Superradiant light scattering from condensed and non...
Transcript of Superradiant light scattering from condensed and non...
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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
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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)
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S. Inouye, et. al., Science 285, 571 (1999)
Superradiant Rayleigh scattering from a Bose-Einstein condensate
35µs 75µs 100µsNa BEC
Off-resonant pump light
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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
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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
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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
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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 )
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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
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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
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Asymmetry of the X-shaped pattern
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Explanation for the asymmetricX-shape
+
=
Pump beamScattered light(endfire modes) Optical grating
Kapitza-Dirac Diffractionof the matter wave
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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
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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
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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!
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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
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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>
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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)
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…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
≈
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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
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Analysis including propagation effects
O. Zobay and Georgios M. Nikolopoulos, PRA 72, 041604R (2005)
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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)
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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 >
+
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Merits of Raman superradiance over Rayleight superradiance
•No backward scattering (K-D scattering)
•No interaction with the pump bean once scattered
Endfire modeRaman Rayleigh
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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)
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Where is a grating?
Pumpbeam
one photon
cloud of atoms
Pumpbeam
PMT
click!
hq
Thermal atoms
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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
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Writing, storing, and reading of a single photon
writebeam
one photon
click!
readbeam
one photon
writing storing reading
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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)
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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
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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
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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
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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
=
=
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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.
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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
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K-D diffraction with a focused pump beam
10 ms TOF
Focusedpump beam
BEC
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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