / m ) k g s i n O.
(18)
Thus, the net rate at which atoms starting in ] N C ( p ) ) is coupled into the excited state is proportional to (p sin O) z. The formal approach used in deriving eq. (18) is presented to clarify the physics of VSCPT in standing 457
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waves, as well as to d e m o n s t r a t e the existence o f additional dark states when ~ = 0. However, it is possible to understand the behavior in eq. ( 18 ) in a relatively simple way by inspection o f fig. 3. If p C 0 (the dashed levels in the figure), each level has an a d d i t i o n a l energy due to the kinetic energy associated with the m o m e n t u m . In the figure, the energy p2/2M is chosen as the energy zero so that the states le, 0 ) and le, p ) are superimposed. If we consider the sum o f the interaction matrix elements coupling the ground states to each o f the excited states, we see that the couplings to the levels le, p - 2 h k ) and le, p+2hk) still vanish but the coupling to the level le, p ) does not. In particular, assume that the a t o m s are in I N C ) at t = 0 , as given by eq. (9). The problem now becomes one o f weighted s u m m a t i o n of the four transition paths to state [e, p ) from the ground states, as shown in fig. 3. Using the relative ground state weights (shown in fig. 3) and the relative transition matrix elements we find that (e, p l N C ( p ) )
~ [q*~+q~*-q~-~l*~*] ~sin(12vt) s i n 0 .
(19)
Note that since q = e x p ( i 0 ) and ~=exp(i£2vt), eq. (19) has the same functional dependence on ~ and £2v as eq. (18), in the limit of small t. Thus, the expression for the efficiency of V S C P T can be derived essentially by inspection o f fig. 3. According to eq. ( 19 ), we expect that the efficiency o f V S C P T would vary roughly as sin20, being m a x i m u m at 0 = ( 2 n + 1 )7r/2 and vanishing at O=nn. This phase d e p e n d e n c e of standing wave VSCPT is also consistent with the semi-classical picture. Consider the basis states I - ) = c o s 0 [ a ) - s i n 0 1 b ) , I+)-sin01a)+ c o s 0 1 b ) , and [ e ) where O(vt)-=tan-[gt(t,t)/g2(vt)]. For l ' = 0 , g~ and g2 are constants, and the 1 - ) state is the dark state, decoupled from I + ) and l e ) . F o r v¢O, I - ) is coupled to I + ) at the rate of vV0~ kv sin 0. Note that the phase dependence o f this coupling rate is also the same as the phase dependence of VSCPT (eqs. (18) and (19) ) since 12v= kv. In particular, for 0 = 0 , g~ =g2 at all positions and therefore O(vt) is a constant, so that this coupling vanishes, and I - ) is a dark state for each velocity. It is also possible to estimate the relative efficiency o f standing wave VSCPT c o m p a r e d to that o f travelling wave VSCPT. C o n s i d e r the case of 0 = (2n + 1 )~z/2. In this case, IN C ( p ) ) is coupled to IC3 ( p ) ) at the rate o f g2v. In turn, states I C~ ( p ) ) , I C 2 ( p ) ) , IC3 ( p ) ) are each coupled to the excited state (through one or more channels) at the rate o f 2g, as given by eq. ( 1 5). If we analyze the travelling wave case in the same way, we find that the coupling rates are the same, except that there are only three states, ( l a , p - h k ) - I b , p + h k ) ), ( l a, p - h k ) + [b, p + hk) and l e, p ) , which have properties analogous to IN C ) , I C3), and 16), respectively. Thus, for ~ = (2n + 1 ) ~ / 2 , we expect the standing wave V S C P T efficiency to be o f the same order o f magnitude as the travelling wave V S C P T efficiency. The V S C P T rate d e r i v e d above (eq. ( 1 8 ) ) is in agreement with numerical simulations o b t a i n e d by integrating the optical Bloch equations for the A system u n d e r standing wave excitation. In particular, we calculated the time evolution o f the m o m e n t u m d i s t r i b u t i o n for different values o f ~. Even though there are no closed families o f states in the SW system, an excited state with m o m e n t u m p is still coupled only to ground states with m o m e n t a p + hk which in turn couple to excited states with m o m e n t a p ++_ 2hk, etc. Thus, there are still families o f states which can be closed by truncation. F o r our calculation, we choose a m o m e n t u m range o f + 5hk. Figure 4 shows the initial ( t o p trace) and final m o m e n t u m distributions at t = 5 0 F - ~ for helium excited by standing waves o f the phase indicated ( m i d d l e traces), and also for travelling wave excitation ( b o t t o m trace), included for comparison. We see that for 0 = n/2 V S C P T does occur and is about half as efficient as for travelling waves. F o r other ~, we see the predicted sin20 dependence. In particular, at 0 = 0 there is no peaking of the m o m e n t u m distribution ~3.
