ABSTRACT
TWO-PHOTON POLARIZATION SPECTROSCOPY OF ATOMIC CESIUM USING
CIRCULARLY POLARIZED LIGHT.
by Dave S. Fisher
Measurements of collisional depolarization cross sections play an important role in gaining
valuable information about the interaction between inert and rare-gas atoms. A key element in understanding collisional dynamics is the detailed analysis of the polarization of
the emitted photons. We have experimentally investigated the collisional cross section of
the excited J=1/2 cesium atoms, by collisions with the ground level argon atoms, from
a study of circular polarization spectra. Orientation in the J=1/2 level was optically induced by a circularly polarized light with a positive helicity. A two-photon double-resonance
(6s 2 S1/2 →6p 2 P1/2 → 10s 2 S1/2 ) condition was achieved using nanosecond pulsed dye lasers,
and the intensity of the cascade fluorescence was monitored in the presence of argon atoms
ranging from 10−4 torr to 100 torr.
TWO-PHOTON POLARIZATION SPECTROSCOPY OF
ATOMIC CESIUM USING CIRCULARLY POLARIZED LIGHT
A Thesis
Submitted to the
faculty of Miami University
in partial partial fulfillment of
the requirements for the degree of
Master of Science
Department of Physics
by
Dave S. Fisher
Miami University
Oxford, OH
2010
Approved:
ADVISOR: Burçin S. Bayram
READER: Samir Bali
READER: Perry Rice
Contents
1 Introduction
1
2 Properties of Cesium
3
3 Excitation Theory
8
3.1
Excitation Scheme and Selection Rules . . . . . . . . . . . . . . . . . . . . .
8
3.2
Linear Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.3
Circular Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.4
Broadening Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.4.1
Line Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.4.2
Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.4.3
Collisional (Pressure) Broadening . . . . . . . . . . . . . . . . . . . .
16
3.5
Alignment and Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.6
Intensity and Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.7
Hyperfine Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.7.1
Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.7.2
Hyperfine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4 Polarization Consideration
25
4.1
Clebsch-Gordon Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Polarization Calculation using Wigner-Eckart
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
25
26
4.3
4.4
Polarization Calculation Using Fluorescence
Radiation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Hyperfine Consideration in Polarization . . . . . . . . . . . . . . . . . . . . .
30
5 Lasers
32
5.1
The Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5.2
Nd:YAG Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
5.3
Dye Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.3.1
Laser 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
5.3.2
Laser 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.4
Free Spectral Range
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Experimental Apparatus
36
37
6.1
Beam Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
6.2
Polarization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6.3
Cesium Oven Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
7 Overview of Measurement
42
7.1
Detecter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
7.2
Boxcar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.3
Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
8 Systematic Effects
45
8.1
Temperature Dependency Runs . . . . . . . . . . . . . . . . . . . . . . . . .
45
8.2
Power Dependency Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
8.3
Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
9 Results
49
9.1
Polarization Dependency on Pressure . . . . . . . . . . . . . . . . . . . . . .
49
9.2
Cross Section Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
10 Conclusions and Future Work
55
iii
A LabView
56
A.1 Meadowlark USB Set Voltage.VI . . . . . . . . . . . . . . . . . . . . . . . .
56
A.2 takedata2 sub.vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
A.3 mircometer subVI2.vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
A.4 Modified-IntensityLCVR4.vi . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
iv
List of Tables
2.1
Properties of Naturally Occurring 133 Cs Atoms [10] . . . . . . . . . . . . . .
4
3.1
Selection rules and Zeeman sublevel transition characterization . . . . . . . .
10
3.2
Associated Polarization for β
. . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.1
Clebsch-Gordon Coefficients Equations use for Experiment [26] . . . . . . . .
26
4.2
Calculated values of orientation, alignment, and polarization for the D line
9.1
excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Polarization degree at various pressures of Ar . . . . . . . . . . . . . . . . .
