1. Resonant magneto-optical effects in atoms
Quantum Optics course
Student: Alexander Gusarov

2. 2. Linear magneto-optical (Faraday rotation) effects.
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

3. Magneto-optical effect
Vapor cell
Linear Polarization
Magnetic
Field
 =(n+-n-) l 
l –optical length of the sample
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.
The magnitude of optical rotation per unit magnetic field and unit length is characterized by the Verdet constant V

4. Linear Magneto-Optical (Faraday) Rotation
Voigt connects Faraday rotation to the Zeeman effect
In the presence of a longitudinal magnetic field, the Zeeman sublevels of the ground state are shifted in energy by gB×M. This leads to a difference in resonance frequencies for left- and right-circularly polarized light (±). s- s+
M= M= M=1
gmB
hn0
An F=1- F'=0 atomic transition.

5. Linear Magneto-Optical (Faraday) Rotation*
2D/G
-10 -8 -6 -4 -2 2 4 6 8 10 1 n + -
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, , 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
Refractive index
n+-n-
Light detuning
D. Budker et al., Rev. Mod. Phys. 74, (2002)

6. Linear Magneto-Optical (Faraday) Rotation
B ~ 400 G
g - Land´e factor
- Bohr magneton
l0 - absorption length
D. Budker et al., Rev. Mod. Phys. 74, (2002)

7. The Bloch Sphere Z Y X  
c0,1 are complex numbers .

8. Two-state system and The system is fully described by the two states:
Hamiltonian has only two stationary states
, k=0,1;
Schrödinger equation
External force W(t), may be represented by its matrix elements Wnm(t):
describes the time evolution of the atomic state under the influence of the external interaction W(t), .

9. Ensemble of quantum states*
Spin operator:
The Bloch vector components, in Cartesian coord. are:
2. The equation of motion for
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
The Bloch vector can represent dissipation and dephasing.
z - the z component of the Bloch vector at thermal equilibrium
* Mabuchi, H., Physics course notes, 2001

10. 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.

11. 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
for coherence effect  ~ 1Hz
for Bennett structure  1-10MHz
Doppler width ~ 1GHz
 - line width

12. Bennett-structure effect
involve the perturbation of populations of atomic states during optical pumping.
The hole in the atomic velocity distribution due to velocity-selective optical pumping. V z o r D
Frequency Detuning D
Number of atoms
Atomic velocity
Linear rotation
Faraday rotation
Bennett structure effect
The hole width typically ~1‑10 MHz

13. Coherence Effects in NMOR
involve the creation and evolution of atomic polarization
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.
Three stages: B0
Probe laser
beam
Pumping laser beam
Spin
Photo diode

14. 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)

15. 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
can be composed of a + - circularly polarized component:
where
and
are unit vectors in the x- and y- direction, respectively.
While passing through the sample the two components experience different refractive indices n+:
A phase difference between the two components
Δφ=(k+-k-)L=(ωL/c)(n+-n-)

16. 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
the corresponding differential equation of motion for a single electron is
Applying the Fourier transform to the equation, we obtain solution for a single Fourier component:
(1)
where 02=k/m, corresponds to the central frequency of an atomic transition from the ground to the excited state

17. An induced dipole moment:
(2)
In a sample with N oscillators, the polarization P
(3)
On the other hand, the polarization can be derived from Maxwell’s equations using the dielectric constant
or the susceptibility .
(4)
The relative dielectric constant
is related to the refractive index n by
(5)
Combining (1-4), the refractive index n can be written as

18. The refraction index is complex and can be written in the form:
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
(6)
The refraction index is complex and can be written in the form:
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 .
The frequency dependence of  and n’ can be obtained by inserting into (6) and separating real and imaginary parts:

19. The changes of absorption  and dispersion n caused by the pump wave are:
where
and 0=(0),
For our case with the relaxation rate of the ground state alignment  and
where g is the Land´e factor, 0 is the Bohr magneton and B is the applied magnetic field.

20. 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 *:
rel – relaxation rate of the atomic polarization
Depolarization rates ~21 Hz
have been observed by Budker et al. Phys.Rev. Lett. 81, for atoms in paraffin-coated cells.
*Budker et al. Rev. Mod. Phys. 74, 1153 (2002)

21. Optical pumping with linearly polarized light*
Light linearly polarized along x can create alignment along x-axis. x
F’ = 0
F = 1
MF = -1
MF = 0
MF = 1
Derek F. Kimball, “NMOR with FM light” course lectures

22. Optical pumping with linearly polarized light*
Light linearly polarized along x can create alignment along x-axis. x
F’ = 0
F = 1
MF = -1
MF = 0
MF = 1
Medium is now transparent to light
with linear polarization along x!
Derek F. Kimball, “NMOR with FM light” course lectures

23. Optical pumping with linearly polarized light*
Light linearly polarized along x can create alignment along x-axis. x
F’ = 0 .
F = 1
MF = -1
MF = 0
MF = 1
Medium strongly absorbs light
polarized in orthogonal direction!
Derek F. Kimball, “NMOR with FM light” course lectures

24. Rotation of light polarization*x Z y
E|| E   j
sin(2)
 - the angle between the transmission axis of the “polarizer” and the direction of light polarization.
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),

25. NMOR Study Set-up  - optical rotation in the sample (for  <<1)B
 - optical rotation in the sample (for  <<1)
P1’ P2 - photodiodes signal
*Budker et al. Rev. Mod. Phys. 74, 1153 (2002)

26. 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).
This technique is very suitable for miniaturization and was successfully implemented in the development of the Chip Scale Atomic Magnetometer at NIST.
Figures from: P.D.D. Schwindt at all., Appl. Phys. Lett. 85, 6409 (2004)

27. 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

28. Chip scale atomic magnetometer*
*P.D.D. Schwindt at all., Appl. Phys. Lett. 85, 6409 (2004)

29. Spin exchange collisions preserve total mF, but change F
Eliminating spin-exchange relaxation*
Spin exchange collisions preserve total mF, but change F
For wTSE<< 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).

30. 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:
I0[am2sin2(wmt)+2amjsin(wmt)+j2],
j is the rotation angle induced by the atoms B
polarizer
Lock-in amplifier signal at wm:
Measurable angle: 10-7 – 10-8 rad
Linearly polarized
probe beam
Ilock-in=I02m~B

31. BGU SERF Magnetometer

32. 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.
A resonance occurs when m = 2L.
1 laser frequency
0 atomic resonance frequency
Picture from Derek F. Kimball, “NMOR with FM light” course lectures

33. 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.

34. 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

35. 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