Transcript Superradiant light scattering from condensed and

US-Japan Seminar, Breckenridge, Aug23-25, 2006
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
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)
Superradiant Rayleigh scattering
from a Bose-Einstein condensate
S. Inouye, et. al., Science 285, 571 (1999)
Na BEC
Off-resonant
pump light
35ms
75ms
100ms
Semi-classical explanation
q
N0
Nq
+
Condensate
2 N 0N q
The amplitude of
density modulation:
V
=
Recoiling atoms
2w
Bragg scattered Light
(end-fire mode)
Matter wave grating
Phase-matching
solid angle

 
 
Pump beam
Power in the
end-fire mode
sin 
P  
RN 0N q 
8 / 3
2
R : single-atom Rayleigh scattering rate
2
2
sin


Nq  R
N 0N q 
8 / 3
Fully-quantum picture
(Fermi’s Golden Rule)
Scattering process:
Recoiling
Condensate atoms
| N 0 , n k ; N q , n k q  H

 | N 0  1, n k  1; N q  1, n k q  1 
ˆ aˆ cˆ  aˆ cˆ
q k q 0 k
Pump beam Endfire mode
Scattering rate:
W  | N 0  1, n k  1; N q  1, n q k  1 | Hˆ | N 0 , n k ; N q , n k q |2
 N 0n 0
N
q
 1n
k q
neglect
 1
2
sin

N q  R
N 0 N q  1 
Summing
8 / 3
over 
Stimulated scattering
(Bosonic enhancement)
Spontaneous
scattering
Dicke’s picture
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
WN   J , M | J  J  | J , M
 (J  M )(J  M  1)
 N e (N g  1)
R. H. Dicke, Phys. Rev. 93, 99 (1954)
M. Gross and S. Haroche, Phys. Rep. 93,
301 (1982)
sin 2 
 R

8 / 3
Ng  N0
Ne  Nq
2
sin


Nj R
N 0 N q  1 
8 / 3
Three different pictures
for 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
Rb BEC
Week pump light
25 ms-3200 ms
63 mW/cm2
detuning: -4.4 GHz
30 ms TOF
D. Schneble, Y.T., M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003)
Superradiant Rayleigh scattering
in the short (strong) pulse regime
Rb BEC
Strong pump light
1 ms-10 ms
63 mW/cm2
detuning: -440 MHz
30 ms TOF
D. Schneble, Y.T., M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003)
Asymmetry of the X-shaped pattern
Explanation for the asymmetric
X-shape
Scattered light
(endfire modes)
Pump beam
Optical grating
+
=
Kapitza-Dirac Diffraction
of the matter wave
Intensity of the endfire mode
experiment
theory
Pn()
1.0
0.5
0.0
0

2R
5

10
8
4
0
2
-2
on
r
de
r
o
n
cti
rf a
dif
Pn ( )  J ( 2R ),  2R 
2
n
-8
-4
p e
2
Intensity of the
 e2
I e  2 2 I s  0.8 mW/cm2 I s  1.6 mW/cm2 
Endfire mode

Scattering rate when
the endfire photon nk-q>>1
Scattering process:
| N 0 , nk ; N p , nk q | N 0  1, nk  1; N p  1, nk q  1 
Reverse scattering process:
| N 0 , nk ; N p , nk q | N 0  1, nk  1; N p  1, nk q  1 
Net scattering rate:
W  N 0n k N q  1n k q  1  N q n k q N 0  1n k  1
 N 0n k N q  n k q  1
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?
W  N 0n 0 N q  nk q  1
nk q  1
W  N 0n 0 N q  1
Stimulation by Nq (atom)
N q  1
nk-q= Nq
W  N 0n 0 nk q  1
Stimulation by nk-q (photon)
Both pictures would give the same scattering rate!
New interpretation of superradiance
(in the long pulse regime)
Endfire beam
Pump 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
Fermi’s Golden Rule
E
2
W  2 |  2R / 2 |2  ( 2 )

|e>
Normalized
Lorentzian

2  1 /  c
e
|g>
p
|q>
2
(2 / 2) / 
2
22  2 / 2 
Width of the two-photon
(Bragg) resonance
Coherence time of
the condensate
At two-photon resonance (2=0)
|0>
p
W  N 0  22R / 2  N 0
 2p  e2
4
2
/ 2
How to express the rate W
in terms of R and nk-q?
W  N0
 2p  e2
4
2
/ 2
Single-atom Rayleigh scattering rate:
 2p
1
1
R  ee    s0
 2
2
2 ( 2 /  )
4

2 2p 
 s0 

2 

 

Saturation parameter
Intensity of the endfire mode:
of the pump beam
2
2 e 
  Saturation intensity
Ie  Is 2 Is 
2  (I = 1.6 mW/cm2 for Rb D line)
2
3  s