50
v
List of Figures
2.1
Calculated Vapor pressure and number density as a function of temperature
for atomic Cs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Grotrian Diagram of Electronic States of Cs [19]. . . . . . . . . . . . . . . .
7
3.1
Electron absorption and emission between two energy levels. . . . . . . . . .
8
3.2
Partial energy level diagram for the two-photon excitation of the electronic
states of atomic cesium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
9
Linear, or plane polarization. E-field travels down the z-axis and is the combination of vector additions of the x and y components. . . . . . . . . . . . .
11
3.4
Allowed transitions to Zeeman levels by m selection rules. . . . . . . . . . . .
11
3.5
Circularly polarized light propagation with time. . . . . . . . . . . . . . . . .
12
3.6
Vector addition of the axis’s and resulting electric field for circularly polarized
light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.7
Kastler Diagram displaying dual helicity circular polarization excitation. . .
13
3.8
Alignment for a transitions to a J=3/2 state (hAi = 52 ) [21, 18]. . . . . . . .
17
3.9
Orientation populating a J=3/2 state (hOi =
√5 )
2 15
[21, 18].
. . . . . . . . .
18
3.10 Population verses magnetic sublevels for (a) a pure monopole moment, (b)
alignment, the magnetic quadrupole moment, and (c) orientation, the electric
dipole moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.11 Reference Frames of excitation and detection for theoretical calculations. . .
20
3.12 Experimental reference frame, detection angle aligned on the y axis [12]. . .
20
3.13 Hyperfine Structure of Ground and Excited States. . . . . . . . . . . . . . .
24
4.1
27
Transitions for Clebsch-Gordon calculations. . . . . . . . . . . . . . . . . . .
vi
5.1
The Stages of stimulated emission due to Population Inversion [4]. . . . . . .
33
5.2
Littman-Metcalf Configuration for tunable dye laser. . . . . . . . . . . . . .
34
5.3
Calibration of Laser 2 with adjustment to the micrometer mount. . . . . . .
36
6.1
Beam overlap time scale for two-photon excitation . . . . . . . . . . . . . . .
38
6.2
The scale of one of the cesium cells used in the experiment. . . . . . . . . . .
40
6.3
The actual oven chamber used in the experiment wrapped in aluminum. . . .
40
6.4
Experiment Optics Table Diagram. . . . . . . . . . . . . . . . . . . . . . . .
41
7.1
The Boxcar averager and integrator with oscilloscope for signal visual check.
44
8.1
Polarization and the effects of Temperature and Power of L1 . . . . . . . . .
46
8.2
Polarization dependency on power of Laser 2. . . . . . . . . . . . . . . . . .
47
9.1
Polarization for Cs cells under various pressures of Ar at 70 ◦ C. . . . . . . .
49
9.2
A picture of rates and the effects of collisions. . . . . . . . . . . . . . . . . .
51
A.1 Meadowlark program for testing LCVR functionability. . . . . . . . . . . . .
58
A.2 Data taking and baseline setting program (takedata2.vi.) . . . . . . . . . . .
58
A.3 Micrometer adjustment program for Laser 2 tuning. . . . . . . . . . . . . . .
59
A.4 Spectrum data collection program (ModifedIntensity.vi.) . . . . . . . . . . .
59
vii
Acknowledgments
Having the opportunity to express my personal feelings and mention a few name-droppingworthy individuals, I would like to describe to you my gratitude and experience that has
shaped the completion of this project. First and foremost I should like to thank God for
His hand in shaping my understanding of Him and His endless mercies through the years
I’ve been in graduate school. I would like to thank my parents for their years of support,
both financially and emotionally, and all their aggravating phone calls about what my future
plans are, full knowing that they are proud of me and have the best interests at heart. I
would like to thank the faculty of Miami’s Physics department for their commitment and
encouragement to students. Shout out to Mike Eldrige for fulfilling a childhood dream by
teaching me how to use a lathe and to weld. Additionally, I would like to thank Judy Eaton
and Teresa Kolb for their logistical help and fascinating conversation. I lift up my brethren
and Sistren in arms, Erik Alquist, Aaron Godfrey, Ian Steward, Todd Van Woerkom, and
Jia Ying for their company through the trenches. Last, I extend tremendous gratitude to
Burçin Bayram for her guidance and encouragement. They say that an Advisor is the person
you most admire and who pushes you toward greatness while in graduate school. Burçin has
fulfilled this role most stellarly and I am eternally grateful.
