Nonlinear spectroscopy of cold atoms in diffuse laser light.
ABSTRACT The nonlinear spectroscopy of cold atoms in the diffuse laser cooling system is studied in this paper. We present the theoretical models of the recoilinduced resonances (RIR) and the electromagneticallyinduced absorption (EIA) of cold atoms in diffuse laser light, and show their signals in an experiment of cooling (87)Rb atomic vapor in an integrating sphere. The theoretical results are in good agreement with the experimental ones when the light intensity distribution in the integrating sphere is considered. The differences between nonlinear spectra of cold atoms in the diffuse laser light and in the optical molasses are also discussed.

Article: Testing the equivalence between the canonical and Minkowski momentum of light with ultracold atoms
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ABSTRACT: We design an experimental system to test the equivalence between the canonical and Minkowski momentum of light, which can provide a judgment on the recent resolution of the centuryold AbrahamMinkowski controversy by S. M. Barnett [ Phys. Rev. Lett. 104 070401 (2010)]. By measuring the recoil momentum of ultracold rubidium atoms in a rubidium BoseEinstein condensate after the electromagnetically induced absorption of a monochromatic laser pulse, the momentum of the pulse in the ultracold atoms can be obtained. If the equivalence is valid, the measured results will coincide with the theoretical values of the canonical momentum of the pulse. Otherwise, if the equivalence is invalid, the measured results will coincide with the theoretical values of the Minkowski momentum, which are significantly different from that of the canonical momentum. Our scheme is amethod to test the equivalence between the canonical and Minkowski momentum of light. It can also be improved to distinguish between the Minkowski and Abraham momenta to contribute to the study of the AbrahamMinkowski controversy in the future.Physical Review A 05/2012; 85(5). · 2.99 Impact Factor
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Nonlinear spectroscopy of cold atoms in
diffuse laser light
WenZhuo Zhang1,2, HuaDong Cheng1, Ling Xiao1, Liang Liu1†, and
YuZhu Wang1
1Key Laboratory of Quantum Optics, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China.
2Graduate University of the Chinese Academy of Sciences, Beijing 100039, China.
†Corresponding author: liang.liu@siom.ac.cn
Abstract:
cooling system is studied in this paper. We present the theoretical models of
the recoilinduced resonances (RIR) and the electromagneticallyinduced
absorption (EIA) of cold atoms in diffuse laser light, and show their signals
in an experiment of cooling87Rb atomic vapor in an integrating sphere.
The theoretical results are in good agreement with the experimental ones
when the light intensity distribution in the integrating sphere is considered.
The differences between nonlinear spectra of cold atoms in the diffuse laser
light and in the optical molasses are also discussed.
The nonlinear spectroscopy of cold atoms in the diffuse laser
© 2009 Optical Society of America
OCIS codes: (300.6210) Spectroscopy, atomic; (190.4180) Multiphoton processes; (140.3320)
Laser cooling.
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1.Introduction
Cooling of atoms in the diffuse laser light is an all optical laser cooling technique. In the diffuse
laser light, an atom with velocity? v resonates with the photons whose propagating directions are
at an angle θ with? v, and
Δ−kvcosθ = 0,
where Δ is the laser detuning. The diffuse laser light can cool more atoms than optical molasses
does due to the large resonant velocityrange (kv≥Δ/k). Because of its all optical configuration
and wide cooling velocityrange, diffuse laser light is not only a laser cooling method besides
the optical molasses and the magneticoptical trap (MOT), but also a good choice in magnetic
fieldfree cases such as the development of a compact coldatom clock [1].
Slowing and cooling atomic beams in diffuse light was first experimentally realized by Ket
terle et al. in 1992 [2] and was succeeded by Batelaan et al. in 1994 [3]. An idea that cooling
atomic beam in an integrating sphere in which the diffuse light field can be formed was first
proposed by Y. Z. Wang in 1979 [4] and was realized in 1994 [5]. In this paper, we produce the
diffuse laser light in a ceramic integrating sphere to have the threedimensional cooling of87Rb
atomic vapor [6]. It can be compared with the 3D cooling experiment of cesium atomic vapor
in a copper integrating sphere [7].
