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Resonant magneto-optical
effects in atoms
203-2-4321 Quantum Optics course
Student: Alexander Gusarov
1. Introduction
2. Linear magneto-optical (Faraday rotation) effects.
3. Two-state system and The Bloch Sphere
4. Nonlinear effects
 Physical mechanisms of nonlinear magneto-optical effects
• Bennett-structure (hole burning) effects
• Coherence effects
 Optical pumping/probing with linear polarized light
 Main relationships and the oscillator model
 NMOE in optically thick vapor (SERF – spin exchange relaxation
free case)
5. Applications and experiments
– Typical NMOE experiment Frequency modulated nonlinear
rotation
– Applications for magnetometry
6. Conclusions
Magneto-optical effect

 =(n+-n-) l

l –optical length of the sample
Vapor cell
Magnetic
Field
This effect is observed when the
polarization plane of the linearly
polarized probe beam is rotated
due to propagation through the
vapor subject to a magnetic field
applied in the direction of light
propagation. The angle of
rotation is proportional to the
external magnetic field.
Linear Polarization
The magnitude of optical rotation per unit
magnetic field and unit length is characterized
by the Verdet constant V
Linear Magneto-Optical (Faraday) Rotation
Voigt connects Faraday rotation
to the Zeeman effect
hn0
s+
s-
gmB
M=-1
M=0
M=1
An F=1- F'=0 atomic transition.
In the presence of a longitudinal magnetic
field, the Zeeman sublevels of the ground
state are shifted in energy by gmB×M.
This leads to a difference in resonance
frequencies for left- and right-circularly
polarized light (s±).
Linear Magneto-Optical (Faraday) Rotation*
2 gmB
G
Refractive index
n
n+
n1
n+-n0
-10 -8
-6
-4
Light detuning
-2
0
2D/G
2
4
6
8
10
When a magnetic field is applied the
Zeeman shifts lead to a difference between
the resonance frequencies for the two
circular polarizations. This displaces the
dispersion curves for the two polarizations.
A characteristic width of these dispersion
curves, G, corresponds to the spectral
width of an absorption line, which under
typical experimental conditions in a vapor
cell is dominated by the Doppler width and
is on the order of 1 GHz for optical
transitions
D. Budker et al., Rev. Mod. Phys. 74, 1153 (2002)
Linear Magneto-Optical (Faraday) Rotation
0.6
0.5
0.4
Rotation angle  (rad)
0.3
0.2
0.1
-0.0
-0.1
DB ~ 400 G
-0.2
-0.3
-0.4
-0.5
-0.6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Normalized magnetic field (b = 2gFm0B / G)
2 g F m0 B / G
l

2l0 1  2 g F m0 B / G 2
Bmax
G

2 gm
g - Land´e factor
m- Bohr magneton
l0 - absorption length
D. Budker et al., Rev. Mod. Phys. 74, 1153 (2002)
The Bloch Sphere
0
Ψ  c0 0  c1 1
Z
c0,1 are complex numbers

c c
*
i i

 1, i  0,1
i

X
θ
θ.
i
Ψ  cos 0  e sin 1
2
2
0  θ  π 0    2
1
Y
Two-state system
The system is fully described by the two states:
0  ψ0
Hamiltonian has only two stationary states
H 0 k  E k k , k=0,1;
and
1  ψ1

i Ψ  H(t)  W(t) Ψ
t
Schrödinger equation
External force W(t), may be represented by its matrix elements Wnm(t):
Wnm (t)  ψn W(t) ψm
.
describes the time evolution of the atomic state
interaction W(t),
Ψ(t)
under the influence of the external
Ensemble of quantum states*
  
 0 1 0  i 1 0 
 
, 
, 
 
S

σ

(
σ̂
,
σ̂
,
σ̂
)

Spin operator:
x
y
z

1
0
i
0
0

1
2
2
2 




The Bloch vector components, in Cartesian coord. are:
ν  (ν x , ν y , ν x )  sin θ cos , sin θ sin  , cosθ 

νB
2. The equation of motion for

 
dν

1
i B  Tr σH, 1  ν B  σ 
dt
2
Example: when a static magnetic field Bz is applied, the solution of this equation shows that the Bloch vector
precesses about the z axis at the Larmor frequency, given by
ωL  γBz
The Bloch vector can represent dissipation and dephasing.
ν x (t)  ν x (0)e
ν y (t)  ν y (0)e


t
T2
nz - the z component of the Bloch vector at thermal equilibrium
t

T2

ν z (t)  ν z (0)  ν 0z e

t
T1
 ν 0z
* Mabuchi, H., Physics course notes, 2001
Nonlinear Magneto-Optical Rotation
• Linear Faraday rotation is independent of light intensity.
• Nonlinear magneto-optical rotation possible when light modifies the properties
of the medium.
Small-field rotation can be enhanced by many orders of magnitude
compared to the linear case due to nonlinear effects.
Two main mechanisms are responsible
for the enhanced rotation:
•Zeeman coherence (small magnetic field).
•Bennett –structure formation in the atomic velocity distribution.
empirical distinction between these two effects is the magnitude of the
magnetic field Bmax at which optical rotation reaches a maximum
Bmax