~3 In fact, we see some very slight alteration in the momentum distribution at ¢=0 due to optical pumping into the relevant dark states. However, after a short time (< 50 F -l ) the momentum redistribution stops, in contrast to the ¢4:0 cases where the + hk peaks continue to grow and sharpen. 458
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INIT
A
v
Z
o
17
~z
0_ c~ Q-
i
-1
I
J
0
1
MOMENTUM
(l~k)
Fig. 4. Numerical simulation of VSCPT in the A system with standing waves after t = 50F- ' for He, where gao= g2o= 0.3F and b = 3 = 0. INIT is the initial momentum distribution: a gaussian distribution in the ground states with a standard half width at e - 1 / 2 of 3hk. The middle 4 traces are for standing wave excitation with the phases ~ as indicated. TW is travelling wave case with Rabi frequencies g~+ =g~_ =0.3/". In all cases, momentum is in units of ilk and population is normalized to 1.
3. Previous experimental evidence of standing wave V S C P T As m e n t i o n e d earlier, b o t h the existence a n d the a p p r o x i m a t e phase d e p e n d e n c e o f one d i m e n s i o n a l V S C P T in R a m a n resonant standing waves have already been observed by Aspect et al. [ 1 ], for the j = l ~ j ' = 1 transition in metastable helium. However, since these results were not explicitly presented in the context o f standing wave phase shifts, we will now review these observations for completeness. In the first part o f the Aspect experiment, V S C P T was observed as a pair o f peaks in the m o m e n t u m distribution when the He* b e a m was excited with a pair o f counterpropagating laser b e a m s having opposite circular polarizations. F o r the A system in the j = 1~--~j'= 1 transition this is o f course equivalent to a pair o f counter-propagating pure travelling waves. In the second part o f the experiment, the He* b e a m was excited by counter-propagating linearly polarized laser beams where the polarization axes o f the forward and backward propagating laser beams had a relative angle o f X. Specifically, Z = 0 and Z = ~z/2 correspond respectively to the so-called l i n - p a r a l l e l - l i n and l i n - p e r p - l i n configuration. It is easy to show that for the j = 1~ j ' = 1 transition these experimental configurations are equivalent to a pair o f opposite-circularly polarized R a m a n resonant standing waves, having a phase shift 0 = Z . As reported, no V S C P T was observed when X - 0 , but the double peaked m o m e n t u m distribution reappeared (as evidence o f V S C P T ) when Z = 7r/2. This is in agreement with the present theory. In addition, the relative peak hights in the Aspect e x p e r i m e n t are consistent (within experimental noise) with our theoretical prediction o f relative efficiencies in fig. 4. Here, it m u s t be acknowledged that for the specific case o f the j = l*-~j' = 1 transition, Kaiser et al. [ 7] correctly explained the presence o f V S C P T for the l i n - p e r p - l i n case a n d the absence t h e r e o f for the l i n - p a r a l l e l lin case by considering transitions between superpositions o f the magnetic sublevels forming closed families o f states for this system. However, we emphasize that the theory presented in this p a p e r is not l i m i t e d to the j = l ~ j ' = 1 transition, a n d is valid for any pair o f R a m a n resonant standing waves exciting a A system. In addition, our analysis gives the detailed phase d e p e n d e n c e o f V S C P T efficiency. This is i m p o r t a n t in the context o f a s t i m u l a t e d R a m a n trap, since b o t h the deflecting force a n d the pol-grad cooling force are m a x i m u m at ¢ = 7 r / 4 a n d vanish [3] at Z = 0 a n d X=~/2.
4. Generalization to three dimensions Finally, we have found [ 3 ] that this process can be generalized to three dimensions. Briefly, the scheme again 459
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employs a j = l ~ j ' = 1 transition, except with three mutually orthogonal pairs of Raman resonant standing waves having direction vectors 2, j) and :~. This is accomplished using left-circularly polarized and right-circularly polarized standing waves, with phase differences o f 0i, i = x , y, z. U n d e r this configuration, VSCPT takes place into a zero velocity three-dimensional dark state o f the form ~,=A+ [ , 7 : = + l ) +Ao Ira: = 0 ) +,4 ]m: = - 1 ), where ,4 +, Ao, and A are the standing wave field a m p l i t u d e s at each point in space, which comprise of summ a t i o n s over m o m e n t u m eigenstates. Note that the m a t h e m a t i c a l form o f this zero velocity three-dimensional dark state is similar to that found by Ol'shanii et al. [8 ], except that standing waves (rather than travelling waves) are used in the present case. To c o m p u t e the efficiency o f V S C P T in three dimensions, it is necessary to consider other states with non-zero net m o m e n t u m as well. Extrapolation o f the one d i m e n s i o n a l analysis suggests that this would yield a V S C P T efficiency as a c o m p l i c a t e d function o f ~x, Oy, and G, and the analysis have not yet been performed. N o t e that there is no phase d e p e n d e n c e in the travelling wave three-dimensional V S C P T considered by Ol'shanii et al. [8].
5. Summary We have shown theoretically that phase sensitive V S C P T takes place when a A system is excited by a pair o f R a m a n resonant standing waves. This corresponds to absence o f diffusion for zero velocity atoms in the t r a p p e d state. We also review previous experimental observations reported by Aspect et al. which agree with our predictions. Finally, we outline the generalization to three dimensions. Given our prior observation of stimulated trapping and cooling forces a n d our theoretical prediction of strong pol-grad cooling in R a m a n resonant standing waves, the simultaneous existence o f V S C P T may make it possible to design a subrecoil t e m p e r a t u r e trap for A atoms.