Nonlinear spectroscopy of cold atoms in an opticalmolasses as well as in a MOT has been
widely studied [8, 9, 10, 11, 12], but in the diffuse laser light case it has not been reported
(1)
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2
1/2
5 S
2
3/2
5 P
F=2
F=3
F
m =
3
2
1
0
1
2
3
F
m =
2
1
0
1
2
Fig. 1. Pump transition and steadystate population of every ground states of87Rbin diffuse
laser lights
before. The diffuse laser light is monochromatic and is generated by reflected laser beams from
Lambertianreflectance surface. It acts as both the cooling light and the pump light to the cold
atoms in the pumpprobe configuration. As we know, diffuse reflectance can not change the
temporal coherence of monochromatic light, but the Lambertian reflectance can disorder the
wavefront of the light and break its spatial coherence [13]. For the pumpprobe configured
nonlinear spectroscopy, the phase of pump and probe lights need to be correlated well, then
their frequency difference (ω1−ω0) can oscillate with the atoms, and twophoton process can
happen. Here ω1is the frequency of probe laser light and ω0is the frequency of diffuse pump
laserlight.Usuallypumplightandprobelightarefromthesamelasersource,sotheyhavesame
timedepended phase shift φ(t) and they are well correlated. Fortunately, the diffuse pump light
hasastableφ(t)tothelasersourcebecausethetemporalcoherenceisnotbrokenbyLambertian
reflection. The reason is the phase shift caused by the mechanical vibration of the Lambertian
surface is tiny compared with φ(t) so it can be neglected, and thus the diffuse pump light is
well correlated to the laser source, as well as the probe light. It is necessary for the nonlinear
spectroscopy of cold atoms in diffuse light.
Themaindifferencesbetweennonlinearspectroscopyofcoldatomsinthediffuselightandin
theopticalmolassesarethelightshiftandthesteadystatepopulationofgroundstatesublevels.
Because the diffuse laser light is depolarized by the Lambertian reflection [14], its polarization
distribution is totally random. In this paper we assume an optimum condition that the probabili
ties of σ+, σ−, and π transition of atoms in the diffuse laser light are equal, and the differences
among the steadystate population of all ground state sublevels are quite small. Figure 1 shows
the steadystate population and the transition of such optimum condition.
We focus on two kinds of twophoton process which are most possible to lead to pump
probe configured nonlinear spectra of cold atoms in diffuse laser light. One is recoilinduced
resonance, whose introduction and theoretical model in diffuse laser light pumped case are
showed in Sec. 2. The other is electromagnetically induced absorption (EIA), whose theoretical
model in diffuse laser light pumped case is showed in Sec. 3. Sec. 4 is our experimental setup
and the suitable condition for nonlinear spectroscopy study of cold87Rb atoms in diffuse laser
light. The experimental results of RIR and EIA are compared with the theoretical ones in Sec.
5, where the influence from the intensity distribution of diffuse laser light is considered. Finally,
the possible stimulated Raman process, and the differences between nonlinear spectra of cold
atoms in diffuse laser light and in an optical molasses as well as in a MOT are discussed.
2.Recoilinduced resonances of cold atoms in diffuse pump light
Recoilinduced resonances (RIR) was first theoretically predicted by Guo et al. in 1992 [15,
16]. Its signal appears as a derivative line shape in pumpprobe spectra, which can be used
to measure the velocity distribution of cold atoms [17]. The first experimental observation of
recoilinduced resonances is obtained by Courtois et al. in 1994 [18], where the pump laser is
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a 1D optical molasses and the probe laser beam has a small angle with respect to it.
Because the diffuse laser light is depolarized by the Lambertian reflection, and the atoms
also have a threedimensional distribution, pump and probe laser light can make atoms have
all the three kinds of transitions (σ+, σ−and π). For a twolevel atom, relative light shift
between every two sublevels of ground state do not infect the line shape of recoilinduced
resonance signal [15], but only infect the detuning of the pump laser a little. So when the
detuning of the diffuse pump laser light ω0is much larger than all the light shift among every
ground state sublevels, the atomic system is approximated to a twolevel system for the recoil
induced resonances. The interaction Hamiltonian of pumpprobe lights interacting with a two
level atomic system is
HI(ω0,1) = ¯ hΩ0,1e??gcos(k0,1·X−ω0,1t)+ ¯ hΩ∗
where Ω0is the Rabi frequency of pump field with wave vector k0and frequency ω0. Ω1is the
Rabi frequency of probe field with wave vector k1and frequency ω1. e? is the excited state and
g? is the ground state. Atomic density matrix can be expanded to the basis of the internal states
a? = e?,g? and external centerofmass momentum states p?
ρ =∑
0,1g??ecos(k0,1·X−ω0,1t),
(2)
a,a?ρaa?a,p??a?,p?.
