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Superradiant light scattering from condensed and
non-condensed atoms
Aug 23, 2006
Yoshio Torii, Yutaka Yoshikawa ,and Takahiro Kuga
Institute of Physics, University of Tokyo, Komaba
US-Japan Seminar, Breckenridge, Aug23-25, 2006
Outline
• Review of superradiance in a BEC (MIT99)
• Superradiance in the short and strong pulse regime (MIT03)
• Raman superradiance (Tokyo04, MIT04)
• Superradiance in a thermal atom cloud(Tokyo05)
S. Inouye, et. al., Science 285, 571 (1999)
Superradiant Rayleigh scattering from a Bose-Einstein condensate
35µs 75µs 100µsNa BEC
Off-resonant pump light
Semi-classical explanationhq
+ =V
NN q02
Power in the end-fire mode
The amplitude ofdensity modulation:
0N qN
Bragg scattered Light(end-fire mode)
Pump beam
Ω= qNRNP 0
2
3/8sin
πθωh
2
≈Ω
ωλ
Ω= qq NNRN 0
2
3/8sin
πθ&
w2
R : single-atom Rayleigh scattering rate
Phase-matchingsolid angle
Condensate Recoiling atoms Matter wave grating
Fully-quantum picture(Fermi’s Golden Rule)
>++−− →> −=− +−
+ 1,1;1,1|,;,| 0ˆˆˆˆˆ00
qkqkcacaHqkqk nNnNnNnNkqkq
( )Ω+= 13/8
sin0
2
qq NNRNπ
θ&
Scattering rate:
Scattering process:
Condensate Recoilingatoms
Endfire modePump beam
( )( )11
|,;,|ˆ|1,1;1,1|
00
200
++=
>++−−<∝
−
−−
qkq
qkqkkqqk
nNnNnNnNHnNnNW
Summing over Ω
Spontaneousscattering
Stimulated scattering(Bosonic enhancement)
neglect
Dicke’s picture
)1()1)((
,||,
+Γ=+−+Γ=
Γ= −+
ge
N
NNMJMJ
MJJJMJW
R. H. Dicke, Phys. Rev. 93, 99 (1954)M. Gross and S. Haroche, Phys. Rep. 93,301 (1982)
N-atom system ⇔ N spin-1/2 system with the total spin J = N/2(assumption: Indiscernability of the atoms with respect to photon emission)
Spontaneous emission rate
Ω→Γ3/8
sin 2
πθR
( )Ω+= 13/8
sin0
2
qj NNRNπ
θ&
qe
g
NNNN
=
= 0
Three different picturesfor superradinace in a BEC
• Semi-classical picture (Bragg diffraction of a pump beam off a matter wave grating)
• Full-quantum picture (Bosonic enhancement by the recoiling atoms)
• Dicke’s picture (enhanced radiation from a symmetric cooparative state )
Superradiant Rayleigh scattering in a Rb BEC
Week pump light
Rb BEC
D. Schneble, Y.T., M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003)
63 mW/cm2
detuning: -4.4 GHz
30 ms TOF
25 µs-3200 µs
D. Schneble, Y.T., M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003)
Strong pump light
Rb BEC
63 mW/cm2
detuning: -440 MHz
Superradiant Rayleigh scatteringin the short (strong) pulse regime
30 ms TOF
1 µs-10 µs
Asymmetry of the X-shaped pattern
Explanation for the asymmetricX-shape
+
=
Pump beamScattered light(endfire modes) Optical grating
Kapitza-Dirac Diffractionof the matter wave
Intensity of the endfire mode
0
5
10
0.0
0.5
1.0
-8-4
84
2
-20
Pn(τ)
diffractio
n order n
Ω2R τ
∆
ΩΩ=ΩΩ=
2),()( ep
222
RRnn JP ττ
experiment theory
( )222
2e
e mW/cm6.1mW/cm8.02 ==ΓΩ
= ss IIIIntensity of the Endfire mode
Scattering rate when the endfire photon nk-q>>1
>++−−>→ −− 1,1;1,1|,;,| 00 qkpkqkpk nNnNnNnN
( )( ) ( )( )( )1
1111
0
00
++=
++−++∝
−
−−
qkqk
kqkqqkqk
nNnNnNnNnNnNW
Scattering process:
Reverse scattering process:
Net scattering rate:
>−−++>→ −− 1,1;1,1|,;,| 00 qkpkqkpk nNnNnNnN
Bosonic stimulation by the sum (not the product) of Nq and nk-q
Bosonic stimulation by the recoiling atoms Nq or the endfire photon nk-q?
