Coherent population oscillation produced by saturating probe and pump fields on the intercombination Line

0.35 MB·

by  A. Vafafard

Coherent population oscillation produced by saturating probe and pump
fields on the intercombination Line
A. Vafafard1,2,∗ M. Mahmoudi2, and G. S. Agarwal1
1Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA and
2Department of Physics, University of Zanjan, University Blvd, 45371-38791, Zanjan, Iran
(Dated: January 29, 2016)
Wepresenta theoretical studyoftheexperimentson coherent population oscillations and coher-
ent population trapping on theintercombination line of 174Yb. The transition involves a change of
6
the spin and thus can not be interpreted in terms of an effective Lambda system. The reported
1
experiments are done in the regime where both pump and probe fields can saturate the transition.
0
We demonstrate by both numerical and analytical calculations the appearance of the interference
2
minimum as both pump and probe start saturating the transition. We present an analytical result

n for the threshold probe power when the interference minimum can appear. We also present de-
a tailed study of the appearance of the interference minimum when magnetic fields are applied. The
J
magnetic fieldsnot only create Zeeman splittings butin addition makethesystem open because of

7 the couplings to other levels. We show the possibility of interference minimum at the position of
2 subharmonicresonances.

PACS numbers: 42.50.Gy, 42.50.Hz, 42.50.Nn
]
h
p
- I. INTRODUCTION and Natarajan studied the intercombination line 1S0
m 3P in 174Yb [19]. They reportedboth CPOandCPT→in
1
o Atomic coherence effects, induced by laser fields, such such a system. It is to be noted that for the observation
at as the coherent population trapping (CPT) [1], the co- of CPO in two level atoms, we need strong dephasing
. herent population oscillation (CPO)[2], the electromag- [14]. An atomic beam has no dephasing, hence observa-
s
c neticallyinducedtransparency(EIT)[3],havebecomein- tion of CPO is quite remarkable. The ground state of
i creasinglypopularbecauseoftheirverywideapplications 174Yb has the configuration 6s2 with the term symbol
s
y in lasing without inversion[4], enhancement of refractive 1S0 and the upper state is 6s6p with the term symbol
h index[5],slowlight[6–8],storageoflight[9,10],nanoscale 3P1. They also reported both CPT and CPO by using
p resolution[11], magnetometry [12], etc. The interference a magnetic field. Note that in the ground state the two
[ resultingfrommultiple pathwaysproducesanarrowdip spins are antiparallel whereas in the excited state the
1 in the absorptionspectra whichtypically are used in ap- two spins are parallel. Thus the situation is quite differ-
v plications. The atomic coherence effects in single elec- entfromthatconsideredbyMompartetal[18]. Wethus
6 tron atoms like Na, 85Rb, and Cs have been extensively needatheoreticalmodeltounderstandthe experimental
8 studies. Results for both coherent population trapping results on the intercombination line, which is forbidden
7 and coherent population oscillations are available [13– in LS coupling, of 174Yb. We also note that in the ex-
7
17]. Compared to single-electron atoms, there are only perimental study the pump and probe had comparable
0
. much fewer studies for two-electron atoms. Mompart et intensities and thus the probe unlike other experiments
1 al. have presented a study of CPT in two-electron atom is not weak compared to the pump and the saturation
0
withalignedspins[18]. Intheinvestigationofatwoelec- of the transition by both pump and probe fields is ex-
6
1 tronsystemoneneedstotakeintoaccountthePauliex- pected to have a major effect. This must be accounted
: clusion principle to obtain the allowed transitions. May- in any theoretical modelling. The organization of this
v nard et al [10] studied the transition 2 3S 2 3P paper is as follows, in Sec II we describe the intercom-
Xi in metastable He. This is a Λ-system. The1y→observed1 bination line as an effective four level system with the
r CPO between two levels involving only change of spin. ground and excited states coupled by pump and probe
a Mompart et al [18] specifically studied the transitions in fields which are orthogonally polarized. The two fields
a two electron system where both ground and excited havecomparableintensities. We derivethe basic density
states had aligned spins (ortho system) so that the or- matrix equations andpresentthe expressionsfor the flu-
bital partof the two electronsystemwas antisymmetric. orescence. In Sec III we presents numerical results for
Usingthistheyshowedaveryinterestingpossibilitythat fluorescence obtained from a Floquet analysis. We show
an ortho system like the transition 4s4p to 4p4p in Ca, results in the absence of the magnetic field and in pres-
whichisaV-systemcanshowCPT.Thisisbecauseofthe enceofthemagneticfield. Ournumericalresultsconfirm
Pauli principle that the V-system becomes equivalent to the behavior as observed in experiments. In Sec IV we
a Lambda system. In a recent experimental work Singh presentanumberofanalyticalresultswhenthemagnetic
field is zero. The analytical results help us understand
the observed behavior of fluorescence.
