Transcript Nonlinear Magneto-Optical Rotation with Frequency

Nonlinear Magneto-Optical Rotation
with
Frequency-Modulated Light
Derek Kimball
Dmitry Budker
Simon Rochester
Valeriy Yashchuk
Max Zolotorev
and many others...
Some of the many others:
Budker Group:
Non-Berkeley Folks:
D. English
K. Kerner
C.-H. Li
T. Millet
A.-T. Nguyen
J. Stalnaker
A. Sushkov
Technical Support:
M. Solarz
A. Vaynberg
G. Weber
J. Davis
E. B. Alexandrov
M. V. Balabas
W. Gawlik
Yu. P. Malakyan
A. B. Matsko
I. Novikova
A. I. Okunevich
S. Pustelny
A. Weis
G. R. Welch
Funding: ONR, NSF
Plan:
• Linear Magneto-Optical (Faraday) Rotation
• Nonlinear Magneto-Optical Rotation (NMOR)
– Polarized atoms
– Paraffin-coated cells
– Experiments
Review: Budker, Gawlik, Kimball, Rochester, Yashchuk,
Weis (2002). Rev. Mod. Phys. 74(4), 1153-1201.
• NMOR with Frequency-Modulated light (FM NMOR)
– Motivation
– Experimental setup
– Data: B-field dependence, spectrum, etc.
• A little mystery...
• Magnetometry
Linear Magneto-Optical (Faraday) Rotation
1846-1855: Faraday discovers magneto-optical rotation
1898,1899: Macaluso and Corbino discover resonant character
of Faraday rotation
Linear Polarization
Medium
0l
 = (n+-n-)
= (n+-n-) l
2c


Circular
Components
Magnetic
Field
Linear Magneto-Optical (Faraday) Rotation
1898: Voigt connects Faraday rotation to the Zeeman effect
Linear Magneto-Optical (Faraday) Rotation
Linear Magneto-Optical (Faraday) Rotation
0 .6
0 .5
0 .4
R otation angle  (rad)
0 .3
0 .2
0 .1
-0 .0
-0 .1
B ~ 400 G
-0 .2
-0 .3
-0 .4
-0 .5
-0 .6
-5
-4
-3
-2
-1
0
1
2
3
4
N orm alized m agnetic field (b = 2 g F  0 B /  )
 
l
2gF 0B / 
2 l 0 1  2 g F  0 B /  
2
5
Nonlinear Magneto-Optical Rotation
• Faraday rotation is a linear effect because rotation is independent
of light intensity.
• Nonlinear magneto-optical rotation possible when light modifies
the properties of the medium:
Number of atoms
1
0.8
0.5
Index of refraction
 Spectral hole-burning:
B0
0
B=0
-0.5
0.6
-1
0.4
Re[n+-n-]
0.2
-2
-2
-1
0
1
Atomic velocity
2
-1
0
1
Light detuning
2
3
Small field NMOR enhanced!
3
Nonlinear Magneto-optical Rotation
due to atomic polarization
Three stage process:
Optical
pumping
Probing
via optical
rotation
Precession
in B-field
Optical pumping
Circularly polarized light propagating in z direction
can create orientation along z.
z
Fluorescence has random
direction and polarization.
F’ = 0
M = 1
MF = -1
MF = 0
MF = 1
Circularly polarized light consists of photons
with angular momentum = 1 ħ along z.
F=1
Optical pumping
Circularly polarized light propagating in z direction
can create orientation along z.
z
F’ = 0
MF = -1
MF = 0
MF = 1
F=1
Medium is now transparent to light
with right circular polarization in z direction!
Optical pumping
Light linearly polarized along z can create alignment along z-axis.
F’ = 0
z
MF = -1
MF = 0
MF = 1
F=1
Optical pumping
Light linearly polarized along z can create alignment along z-axis.
F’ = 0
z
MF = -1
MF = 0
MF = 1
F=1
Medium is now transparent to light
with linear polarization along z!
Optical pumping
Light linearly polarized along z can create alignment along z-axis.
F’ = 0
z
.
MF = -1
MF = 0
MF = 1
F=1
Medium strongly absorbs light
polarized in orthogonal direction!
Visualization of Atomic Polarization
Draw 3D surface where distance from origin equals the probability
to be found in a stretched state (M=F) along this direction.
z
z
y
z
y
x
y
x
x
Unpolarized
Oriented
Aligned
Sphere centered
at origin,
equal probability
in all directions.
“Pumpkin” pointing
in z-direction 
preferred direction.
“Peanut” with axis
along z 
preferred axis.
Optical pumping
Optical pumping process polarizes atoms.
Optical pumping is most efficient when
laser frequency (l) is tuned to
atomic resonance frequency (0).
Precession in Magnetic Field
Interaction of the magnetic dipole moment
with a magnetic field causes the angular momentum
to precess – just like a gyroscope!