(3)
Then we obtain the time evolution equations for all atomic density matrix elements [15]:
d
dt˜ ρee(p,p?) = −(Γ+γ)˜ ρee(p,p?)
1
∑
a=0
1
∑
a=0
+i
Ω∗
aexp[i(Δa+ωr−ka·p?
m
)t]ρeg(p,p?− ¯ hka)
−i
Ω∗
aexp[i(−Δa−ωr+ka·p
m
)t]ρge(p− ¯ hka,p?),
(4)
d
dt˜ ρgg(p,p?) = −
?
ΓN(q)dq˜ ρee(p+ ¯ hq,p?+ ¯ hq)exp(ip−p?
m
·qt)
−γ ˜ ρgg(p,p?)+γW(p,p?)
+i
∑
a=0
1
∑
a=0
dt˜ ρge(p,p?) = −Γ
1
∑
a=0
1
∑
a=0
d
dt˜ ρeg(p,p?) = [d
1
Ω∗
aexp[i(Δa+ωr−ka·p?
m
)t]ρgg(p,p?− ¯ hka)
−i
Ω∗
aexp[i(Δa−ωr−ka·p
m
)t]ρee(p+ ¯ hka,p?),
(5)
d
2˜ ρge(p,p?)
aexp[i(Δa+ωr−ka·p?
+i
Ω∗
m
)t]ρgg(p,p?− ¯ hka)
−i
Ω∗
aexp[i(Δa−ωr−ka·p
m
)t]ρee(p+ ¯ hka,p?),
(6)
dt˜ ρge(p?),p]∗.
(7)
Here
˜ ρaa?(p,p?) = ρaa?(p,p’)e[(p2−p?2)t/2m¯ h]eiωaa?t,
(8)
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Δ0is the detuning of pump laser and Δ1is the detuning of probe laser, which are given by
Δ0,1= ω0,1−ωeg,
(9)
ωegisthe transition frequency from ground state g? to excited state e?,and the recoil frequency
is
ωr=¯ hk2
2m.
(10)
Γ is the decay rate of excited state and γ is the decay rate due to time of flight which is much
smallerthanΓ.W(p,p?)isthemomentumdistributionofatomsatgroundstate,whichisusually
considered as a MaxwellBoltzman distribution. Another momentum distribution N(q) is the
normalized probability density for emitting a photon with momentum ¯ hq, which for a two level
atom is given by [15]
N(q) =
8πsin2θ,
where θ is the angle between ¯ hq and field polarization direction.
The absorption signal is proportional to the imaginary part of the component of ˜ ρge, which is
3
(11)
˜ ρge= ρge(x,t)exp(ik1·x−iω1t)
=
(2π¯ h)3
1
? ?
dpdp?
?
i(p−p?)·x
¯ h
−i(p2− p?2)
2m¯ h
t +iωt
?
˜ ρge(p,p?)exp(ik1·x−iω1t).
(12)
It can be solved from the thirdorder perturbation solutions of Eqs. (4)(7) in the limit that
perturbation treatment is valid [15].
Ω2
γΔ1
ωr
k1v< 1 (13)
In diffuse laser cooling, the pump laser light is isotropic so the interaction range only depends
on the spot size of probe beam. Condition Eq. (13) needs that the spot size of probe laser is
small. It is because only if the interaction range is small enough does the decay rate due to time
of flight γ can be large enough to fit limit Eq. (13).
Under the limits
Δ ? Γ/2 ? ω1−ω0
kv ? γ
¯ hΔ ? kbT
θ ≈ 0,
(14)
where kbis Boltzman constant, the simplified general expression of ˜ ρgefor arbitrary wave
vector of pump laser k0= k0 and probe laser k1= k1 is expressed by
˜ ρge= −2iΩ∗
Γ+2iΔ1
Γ2+4Δ2
0
(k2
1N0
4Ω02
?
8iπωrΔ0
1+k2
0+2k1k0cosθ)v2p2W?(y)−2
Γ−
4iΓωr
(Γ+2iΔ1)γ−2
?
, (15)
where N0is the atomic number, θ is the angle between the pump and the probe laser beams, v
is the atomic velocity, and
y = −
2p(ω1−ω0)
1+k2
?
k2
0+2k1k0cosθ ·v
.
(16)
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?
??