Number of photons emitted in the coherence time c=1/2:
n q k
I e A c 2A e2

 2

3 2
…continued
2
4

 2p  R

W  N0
 2p  e2
42
 e2  n k q
32
 2
2A
32
/ 2  R
N 0n k q
2A


  
A
w
2
3
W R
N 0n k q 
2
2

Semi-classical expression based
on the matter wave grating
2
sin


Nj R
N 0N q 
8 / 3
Four different pictures
for 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
F=2
F=1
Polarization
Pump
BEC
Polarization
Pump
Detuning: -2.6 GHz
Intensity: 40 mW/cm2
Pulse duration: 100 ms
Y. Yoshikawa, et al., PRA 69 041603 (2004)
D. Schneble, et at., PRA 69 041601 (2004)
Raman superradiance
The only condition for Raman superradiance:
Raman scattering gain > Rayleigh scattering gain
3
sin 2 
R Raman
 R Ray leigh
2
16 (1  cos  )
8 / 3
PMT
End-fire
mode

-pol.
BEC

End-fire
mode
Pump beam
-pol.
|F = 2, mF = 2>′
Pump beam
D1 line: 795 nm
|F = 2, mF = 2>
x
6.8 GHz
y
z
|F = 1, mF = 1>
Merits of Raman superradiance
over Rayleight superradiance
Raman
Rayleigh
Endfire mode
•No backward scattering (K-D scattering)
•No interaction with the pump bean once scattered
Exponential growth of
the Raman scattered atoms
=－2.3 GHz
40
3
－3.5 GHz
2
1
0
0
20
40
60
80
100
Pump time [ms]
3
N 0  N q

N q  R N 0N q   N q  e Gt
8
R: single-atom Raman scattering rate
Growth
rateG[kHz]
gain
Small-signal
[kHz]
－2.9 GHz
5
Number of atoms [10 ]
4
30
20
gradient: 1022
10
0
0
10
20
30
40
Optical
pumping
rate [Hz]
single-atom
Raman
scattering
rate R [Hz]
Small-signal gain:
G
3
R N 0   890 R
8
Y. Yoshikawa, et al., PRA 69 041603 (2004)
Where is a grating?
cloud of atoms
atoms
Thermal
PMT
Pump
beam
q
click!
one photon
Pump
beam
The origin of a grating
(Collective mode excitation)
q
One atom is excited to the collective
atomic mode defined by S+
q
q
+
q
1/ N 0 
q
q
+
1
+
+
+
(N 0  1) 
| J , M  J  1  S  | J , M  J 
+
....
 
S  1

N0


| q i  0 | 

i 1

N0

Writing, storing, and reading of
a single photon
writing
storing
reading
read
b eam
click!
one photon
write
beam
one photon
Superradiance in a Thermal gas
At om cloud
End-fire mode
B
N A=0.19
Pump
b eam
PMT
Growth rate [kHz]
Intensity [a.u.]
Pure condensate
Thermal gas
(560 nK)
Pure condensate
300
200
Thermal gas
(T = 560 nK)
100
0
0
40
80
120
Pump
pulse
duration
Pump
time
[ms] [ms]
0
50
100
150
Optical
pumping
rate [Hz]
single-atom
Raman
scattering
rate R [Hz]
Y. Yoshikawa, Y. T. and T. Kuga, PRL 94 083602 (2005)
The origin of the threshold
Loss term
(G  L )t

N q  (G  L )N q  N q  e
G-L
L G
G L
: No exponential growth
: exponential growth with
a growth rate of G-L
L  1/ c
coherent time of the system
Rth
-L
R
What determines the coherence time?
a) The endfire mode
Doppler width:
 D  q v

v  k BT

m

RMS velocity
1
1
c 

D qv
b) matter wave grating
(overlap of the wave packets)




q
v
m
p mv
x 



x
1
c 

v
qv
Coherence time is given by the inverse of the Doppler width
Measurement of the coherence time
1.0
0.5
T = 280 nK
Signal ratio
0.8
0
0.6
0
200
400
600
0.4
0.2
0
T = 280 nK
T = 760 nK
0
10
20
30
Duration of the dark period [ms]
40
Coherence time vs. temperature
Pure condensate
1000
bimodal
Pure thermal
280 ms
100
c [ms]
c 

10
Tc
1
0
200
400
600
800
1000
Temperature [nK]
1200
1400
1
2kv

4
m
k BT
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
scattering a pump photon
into the endfire mode
(stimulated) scattering of a pump
photon into the pump beam
E = p2 /2m
E = p 2 /2 m
|e >
Forward
peak
|e >

p

e
p
e
|g>
|g>
|q>
|-q>
backward
peak

|0>
|0>
p
p
E  2 rec 
h  15 kHz (Rb)
h  100 kHz (Na)
K-D diffraction with a focused
pump beam
BEC
Focused
pump beam
10 ms TOF
Velocity and spatial distribution
before and after the SRS
Velocity distribution
spatial distribution
before SRS
(a)
y
(c)
5
z
2
F=2
(O.D.0.5)
(b)
Optical density
4
before SRS
3
T = 473 nK
2
F=1
1
T = 544 nK
after SRS
after SRS
4
0
0
0.5
1.0
1.5
position [mm]
2.0
O.D0.1)
0
0
0.2
0.4
0.6
z - position [mm]
0.8
1.0