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