??
k1
k0
xe
ye
y p
xp
Fig. 2. Scheme of recoilinduced resonance. Probe laser (k1,ω1) travels along the direction
ex, The another beam (k0,ω0) is the one of the isotropic laser lights which can cause recoil
induced resonance of the atom with the probe laser.
Here W?(y) is the firstorder derivative of onedimensional momentum distribution W(y).
We are interested in the case of diffuse laser light pumped system, where the propagating di
rection of pump light is isotropic. The imaginary part of Eq. (15) is approximate to Eq. (17)un
der the condition of Eq. (13) and Eq. (14), where m is the atomic mass.
Im(˜ ρge) =2πN0Ω∗
1Ω02ωrm2
1k2(1+cosθ)
Δ2
W??−pω1−ω0
2kv√1+cosθ
?
−2N0Γ2Ω∗
1Ω02ωr
Δ5
1γ
.
(17)
Because of the isotropic distribution of the pump laser, we need to find a critical angle θc
that the pump beams within the range of θ < θccan have recoilinduced resonance with the
majority of the cold atoms. θccan be calculated by the energy conservation of initial and final
states of the recoilinduced resonance. Recoilinduced resonance is a twophoton process with
one photon being absorbed and the other being stimulatingly emitted. Its scheme is showed
in Fig. 2. Probe laser beam (k1,ω1) propagates through the direction ˆ ex, and one beam of the
isotropic pump laser light (k0,ω0) has the angle θ to it. For an atom with initial momentum px
in direction ˆ exand momentum pyin direction ˆ ey, pxand pxare all positive. The momentum
change of the twophoton process is (¯ hk1+ ¯ hk0cosθ)ˆ ex+ ¯ hk0sinθ ˆ ey. Because of the energy
conservation, its corresponding energy change needs to be equal to the energy change of the
light field 2π¯ h(ω1−ω0). Then we obtain
4πm(ω1−ω0) =2k1px+2k0(pxcosθ − pysinθ)
+ ¯ h(k2
1+k2
0+2k1k0cosθ).
(18)
For atoms with temperature being above Doppler cooling limit, ¯ h(k2
small compared with k0,1px,yand can be neglected. Because ω = ck/2π, where c is the speed
of light, k0 and k1 need to be very close to each other, then 4πm(ω1−ω0) may have the same
order with 2k(0,1)p(x,y).
The importance of angle θ can be analyzed with Eq. (18). In the recoilinduced res
onance range, k0≈ k1. If we make the transform px→ −pysinθ/(1 + cosθ) and py→
−px(1+cosθ)/sinθ, Eq. (18) is approximately unchanged. When θ = π/2, the transform
means px= −pyand W(px)−W(py) = W(px)−W(−px) = 0. If we assume px> 0, mo
mentum distribution W(−px) > W(−px+ ¯ hk1) so the transform W(−px) → W(−px+ ¯ hk1)
means the strengthened absorption. However, w(px) → w(px+ ¯ hk1) means the stimulated am
plification of probe beam. The two signals are just canceled by each other so the total sig
nal of recoilinduced resonance can not be observed. When θ ≈ 0, the result is px? −py
and W(px)−W(py) = W(px)−W[−px(1+cosθ)/sinθ] is always nonvanishing, then we
can observe the RIR signal. We need to determine a critical angle θc, for which the condi
tion px ? − pysinθ is satisfied in the region of 0 < θ < θcand the RIR signal is evi
dent. We choose the minimum momentum limit px= 2¯ hk, which is recoil momentum of two
photons, and choose pyequals to twice of the most probable momentum√2mkbT, with which
1+k2
0+2k1k0cosθ) is quite
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Fig. 3. Calculated probe absorption signal of recoil induced resonance in diffuse pump
field. Γ = 6.056MHz (87Rb, Fg= 2 → Fe= 3), γ = 0.05Γ, Δ0= −3Γ, T = 200μK, S0= 2
and S1= 0.01.
0 < ? v < 2√2mkbT contains the majority of the cold atoms. Then θccan be calculated from
sinθc
1+cosθc
=
¯ hk1
2√2mkbT
(19)
So the solution of θcis small, which just meet the condition θc≈ 0 of deriving Eq. (17). In
the range 0 < θ < θc, for all px> 2¯ hk and py<√8mkbT, the W(px)−W(py) is always non
vanishing. The total signal for isotropic pumped recoilinduced resonance is an integration of
Eq. (17) over −θcto θc. Under the condition of Eq. (14), we calculated the average of the
recoilinduced resonances over the range −θc≤ θ ≤ θc. The result is showed in Fig. 3 with m
being chosen as the mass of a87Rb atom.