2 gm
 - line width
for coherence effect  ~ 1Hz
for Bennett structure  1-10MHz
Doppler width ~ 1GHz
Bennett-structure effect
Number of atoms
involve the perturbation of populations of atomic states during optical pumping.
The hole in the atomic velocity
distribution due to velocity-selective
optical pumping.
Vz or D
Faraday rotation
Atomic velocity
Linear rotation
Frequency Detuning D
The hole width typically ~1-10 MHz
Bennett structure effect
Coherence Effects in NMOR
involve the creation and evolution of atomic polarization
Three stages:
1.
2.
3.
Resonant light polarizes atomic sample via optical pumping.
Polarized atoms precess in the magnetic field.
This changes the optical properties of the medium
 rotation of light polarization plane.
B0
Probe laser
beam
Photo diode
Spin
Pumping laser beam
Optical pumping*
Simplified level diagram
•
•
•
A magnetic field has to be applied
in the direction of the pumping
light to split the Zeeman
sublevels.
A circularly polarized light is
applied to the sample at the
frequency of an electronic
transition from the ground state
Pumping aligns atomic spins in
the direction of the pump beam
*W. Happer, Rev. Mod. Phys., 44, 169 (1972)
Magneto-Optical Rotation
Here we consider the interaction of linearly polarized probe light with optically polarized
atoms and derive analytical expressions for the polarization rotation angle of the
electromagnetic field.
The linearly polarized probe wave
E  E0ei (t  kz )
E 0  E 0 ,0,0
can be composed of a s+ s- circularly polarized component:

 i (t  k  z )
0
1

, E  E0 (xˆ  iy ),
2

 i (t  k  z )
0
1

, E  E0 (xˆ  iy ),
2
E E e
E E e

0
where
x̂
and

y
are unit vectors in the x- and y- direction,
respectively.

0
While passing through the sample the two components experience different refractive indices
n +:


0
E E e
i (t  k  L )
,

E E e
A phase difference between the two components
Δφ=(k+-k-)L=(ωL/c)(n+-n-)
 i (t  k  L )
0
.
An excited atomic electron can be described by the classical model of a
damping harmonic oscillator with charge q, frequency ω, mass m and
damping constant .
Under the influence of driving force qE, caused by the incident
electromagnetic wave with amplitude
E  E0eit ,
the corresponding differential equation of motion for a single electron is
q
x  x   x  E0eit .
m
2
0
Applying the Fourier transform to the equation, we obtain solution for a
single Fourier component:
qE0eit
x
,
2
2
m(0    i)
(1)
where 02=k/m, corresponds to the central frequency of an atomic
transition from the ground to the excited state
An induced dipole moment:
q 2 E0eit
p  qx 
m(02   2  i)
(2)
In a sample with N oscillators, the polarization P
P  Nqx
(3)
On the other hand, the polarization can be derived from Maxwell’s equations using
the dielectric constant 
0
or the susceptibility

.
P   0 (  1) E   0 E ,
The relative dielectric constant

Combining (1-4), the refractive index n can be written as
(4)
is related to the refractive index n by
n   1/ 2
2
Nq
n2  1 
.
2
2
 0 m(0    i)
(5)
In alkali-metal vapor at sufficiently low pressures, the index of refraction is close to
unity. Therefore n2-1=(n+1)(n-1)≈2(n-1), and (5) can be reduced to
Nq 2
n 1
.
2
2
2 0 m(0    i)
(6)
The refraction index is complex and can be written in the form:
n  n  ik ,
where the real part n’() represents the dispersion of the electro-magnetic, passing
through a medium with the refractive index n,
and the imaginary part k() describes the absorption of the wave with the absorption
coefficient
  4k / 0
.
The frequency dependence of  and n’ can be obtained by inserting into (6) and separating
real and imaginary parts:
Nq20


,
2
2 2
2 2
c 0 m (0   )   
Nq 2
02   2
n  1 
.
2
2 2
2 2
2 0 m (0   )   
The changes of absorption D and dispersion Dn caused by the pump wave are:
D ( ) 
where
D 0
,
2
1 
Dn( ) 
c D 0
,
2
0 1  
 0 


and 0=(0),
For our case with the relaxation rate of the ground state alignment  and
0    2 gmB.
LD 0 2 gm0 B / 
D 
20 1  2 gm0 B /  2
where g is the Land´e factor, m0 is the Bohr magneton and B is the applied magnetic field.
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 G  rel *:

