Resonant nonlinear magneto-optical effects in atoms∗ - The Budker ...

2
1. Experiments 29
2. Theoretical analysis 30
F. Gas discharge 30
G. Atoms trapped in solid and liquid helium 30
H. Laser-cooled and trapped atoms 31
IX. Magneto-optical effects in selective reflection 31
1. Linear effects 31
2. Nonlinear effects 32
k
Polarizer
Linear
polarization
Circular
components
B ⊥ B||
φ
Medium
Analyzer
Detector
X. Linear and nonlinear electro-optical effects 32
XI. Experimental techniques 33
A. A typical NMOE experiment 33
B. Polarimetry 34
C. Nonlinear magneto-optical rotation with
frequency-modulated light 34
D. Magnetic shielding 35
E. Laser-frequency stabilization using magneto-optical
effects 35
XII. Applications 36
A. Magnetometry 36
1. Quantum noise limits 36
2. Experiments 37
B. Electric-dipole moment searches 38
C. The Aharonov-Casher phase shift 39
D. Measurement of tensor electric polarizabilities 39
E. Electromagnetic field tomography 40
F. Parity violation in atoms 40
XIII. Closely-related phenomena and techniques 41
A. Dark and bright resonances 41
B. “Slow” and “fast” light 41
C. Self-rotation 42
XIV. Conclusion 42
Acknowledgments 42
Appendices 42
A. Description of light polarization in terms of the
Stokes parameters 42
B. Description of atomic polarization 43
1. State multipoles 43
2. Visualization of atomic polarization 43
C. Abbreviations 44
References 44
I. INTRODUCTION
Magneto-optical effects arise when light interacts with
a medium in the presence of a magnetic field. These effects
have been studied and used since the dawn of modern
physics and have had a profound impact on its development.
1 Most prominent among the magneto-optical
effects are the Faraday (1846a,b, 1855) and the Voigt
1 Magneto-optics were listed among the most important topics in
Physics at the World Congress of Physics in Paris in 1900 (Guillaume
and Poincaré, 1900).
FIG. 1 The Faraday rotation effect. Light, after passing
through a linear polarizer, enters a medium subjected to a
longitudinal magnetic field B ‖ . Left- and right-circularly polarized
components of the light (equal in amplitude for linearly
polarized light) acquire different phase shifts, leading
to optical rotation. A difference in absorption between the
two components induces ellipticity in the output light. A
particular polarization of the transmitted light, depending on
the orientation of the analyzer relative to the polarizer, is
detected. Analyzer orientation varies with the type of experiment
being performed. In forward-scattering experiments
(Sec. II.B), the analyzer is crossed with the input polarizer,
so that only light of the orthogonal polarization is detected.
In the “balanced polarimeter” arrangement (Sec. XI.B), a polarizing
beam splitter oriented at π/4 to the input polarizer
is used as an analyzer. The normalized differential signal
between the two channels of the analyzer depends on the rotation
of light polarization while being insensitive to induced
ellipticity. The Voigt effect is similar except that instead of a
longitudinal magnetic field, a transverse field B ⊥ is applied.
Here optical rotation and induced ellipticity are due to differential
absorption and phase shifts of orthogonal linearly
polarized components of the input light (Sec. VI).
(1901) effects, i.e., rotation of light’s polarization plane as
it propagates through a medium placed in a longitudinal
or transverse magnetic field, respectively (Fig. I). The
linear (Secs. II, III) near-resonance Faraday effect is also
known as the Macaluso-Corbino (1898a; 1899; 1898b) effect.
The Voigt effect is sometimes called the Cotton-
Mouton (1907; 1911) effect, particularly in condensedmatter
physics.
The remarkable properties of resonant (and, particularly,
nonlinear) magneto-optical systems—as compared
to the well-known transparent condensed-matter
magneto-optical materials such as glasses and liquids—
can be illustrated with the Faraday effect. The magnitude
of optical rotation per unit magnetic field and
unit length is characterized by the Verdet constant V .
For typical dense flint glasses that are used in commercial
Faraday polarization rotators and optical isolators,
V ≃ 3 × 10 −5 rad G −1 cm −1 . In subsequent sections, we
will describe experiments in which nonlinear magnetooptical
rotation corresponding to V ≃ 10 4 rad G −1 cm −1
is observed in resonant rubidium vapor (whose density,
∼3 × 10 9 cm −3 , satisfies the definition of very high vacuum).
Taking into account the difference in density between
glass and the rarified atomic vapor, the latter can
be thought of as a magneto-optical material with some