Theory of Dicke narrowing in coherent population trapping

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by  O. Firstenberg

Theory of Dicke narrowing in coherent population trapping
O. Firstenberg,1 M. Shuker,1 A. Ben-Kish,1 D. R. Fredkin,2 N. Davidson,3 and A. Ron1
1Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
2Department of Physics, University of California, San Diego, La Jolla, California 92093
3Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
TheDopplereffectisoneofthedominantbroadeningmechanismsinthermalvaporspectroscopy.
For two-photon transitions one would naively expect the Doppler effect to cause a residual broad-
ening, proportional to the wave-vector difference. In coherent population trapping (CPT), which
isatwo-photonnarrow-band phenomenon,suchbroadeningwas notobserved experimentally. This
has been commonly attributed to frequent velocity-changing collisions, known to narrow Doppler-
broadened one-photon absorption lines (Dicke narrowing). Here we show theoretically that such a
narrowing mechanism indeedexists forCPT resonances. Thenarrowing factor istheratiobetween
7 theatom’smeanfreepathandthewavelengthassociatedwiththewave-vectordifferenceofthetwo
0 radiation fields. A possible experiment to verify the theory is suggested.
0
2
n I. INTRODUCTION
a
J
The spectral line shape of atomic transitions is de-
5
1 termined by many different mechanisms [1]. Specifi-
cally,aDoppler-broadenedspectrumcanbedramatically

2 narrowed due to frequent velocity-changing collisions —
v Dicke narrowing [2, 3]. The narrowing factor is pro-
8
portional to the ratio between the collisions mean free
0
pathandthe radiationwavelength. Dickenarrowingwas
0
1 observed for microwave transitions [4] and recently also FIG. 1: a. Energy levels scheme of CPT system. Two laser
0 for optical transitions [5]. For two-photon transitions, fields with frequencies ω1,ω2 excite two levels to a common
7 such as coherent population trapping (CPT) [6], a dra- upper level. b. Typical measurement of a CPT resonance in
0 matic narrowingof the expected Doppler-width was also room temperature 87Rb vapor, with frequency difference of
/ observed and was attributed to a Dicke-like narrowing ωHF =6.834GHz. The residual Doppler broadening is about
h
10KHz, while the measured width is only about 300Hz.
p [7, 8, 9, 10]. CPT is a light-matter interaction involv-
- ing three-level atoms and two resonant radiation fields
t
n (see figure 1.a). The atoms are driven into a coherent
a superposition of the lower levels, |1i and |2i which does
ample, the clock-transitionCPT resonanceof roomtem-
u not absorb the radiation. The efficiency of this process
q perature Rubidium vapor is expected to have a residual
strongly depends on the Raman (two-photon) detuning,
: Doppler width of
v definedas∆ =ω −(ω −ω ). Therefore,whenscan-
R HF 1 2
Xi ningthefrequencyofoneofthelasers,anabsorptiondip Γres = vthω ≈10KHz, (1)
is observed, which we will refer to as a CPT resonance D c HF
r
a (see figure 1.b). where v is the thermal velocity, ω is the hyperfine
th HF
The narrowwidth ofCPT resonancesis usefulforsev- energy-gap,andcisthespeedoflight. However,atypical
eral applications such as frequency standards [7, 11], measurement of this CPT resonance, depicted in figure