3.Electromagnetically induced absorption of cold atoms in diffuse light
Electromagnetically induced absorption (EIA) [19] is a nonlinear optical effect due to the
atomic coherence. Pumpprobe configured EIA [20, 21, 22] can happen when a strong pump
laser and a weak probe laser are interacting with a degenerated twolevel atomic system which
satisfies Fe> Fg> 0 [19, 20]. It leads to a sharppeak enhancement effect in the absorption
spectrum at the position where probe laser resonates with the pump one. It is just opposite to
the electromagnetically induced transparency (EIT) which leads to a sharpdip attenuation at
the pumpprobe resonating position. Pumpprobe configured EIA was studied in details and
classified into two cases: EIATOC (transfer of coherence) [23] and EIATOP (transfer of pop
ulation) [24]. The EIATOC requires the pump and probe laser have different polarizations and
the EIATOP requires that they have the same polarization. Some interesting phenomenon in
Hanle configured EIA was also studied recently [25, 26].
Most experimental researches of EIA were carried in atomic vapor cells at room temperature.
In fact, the cold atoms are more suitable to study atomic coherence. The first EIA of cold atoms
was observed by Lipsich et al. in a MOT [27]. Although the MOT is the most common method
to obtain cold atoms, the small interaction region and the strong magnetic field restrict itself to
be an optimum choice for pure pumpprobe configured EIA studies.
In diffuse laser cooling, atoms can easily be cooled to the temperature near its Doppler limit,
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and no magnetic field is added. The diffuse laser light then becomes a strong pump field with
random polarization. When an arbitrary polarized probe beam is added, the EIATOC happens
easily. For theoretical study, the model of pumpprobe configured EIA of stationary atoms is
also more suitable for cold atoms in diffuse laser light than roomtemperature atoms.
In our model, we select the Fg=2→Fe=3 transition of87Rb atoms as the cooling transition
because this transition is used for the diffuse laser cooling of87Rb atomic vapor in our exper
iment. The cooling laser, which is also the pump one, interacts with the cold atoms together
with the probe laser, then the interaction Hamiltonian for every two Zeeman sublevels in the
rotating wave approximation are
HI
eigj= HI
eigj(ω0)e−iω0t+HI
eigj(ω1)e−iω1t,
(20)
where HI
eigj(ω0,1) is the Rabi frequency for ei→ gjtransition.
HI
eigj(ω0,1) = μeigjE0,1= ¯ h(−1)Fe−me
?
Fe
−me
1
q
Fg
mg
?
Ω0,1.
(21)
Here ω0is the frequency of pump laser and ω1is the frequency of probe laser. Ω0,1are Rabi
frequency caused by the two laser.
Time evolution equations of every density matrix element can be solved in two stages [23].
The first stage contains only the strong pump laser, and the equations are showed in Eqs. (22)
(24).
d
dtρeiej(ω0) =−(Γ+iωeiej)ρeiej(ω0)
+i
¯ h
2
∑
k=−2
[ρeigk(ω0)HI
gkei(−ω0)−HI
eigk(ω0)ρgkei(−ω0)],
(22)
d
dtρeigj(ω0) =−[Γ+Γgi
2
3
+i(ωeigj−ω0)]ρeiej(ω0)
+i
¯ h
∑
k=−3
[ρeiek(ω0)HI
ekgj(−ω0)−i
2
∑
k=−2
HI
eigk(ω0)ρgkei(−ω0)],
(23)
d
dtρgigj(ω0) = −iωgigjρgigj(ω0)+7Γρs
+i
¯ h
k=−3
gigj(ω0)
3
∑
[HI
eigk(ω0)ρgkei(−ω0)−ρeigk(ω0)HI
gkei(−ω0)],
(24)
where
ρeiej(ω0) = ρeiejexp(iω0t),
(25)
and
ρs
gigj(ω0,1) =
1
∑
q=−1
?
3
∑
k,l=−3
Fg
−mgi
(−1)−mk−ml
×
1
q
Fe
me
?
ρekel(ω0,1)
?
Fe
−me
1
q
Fg
mgi
?