∗ [email protected]
2
written as
(cid:3)
(cid:4)
3P1 (cid:3)(cid:4) ∆(cid:2) Ωp+ =Ωp− = √Ap2sinθ, Ωp0 =Apcosθ,
A
l
(cid:3)(cid:4) − Ωl+ =Ωl− = √2cosθ, Ωl0 =−Alsinθ, (3)
,
E E
l p
whereA = dε andA = dε . Weusethesamegeom-
p p l l
| | | |
etry as in [19] i.e. atomic beam is moving in direction z,
the lasersarepropagatingperpendicular to the direction
of the atomic beam. The Hamiltonian of the system in-
teracting with two laser fields in the dipole and rotating
1 wave approximations is given by:
S
0
Hin = ~[( (Ωpie−iωpt+Ωlie−iωlt) j g )+c.c],
FIG. 1. Two-electron four-level quantum system driven by − | ih |
→ → j=X+,0,−
two orthogonally polarized probe Ep (solid) and pump El (4)
(dashed)fields. Asindicatedinthetextthesecouplingswith where ω and ω denote the frequencies of the applied
p l
thelasers are effectivecouplings.
fields. In the rotating frame, the density matrix equa-
tions ,which show the response of the medium to the
field, are given by:
II. MODEL AND EQUATIONS
ρ˙ =ρ Ω e−i∆t+Ω iρ Ω ei∆t+Ω
++ g+ p+ l+ +g p+ l+
Consider an atomic model as shown in Fig. 1. The −
γ(cid:0) ρ , (cid:1) (cid:0) (cid:1)
gtp(tiFrreoaornnu=slnleti1dnvio,eenMslstouaffatstere=hedei0s|ie−)sv|,gtiehiane=n=id1sSo31tP0So|1+→p0(i(F1F3=74P==Y31P1ba01,ta,M(Mt5Fo5fmf6==n=.m1T−0,h1)Mine)a,tfsent|r=d0acito1etmh)=3.ebPTi3unPhpai1es-- ρ˙ρ−˙0−0 ==ii−−ρρggγ0−+0(cid:0)gΩgΩρp0p+00−e+,e−−i∆i∆tt++ΩΩl0l(cid:1)−−−iρi0ρg−(cid:0)gΩpΩ0ep−i∆eti∆+tΩ+l0Ω(cid:1)l−
weakly mixed to transition1P which has two spins 1an- γ−(cid:0)gρ−−, (cid:1) (cid:0) (cid:1)
1 −
tiparallel. Thus effectively, we can think that the level ρ˙ = iδ ρ +i(ρ ρ ) Ω ei∆t+Ω
g+ + g+ ++ gg p+ l+
3P1 is coupled to the level 1S0 via→the laser field. Two −+iρ0+ Ωp0ei∆t+−Ωl0 +(cid:0)iρ−+ Ωp−ei∆t(cid:1)+Ωl−
orthogonal linarly polarized fields, E as the probe field
p Γ ρ(cid:0) , (cid:1) (cid:0) (cid:1)
→ g+ g+
−
and El as the pump field, couple the ground level to the ρ˙ = iδ ρ +i(ρ ρ ) Ω ei∆t+Ω
g0 0 g0 00 gg p0 l0
upper levels : − −
+iρ+0 Ωp+ei∆t+Ωl+(cid:0) +iρ−0 Ωp−e(cid:1)i∆t+Ωl−
→
Ep =εp(sinθxˆ+cosθzˆ)e−iωpt+ikpy +c.c, Γg0ρg(cid:0)0, (cid:1) (cid:0) (cid:1)
−
E→l =εl(cosθxˆ sinθzˆ)e−iωlt+ikly+c.c, ρ˙g− =−iδ−ρg−+i(ρ−−−ρgg) Ωp−ei∆t+Ωl−
E→ . E→=0, − (1) +iρ+− Ωp+ei∆t+Ωl+ (cid:0)+iρ0− Ωp0ei∆t(cid:1)+Ωl0
p l
Γg−ρg(cid:0)−, (cid:1) (cid:0) (cid:1)
−
where εp and εl are the probe and pump fields am- ρ˙ = i(δ δ )ρ iρ Ω ei∆t+Ω
0+ + 0 0+ 0g p+ l+
plitude, respectively. Moreover θ showes the angle be- − − −
tween the polarization and direction of the propagation. +iρg+ Ωp0e−i∆t+Ωl0(cid:0) Γ0+ρg0, (cid:1)