Acknowledgements We wish to acknowledge the helpful discussions with Prof. S. Ezekiel o f MIT, and A. Aspect and R. Kaiser o f Institut d ' O p t i q u e Th6orique et Appliqu6e. This work was s u p p o r t e d by R o m e L a b o r a t o r y Contract No. F19628-92-K-0013 a n d Office o f N a v a l Research G r a n t No. ONR-N0014-91-HJ-1808. N P B is an Alfred P. Sloan Research Fellow.
References [ 1] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste and C. Cohen-Tannoudji, Phys. Rev. Lett. 61 ( 1988 ) 826. [2] F. Mauri and E. Arimondo, Appl. Phys. B 54 (1992) 2408. [ 3 ] M.S. Shahriar, P.R. Hemmer, M.G. Prentiss, J. Mervis, D.P. Katz, T. Cai and N. Bigelow, Bulletin of the American Physical Society, Series II, Vol. 38, No. 3, p. 1122 (1993), presented in the 1993 Annual Meeting of the Division of Atomic, Molecular, and Optical Physics, Reno, NV, 1993; P. Hemmer, M. Shahriar, M. Prentiss, D. Katz, K. Berggren, J. Mervis and N. Bigelow, Bulletin of the American Physical Society, Series II, VO1. 37, No. 4, p. 1208 ( 1992 ) presented in the InterdisciplinaryLaser Science Conf., Albuquerque, NM, 1992. [ 4 ] P. Hemmer, M. Shahriar, M. Prentiss, D. Katz, K. Berggren, J. Mervis and N. Bigelow, Phys. Rev. Lett. 68 ( 1992 ) 3148. [ 5 ] S. Chang, B. Garraway and V. Minogin, Optics Comm. 77 (1990) 19. [ 6 ] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 (1989) 2112. [ 7 ] R. Kaiser, These de Doctorat, Ecole Normale Superieure, 1990 (unpublished). [ 8 ] M. Ol'shanii and V. Minogin, Proc. Int. Workshop on Light induced kinetic effects on atoms, ions, and molecules, Marciana Marina, Italy, eds. L. Moi et al. (ETS Editrice, Pisa, 1991 ) p. 99.
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F u l l length article
Phase-dependent velocity selective coherent population trapping in a folded three-level ( A ) system under standing wave excitation M.S. Shahriar a b c a
a, P.R. H e m m e r b, M . G . Prentiss c, A. C h u c, D . P . K a t z a a n d N.P. B i g e l o w e
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Rome Laboratory/EROP, Hanscom AFB, MA O1731 USA Department of Physics, Harvard University, Cambridge, MA 02138, USA Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA Department of Physics and Astronomy and the Laboratory for Laser Energetic& University of Rochester, Rochester, N Y 14627, USA
Received 23 February 1993; revised manuscript received 17 June 1993
Theoretically identified are the conditions under which velocity selective coherent population trapping (VSCPT) takes place when a three-level A system is excited by a pair of Raman resonant standing waves. The efficiency of the VSCPT depends on the relative phase, O, between the standing waves, and vanishes when ~ = 0. Also reviewed are previous experimental observations by Aspect et al. [Phys. Rev. Lett. 61 ( 1988 ) 826 ] which qualitatively validate the predictions. Finally, the implication of this result to the feasibility of a dense trap of sub-recoil temperature A atoms is discussed.