,
(26)
which is the spontaneous emission term. In Eqs. (22)(24), Γ is the decay rate of excited state
Fe= 3, and Γgiis the decay rate from ground state sublevel gi. Because87Rb atom has another
ground state hyperfine level Fg= 1 whose energy is 6.8GHz lower than Fg= 2, the Zeeman
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sublevels of Fg= 2 can decay to Fg= 1 with rate gi. The repumping laser is needed to pump
the population from Fg=1 back to Fg=2. Another decay rate is the γ due to time of flight in the
interaction range between atoms and pumpprobe light. In diffuse cooling system the diffuse
light and the cold atoms distribute all over the cavity, so atoms can always in the interaction
range of the diffuse laser light. After the probe laser beam is added, the interaction range is only
determined by the probe laser.
In the second stage, a weak probe laser is added as a perturbation to the system. The density
matrix elements of population ρeiejand ρgigjcan oscillates with frequencies ω1−ω0and ω0−
ω1, while the coherence term of ground and excited state is
ρeigj= ρeigj(ω0)exp[−iω0t +iφ]+ρeigj(ω1)exp[−iω1t].
Then we can get the optical Bloch equations of the second stage
(27)
d
dtρeiej(ω1−ω0) =−[Γ+γδeiej−i(ω1−ω0−ωeiej)]ρeiej(ω1−ω0)
+i
¯ h
k=−2
dtρeigj(ω1) =−[Γ+Γgj
2
+i
¯ h
k=−3
+i
¯ h
k=−3
d
dtρgigj(ω1−ω0) =−[Γgi+Γgj
2
2
∑
[ρeigk(ω1)HI
gkei(−ω0)−HI
eigk(ω1)ρgkei(−ω0)],
(28)
d
+i(ωeigj−ω1)]ρeiej(ω1)
3
∑
ρeiek(ω0)HI
ekgj(ω1)−i
2
∑
k=−2
HI
eigk(ω1)ρgkei(−ω0)
3
∑
ρeiek(ω1−ω0)HI
ekgj(ω0)−i
2
∑
k=−2
HI
eigk(ω0)ρgkei(ω1−ω0),
(29)
−i(ω1−ω0−ωgigj)]ρgigj(ω1−ω0)
2
∑
k=−2
+7Γρs
gigj(ω1−ω0)+
Γgigkδgigjρgkgk(ω1−ω0)
+i
¯ h
3
∑
k=−3
[HI
eigk(ω1)ρgkei(−ω0)−ρeigk(ω1)HI
gkei(−ω0)].
(30)
There are also another three equations that describe the time evolution of the density matrix
elements at the frequency ω0−ω1, which can be written via replacing ω1−ω0by ω0−ω1in
Eqs. (28)(30).
The probe absorption intensity is proportional to
Im[∑
i,j
HI
eigj(ω1)ρeigj(ω1)],
(31)
whose steadystate value can be solved from Eqs.(28)(30). There are relative light shifts ωgigj
between each two groundstate Zeeman sublevels in Eq. (30), which can shift the position
of EIA signal. For diffuse laser light, we have assumed that the atoms have same probabil
ities for σ+, σ−, and π transitions, then the light shift of every groundstate Zeeman sub
level δgican be calculated from the ClebschGordan coefficient given in Fig. 4. The square
of these coefficients give the probabilities of corresponding transitions. Relative light shifts of
groundstate Zeeman sublevels are ωgigj= ¯ h(
(Ωgie(i−1)+Ωgiei+Ωgie(i+1))/3 is the average Rabi frequency of σ+, σ−, and π transitions. Rabi
?
Δ2+Ω2gi−
?
Δ2+Ω2gj)/2 [28], where Ωgi=
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2
1/2
5 S
2
3/2
5 P
F=2
F=3
F
m =
3
2
1
0
1
2
3
F
m =
2
1
0
1
2
1
30
1
30
1
2
1
2
1
6
1
6
1
3
1
3
1
5
1
5
4
15
4
15
3
10
1
10
1
10
Fig. 4. ClebschGordan coefficients and light shifts of every sublevels for Fg= 2−Fe= 3
transition.
Fig. 5. Calculation results of probe absorption. Γ = 6.056MHz (87Rb, Fg= 2 → Fe= 3),
γ = Γg= 0.05Γ, Δ0= −3Γ, S1= 0.1, S0= 1,5,10.
frequencies Ωgie(i,i±1)are proportional to the square of ClebschGordan coefficients of corre
sponding transitions, so after the calculation we find that Ωg1= Ωg−1= Ωg2= Ωg−2< Ωg1,
then the terms of ωg(±1,±2)g(±1,±2)are all equal to zero. Only ωg0g(±1,±2)is nonvanishing and
contributes to the relative light shifts of groundstate Zeeman sublevels in Eq. (30).