−
The Rabi frequensies of the probe and pump→fields can ρ˙−+ =i(δ−−δ(cid:0)+)ρ−++iρg+ (cid:1)Ωp−e−i∆t+Ωl−
be defind as 2Ωpi = 2[εp(sin→θxˆ+cosθzˆ)]· dig /~ and −iρ−g Ωp+ei∆t+Ωl+(cid:0) −Γ−+ρ−+, (cid:1)
2→Ωli = 2[εl(cosθxˆ−sinθzˆ)]. dig /~ (i=+,0,−). The ρ˙−0 =i(δ−−δ(cid:0)0)ρ−0+iρg0 Ω(cid:1)p−e−i∆t+Ωl−
dig givesthe dipolematrix elementwhichcanbe written iρ−g Ωp0ei∆t+Ωl0(cid:0) Γ−0ρ−0, (cid:1) (5)
using Clebsch-Gordan coefficients as − −
(cid:0) (cid:1)
where γ is the spontaneous decay rate from level i to
→ → → ig
| i
d+g= d ˆǫ+, d0g= d zˆ, d−g= d ǫˆ−, (2) level g . Theoffdiagonalelement,ρij decaysattherate
| | | | | | | i
Γ =(γ +γ )/2. The parameter∆=ω ω denotes
ij ig jg p l
−
where d is the reduced dipole matrix element and ǫˆ± = the probe field detuning with respect to the pump field.
| |
(xˆ iyˆ/√2). Itshouldbeborneinmindthattheparam- Alsoδ =ω ω (i=+,0, )isthepumpfielddetuning
i l ig
± − −
eter d would include the coefficient of mixing with the with the atomic resonance transition. Because of the
level 1P . By using Eq. 2, the Rabi frequencies can be explicittime dependence inEq. (5),the steadystatehas
1
3

the Floquet expansion:
(a)
∞ 0.4 A =0.01 γ
ρ = ρ(m)e−im∆t, (6) p
ij ij A =0.1 γ
mX=−∞ 0.35 p
A =0.5 γ
where m = 0 denotes the time-independent part of the 0.3 p
A =1 γ
density matrix elements. The positive frequency part of I p
the electric field operator at detector is [20] 0.25 A =2 γ
p
E→(+) (→r,t)= ωl2 →n [→n →d]eiklr−iωlte−ikl→n.→R,(7) 0.2 Ap=3 γ
−c2 × × r
0.15
where the dipole moment operator is given by
→ → 0.1
d= dgi g i , (8) 0 5 10 15 20
| ih |
i=X+,0,− 0.28
(b)
→
and r denotes the point at which the fluorescence be
→
measured. Atomic sample is located at R. Moreover 0.27
→ →
n= r is the direction of the observation. According to
r
theexperimentsetup,thefluorescenceiscollectedinthe
directionperpendiculartotheatomicbeamandthelaser I 0.26
beams i.e. in x direction. The fluorescence is given by
→(−) → →(+) →
I = E (r,t) . E (r,t) , (9) 0.25
(cid:28) (cid:29)
→(−)
where E denotes the negative frequency part of the 0.24
→
electric fieldoperatoratthe detector. With n=xˆ inEq. 0 5 10 15 20
(8), the dc component of fluorescene can be written as 0.4
I
I = 20[2ρ000+ρ0+++ρ0−−−ρ0+−−ρ0−+], (10) 0.35
where the constant I will depend on ω and d. In
0 l
| | 0.3
our further consideration, we will refer to fluorescence
in units of I0. Note that the fluorescence has contribu- I
tionsfrombothexcitedstatepopulationsI = I0(2ρ0 +
p 2 00 0.25
ρ0 +ρ0 ), aswellascoherencesI = I0(ρ0 +ρ0 ).