magnetometers [12], slow and stored light [13, 14]. In 1.b, shows a width of only 300Hz. In order to attribute
alltheapplicationsakeyparameteristhespectralwidth this dramatic narrowingto a Dicke-like narrowingeffect,
of the CPT resonance. When the CPT is performed in onecommonlyassumesthatthewavelengthwhichdeter-
room-temperature vapor, the two radiation fields expe- minesthenarrowingfactoristheoneassociatedwiththe
rience a Doppler shift which is different for each atomic frequency difference of the lasers(i.e. the microwavefre-
velocity group. In the case of non-degenerate lower lev- quency) [7]. In Ref. [15] a numerical model was used to
els (e.g. two hyperfine levels in the ground state of an calculatetheCPTline-shape,introducingdiscreteveloc-
alkaliatom)or for non-collinearlaserbeams, the two ra- ity groups, and indeed demonstrated the expected nar-
diation fields experience slightly different Doppler shifts. rowing. Inthepresentworkwederiveananalyticexpres-
This difference, denoted as the residual Doppler shift, sion for the line-shape of CPT resonances, that demon-
results in an effective Raman detuning which is different strates both Doppler broadening and collisions-induced
for each velocity group. Therefore, one would naively Dicke narrowing. We show that in a regime relevant to
expect a residual Doppler broadening of the CPT reso- mostCPTexperiments,thenarrowingisgovernedbythe
nance. However, the measured CPT resonance width is ratiobetweenthe meanfreepathandthe wavelengthas-
well below the expected residual Doppler width. For ex- sociated with the wave-vectordifference of the lasers. In
Typeset by REVTEX
2
section II we review the two-level Dicke narrowing. In with the formal solution
section III we derive the theory for Dicke narrowing of
t
CPT resonances. Section IV contains a discussion of the ρ(1)(t)=iΩ dt′e−i(ω0−iΓ)(t−t′)eiq·r(t′)−iωt′. (7a)
resultsandpresentsapossibleexperimentalsetuptover- 3,1 Z0
ify this result.
Substituting ρ(1)(t) in Eq.(5), denoting τ = t−t′ and
3,1
taking the upper limit of the τ−integral to infinity, we
II. DICKE NARROWING OF TWO-LEVEL obtain in steady-state,
ATOMS ABSORPTION LINE
∞
S(ω)=Re dτe−i(∆−iΓ)τeiΦ(τ), (8)
Z
Inthissection,asanintroductiontotheCPTcase,we 0
develop a theory for the line-shape of a two-level atom, where ∆=ω−ω is the detuning from resonance,
0
includingbothDopplerbroadeningandDickenarrowing.
OurapproachissimilartothatpresentedbyGalatry[3]. Φ(τ)=q·[r(τ)−r(0)], (9)
We consider a two-level atom with the upper and lower
states, |3i and |1i, and an optical energy gap of ~ω . isthephaseaccumulatedduringτ andtheaverageisover
0
atom trajectories,
We assume that the kinetic energy of the atom is much
smaller than ~ω and take the motion of the center of
0 1 T
massoftheatomtobeclassical. Theatominteractswith eiΦ(τ) = lim dteiq·[r(t)−r(t−τ)]. (10)
an external, classical electromagnetic field, with wave- T→∞T Z0
vectorqandfrequencyω =c|q|. Weassumethatω isof
Notice that the accumulated phase includes both the
theorderofω andexpresstheHamiltonianinthedipole
0 Doppler effect and collisions. Assuming that Φ(τ) is a
approximation and the rotating wave approximationas
randomGaussianvariableandfollowingthecumulantex-
pansionprocedure [17], we substitute eiΦ(τ) =e−Φ(τ)2/2.