( )100 ++∝ −qkq nNnNW
( )100 +∝ qNnNW ( )100 +∝ −qknnNW
1<<qN1<<−qkn
Stimulation by Nq (atom) Stimulation by nk-q (photon)
nk-q= Nq
Both pictures would give the same scattering rate!
New interpretation of superradiance (in the long pulse regime)
+
=
Endfire beamPump beam Moving optical grating
Bragg diffraction of the matter wave
Superradiant Rayleigh scattering regarded as (self-stimulated)Bragg diffraction of a matter wave off a moving optical grating
Semi-classical derivation of the Bragg scattering rate
| >g
| >e
Ωp
Ωe
)(|2/|22
222 ∆Ω= δπRW h
h
Fermi’s Golden Rule
( )22
22
2
2//)2/(
Γ+∆Γ π
cτ/12 ≡Γ Width of the two-photon (Bragg) resonance
Coherence time of the condensate
22
2e
2p
02220 /
4/ Γ
∆
ΩΩ=ΓΩ= NNW R
At two-photon resonance (∆2=0)
Normalized Lorentzian
p
E
∆
∆2
|hq>
|0>
How to express the rate W in terms of R and nk-q?
2
2p
20 4)/2(1
21
∆
ΩΓ=
Γ∆⋅Γ≅Γ= sR eeρ
Single-atom Rayleigh scattering rate:
22
2e
2p
0 /4
Γ∆
ΩΩ= NW
Γ
Ω≡ 2
2p
0
2s
Intensity of the endfire mode:
2
2e
e2ΓΩ
= sII
Saturation parameterof the pump beam
Number of photons emitted in the coherence time τc=1/Γ2:
2
2e
2e
32
ΓΓΩ
==− λπ
ωτ AAIn c
kqh
Γ
≡ 23λωπh
sI Saturation intensity(Is = 1.6 mW/cm2 for Rb D2 line)
…continued
Γ∆
=Ω2
2p
4R 2
22e A2
3ΓΓ=Ω − π
λqkn
qknNA
RNW −=Γ∆
ΩΩ= 0
2
22
2e
2p
0 23/
4 πλ
Aw
22 λλ≈
≈Ω
Ω= −qknNRW 023π
Ω= qj NNRN 0
2
3/8sin
πθ&
Semi-classical expression based on the matter wave grating
≈
Four different picturesfor superradinace in a BEC
• Semi-classical picture (Bragg diffraction of a pump beam off a matter wave grating)
• Full-quantum picture (Bosonically enhanced scattering by the recoiling atoms)
• Dicke’s picture (enhanced radiation from a symmetric cooparative state )
• self-stimulating Bragg diffraction of the matter wave off the optical grating
Analysis including propagation effects
O. Zobay and Georgios M. Nikolopoulos, PRA 72, 041604R (2005)
Changing the polarization of the pump beam
Polarization
BECPump
Pump
Polarization
F = 2 F = 1
Detuning: -2.6 GHzIntensity: 40 mW/cm2
Pulse duration: 100 µs
Y. Yoshikawa, et al., PRA 69 041603 (2004)D. Schneble, et at., PRA 69 041601 (2004)
Raman superradiance
End-firemode
| = 2, = 2F m >′
| = 2, = 2F m >
| = 1, = 1F m >
Pump beamD line: 795 nm1
δ
F
F
F
6.8 GHz
π-pol. x
y z
σ -pol.+BEC
Pump beam
End-firemode
PMT
Raman scattering gain > Rayleigh scattering gainThe only condition for Raman superradiance:
3/8sin
)cos1(163 2
Rayleigh2Raman πθ
θπRR >
+
Merits of Raman superradiance over Rayleight superradiance
•No backward scattering (K-D scattering)
•No interaction with the pump bean once scattered
Endfire modeRaman Rayleigh
Exponential growth of the Raman scattered atoms
0 10 20 30 40
0
10
20
30
40
Gro
wth
rate
[kH
z]Optical pumping rate [Hz]
0 20 40 60 80 100
0
1
2
3
4
Num
ber o
f ato
ms [
105 ]
Pump time [µs]
-2.9 GHz
-3.5 GHz
Small-signal gain:
RNRG 89083
0 =Ω=π
R: single-atom Raman scattering rate
gradient: 1022
∆ =-2.3 GHz
single-atom Raman scattering rate R [Hz]
Gtq
NNqq eNNNRN q ≈ →Ω≈ <<0
083π
&
Smal
l-sig
nal g
ain
G[k
Hz]
Y. Yoshikawa, et al., PRA 69 041603 (2004)
Where is a grating?