1. Introduction Recently, there has been a great deal o f interest in the forces experienced by a folded three-level ( A ) atom. Aspect et al. [ 1 ] first d e m o n s t r a t e d that such an a t o m can be cooled below the recoil limit via velocity selective coherent p o p u l a t i o n t r a p p i n g ( V S C P T ) , when excited by a pair o f counter-propagating travelling waves. In this paper, we show that V S C P T occurs in a A a t o m excited by a pair o f R a m a n resonant standing waves, and its efficiency d e p e n d s on the relative phase ~l, ¢, between the standing waves. We will also review previous experimental observations m a d e by Aspect et al. [ 1 ] in R a m a n resonant standing waves which are consistent with our theoretical predictions. Finally, we briefly describe the generalization of V S C P T in standing waves to three d i m e n s i o n s [ 3 ]. The interest in V S C P T in R a m a n resonant standing waves stems from our earlier experimental [4] observation o f deflection and cooling o f A sodium a t o m s in an atomic beam. This experiment suggests that these standing wave forces could be used to design a stimulated force trap [4]. In addition, we have developed a theory [3 ] which predicts that the cooling can be m a d e to have characteristics very similar to those o f conventional polarization gradient ( p o l - g r a d ) cooling, so that such a trap should have a sub-Doppler temperature. The results derived in the present p a p e r suggest that it m a y be possible to reach a sub-recoil t e m p e r a t u r e in a R a m a n force trap. The existence o f V S C P T corresponds to absence o f diffusion for zero velocity a t o m s in the t r a p p e d state. However, Chang et al. [5] have previously c o m p u t e d a non-zero diffusion coefficient in R a m a n resonant standing waves by a p e r t u r b a t i v e numerical solution o f the Wigner density matrix equation o f motion. O u r disagreement with Chang et al. seems to stem from the fact that they assumed, a priori, that the distribution o f a t o m s in m o m e n t u m space is smooth, which is in sharp contrast with the result o f VSCPT. at Mauri et al. [2 ] also found a process of VSCPT with a phase-dependent efficiency. 0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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2. Theory of standing wave V S C P T In general, it is rather difficult to find an ideal /k system. One example is when a j = l~-,j' = 1 transition is excited by a pair of left- and right-circularly polarized waves, which turns into a A system after a few optical p u m p i n g cycles [ 1 ]. This is illustrated in fig. la, for the case where each circular polarization forms a standing wave. The lines j o i n i n g the magnetic sublevels indicate the allowed electric-dipole transitions. As can be seen, the m=O~m' = 0 transition is not allowed. In addition, the fields do not excite any z transitions (the dashed lines). As a result, after a few spontaneous lifetimes, the a t o m s originally in the m = O sublevel get completely depleted, and we get a perfect A system, as indicated by the thick lines. In the coming analysis, we will use this j = l ~ j ' = 1 system. We emphasize, however, that the analysis is valid for any ideal A system, independent o f the polarizations o f the excitation fields. The three atomic states ]a>, ]b>, and ]e> in fig. l b correspond respectively to the i n = - 1, m = + 1, and m' = 0 states. The electric field is expressed as E = ½El (a) [exp( - i ~ o , t) +c.c. ] +
½E2(z) [exp( - i ( 0 2 t ) +c.c. ] ,
( 1)
where
El(z)=6-+Emcos(kiz),
E2(z)=6-_E2ocos(k2z+O),
(2)
z is the position o f the center o f mass o f the atom, ~ is the phase difference between the standing waves, and 6-+ and 6 - are the polarization vectors, corresponding respectively to right- and left-circular polarizations. We define the Rabi frequencies as gl (Z)=--,Uae " E l / f l ~ g , o
COS(k, Z) ,
g 2 ( Z ) =,/Abe
"Ez/h=-g2o C O S ( k 2 2 + q ) )
(3)
.
Ignoring the anti-resonant terms, and making the rotating-wave transformation, we get the h a m i l t o n i a n for the system:
-~ H=7
-g,(z)
-g~(=)/
-g2(z)
-2~
]
=(h/2)[A(ta>
.
(4)
Here, the R a b i frequencies are real, and the c o m m o n detuning c~and the difference detuning A are as defined in fig. lb. In what follows, we will consider p r i m a r i l y the case where A = 0 . We also assume that states la> and ]b > are degenerate, so that 0), = o)2 = o), 6, = ~ 2 - 6, and k, = k 2 - k. In addition, for simplicity we consider only the case ofglo=g2o-4g. So far, we have assumed that z, the position of the center o f mass of the atom, can be treated as a classical
ZZ ZZZ~A
J/= i m'=-i
@
I
I
o-'! / \
/\
I I
m=-I
@ d=l
(a) 454
81 . . . .
l
I I
1
la>
(b)
Ib>
Fig. 1. (a) A J = l ~ J ' = 1 transition, excited by a pair of circularly polarized standing waves. (b) The resulting ideal A system: is the average detuning, A is the differential detuning, and F is the excited state decay rate.
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variable. However, we want to allow for the possibility that the atoms become cold enough that the de Broglie wavelengths become comparable to or bigger than the optical wavelength [6]. In this regime, it is necessary to treat z as a quantum mechanical operator, ~, with the atom's linear momentum in the z-direction,/~, being the canonical conjugate. Denoting by IP) the eigenstate of/~ (i.e.,/~IP) = P I P ) ), the basis states can now be written as la, p a ) = - l a ) ® l P a ) ,
Ib, P b ) - l b ) ® l P b ) ,
le, p c ) - - l e ) ® l p ~ ) ,
(5)
where the linear m o m e n t u m is a continuous variable: - o e < p ~ < o e , ( i = a , b, e). In order to find the hamiltonian in this basis, we note first that +oo
exp(+ik:?)=
j dplp)(p~hk[,
(6)
--oo
which simply means that a travelling wave optical field with a wavenumber k couples a state with a momentum
p-hk to a state with a momentum p, conserving linear momentum. Given eq. (6), and noting that the kinetic energy of a state IP) is p2/2M, where M is the mass of the atom, we get the following form for the hamiltonian: H= p a , P~b , p e L2Mla, Pa)(a, Pal+-~-~lb, pb)(b, Pbl+ \2M -hg j dp ( I P ) (p+hkl + IP) (p-hkl) ( [ a ) (el +h.c.) --oo
-Foo
-hg J dp [ I P ) (p+hklexp( - i 0 ) + IP) (p-hklexp( + i ~ ) ] ( I b ) (el + h . c . ) .