Figure 5 shows the calculated result of EIA signal of cold atoms (v ≈ 0) under the relative
light shifts ωgigjand an absolute light shift δ in Fg= 2?
is the saturation parameter, subscript “0” denotes to the pump light and ”1” denotes to the probe
light. As we know the intensity of isotropic pump laser lights is much higher than the probe
laser light, so the light shift is mainly determined by the pump laser light. Ωgie(i,i±1)equals to
Ω0multiplied by ClebschGordan coefficients of corresponding transitions in Fig. 4, while δ
can be obtain directly from Ω0in dressed atom picture, that is
shifts can be calculated from S0.
? Fe= 3 transition. S0,1= 2Ω0,12/Γ2
?
Ω2
0+Δ2
0−Δ0[29], so all light
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Digital
oscilloscope
Integrating
sphere
Glass
cavity
Lightbalance
circuit
LF356

+
Mutimode fibers
Laser
for cooling and probe
Laser
for repumping
???
???
???
Fig. 6. Experimental setup of the diffuse cooling87Rb atomic vapor in an integrating
sphere.
4.Experimental setup
Figure 6 shows the experimental setup of diffuse laser cooling of87Rb atoms. The87Rb atomic
vapor is filled in a spherical glass cavity which connects to an ion pump. The vacuum in the
cavity is about 10−9Torr. A ceramic integrating sphere is settled surrounding the spherical glass
cavity. The cooling and probe lasers for are from one Toptica TA100 semiconductor laser, while
the repumping beam is from a Toptica DL100 laser. The cooling and repumping beam enter
the integrating sphere from two multimode fibers and generates a diffuse light field via being
reflected by the integrating sphere. The inner diameter of the integrating sphere is 48mm and the
diameter of the spherical glass cavity is 45mm. The cooling laser is locked with detuning Δ0to
the Fg=2→Fe=3 transition frequency, and the repumping laser is lock to the Fg=1→Fe=2
transition.
The probe system is made up of a probe laser beam, a detector, and a digital oscilloscope.
The probe laser is split from cooling laser so the phase of pump and probe lights are highly
correlated. The detector is a lightbalanced amplification circuit with two photodiodes, one
receives the probe laser beam that propagates through the spherical glass cavity and the other
receives the laser beam from probe laser directly. We sweep the frequency of the probe laser
with AOM to cover the transmission signals. With the detector, the amplified signals can be
obtained and seen in the digital oscilloscope.
The reflectance of our integrating sphere is about 98%, and the aperture area of the integrat
ing sphere is 2.2% of its whole inner surface area. It means a photon can be averagely reflected
about 24 times when its remanent probability decays to 1/e. Because the diameter of our in
tegrating sphere is 48mm, a photon can travel at most 116cm for 24 time reflections, which is
very small compared with the coherent length of semiconductor laser (usually several hundred
meters), and the temporal coherence of diffuse pump light is then well maintained. The inte
grating sphere here can also randomize the polarization of diffuse cooling laser [30], so atoms
have all σ+, σ−, and π transitions, which are necessary for the EIATOC [23].
5.Results and discussions
Figure 7 is the absorption signals varying with detuning of the probe laser Δ1. The large ab
sorption peak around Δ1−Δ0= 0 is the signal of F = 2 → F = 3, which is not exactly on the
zero position due to the light shift caused by the diffuse laser light. The position of the non
linear spectra is around the detuning of cooling laser Δ0. The strength of the F = 2 → F = 3
absorption signal and the nonlinear signal are highest at Δ0= −3Γ because at this detuning the
largest number of87Rb atoms are cooled and captured in our experiment system [6]. Figure
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Fig. 7. Experimental signal varying with the detuning of probe laser light at three different
diffuse light detunings: (a) Δ0= −2Γ, (b) Δ0= −3Γ, (c) Δ0= −2Γ. Power of injected
cooling laser beams are 40 mW/cm2.
Fig. 8. Experimental signal varying with power of injected cooling laser: (a) 40 mW/cm2,
(b) 32 mW/cm2, (c) 24 mW/cm2, (d) 16 mW/cm2. Δ0= −3Γ.
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Fig. 9. (a) The two dot line are the calculated signals of recoilinduced resonances when
S0= 2.5, S1= 0.03 and EIA when S0= 3.80, S1= 0.03. Their sum is the solid line. (b)
Experimentally observed signal under Δ0= −3Γ when the total power of injected cooling
laser beams is 40 mW/cm2.