H =~ω |3ih3|− ~Ωei(q·r−ωt)|3ih1|+H.c. , (2) Following Eq.(9) we write
0
h i
τ
whereΩistheRabifrequencyofthefieldandr=r(t)is Φ(τ)= qα dt1uα(t1), (11)
Z
the time-dependent center of mass position of the atom. Xα o
The equations of motion of the density matrix elements
whereuα arethe velocity cartesiancomponents andα=
ρ and ρ are given by:
3,3 3,1 x,y,z, to get
ρ˙ =−i(ω −iΓ)ρ −iΩei(q·r−ωt)(2ρ −1) (3a) τ τ
3,1 0 3,1 3,3 Φ(τ)2 = qαqα′ dt dt uα(t )uα′(t )
ρ˙ =−2Γρ +2ImΩ∗e−i(q·r−ωt)ρ . (3b) Z 1Z 2 1 2
3,3 3,3 3,1 αX,α′ 0 0
τ
Here we assumed a single relaxation term (radiation =2 qαqα′ dt(τ −t)uα(t)uα′(0). (12)
bath), whichinduces transitionsbetweenthe atomiclev- Z
αX,α′ 0
els(|3i→|1i),withtherateΓ[16]. Inordertocalculate
theabsorptionspectrum,wewritetheenergyabsorption As shown in the appendix, the velocity-velocity corre-
W˙ (t),intermsofthepopulationresponsetotheapplied lation function is given approximately by
field:
uα(t)uα′(0)=δα′αvt2he−γ|t|, (13)
W˙ (t)=~ω0{ρ˙3,3}field =2~ω0Im Ω∗e−i(q·r−ωt)ρ3,1 . where δα,α′ is the Kronecker Delta, vth is the thermal
(cid:16) (cid:17) velocity and γ is interpreted as the velocity relaxation
(4)
rate(fortheBrownianmotionregime)orasthecollisions
In the steady-state we take the temporal average, and
rate (for the strong collisions regime). Substituting this
find the absorption spectrum S(ω) to be:
in Eq.(12) gives
S(ω)= W˙ (t) =Im lim T dt ρ3,1(t) . (5) Φ(τ)2 =2q2v2 τdt(τ −t)e−γt = 2q2vt2hG(γτ) (14)
2~ω0|Ω|2 T→∞Z0 T Ωei(q·r−ωt) thZ0 γ2
We consider the unsaturated case in which Ω ≪ Γ, where
and use the perturbation expansion of ρ˙(t). Taking the
initial state of the atom to be the ground state, we find G(x)=x−1+e−x. (15)
to zero order in Ω that ρ(0)(t)=ρ(0)(t)=0, and to the
3,3 3,1 Then the absorption spectrum, Eq.(8), is written as
first order that
∞
ρ˙(1) =−i(ω −iΓ)ρ(1)−iΩei(q·r−ωt) 2ρ(0)−1 , (6) S(ω)= dτe−Γτ−Γ2DG(γτ)/γ2cos(∆τ), (16)
3,1 0 3,1 3,3 Z
(cid:16) (cid:17) 0
3
Λ−configuration, with the energy levels E > E > E
3 2 1
correspondingto anupper state |3iand two lowerstates
|2i,|1i (see figure 1.a). We will denote the atomic fre-
quencies ωnn′ =(En−En′)/~. The atom interacts with
external probe and pump electric fields, close to reso-
nance with the |3i → |1i and |3i → |2i transitions, re-
spectively. The coupling Hamiltonian is
H =−~ Ω eiqn·r(t)−iωnt|3ihn|+H.c. (19)
C n
nX=1,2
where q are the wave-vectors of the fields and ω =
n n
c|q |. For brevity, we denote
n
Ω (t)=Ω eiqn·r(t)−iωnt n=1,2. (20)
n n
We consider both dipole relaxation terms, which induce
transitions between the excited and ground levels |3i→
|niwithratesΓ (n=1,2),andwithinthe groundstate
n
FIG.2: TheeffectofDickenarrowingontheabsorptionspec- |1i ↔ |2i with rate Γ ; and a spin-boson relaxation
2↔1
trum: anumericalsolutionofEq.(16)forseveralvaluesofthe term, which induces adiabatic transitions (decoherence)
Dickeparameter 2πΛ/λ, with ΓD =5Γ. between |1i ↔ |2i with rate Γ . For an atomic vapor
ad
in the presence of a buffer gas, Γ will be the pressure-
n
broadened homogenous width. The equations of motion
for the density matrix are thus
where Γ = qv = ωv /c is the Doppler width. Re-