(7)
-oo
To illustrate this, consider some representative matrix elements. For example, the energy of the state l a, Pa) is given by (a, pa}Hla, Pa) =PZ/2M, while the coupling between states la, Pa) and le, Pe) is given by
(a,p, JHle, pc)=-hg[~.po_hk+~p~,po+~k], ~ , j = 0 i f i ~ j a n d l i f i = j .
(8)
This hamiltonian is illustrated in fig. 2 for a few of the infinite number of basis states. As can be seen, each of the ground states is connected to two excited states. Therefore, it is not possible to find closed families of states, as is possible for two travelling waves. Nevertheless, we will be able to find a velocity selective dark state and calculate the VSCPT rate into this state (as a function of ~) both perturbatively and by numerically integrating the optical Bloch equations [6]. In analogy to the travelling wave VSCPT, we are interested in finding a state (the dark state) which satisfies the following conditions: (i) it does not contain any excited state, so that it is completely decoupled from the vacuum fields, and (ii) the net amplitude for coupling this state to any of the excited states must vanish. It can be shown that if p = 0 a n d / o r ~ = 0 then the state [NC(p))
=½[ la, p - h k ) e x p ( - i ~ ) + la, p+hk) e x p ( i 0 ) - I b , p - h k ) - Ib, p+hk) ]
(9)
does not couple to any excited state. This is illustrated in fig. 3 for two cases: p = 0 (levels denoted by solid lines) and p ~ 0 (levels denoted by dashed lines). In either case, m o m e n t u m conservation allows for only three excited states to be coupled by the hamiltonian directly to INC ( p ) ) . We will show that the coupling to each of these states vanishes (for p = 0 a n d / o r 0 = 0 ) by showing that if the system is in the state I N C ) at t = 0 , then it remains in INC ) after an infinitesimal amount of time, dt. We introduce the following notation for the seven levels of fig. 3: 455
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le,-aT~k >
(~_2nv%l
le, O>
le,ahk >
le,p+ahk>
le,p>
(3
e
I
e,p-~'hk >
//I
J aav 1 ~b,p-hk >
Ia,p-hk >
• "q*la,-hk>
p-hk)
,
15)-Ie,
p-2hk)
13)=-Ib, p-hk)
12)-=la, p+hk),
,
[6)-Ie.,p),
17)-Ie,
÷~la,hk >
-Ib,]hk >
Fig. 3. Illustration of the dark state, INC(0) ) (levels denoted by solid lines) and the state INC(p)) (levels denoted by dashed lines). Here, q_=exp(i¢), 0=-exp(i4g2d), and ~=-exp(i-Qvt). The energy zero is chosen to be (p2)/(2M). Note that this causes overlap of the states le, p> and le, 0). The symbols in the transition lines represent the effects of the standing wave phase shift, whereas those next to the solid ( p = 0 ) levels are associated with the phases produced by the motional energy shifts. The symbols next to the dashed (p # 0) levels are the additional phases associated with the center-of-mass momentum, p. Finally, the weights for the state INC ) are shown to the ground state labels.
Fig. 2. Illustration of the energy levels of and couplings between a typical set of momentum states in a A system excited by standing wave fields. Here, we have chosen ~= -t2~.
II)-Ia,
-Ib,-Xk>
p+2hk)
,
14)=-Ib, p + h k )
, (10)
.
T h e w a v e f u n c t i o n ~2 c a n b e w r i t t e n as Iq/)=
~Ai[i),
(11)
i = 1 , 2 ..... 7 .
i
T h e r e d u c e d h a m i l t o n i a n f o r t h i s s u b s e t o f s t a t e s is g i v e n , f r o m eq. ( 7 ) b y 0
H=h
_
0
0
g
g
0
0
12v
0
0
0
g
g
0
0 g g
0 0 0 g
- g2v 0 grl grl*
0 .(2, 0 g~l
grl* 0 - 2Y2,, + 4Y2r 0
gq grl* 0 0
0 grl 0 0
0
g
0
gr/*
0
0
-
(12)
2Y2~+ 41"2r _
H e r e , d e n o t i n g b y v t h e v e l o c i t y c o r r e s p o n d i n g to t h e m o m e n t u m p, we d e f i n e t h e D o p p l e r shift, Y 2 v - P k / m = kv, t h e recoil energy, £2r = h k 2 / 2 A L a n d q-= e x p ( i 0 ) . N o t e t h a t f o r s i m p l i c i t y , we h a v e c h o s e n f i = -Y2r, a n d h a v e shifted the zero of energy to coincide with the energy of state 16). ~2 The use of wave function is exactly valid for the case where [NC) is a dark state, which implies that As=A6=A7=O for t~>0. When INC } is not a dark state, the wave function approach is still valid as long as dt << 1/1-'. 456
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Note now that the dark state, INC ( p ) ) , is a linear combination of states I 1 ) through 14). We define IC~ ( p ) ) , [C2(p)) and IC3(p)) as three other mutually orthogonal states such that IC, ) [ = 1 r/* IC2)/ 2 / q* IC3)d L-,•*
q r/
1 1 1
12) 13) 145
1
(13)
The wavefunction can now be written as
I~')=ANcINC)+Ac, ICI)+Ac2ICz)+Ac3 C3) +As 15) +A616) +A7 [7) .