=B

dF

=
dt

 
dF  
dt =   B = gF B F  B

B


, F


L = gF B B
Precession in Magnetic Field


  B torque causes polarized atoms to precess:
Relaxation and probing of atomic polarization
(polarized atoms only)
• Relaxation of atomic polarization:
• Equilibrium conditions result in net atomic
polarization at an angle to initial light polarization.
• Plane of light polarization is rotated,
just as if light had propagated through
a set of “polaroid” films.
Coherence Effects in NMOR
Magnetic-field dependence of NMOR due to atomic polarization
can be described by the same formula we used for linear Faraday
rotation, but   rel :
 
l
2l0

 dt e
t0
  rel t
sin  2 g F  0 Bt  
l
2 g F  0 B /  rel
2 l 0 1   2 g F  0 B /  rel 
How can we get slowest possible rel?
2
Paraffin-coated cells
Academician Alexandrov has
brought us some beautiful
“holiday ornaments”...
Paraffin-coated cells
F=3
Alkali atoms work best with
paraffin coating...
F=2
F=1
2
5 P 3 /2
496 M H z
D 2 (78 0.0 nm )
F=0
Most of our work involves Rb:
~
~
F=2
2
5 P 1 /2
D 1 (7 94 .8 n m )
87Rb
812 M H z
(I = 3/2)
F=1
~
~
F=2
2
5 S 1 /2
F=1
6835 M H z
Paraffin-coated cells
Polarized atoms can bounce off the walls of a paraffin-coated
cell ~10,000 times before depolarizing!
This can be seen using the method of “relaxation in the dark.”

4

B
Relaxation in the Dark
Light can be used to probe ground state atomic polarization:
Photodiode
F’ = 0
z
MF = -1
MF = 0
MF = 1
F=1
No absorption of
right circularly polarized light.
Relaxation in the Dark
Light can be used to probe ground state atomic polarization:
Photodiode
F’ = 0
z
MF = -1
MF = 0
MF = 1
F=1
Significant absorption
of left circularly polarized light.
Paraffin-coated cells
P ro b e tra n s m is s io n (a rb . u n its )
0 .2 6
0 .2 5
Bx = 100 G
 re l = 2   1 .0 0 4 (2 ) H z
0 .2 4
0 .2 3
0 .2 2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
T im e (s )
1 .2
1 .4
1 .6
1 .8
2 .0
Experimental Setup
lock-in
magnetic
coil
reference
DC
polarimeter
calibration
magnetic
shield
Rb-cell
pre-amplifier
polarizer
analyzer
polarizationmodulator
PD1
PD2
magnetic
field
current
absorption
first harmonic
attenuator
light-pipe
PD
control
and data
acquisition
spectrum analyzer
fluorescence
laser frequency control
feedback
differential
amplifier
polarizationrotator
Dichroic Atomic Vapor Laser Lock
PD
BS
PD
/4
uncoated Rb cell
in magnetic field
P
diode laser
Magnetic Shielding
ø 18"
ø 21"
16"
20"
25"
ø 2 4 .5 "
12"
Four-layer ferromagnetic magnetic shielding with nearly
spherical geometry reduces fields in all directions
by a factor of 106!
Magnetic Shielding
3-D coils allow control
and cancellation of fields
and gradients inside shields.
NMOR Coherence Effect in Paraffin-coated Cell
85Rb
D2 Line, I = 50 W/cm2,
F=3  F’=4 component
10
8
R o ta tio n A n g le (m ra d )
6
4
2
0
-2
-4
-6
-8
-1 0
-1 0
-8
-6
-4
-2
0
2
4
M a g n e tic F ie ld (  G )
Kanorsky, Weis, Skalla (1995). Appl. Phys. B
60, 165.
Budker, Yashchuk, Zolotorev (1998). PRL
rel
81, 5788.
Budker, Kimball, Rochester, Yashchuk, Zolotorev (2000). PRA 62, 043403.

