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Quantum Optics II – Cozumel, Dec. 6-9, 2004
Coherent Spectroscopy of Cold Cesium Atoms Using Light
Carrying Orbital Angular Momentum
J. W. Tabosa
Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife,
PE, Brazil [email protected]
Financial support: FINEP, CNPq: Pronex, Instituto do Milenio
Collaborators:
Prof. A. Lezama (Inst. of Phys. Montevideo, Uruguay)
Students:
George C. Cardoso (PhD, Post-doc/USA)
Sergio Barreiro (PhD)
OUTLINE:
 First Part:
• Introduction: The Orbital Angular Momentum (OAM) of Light
(Entanglement of OAM states in Parametric Down Conversion)
• Electromagnetically Induced Transparency (EIT) and
Electromagnetically Induced Grating (EIG)
• Coherent Four-Wave Mixing Using Light with OAM: Generation of
Superposition of OAM states
 Second Part:
• Population Grating Transfer in Cold Atoms: Applications for Cold Atoms
Velocimetry
• Coherence Grating Transfer in Cold Atoms.
 Summary and Conclusions
Phase Dislocation in an Optical Field


E ( r ,  , z )  A( r , z ) exp[ i ( r ,  , z )]
Dislocations are located in positions where:


Re{ E (r ,  , z )}  Im{ E (r ,  , z )}  0
Screw dislocation (Vortex):
Intensity
 

Re{ E ( r0 )}  0
 


Im{ E ( r0 )}  0
Phase
Orbital and Spin Angular Momentum of Light with Screw
Phase Dislocation
Laguerre-Gaussian
 Mode:

E ( r ,  , z )  A( r , z ) exp i[ m   ( r , z )]
Phase front:
Topological charge
(L. Allen et al. PRA, (1992))



 

3
J   0  d r {r  E  B}  L  S
Total Angular Momentum:
Total Energy:
W 
0
2
 
3
d
r
{
E
 E}

Jz
1

(m   )
For Laguerre-Gauss Mode: W

OAM per Photon:
Lz
Photon
Orbital
Spin
  0, linear polarizati on
  1, circular polarizati on
 m
Generation of Vortex Beams
•Mode Converter:
•Computer-generated holograms:
m=1
m=2
Measuring the topological charge:
Mask
Vortex beam
C
C
D
Reference wave
Interferograms for a single charged beam
Collinear
Noncollinear
Mach-Zehnder Interferometer: mode selector
m=even
m=odd
Phase shift :   m
Arbitrary mode selector:
m
Nature (2002), Zeilinger et at.
Spontaneous Parametric dow-conversion:

/2
/2
OAM Conservation:
Emitted photon state:
  C0,0 0 0  C1, 1 1  1  C1,1  1 1  ...
OUTLINE:
 First Part:
• Introduction: The Orbital Angular Momentum (OAM) of Light
(Entanglement of OAM states in Parametric Down Conversion)
•
Electromagnetically Induced Transparency (EIT) and Electromagnetically
Induced Grating (EIG)
• Coherent Four-Wave Mixing Using Light with OAM: Generation of
Superposition of OAM states
 Second Part:
• Population Grating Transfer in Cold Atoms: Applications for Cold Atoms
Velocimetry
• Coherence Grating Transfer in Cold Atoms.
 Summary and Conclusions
Coherent Effects in Three-Level Systems:
Electromagnetically Induced Transparency (EIT)
Boller, Imamoglu, Harris, PRL (1991)
b
d

Coupling field
WC
Probe field
WP
a

R
c

Group Velocity:


Vg 
d
c

dn
dk
n 
d
- Light storage: C. Liu et al, Nature (2001); M.D.
Lukin, Rev. Mod. Phys. (2003)
d0 (Raman resonance)
Transparency window n:ground states dephasing rate g
Electromagnetically Induced Grating (EIG)
Ling, Li, and Xial, PRA (1998)
Spatially Modulated Coherence:
FWM
Spatially Modulated EIT
b

d

W C (r )
WP
WP
c
EIT
-2
~
WC
EIG signal
Absorption Spectrum
a
EIT grating
~
WC
-1
0
d/G
1
2
EIG
-2
-1
0
d/G
1
2
Coherent spectroscopy in Recife
Cold Atoms:

The Magneto-Optical Trap
(MOT)