(14)
Using eqs. (12) and (13), we get the hamiltonian for this basis:
-0 0 0 t2v 0 0
H=h
_ 0
0 0 --Qv 0 grl 0
0 -g2v 0 0 gq 2Cg
Ov 0 0 0 0 -i2Sg
-gr/*
g~/*
0
0 gq* gq* 0
0 0 2Cg i2Sg
0 -grl g~l 0
2(2~r -£2v)
0
0
-
0
0
0
0
0
2(2£2r+f2v)_
(15)
Here, we have defined S -= sin ~ and C - c o s ~ so that q= C + iS. For p = 0 , so that .Qv=0, it is clear from eq. (15) that the state ]NC(0) ) is completely decoupled from any state. Therefore, INC ( 0 ) ) is a pure dark state. For p :~ 0, consider first the case of ~ # m z, where n is an integer. Note that INC ( p ) ) is still not coupled directly to any of the excited states. However, it is coupled to IC3 ( p ) ) , which in turn is coupled to one of the excited states, 16 ( p ) ) . Thus, for ~ ¢ mr, atoms with non-zero velocity will be optically pumped into INC ( 0 ) ) , which is the zero velocity dark state. This is of course velocity selective coherent population trapping (VSCPT). Consider next the case of ~=nn, with p:~0. In this case, S = s i n ~ = 0 , so that I N C ( p ) ) is coupled only to IC3 ( p ) ) , which in turn is coupled only to INC ( p ) ) . Thus, both ]NC ( p ) ) and IC3 ( p ) ) are pure trapped states, independent of the value of p, as long as
A6(t)= i dt' i dt"A'6(t"). 0
(16)
o
In the perturbative limit, where at t = 0 all the atoms are in the state I N C ) , the integrand can be found from applying the Schroedinger equation twice:
]t'6( t" ) = ( - i / h )2 ( NC IHIC3 ) (C3 IHI6 ) ANc( t" ) .
(17)
Since the hamiltonian is independent of time, and ANc(t" ) ~ 1 is a constant in the perturbative limit, we find that the amplitude of the state 16) after a differential amount of time, dt, is given by rim6(P)
=
(dt)Z ( - i / h ) 2 ( N C IHI C3 ) (C3 IHI 6) = - 2i(dt)2g2vg sin 0 = - 2 i ( d t ) 2 ( p / m ) k g s i n O.
(18)
Thus, the net rate at which atoms starting in ] N C ( p ) ) is coupled into the excited state is proportional to (p sin O) z. The formal approach used in deriving eq. (18) is presented to clarify the physics of VSCPT in standing 457
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waves, as well as to d e m o n s t r a t e the existence o f additional dark states when ~ = 0. However, it is possible to understand the behavior in eq. ( 18 ) in a relatively simple way by inspection o f fig. 3. If p C 0 (the dashed levels in the figure), each level has an a d d i t i o n a l energy due to the kinetic energy associated with the m o m e n t u m . In the figure, the energy p2/2M is chosen as the energy zero so that the states le, 0 ) and le, p ) are superimposed. If we consider the sum o f the interaction matrix elements coupling the ground states to each o f the excited states, we see that the couplings to the levels le, p - 2 h k ) and le, p+2hk) still vanish but the coupling to the level le, p ) does not. In particular, assume that the a t o m s are in I N C ) at t = 0 , as given by eq. (9). The problem now becomes one o f weighted s u m m a t i o n of the four transition paths to state [e, p ) from the ground states, as shown in fig. 3. Using the relative ground state weights (shown in fig. 3) and the relative transition matrix elements we find that (e, p l N C ( p ) )
~ [q*~+q~*-q~-~l*~*] ~sin(12vt) s i n 0 .