++ −− c −2 +− −+
0.2
(c)
III. NUMERICAL RESULTS

0 5 10 15 20
The Eq. 5 for the density matrix elements are nu- ∆/γ
merically solved using the Eq. 6. Note that equations
for ρ(m) get coupled to ρ(m±1). The convergence of the
ij ij
truncation is tested for every set of parameters. As in FIG. 2. Fluorescence for different values of probe field. Se-
lected parameters are A =2γ, θ=π/6 (a), θ=π/4 (b) and
the experiment we choose pump on resonance i. e. we l
θ=π/3 (c).
set δ = 0. The probe is scaned i. e. the detuning pa-
0
rameter ∆ is varied. We scale all parameters in units of
the natural line width γ = 2π 185KHz for the 174Yb
×
intercombination line. Further γ±g =γ0g =γ. first fix the Rabi frequency of the pump value at which
We first describe the results in the absence of the the transition can be saturated (A =2γ). We show flu-
l
magnetic field. In figures, we show the fluorescence orescence I (in units of I ) in Fig. 2 for different values
0
as a function of the probe detuning ∆ 0 only as of the strength of the probe. Note that for θ = π/4, the
≥ 6
I( ∆) = I(+∆). Typically pump and probe experi- strength of the pump and probe for different transitions
−
ments are done for a weak probe and strong pump. We are different, for example g 0 transition has probe
| i →| i
4
0.5
(a) 0.55
0.4
0.5
0.3
Ip 0.45
0.2
A=0 I
l
A=0.5 γ
l 0.4
0.1 A=1 γ
l A =A=3 γ
A=2 γ p l
l
0.35 Ap=Al=4 γ
0
0 5 10 15 20 Ap=Al=5 γ
0

(b) 0 5 10 15 20
∆/γ
−0.05
FIG. 4. The fluorescence in presence of magnetic field. Pa-
rameters used are θ = π/4 and ∆B = 4γ. The pump and
probeare of equal intensities.
c −0.1
I
pronounced when pump and probe strengths are com-
−0.15
parable. The fluorescence behavior in Fig. 2 is quite
comparable to that reported in [19]. The results of this
figure clearly show that the experimentally observed dip
−0.2 at∆=0isduetothesaturationproducedbybothpump
0 5 10 15 20
and probe fields. We also present the results for fluores-
0.7
cencewhentheprobesaturatestheatomictransitionand
(c)
thestrengthofthepumpfieldisincreased. Asmentioned
0.6
earlier, I has contributions from populations and coher-
0.5 ences. These contributions are shownseparatelyin Figs.
3a, b. The Fig. 3c givesthe interferencecontributionto
0.4 I. For the direction of observation under consideration,
the interference terms are destructive. It is shown that
I
0.3 the observed dip in I at ∆= 0 is the result of the satu-
ration of the transitions by both pump and probe fields.
0.2 Further both populations and coherences contribute to
the dip.
0.1 In the presence of magnetic field, the excited levels
|±i
split by ∆ = gµB, where g is the Lande g factor, µ
B
±
0 is the bohr magneton and B is the magnetic field. Now
0 5 10 15 20
the CPT resonances and CPO resonances separate out.