D th th
stricting the discussion to the regime where Γ ≫ Γ, ρ˙ =−2ImΩ∗(t)ρ +Γ ρ +Γ (ρ −ρ )
D 1,1 1 3,1 1 3,3 2↔1 2,2 1,1
one finds two limits [3]: ρ˙ =−2ImΩ∗(t)ρ +Γ ρ −Γ (ρ −ρ )
2,2 2 3,2 2 3,3 2↔1 2,2 1,1
(i) The Doppler limit, γ ≪ Γ , where the quadratic
D ρ˙ =2Im[Ω∗(t)ρ +Ω∗(t)ρ ]−(Γ +Γ )ρ
term G(γτ) ≈ γ2τ2/2 is dominant and the absorption 3,3 1 3,1 2 3,2 1 2 3,3
ρ˙ =iΩ∗(t)ρ −iΩ (t)ρ −i(ω −iΓ )ρ
spectrum, 2,1 2 3,1 1 2,3 21 21 2,1
ρ˙ =−iΩ (t)(ρ −ρ )+iΩ (t)ρ
3,1 1 3,3 1,1 2 2,1
SDoppler(ω)=rπ2Γ1 e−2∆Γ2D2 , (17) −i(ω31−iΓC)ρ3,1
D ρ˙ =−iΩ (t)(ρ −ρ )+iΩ (t)ρ
3,2 2 3,3 2,2 1 1,2
is the well-known Doppler-broadened spectrum. The −i(ω −iΓ )ρ (21)
32 C 3,2
same expression is usually derived by a convolution of
where Γ = (Γ +Γ +Γ +Γ )/2 , Γ = Γ +
thehomogenouslineshapewiththethermalvelocitydis- C 1 2 2↔1 ad 21 2↔1
2Γ . Tofindtheabsorptionspectrumoftheprobefield,
tribution. ad
we follow Eqs.(4)-(5) using the response terms of the
(ii) The Dicke limit, γ ≫ Γ , where the linear term
D probe and get,
G(γτ)≈γτ is dominant and the absorption spectrum,
T dtρ (t)
3,1
Γ+Γ2 /γ S(ω1)=Im lim . (22)
SDicke(ω)= ∆2+(Γ+DΓ2 /γ)2, (18) T→∞Z0 T Ω1(t)
D We consider the standard weak probe case, namely
Ω ≪ Ω , and avoid saturation by assuming Ω smaller
is a Lorentzian with natural-width Γ, broadened by 1 2 2
Γ2 /γ. Denoting the mean free path Λ = v /γ, and the than any of the relaxation rates Γn→n′. The perturba-
D th tion solution of Eqs.(21) can easily be done, following
field’s wavelength λ = 2π/q, we find the line-width to
the same procedure as in the two-level case. To zero or-
be Γ+(2πΛ/λ)Γ , showing that the Doppler broaden-
D der in Ω , with the initial state ρ(0) = |1ih1|, we find
ing is effectively reduced by a factor known as the Dicke 1
ρ(0) = 1 and all other matrix elements vanish. To the
parameter [2]. A numerical solution of Eq.(16), showing 1,1
first order in Ω ,
the transitionbetween the Doppler andthe Dicke limits, 1
is shown in Fig.(2). ρ˙(1) =iΩ (t)ρ(1)+iΩ (t)−i(ω −iΓ )ρ(1) (23a)
3,1 2 2,1 1 31 1 3,1
ρ˙(1) =iΩ∗(t)ρ(1)−i(ω −iΓ )ρ(1) (23b)
2,1 2 3,1 21 21 2,1
III. DICKE NARROWING OF CPT
and all other matrix elements vanish. We solve formally
RESONANCES
Eq.(23b), assuming the initial state vanishes,
t
We now turn to analyze the line-shape of CPT reso- ρ(1)(t)=i dt Ω∗(t )e(−iω21−Γ21)(t−t1)ρ(1)(t ),
nances in the presence of thermal motion and velocity- 2,1 Z 1 2 1 3,1 1
0
changing collisions. Consider a three-level atom in a (24)
4
and substitute it back into Eq.(23a), and Φ (t,τ) ≡ q ·[r(t)−r(t−τ)]. Both the integra-
n n
tions over τ and τ contain exponential decay terms, al-
1
ρ˙(1) =iΩ (t)−i(ω −iΓ )ρ(1) (25) lowing us to take their upper limits to infinity and aver-
3,1 1 31 1 3,1
age the phase-lag term over t,
t
−Ω (t) dt Ω∗(t )e(−iω21−Γ21)(t−t1)ρ(1)(t ),
2 Z 1 2 1 3,1 1 ∞
0 S (∆ )=−|Ω |2Re dτe(i∆R−Γ21)τ
2 R 2
Z
0
whoseformalsolutionis anintegralequationforρ(31,1)(t), ∞dτ e−Γ1τ1 τ1dτ eiK. (30)
1 3
Z Z
t 0 0
ρ(1)(t)= dt e(−iω31−Γ1)(t−t2)[iΩ (t )− (26)
3,1 Z 2 1 2 In a similar manner to the two-levelanalysis,we use the
0
t2 cumulant expansion to get
Ω (t ) dt Ω∗(t )e(−iω21−Γ21)(t2−t1)ρ(1)(t ) .