(19)
Note that since q = e x p ( i 0 ) and ~=exp(i£2vt), eq. (19) has the same functional dependence on ~ and £2v as eq. (18), in the limit of small t. Thus, the expression for the efficiency of V S C P T can be derived essentially by inspection o f fig. 3. According to eq. ( 19 ), we expect that the efficiency o f V S C P T would vary roughly as sin20, being m a x i m u m at 0 = ( 2 n + 1 )7r/2 and vanishing at O=nn. This phase d e p e n d e n c e of standing wave VSCPT is also consistent with the semi-classical picture. Consider the basis states I - ) = c o s 0 [ a ) - s i n 0 1 b ) , I+)-sin01a)+ c o s 0 1 b ) , and [ e ) where O(vt)-=tan-[gt(t,t)/g2(vt)]. For l ' = 0 , g~ and g2 are constants, and the 1 - ) state is the dark state, decoupled from I + ) and l e ) . F o r v¢O, I - ) is coupled to I + ) at the rate of vV0~ kv sin 0. Note that the phase dependence o f this coupling rate is also the same as the phase dependence of VSCPT (eqs. (18) and (19) ) since 12v= kv. In particular, for 0 = 0 , g~ =g2 at all positions and therefore O(vt) is a constant, so that this coupling vanishes, and I - ) is a dark state for each velocity. It is also possible to estimate the relative efficiency o f standing wave VSCPT c o m p a r e d to that o f travelling wave VSCPT. C o n s i d e r the case of 0 = (2n + 1 )~z/2. In this case, IN C ( p ) ) is coupled to IC3 ( p ) ) at the rate o f g2v. In turn, states I C~ ( p ) ) , I C 2 ( p ) ) , IC3 ( p ) ) are each coupled to the excited state (through one or more channels) at the rate o f 2g, as given by eq. ( 1 5). If we analyze the travelling wave case in the same way, we find that the coupling rates are the same, except that there are only three states, ( l a , p - h k ) - I b , p + h k ) ), ( l a, p - h k ) + [b, p + hk) and l e, p ) , which have properties analogous to IN C ) , I C3), and 16), respectively. Thus, for ~ = (2n + 1 ) ~ / 2 , we expect the standing wave V S C P T efficiency to be o f the same order o f magnitude as the travelling wave V S C P T efficiency. The V S C P T rate d e r i v e d above (eq. ( 1 8 ) ) is in agreement with numerical simulations o b t a i n e d by integrating the optical Bloch equations for the A system u n d e r standing wave excitation. In particular, we calculated the time evolution o f the m o m e n t u m d i s t r i b u t i o n for different values o f ~. Even though there are no closed families o f states in the SW system, an excited state with m o m e n t u m p is still coupled only to ground states with m o m e n t a p + hk which in turn couple to excited states with m o m e n t a p ++_ 2hk, etc. Thus, there are still families o f states which can be closed by truncation. F o r our calculation, we choose a m o m e n t u m range o f + 5hk. Figure 4 shows the initial ( t o p trace) and final m o m e n t u m distributions at t = 5 0 F - ~ for helium excited by standing waves o f the phase indicated ( m i d d l e traces), and also for travelling wave excitation ( b o t t o m trace), included for comparison. We see that for 0 = n/2 V S C P T does occur and is about half as efficient as for travelling waves. F o r other ~, we see the predicted sin20 dependence. In particular, at 0 = 0 there is no peaking of the m o m e n t u m distribution ~3.
~3 In fact, we see some very slight alteration in the momentum distribution at ¢=0 due to optical pumping into the relevant dark states. However, after a short time (< 50 F -l ) the momentum redistribution stops, in contrast to the ¢4:0 cases where the + hk peaks continue to grow and sharpen. 458
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INIT
A
v
Z
o
17
~z
0_ c~ Q-
i
-1
I
J
0
1
MOMENTUM
(l~k)
Fig. 4. Numerical simulation of VSCPT in the A system with standing waves after t = 50F- ' for He, where gao= g2o= 0.3F and b = 3 = 0. INIT is the initial momentum distribution: a gaussian distribution in the ground states with a standard half width at e - 1 / 2 of 3hk. The middle 4 traces are for standing wave excitation with the phases ~ as indicated. TW is travelling wave case with Rabi frequencies g~+ =g~_ =0.3/". In all cases, momentum is in units of ilk and population is normalized to 1.
3. Previous experimental evidence of standing wave V S C P T As m e n t i o n e d earlier, b o t h the existence a n d the a p p r o x i m a t e phase d e p e n d e n c e o f one d i m e n s i o n a l V S C P T in R a m a n resonant standing waves have already been observed by Aspect et al. [ 1 ], for the j = l ~ j ' = 1 transition in metastable helium. However, since these results were not explicitly presented in the context o f standing wave phase shifts, we will now review these observations for completeness. In the first part o f the Aspect experiment, V S C P T was observed as a pair o f peaks in the m o m e n t u m distribution when the He* b e a m was excited with a pair o f counterpropagating laser b e a m s having opposite circular polarizations. F o r the A system in the j = 1~--~j'= 1 transition this is o f course equivalent to a pair o f counter-propagating pure travelling waves. In the second part o f the experiment, the He* b e a m was excited by counter-propagating linearly polarized laser beams where the polarization axes o f the forward and backward propagating laser beams had a relative angle o f X. Specifically, Z = 0 and Z = ~z/2 correspond respectively to the so-called l i n - p a r a l l e l - l i n and l i n - p e r p - l i n configuration. It is easy to show that for the j = 1~ j ' = 1 transition these experimental configurations are equivalent to a pair o f opposite-circularly polarized R a m a n resonant standing waves, having a phase shift 0 = Z . As reported, no V S C P T was observed when X - 0 , but the double peaked m o m e n t u m distribution reappeared (as evidence o f V S C P T ) when Z = 7r/2. This is in agreement with the present theory. In addition, the relative peak hights in the Aspect e x p e r i m e n t are consistent (within experimental noise) with our theoretical prediction o f relative efficiencies in fig. 4. Here, it m u s t be acknowledged that for the specific case o f the j = l*-~j' = 1 transition, Kaiser et al. [ 7] correctly explained the presence o f V S C P T for the l i n - p e r p - l i n case a n d the absence t h e r e o f for the l i n - p a r a l l e l lin case by considering transitions between superpositions o f the magnetic sublevels forming closed families o f states for this system. However, we emphasize that the theory presented in this p a p e r is not l i m i t e d to the j = l ~ j ' = 1 transition, a n d is valid for any pair o f R a m a n resonant standing waves exciting a A system. In addition, our analysis gives the detailed phase d e p e n d e n c e o f V S C P T efficiency. This is i m p o r t a n t in the context o f a s t i m u l a t e d R a m a n trap, since b o t h the deflecting force a n d the pol-grad cooling force are m a x i m u m at ¢ = 7 r / 4 a n d vanish [3] at Z = 0 a n d X=~/2.