E. Raab, et al (1987)


N  10 atoms
7
 1010 atoms / cm3
T  mK  K
I

Repumping

I
Electromagnetically Induced gratings in degenerate
two-level system
G.C.Cardoso and JWT, PRA (2002)
DTLS of Cesium:
Transition: 6S1/2 , Fg=4-6P3/2 , Fe=4
Beams polarization:
( F || P )  B
F
Scanning frequency:
P
F
P S
B
OUTLINE:
 First Part:
• Introduction: The Orbital Angular Momentum (OAM) of Light
(Entanglement of OAM states in Parametric Down Conversion)
•
Electromagnetically Induced Transparency (EIT) and Electromagnetically
Induced Grating (EIG)
• Coherent Four-Wave Mixing Using Light with OAM: Generation of
Superposition of OAM states
 Second Part:
• Population Grating Transfer in Cold Atoms: Applications for Cold Atoms
Velocimetry
• Coherence Grating Transfer in Cold Atoms.
 Summary and Conclusions
FWM in a closed degenerate two-level
system
DTLS of Cesium: Fg=4 – Fe=5

F d
P
S
Fe=5
d

d
B
Fg=4
Two-level systems
F
B
F || P || B
P S
Zeeman coherence grating
B
F
S
P
( F || B )  P
Theoretical Model
DTLS
Fe=2
Fg=1
Master equation:
Measured and Calculated FWM Spectra
Lezama, Cardoso, JWT, PRA (2001)
Parallel polarization
F || P || B
•Narrow dip: quantum interference of
Zeeman pairs with different values of mi
Orthogonal polarization
( F || B )  P
•Zeeman coherence grating
Generation of ligth with OAM via coherence induced grating
S. Brarreiro and JWT, PRL (2003)
Experimental Scheme
S
P
B
F
(F||B) P
200 KHz
Coherently Generated Beam with OAM
Topological charge: m=1
out
in
Generated reference wave
Topological charge: m=2
in
out
Energy and linear momentum conservation:
S  F  B  P    d





KS  K F  K B  K S  K P
OAM conservation:





LS  LF  LB  LP   LP
Generation of coherent Superposition of OAM States
Incident and generated superposition of OAM
IN
m=1+m=2
m=0+m=1+m=2
OUT
 ab  exp( 2i )
OUTLINE:
 First Part:
• Introduction: The Orbital Angular Momentum (OAM) of Light
(Entanglement of OAM states in Parametric Down Conversion)
•
Electromagnetically Induced Transparency (EIT) and Electromagnetically
Induced Grating (EIG)
• Coherent Four-Wave Mixing Using Light with OAM: Generation of
Superposition of OAM states
 Second Part:
• Population Grating Transfer in Cold Atoms: Applications for Cold Atoms
Velocimetry
• Coherence Grating Transfer in Cold Atoms.
 Summary and Conclusions
Population Grating Transfer in Cold Cesium Atoms
Bragg diffraction into a transferred population grating
Observed spectra
(a)
Cesium D2 Line
F,1
gT
P,1
gR
B, 2
D
F'=3
F//P//B
(c)
(F//P)_B
P__
(F//B)
(d)
1
Grating and repumping: Off
Diffraction Decay:
L27m
I D (t )  e



t
F'=5
(b)
Cold atoms velocimetry:
Gaussian
fitting
F'=4
G



2
Coherence transfer:
•theoretical model (V-Lscheme)
W ,W 
 e,1; e1
SE
Master Equation: (Barrat and C. Cohen-Tannoudji)
 g  , 1; g 1
R
 g  , 1; e ,0
Diffracted spectrum:
D
Observation of grating coherence transfer:
S. Barreiro and JWT, to appear in PRA (2005)
Cesium level scheme:
Experimental setup:
- MOT beams and
magnetic field are
switched off.
Experimental observation of Zeeman grating transfer
Cesium level scheme:
Observed Bragg
diffraction spectra:
Summary and Conclusions
 Generation of superposition states of OAM
via coherence grating: possibility to store a
multidimensional quantum state of light in a
long-lived atomic coherence.
 Theoretical and experimental demonstration of coherence
grating transfer between different pairs of Zeeman sublevels.
•
Current research:
•
Storage of light carrying OAM in an atomic
coherence.
•
Squeezing via four-wave mixing in an EIT
medium.