∆/γ
The CPO resonances still occur at ∆ = 0. The CPT
likeresonanceswouldoccuratpositiondetermindbythe
FIG. 3. Population (a), coherence (b) terms and total mag- magnetic field. In Fig. 4, we show the behavior of I as a
nitude of the fluorescence (c) for different values of pump functionof∆for∆B =4γ andfordifferentvaluesofthe
field. Parameters used are γ+g = γ−g = γ0g = γ, Ap = 2γ, Rabi frequencies. The numerical results in Fig. 4 show
δ± =δ0 =0 and θ=π/4. thegeneraltrendseenintheexperimentaldata. InFig. 5
we presentadditionalresults for largervalue ofthe mag-
netic field. We see a resonant structure at ∆ ∆ for
B
∼
(pump) Rabi frequency proportionalto cosθ (sinθ). We low Rabi frequencies. As the pump and probe saturate
show the results for different polarization angles in Fig. theatomictransition,awelldefindinterferenceminimum
2. This figure shows pronounce differences between the is seen at ∆ ∆ . This is in agreementwith the exper-
B
∼
cases θ = π/3, θ = π/6, and θ = π/4. As the strength imentalobservation. Wealsoseeanadditionalminimum
of the probe increases, a minimum at ∆ = 0 starts ap- at ∆ ∆ /2. This additional minimum can be inter-
B
∼
pearing. For θ = π/4, the minimum at ∆ = 0 is most preted as a subharmonic resonance. We note that sub-
5
0.5
(a)
0.48
0.46

E
0.44 p El
I 0.42
0.4
0.38 FIG. 6. Schematic diagram of quantumsystem in new basis.
A =A=1 γ
p l
0.36 A =A=2 γ
p l a phase perturbation and can give rise to interferences
0.34 similar to collision induced effects [25].
0 5 10 15 20
IV. ANALYTICAL RESULTS FOR
0.19 FLUORESCENCE IN THE ABSENCE OF
(b) A =A=0.3 γ MAGNETIC FIELD
p l
0.18
Remarkably enough, the set of Eqs. 5 can be solved
analytically in the absence of the magnetic field. Let us
0.17 make a transformation to a new basis defined by
+ +
ψ =(| i |−i)cosθ 0 sinθ,
1
I 0.16 | i √2 −| i
+ +
ψ =(| i |−i)sinθ+ 0 cosθ,
2
| i √2 | i
0.15
+
ψ = | i−|−i. (11)
3
| i √2
0.14
The choice of these basis is determined by the po-
larization of the pump and probe fields. The states
0.13 ψ (i = 1,2,3) form an orthogonal set. The ψ does
0 5 10 15 20 | ii | 3i
∆ /γ not couple to either probe or pump fields. The level
ψ (ψ ) couples only to the level g by the probe
2 1
| i | i | i
(pump) field. It can also be shown that all the decay
rate γ ,γ and γ are equalto γ. In the new basis
ψ1g ψ2g ψ3g
the pump, probe fields and spontaneaus emission tran-
FIG. 5. The fluorescence in presence of magnetic field. Pa- sitions are shown in Fig. 6. We now rewrite Eqs. 5 in
rameters usedare θ=π/4and ∆B =8γ. The fieldstrengths terms of the density matrix elements in the new basis
are as shown in theboxes.