= 2  0.9 Hz
6
8
10
Sensitive measurement of magnetic fields
85Rb
D2 line, F=3  F’ component,
I = 4.5 mW/cm2
 B z ) (1 0 -1 2 G /H z 1 /2 )
50
40
30
3  10
20
 12
10
0
T ra n sm issio n
1 .0
0 .9
0 .8
0 .7
-1 .2
-0 .8
-0 .4
-0 .0
0 .4
R e la tive F re q u e n cy (G H z)
0 .8
1 .2
G/
Hz
The dynamic range of an NMOR-based magnetometer is
limited by the width of the resonance:
10
8
R o ta tio n A n g le (m ra d )
6
4
2
0
-2
B ~ 2 G
-4
-6
-8
-1 0
-1 0
-8
-6
-4
-2
0
2
4
6
8
10
M a g n e tic F ie ld (  G )
How can we increase the dynamic range?
NMOR with Frequency-Modulated Light
• Magnetic field modulates optical properties of medium at 2L.
• There should be a resonance when the frequency of light is
modulated at the same rate!
Experimental
Setup:
Inspired by:
Barkov, Zolotorev (1978).
JETP Lett. 28, 503.
Barkov, Zolotorev, Melik-Pashaev (1988).
JETP Lett. 48, 134.
Nonlinear Magneto-optical Rotation
1 .0 0
In-phase component
0 .7 5
87Rb
D1 Line
F=21
0 .5 0
F irst H a rm o n ic A m p litu d e (m ra d )
0 .2 5
0 .0 0
-0 .2 5
-0 .5 0
m = 21 kHz
-0 .7 5
-1 .0 0
1 .0 0
Out-of-phase (quadrature) component
0 .7 5
 = 2220 MHz
0 .5 0
P  15 W
0 .2 5
0 .0 0
-0 .2 5
Budker, Kimball,
Yashchuk, Zolotorev (2002).
PRA 65, 055403.
-0 .5 0
-0 .7 5
-1 .0 0
-1 6 0 0
-1 2 0 0
-8 0 0
-4 0 0
0
400
L o n g itu d in a l M a g n e tic F ie ld ( G )
800
1200
1600
Nonlinear Magneto-optical Rotation
Low field resonance:
L  rel
On resonance:
Light polarized along
atomic polarization is transmitted,
light of orthogonal polarization
is absorbed.
Low-field resonance is due to equilibrium
rotated atomic polarization – at constant
angle due to balance of pumping, precession, and relaxation.
Nonlinear Magneto-optical Rotation
1 .0 0
In-phase component
0 .7 5
87Rb
D1 Line
F=21
0 .5 0
F irst H a rm o n ic A m p litu d e (m ra d )
0 .2 5
0 .0 0
-0 .2 5
-0 .5 0
m = 21 kHz
-0 .7 5
-1 .0 0
1 .0 0
Out-of-phase (quadrature) component
0 .7 5
 = 2220 MHz
0 .5 0
P  15 W
0 .2 5
0 .0 0
-0 .2 5
-0 .5 0
-0 .7 5
-1 .0 0
-1 6 0 0
-1 2 0 0
-8 0 0
-4 0 0
0
400
L o n g itu d in a l M a g n e tic F ie ld ( G )
800
1200
1600
Nonlinear Magneto-optical Rotation
High field resonances:
L >> rel
• Laser frequency modulation  modulation of optical pumping.
• If periodicity of pumping is synchronized with Larmor precession,
atoms are pumped into aligned states rotating at L.
Nonlinear Magneto-optical Rotation
• Optical properties of the atomic medium are modulated at 2L.
• A resonance occurs when m = 2L.
Nonlinear Magneto-optical Rotation
• Quadrature signals arise due to
difference in phase between
rotating medium and probe light.
• Second harmonic signals appear for
m = L.
NMOR with Frequency-Modulated Light
(a)
F irst H arm o nic A m plitud e (m rad )
2
Low field resonance
1
0
-1
-2
(b)
2
High field resonance
1
0
Note that spectrum of
FM NMOR First Harmonic
is related to NMOR spectrum:
f 
-1
R otatio n (m rad )
-2
8
 
2
s
0
87
87
R b, F = 2
0.9
85
R b, F = 1
,
F =1
Rb
0.8
0.7

For 2nd harmonic (not shown):
4
1.0
Transm issio n
(c)
12

,
F =1
,
F =2
,
F =2
L aser F requency D etuning (G H z)
(d)

2
Magnetometry
M e a s u re m e n t o f m a g n e tic fie ld w ith F M N M O R
1 .0
F irs t H a rm o n ic (m ra d )
0 .9
0 .8
0 .7
0 .6
1 G
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
-0 .1
0
20
40
60
80
100
120
140
160
180
200
T im e (s )
Demonstrated sensitivity ~ 510-10 G /
Hz
Magnetometry
Magnetic resonance imaging (MRI) in Earth field?
Measurement of Xe nuclear spins.
Magnetometry
Magnetic resonance imaging (MRI) in Earth field?
M agnetic F ield (nG )
20
15
10
5
0
-5
Tim e (m in)
129Xe
26% natural abundance, pressure = 5 bar
A mystery...
Q u ad ratu re S ig n al
(arb . u n .)
In-p hase S ig nal
(arb. u n .)
In -p hase S ig n al
(arb . un .)
L arm o r F req u en cy (H z)
400
P = 210 W
(a)
200
0
-2 0 0
m = 200 Hz
-4 0 0
400
P = 800 W
(b )
m = 4 L
200
0
-2 0 0
P = 800 W
(c)
400
0
-4 0 0
for high light power!
x5
-4 0 0
800
See new resonances at
x5
-8 0 0
L o n g itu d in al M ag n etic F ied (  G )
Hexadecapole Resonance
Arises due to creation and probing of
hexadecapole moment ( = 4).
Yashchuk, Budker, Gawlik, Kimball,
Malakyan, Rochester (2003). PRL 90,
253001.
Hexadecapole Resonance
Highest moment possible:
 = 2F
No resonance
for F=1
Hexadecapole Resonance
At low light powers:
Quadrupole signal  I2
Hexadecapole signal  I4