2 2 Z0 1 2 1 3,1 1 (cid:21) eiK ≈e−K2/2. (31)
An approximate solution for Eq.(26) is obtained by it-
Then
erations up to first order in ρ . It can be shown, by
3,1
comparing the approximate solution to the exact solu- K2 =Φ2(0,τ)+Φ2(0,τ +τ)−2Φ Φ (32)
2 1 1 2 1
tion for an atom at rest, that the approximationis valid
in the low-contrast regime, when |Ω2|2 ≪ Γ12Γ, i.e. in wherethe firsttwoterms canbe foundfromEq.(14)and
the limit of smallrelative increase in transparency. Note the third term is
that the power-broadening in this regime is small com-
pared to Γ . Φ Φ =Φ (−τ ,τ)Φ (0,τ +τ). (33)
12 2 1 2 3 1 1
Performing two iterations of Eq.(26) and returning to
Eq.(22),wefindthespectrumtobethesumoftwoterms: Using Eqs.(11),(13) we find
(i) The one-photon absorption spectrum,
S1 =Z0∞dτe−Γ1τ−Γ2DG(γτ)/γ2cos(∆1τ) (27) Φ2Φ1 =αX,α′q1αq2α′Z−0τ1−τdt1Z−−τ3τ−3τdt2uα(t1)uα′(t2)
where Γ = q v and ∆ = ω −ω is the one-photon 0 −τ3
D 1 th 1 31 1 =q ·q v2 dt dt e−γ|t2−t1|. (34)
detuning of the probe transition. This is essentially the 1 2 thZ 1Z 2
−τ1−τ −τ3−τ
two-levelabsorptionspectrum,Eq.(16),andisequivalent
tothesmall-signalabsorptionspectrumintheabsenceof Performingtheintegralandsubstitutingtheresultsback
the pump. (ii)The two-photonabsorptiondip (the CPT into Eq.(30), we get
line-shape)
∞
S2(∆R)=−|Ω2|2Re dτe(i∆R−Γ21)τe−(ΓrDes/γ)2G(γτ)
S2 =−|Ω2|2Re lim T dt tdt1 t1dt2 t2dt3 ∞ Z0
T→∞Z0 T Z0 Z0 Z0 dτ e−Γ1τ1e−(ΓD/γ)2(γτ1−e−γτ+e−γτe−γτ1)
1
e(−i∆1−Γ1)(t−t1+t2−t3)e(i∆R−Γ21)(t1−t2)× Z0
eiq2·[r(t1)−r(t2)]−iq1·[r(t)−r(t3)] (28) τ1dτ eq1·q2vt2h(e−γτ−1)(e−γτ1eγτ3+e−γτ3−2)/γ2
3
Z
0
(35)
where ∆ = ω −ω −ω is the Raman detuning. For
R 1 21 2
brevity, we will take ∆ = 0 and write the spectrum in
1 where G(x) was defined in Eq.(15) and
terms of ∆ , i.e. assuming the CPT resonance to occur
R
aroundthecenteroftheone-photonabsorptionline. The Γres =|q −q |v (36)
calculation can easily be generalized to ∆ 6=0. D 1 2 th
1
To calculate the shape of the absorption dip, we write is the residual Doppler width.