ρ = ψ ρψ . (12)
αβ α β
h | | i
Since the level ψ is decoupled, it can be dropped from
3
| i
harmonic resonanceswere extensively studied in context further consideration. The relevantdensity matrixequa-
of stimulated Raman scattering [21, 22]. The subhar- tions are
monic resonances arise from strong saturation by both
ρ˙ =iρ A iρ A γ ρ ,
pump and probe fields. However the numerical results ψ1ψ1 gψ1 l− ψ1g l− ψ1g ψ1ψ1
ρ˙ =iρ A iρ A γ ρ ,
do not quite yield the CPT resonance at ∆ = 2∆B, as ψ2ψ2 gψ2 p− ψ2g p− ψ2g ψ2ψ2
observed in the experiment. It should be borne in mind ρ˙ψ1ψ2 =−i∆ρψ1ψ2 −iρψ1gAp+iρgψ2Al
athnadt3tPhelmevaeglnse[t2i3c,fi2e4l]dacnadntchoiuspcloeutphleinlgevieslm3Pos1ttloiktehley1tPhe0 −Γψ1ψ2ρψ1ψ2,
2 ρ˙ =i(ρ ρ )A +iρ A Γ ρ ,
reason why our numerical results do not show the inter- gψ1 ψ1ψ1 − gg l ψ2ψ1 p− gψ1 gψ2
ρ˙ = i∆ρ +i(ρ ρ )A +iρ A
ference minimum at ∆ = 2∆B. It may be note that gψ2 − gψ2 ψ2ψ2 − gg p ψ1ψ2 l
the magnetic field coupling to such states would be like Γ ρ . (13)
− gψ2 gψ2
6
In driving Eq. 13, we use a different rotating frame so Clearly for no pump B = 0, I has a peak at ∆ = 0, as
thatnoexplicittimedependenceappearsinEq. 13. The can be seen from inspection or from ∂2I <0. The peak
∂∆2
state ψ1 is rotated with the pump frequency and the crosses over to a dip at a pump power given by
| i
state ψ is rotated with the probe frequency. The fluo-
2
| i
rescence in new basis is
2A2+γ2
I = I0[ρ (1 cos2θ)+ρ (1+cos2θ)]. (14) A2l > p6 (17)
2 ψ1ψ1 − ψ2ψ2
The Eqs. 13 are for V-system and one can solve for Our numerical results in Figs. 3 and 2 are in conformity
arbitary strenghs of the pump and probe fields. The full with the analytical result, Eq. 17.
solutions for ρ and ρ are
ψ1ψ1 ψ2ψ2
4A2(N +N )
ρ = l 1 2 ,
ψ1ψ1 M
V. CONCLUSIONS
4A2(N +N )
p 3 4
ρ = , (15)
ψ2ψ2 M
We have presented theoretical modelling of the ex-
where
periments on coherent population oscillations and co-
N =4A4+4A4+γ4+5γ2∆2+4∆4, herent population trapping on the intercombination line
1 p l
of 174Yb. The transition involves a change of the spin
N =4A2(γ2 2∆2)+4A2(2A2+γ2+4∆2),
2 l − p l and thus can not be interpreted in terms of an effective
N =4A4+4A4+4A2(2A2+γ2), Lambda system which was suggested in [19] using the
3 p l p l
theoretical framework of [18]. The reported experiments
N =γ2(γ2+∆2)+4A2(γ2+∆2),
4 l aredoneintheregimewherebothpumpandprobefields
M =32A6p+4A4p(24A2l +9γ2+4∆2)+(8A2l +γ2) can saturate the transition. We have shown by both nu-
(4A4+γ4+5γ2∆2+4∆4+4A2(γ2 2∆2)) mericalandanalyticalcalculationstheappearanceofthe
l l − interferenceminimumasbothpumpandprobestartsat-
+12A2(8A4+γ4+2γ2∆2+6A2(γ2+2∆2)).
p l l uratingthetransition. Wepresentananalyticalresultfor
Using analytical results Eqs. 16 and 15 we have repro- the threshold probe power when the interference mini-
duced the numerical results of Figs. 2 and 3. Now the mum can appear. We also present detailed study of the
analyticalresultisusedto find the strengthofthe pump appearance of the interference minimum when magnetic
andprobeforwhichtheinterferenceminimumwouldap- fieldsareapplied. Ourstudiesshowthenewerpossibility
pear. For θ =π/4,and ∆in the neighborhoodofzero, I oftheappearanceofthesubharmonicresonancesinsuit-
becomes ably chosen range of the pump and probe powers. The
magnetic fields not only create Zeeman splittings but in
4(A2+A2)
I = p l addition make the system open because of the couplings
8A2+8A2+γ2 tootherlevels. Suchacouplingcangiverisetoadditional
p l
16A2(2A2 6A2+γ2)∆2 resonancesinawaysimilartothedephasinginducedres-
p p− l . (16) onances.
− (2A2+2A2+γ2)(8A2+8A2+γ2)2
p l p l
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