8 is the absorption signal varying with the power of cooling laser that injected into the inte
grating sphere. We can see two phenomena in the nonlinear spectra. First, the position of the
amplification peak and the small absorption peak do not change with the power of the diffuse
light, which is the feature of recoilinduced resonances signal because the width of RIR signal
only depends on the velocitydistribution of cold atoms [15]. Second, the light shift leads to
a deviation of the absorption peak of F = 2 → F = 3 transition, as well as the Δ1−Δ0= 0
position. The deviation is proportion to the power of diffuse laser light. This is just the feature
of EIA that discussed in Sec. 3.
Figure 9 compares the theoretical and experimental results of nonlinear spectra in diffuse
laser light. The two dot line in Fig. 9(a) are calculated from theoretical model of RIR in Sec. 2
and EIA in Sec. 3 by BeerLambert law. Solid line in Fig. 9(b) is our experimentally observed
signal. We see an interesting result that when the power of pump laser of EIA is about 1.5
times to that of recoilinduced resonances, the theoretical result matches the experimental data
well. The reason is in our experiment, the two injected cooling laser beams are not diffuse light
before the firsttime reflection by the inner surface of integrating sphere. Figure 10 describes
the distribution of lightfield intensity in the integrating sphere. Intensity of the two injected
cooling laser beam is higher than that of the diffuse laser light in the center of the sphere in
our experiment. The two beams are approximately vertical to the probe laser beam, so they do
not participate in the RIR, but they still participate in the EIA. Then we can see the signal is
observed under the condition that intensity of pump laser in EIA is about 1.5 times to it in RIR.
The two counterpropagating injected laser beams are linearly polarized. If we replace them
by two σ+σ−configured laser beams, an onedimensional optical molasses will be formed.
The light shift caused by the σ+σ−onedimensional optical molasses will be large enough
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Probe laser
Cooling laser
Cooling laser
b
b
a
Fig. 10. lightfield distribution in the integrating sphere, the injected laser are reflected by
the inner surface of the integrating sphere to create diffuse laser light for laser cooling.
Before the firsttime reflection, the injected laser are two expanded beams due to the fibers
have a the numerical aperture. Through the light path of the probe beam, the two expanded
beams and the diffuse laser light are all pump light in region (a), while only diffuse laser
light are the pump light in region (b).
to cause significant population difference among all groundstate Zeeman sublevels of cold
atoms [9]. For cold87Rb atoms in onedimensional optical molasses, Fg= 2,mF= 0 has the
largest weight of population and the lowest energy, so stimulated Raman process can happen
and its signal may be observed [9, 10].
Another feature of pumpprobed nonlinear spectra of cold atoms in diffuse laser cooling
system is the much stronger signals than that in optical molasses. The reason is that their inter
action ranges of the cold atoms and pumpprobe laser lights are different. In our experiment,
the diffuse laser light, as well as the cold atoms, distribute all over the integrating sphere, so
cold atoms can be pumped and probed coherently through the whole light path of the probe
laser within the integrating sphere. However, the pumpprobed interaction range in an optical
molasses is the overlap range of probe beam and 3D optical molasses. For same spot size probe
beams, the pumpprobe interaction range in diffuse laser light is much larger than it is in opti
cal molasses, so it makes diffuse laser cooling method as an optimum technique in studying the
nonlinear spectroscopy of cold atoms, as well as their application.
6.Conclusions
In conclusion, we have studied the recoilinduced resonances (RIR) and the electromagnetic
induced absorption (EIA) of cold atoms in diffuse laser light. We present their completed the
oretical models and observe their compound signal of cold87Rb atoms which are cooled and
pumped by the diffuse laser light in an integrating sphere. Theoretical result can match the ex
perimental one well when the intensity of pump laser light of EIA needs to be 1.5 times to that
of RIR, which is because the injected cooling laser before firsttime diffuse reflectance only
participates in the EIA, while the diffuse laser light participates in both EIA and RIR. Compar
ing with such two nonlinear spectra of cold atoms in optical molasses, we show the feature of
diffuse light case is the much larger pumpprobe interaction range. It can provide much stronger
signals of the RIR, as well as EIA, which may benefit their future applications.
Acknowledgment
This work is supported by the National Nature Science Foundation of China under Grant No.
10604057 and National HighTech Programme under Grant No. 2006AA12Z311.
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