it as Eq.(35) is a general analytic expression for the CPT
line-shape. In what follows we limit the discussion to a
T dt t
S (∆ )=−|Ω |2Re lim dτe(i∆R−Γ21)τ specific realistic regime, which is relevent to most CPT
2 R 2
T→∞Z0 T Z0 experiments. In realizations of CPT in vapor medium,
t−τ τ1 the one-photon processes are usually in the far Doppler
dτ e−Γ1τ1 dτ eiK (29)
1 3 regime(Γ ≫Γ andΓ ≫γ)andinmostapplications
Z Z D 1 D
0 0
thevaporcellcontainsbuffergas,forcingthetwo-photon
where processes into the Dicke regime (γ ≫ΓrDes ≫Γ12). Since
this regime includes both Doppler and Dicke regimes we
K ≡Φ (t−τ ,τ)−Φ (t,τ +τ) denoteitasthe intermediate regime. Asdescribedinthe
2 3 1 1
5
previoussection,fortheDopplerregimewetakee−γτ1 ≈ broadening,whichisoftheorderofafewKHz,isstrongly
1 − γτ , and thus e−γτ3 ≈ 1 − γτ and for the Dicke reducedandisnotmeasurable(comparedtootherbroad-
1 3
regime we take e−γτ ≪ 1 and G(γτ) ≈ γτ. With these eningmechanisms). Inordertoverifythetheoreticalpre-
approximations, a simple expression for the CPT line- diction it is necessary to increase either the CPT-Dicke
shape is obtained parameteror the residualDoppler width. An increase of
the CPT-Dicke parameter can be achieved by decreas-
S (∆ )= −|Ω2|2 Γ12+ηΓrDes , ing the effective wavelength or by increasing the mean
2 R [Γ +q ·(q −q )v2 /γ]2∆2 +[Γ +ηΓres]2 free path between collisions. However, changing the ef-
1 1 1 2 th R 12 D
(37) fective wavelength is limited by the atomic structure of
where the parameter the active atoms and changing the mean free path will
result in a significant diffusion-broadening. We propose
η = ΓrDes =2π Λ (38) to increase the residual Doppler width by introducing a
γ λCPT smallangulardeviation, θ, betweenthe pump andprobe
beams. For CPT performed with two degenerate lower
is proportional to the ratio between the mean free path,
levels(|q |=|q |)andsmallθ,boththeresidualDoppler
Λ, and the wavelength associated with the wave-vector 1 2
width andthe CPT-Dickeparameterare proportionalto
difference, λ = 2π/|q −q |. Equation (37) is the
CPT 1 2 θ. Therefore,the resulting broadeningis proportionalto
main result of the present work, showing that the CPT
θ2 and can be increased to a measurable level.
line-shape is the product of two terms, which are both
functions of the wave-vector difference. The first term
determines the line’s amplitude and the second is a
Acknowledgments
Lorentzianthatdeterminesthewidth. Sinceη multiplies
the residualDoppler width, it acts as a narrowingfactor
We thank Nitsan Aizenshtark for reading the
andwedenoteitastheCPT-Dickeparameter. Finally,in
manuscript and helpful suggestions. This work was par-
a typical setup of a CPT-based frequency standard [11],
tiallysupportedbyDDRNDandthefundforencourage-
thelaserbeamsarecollinear(q kq )andtheline-shape
1 2 ment of research in the Technion.