Pumpbeam
one photon
cloud of atoms
Pumpbeam
PMT
click!
hq
Thermal atoms
The origin of a grating (Collective mode excitation)
+
+
+
+
+... .
×0/1 N
×− )1( 0N
+
×1
hq
<>≡
>−=>=+−=
∑=
+
+
0
10
|0|1
,|1,|N
iiq
NS
JMJSJMJ
h
One atom is excited to the collectiveatomic mode defined by S+
hq
hq
hq
hq
hq
Writing, storing, and reading of a single photon
writebeam
one photon
click!
readbeam
one photon
writing storing reading
Superradiance in a Thermal gas
Y. Yoshikawa, Y. T. and T. Kuga, PRL 94 083602 (2005)
0 50 100 1500
100
200
300
Gro
wth
rate
[kH
z]
Optical pumping rate [Hz]single-atom Raman scattering rate R [Hz]
Atom cloud
PumpbeamPumpbeam
End-fire mode
PMTNA=0.19
B
0 40 80 120
Inte
nsity
[a.u
.]
Pump time [µs]
Pure condensate
Thermal gas(560 nK)
Pump pulse duration [µs]
Pure condensate
Thermal gas(T = 560 nK)
The origin of the threshold
tLGqqq eNNLGN )()( −∝→−=&
GL >
LG >R
G-L
-L
Rth
: No exponential growth
Loss term
: exponential growth with a growth rate of G-L
cL τ/1=
coherent time of the system
What determines the coherence time?
a) The endfire mode b) matter wave grating(overlap of the wave packets)
Coherence time is given by the inverse of the Doppler width
Doppler width:
vq=∆ Dω
=
mTkv B
RMS velocity
mqv h
=
hh
vmpx =∆
=∆
vqvx 1
c =∆
=τvq11
Dc =
∆=
ωτ
Measurement of the coherence time
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1.0
0 200 400 6000
0.5
T = 280 nK
T = 280 nK
Sign
al ra
tio
Duration of the dark period [µs]
T = 760 nK
Coherence time vs. temperature
0 200 400 600 800 1000 1200 14001
10
100
1000
τ c [µs]
Temperature [nK]
Tc
280 µsPure thermalbimodal
Pure condensate
Tkm
vk
B
c
4
21
πλ
τ
=
=
Conclusion
• The behavior of superradiance in the short and strong pulse regime has led to a new picture of superradiance (optical stimulation)
• The study of superradiance in a thermal gas showed that a thermal gas will act as a pure condensate within a time scale shorter than the coherence time, which is determined by the Doppler effect.
spacer
Photon picture of K-D diffraction
p
∆
| >hq
|0>
| >g
| >eE = p / m2 2
Ωp Ωe
p
|0>
| >g
| >eE = p / m2 2
|- >hq
∆
ΩeΩp
∆Ε
scattering a pump photoninto the endfire mode
(Na) kHz100(Rb) kHz15
2 rec ××
==∆hh
E ωh
(stimulated) scattering of a pumpphoton into the pump beam
Forwardpeak
backwardpeak
K-D diffraction with a focused pump beam
10 ms TOF
Focusedpump beam
BEC
Velocity and spatial distribution before and after the SRS
0 0.5 1.0 1.5 2.0
0
1
2
3
4
5
F = 1
position [mm]
F = 2(O.D.×0.5)
(b)
(a) (c)yz
Opt
ical
den
sity
T = 473 nK
T = 544 nK
0 0.2 0.4 0.6 0.8 1.00
2
4
z - position [mm]
(O.D.×0.1)
Velocity distribution spatial distribution
before SRS
after SRS
before SRSafter SRS