2 g F m0 B /  rel
l
l
 rel t

dt e
sin 2 g F m0 Bt  

2l0 t 0
2l0 1  2 g F m0 B /  rel 2
rel – relaxation rate of the atomic polarization
Depolarization rates ~21 Hz
have been observed by Budker et al. Phys.Rev. Lett. 81, 5788.
for atoms in paraffin-coated cells.
*Budker et al. Rev. Mod. Phys. 74, 1153 (2002)
Optical pumping with linearly polarized light*
Light linearly polarized along x can create alignment along x-axis.
x
F’ = 0
MF = -1
Derek F. Kimball, “NMOR with FM light” course lectures
MF = 0
MF = 1
F=1
Optical pumping with linearly polarized light*
Light linearly polarized along x can create alignment along x-axis.
x
F’ = 0
MF = -1
MF = 0
MF = 1
F=1
Medium is now transparent to light
with linear polarization along x!
Derek F. Kimball, “NMOR with FM light” course lectures
Optical pumping with linearly polarized light*
Light linearly polarized along x can create alignment along x-axis.
x
F’ = 0
.
MF = -1
MF = 0
MF = 1
F=1
Medium strongly absorbs light
polarized in orthogonal direction!
Derek F. Kimball, “NMOR with FM light” course lectures
Rotation of light polarization*
x

sin(2)

 - the angle between the transmission axis
of the “polarizer” and the direction of light
polarization.
E
Rotating “Polarizer”
E||
E
Probe Laser Beam
y
Z
Thin rotating Polaroid film is transparent to light polarized
along its axis (E||), and slightly absorbent for the orthogonal
polarization (E).
*Budker et al., Am. J. Phys. 67(7), 584. 1999
NMOR Study Set-up
Workstation
analyzer
B
Rb Cell
Laser current
controller
/2
Laser
polar.
P1  P2
* 
2  ( P1  P2 )
/2
 - optical rotation in the sample (for  <<1)
P1’ P2 - photodiodes signal
Laser Lock
*Budker et al. Rev. Mod. Phys. 74, 1153 (2002)
CPT
CPT
The technique uses two coherent
beams that simultaneously couple the
two hyperfine ground states to the
excited state, creating a “dark state”
and a very narrow transparency
window.
The detection of this sharp
transparency window enable
measuring changes in the Zeeman
levels caused by weak magnetic fields
(pico-Tesla).
Figures from: P.D.D. Schwindt at all., Appl. Phys.
Lett. 85, 6409 (2004)
This technique is very suitable for
miniaturization and was successfully
implemented in the development of the
Chip Scale Atomic Magnetometer at
NIST.
Energy level diagram for 87Rb
The laser is tuned to the D1 line
The current to the laser is modulated at
half the hyperfine splitting of the Rb
ground state (3.4 GHz)
Energy difference between two
Zeeman states:
m is an azimuthal quantum number
When the frequency difference
between the first-order sidebands is
equal to the Zeeman splitting,
reduction of the absorbed light power
is observed
Chip scale atomic magnetometer*
*P.D.D. Schwindt at all., Appl. Phys. Lett. 85, 6409 (2004)
Eliminating spin-exchange relaxation*
Spin exchange collisions preserve total mF, but change F
For TSE<< 1
Low external magnetic field (B<<0.1G)
High vapor density 1014 cm-3
No relaxation due
to spin exchange
*W. Happer and H. Tang, Phys. Rev. Lett. 31, (1973).
NMOR Faraday modulation
technique
Analyzer is set at 90° with respect to the initial plane of polarization (extinction).
photodiode
DAQ
analyzer
y
Vapor cell
x
z
Lock-in
amplifier
Pump
beam
Faraday
modulator
Transmitted intensity:
I=I0sin2[msin(mt)+] @ I0[m2sin2(mt)+2msin(mt)+2],
B
polarizer
 is the rotation angle induced by the atoms
Lock-in amplifier signal at m:
Ilock-in=I02m~B
Measurable angle:
10-7 – 10-8 rad
Linearly polarized
probe beam
BGU SERF Magnetometer
NMOR with Frequency-Modulated Light
The dynamic range of an NMOR-based magnetometer is limited by the width of the
resonance.
The atomic alignment precesses at the Larmor frequency due to the magnetic
field.
If periodicity of pumping is synchronized with Larmor precession, atoms are
pumped into aligned states rotating at L
Because the state of atomic polarization is
symmetric, atomic alignment returns to the
same state after half the Larmor precession
period
• Optical properties of the atomic
medium are modulated at 2L.
1 laser frequency
0 atomic resonance frequency
• A resonance occurs when m = 2L.
Picture from Derek F. Kimball, “NMOR with FM light” course lectures
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.
NMOR with Frequency-Modulated Light
• Quadrature signals arise due to
difference in phase between
rotating medium and probe light.
• Second harmonic signals appear for
m = L.
Derek F. Kimball, “NMOR with FM light” course lectures
Conclusions
We have described the history and recent developments in the study and
application of resonant nonlinear magneto-optical effects.
We have discussed the connections and parallels between this and other
subfields of modern spectroscopy, and pointed out open questions and
directions for future work.
Numerous and diverse applications of NMOE include precision
magnetometry, very high resolution measurements of atomic parameters