can be written as
−|Ω |2 Γ +ηΓres
S2k(∆R)= (Γ +η2Γ )2∆2 +1(2Γ +ηDΓres)2. (39) APPENDIX
1 D R 12 D
To obtain the velocity-velocity correlation function of
IV. DISCUSSION AND CONCLUSIONS Eq.(12) we review two simple models:
(i)IntheBrownianmotion casetheequationofmotion
of the α component of the velocity is
CPT is an inherently narrow-band phenomena usu-
ally limited by effective broadening mechanisms. Non- d
degenerateCPTresonancesinahotvaporcellwithbuffer uα(t)=−γuα(t)+aα(t), (40)
dt
gas would naively be broadened by a residual Doppler
broadening. However, the measured line-width of CPT whereγuα(t)isthedynamicfrictionalforceontheatom,
resonances are far below the expected residual Doppler γ isthevelocityrelaxationrate,andaα(t)istherandom
width. This effect was attributed to the frequent veloc- acceleration [18]. For the correlation function we thus
ity changing collisions with the buffer gas, that are well obtain
known to narrow atomic absorption transitions in two-
d
level atoms. It was also demonstrated using a numerical uα(t)uα′(0)=−γuα(t)uα′(0)+aα(t)uα′(0), (41)
dt
simulationthatwhenfrequentcollisionsoccurnoDoppler
broadening is evident in the CPT line [15]. In this work where the bar indicates ensemble average. Since the ac-
wedevelopedthetheoryofDickenarrowingforCPTres- celeration is not correlated with the initial velocity, the
onancesinthreelevelatoms. Themainresultisthatthe last term on the right vanishes, and we have
residual Doppler width (that would be observable in an
apparatuswithnocollisions)isdiminishedbytheratioof uα(t)uα′(0)=uα(0)uα′(0)e−γ|t|. (42)
themeanfreepathbetweencollisionsandthewavelength
InthermalequilibriumwithtemperatureT,theensemble
associated with the wave-vectordifference of the two ra-
average is taken over the velocity distribution function
diation fields. This theory can be readily extended to
describeatomsinconfinedgeometriessuchasthinvapor F(u)=(m/2πk T)3/2e−mu2/2kBT, (43)
cells and cold atoms traps. B
For hyperfine CPT experiments, performed in vapor where m is the mass of the atom, and k is the Boltz-
B
cells with Alkali atoms and several Torrs of buffer gas, mann factor, and we get
the typical CPT-Dicke parameter is η ≈ 10−4 (i.e. very
strong Dicke narrowing). Hence the residual Doppler uα(0)uα′(0)=δα′αvt2h, (44)
6
where δα,α′ is the Kronecker Delta, and vth = kBT/m and, with Eq.(46), the velocity of the atom at time t, is
is the thermal velocity. p
(ii) In the Strong Collisions case the atom is colliding
u(t)= d3uuf(u,t)=u(0)e−γ|t|. (49)
with the dilute buffer gas in thermal equilibrium. The
Z
conditional probability density, f(u,t), to find the atom
with velocity u at time t, given that at time t = 0 its
The velocity-velocity correlationfunction is
velocityisu(0),isgivenbytheBoltzmanncollisionterm,
uα(t)uα′(0)=uα(0)uα′(0)e−γ|t|. (50)
d d
f(u,t)=− f(u,t) , (45)
dt (cid:20)dt (cid:21)
coll and since
with the initial distribution
f(u,t→∞)=F(u), (51)
f(u,0)=δ(u−u(0)). (46)
for consistency
The simplest model is the single relaxation rate approx-
imation, where
uα(0)uα′(0)= d3uuαuα′F(u)= kBT. (52)
d Z m
f(u,t) =γ(f(u,t)−F(u)), (47)
(cid:20)dt (cid:21)
coll We end up in both these models with
F(u) is the equilibrium distribution of Eq.(43), and γ
is the collision relaxation rate. The solution of Eq.(45), uα(t)uα′(0)=δα′αvt2he−γ|t|. (53)
with Eq.(47), is simply
where the interpretationof γ is either the velocity relax-
f(u,t)=F(u) 1−e−γ|t| +e−γ|t|f(u,0), (48) ation rate or the collision relaxation rate.
(cid:16) (cid:17)
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