4. Generalization to three dimensions Finally, we have found [ 3 ] that this process can be generalized to three dimensions. Briefly, the scheme again 459
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1 December 1993
employs a j = l ~ j ' = 1 transition, except with three mutually orthogonal pairs of Raman resonant standing waves having direction vectors 2, j) and :~. This is accomplished using left-circularly polarized and right-circularly polarized standing waves, with phase differences o f 0i, i = x , y, z. U n d e r this configuration, VSCPT takes place into a zero velocity three-dimensional dark state o f the form ~,=A+ [ , 7 : = + l ) +Ao Ira: = 0 ) +,4 ]m: = - 1 ), where ,4 +, Ao, and A are the standing wave field a m p l i t u d e s at each point in space, which comprise of summ a t i o n s over m o m e n t u m eigenstates. Note that the m a t h e m a t i c a l form o f this zero velocity three-dimensional dark state is similar to that found by Ol'shanii et al. [8 ], except that standing waves (rather than travelling waves) are used in the present case. To c o m p u t e the efficiency o f V S C P T in three dimensions, it is necessary to consider other states with non-zero net m o m e n t u m as well. Extrapolation o f the one d i m e n s i o n a l analysis suggests that this would yield a V S C P T efficiency as a c o m p l i c a t e d function o f ~x, Oy, and G, and the analysis have not yet been performed. N o t e that there is no phase d e p e n d e n c e in the travelling wave three-dimensional V S C P T considered by Ol'shanii et al. [8].
5. Summary We have shown theoretically that phase sensitive V S C P T takes place when a A system is excited by a pair o f R a m a n resonant standing waves. This corresponds to absence o f diffusion for zero velocity atoms in the t r a p p e d state. We also review previous experimental observations reported by Aspect et al. which agree with our predictions. Finally, we outline the generalization to three dimensions. Given our prior observation of stimulated trapping and cooling forces a n d our theoretical prediction of strong pol-grad cooling in R a m a n resonant standing waves, the simultaneous existence o f V S C P T may make it possible to design a subrecoil t e m p e r a t u r e trap for A atoms.
Acknowledgements We wish to acknowledge the helpful discussions with Prof. S. Ezekiel o f MIT, and A. Aspect and R. Kaiser o f Institut d ' O p t i q u e Th6orique et Appliqu6e. This work was s u p p o r t e d by R o m e L a b o r a t o r y Contract No. F19628-92-K-0013 a n d Office o f N a v a l Research G r a n t No. ONR-N0014-91-HJ-1808. N P B is an Alfred P. Sloan Research Fellow.
References [ 1] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste and C. Cohen-Tannoudji, Phys. Rev. Lett. 61 ( 1988 ) 826. [2] F. Mauri and E. Arimondo, Appl. Phys. B 54 (1992) 2408. [ 3 ] M.S. Shahriar, P.R. Hemmer, M.G. Prentiss, J. Mervis, D.P. Katz, T. Cai and N. Bigelow, Bulletin of the American Physical Society, Series II, Vol. 38, No. 3, p. 1122 (1993), presented in the 1993 Annual Meeting of the Division of Atomic, Molecular, and Optical Physics, Reno, NV, 1993; P. Hemmer, M. Shahriar, M. Prentiss, D. Katz, K. Berggren, J. Mervis and N. Bigelow, Bulletin of the American Physical Society, Series II, VO1. 37, No. 4, p. 1208 ( 1992 ) presented in the InterdisciplinaryLaser Science Conf., Albuquerque, NM, 1992. [ 4 ] P. Hemmer, M. Shahriar, M. Prentiss, D. Katz, K. Berggren, J. Mervis and N. Bigelow, Phys. Rev. Lett. 68 ( 1992 ) 3148. [ 5 ] S. Chang, B. Garraway and V. Minogin, Optics Comm. 77 (1990) 19. [ 6 ] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 (1989) 2112. [ 7 ] R. Kaiser, These de Doctorat, Ecole Normale Superieure, 1990 (unpublished). [ 8 ] M. Ol'shanii and V. Minogin, Proc. Int. Workshop on Light induced kinetic effects on atoms, ions, and molecules, Marciana Marina, Italy, eds. L. Moi et al. (ETS Editrice, Pisa, 1991 ) p. 99.
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