viii
Chapter 1
Introduction
Polarization spectroscopy is a much extremely useful tool in the analysis of the much undiscovered realm of Atomic and Molecular physics. Scientists have grasped the Bohr model for
an atom, but this simplistic case only describes trends in one of the 100+ known atoms.
Experimentation is still needed to better understand atomic and molecular behavior for
specified cases. In addition, advancement in testing techniques for the innovation of technological knowhow in the development and applications for further research and industry
needs such as techniques that could lead to advances in other areas of optical physics, such
as Electromagnetically Induced Transparency (EIT) [1, 2].
This experiment has been designed to investigate and to confirm the collisional cross-section
collision between cesium and argon atoms and the depolarization effect of the 6p2 P1/2 energy
state from collisions by using a pump-probe two-photon two-color excitation method. This
state is the first destination of a resonant energy transition from the ground state. Studies of
cross-section collisions with noble gases and the resulting depolarization effect are not new
[3, 4, 5], but different techniques could lead to advances in other areas of optical physics,
such as Electromagnetically Induced Transparency (EIT) [1, 2]. A previous experiment investigating collisional cross sections was done by Guiry and Krause [3]. They investigated
the cross-section of collisions between cesium and various noble gases and the depolarization
effect of the 6p2 P1/2 state by using a magnetically induced Zeeman-scanning technique. The
Zeeman-scanning technique involves placing a cell of cesium and 10−6 torr pressure of argon
1
in a weak field (10 kG) electromagnet. Atoms were excited to the 6p2 P1/2 state by way of
a cesium RF lamp attached with a 8943 Å(894.3 nm) interference filter. Circular analyzers
were used to separate the observation of emitted σ + and σ − , the two helicities of circularly
polarized light. Their experiment was run with a cell temperature of 43 ◦ C. The results they
produced showed trends of depolarization of transitions to 6p2 P1/2 as pressure increases from
10−6 torr to 5 torr. The depolarization trend is non-linear. In addition, they extracted a
cross-section of 10.7 ± 0.6 Å2 at 10 kG between cesium and argon atoms. They also report
the cross-section to be 5.0 Å2 for a zero field experiment performed by a A. Gallagher [7],
and Gordeev’s theoretical calculated cross-section of 9.3 Å2 [6].
Previous experiments in Miami University’s Atomic Molecular and Optical Physics lab have
examined the cross-section of cesium and argon and the depolarization of the 6p2 P3/2 transition with circularly polarized light [18], collisional depolarization of the 6p2 P1/2 transition
using linearly polarized light to populate the excited level [16], and depolarization of 6p2 P3/2
transitions with krypton as a buffer gas [17]. This experiment tested the cross-section between Cs-Ar collisions, and categorizes the polarization changes as pressure of buffer gas
increases using the two-photon, two-color pump-probe laser technique. Results will be compared with Guiry’s and Krause’s suggested results.
2
Chapter 2
Properties of Cesium
Alkali metals are important in the spectroscopy field due to their single valence electron for
use in excitation and ionization. Cesium falls on the 1A column of the periodic table and
has an atomic number of 55 and an atomic mass of 133 a.m.u.s. Cesium is the 5th alkali
metal and has a pale golden color which sets it apart from the other alkali metals. Only the
single isotope of
133
Cs is stable and naturally occurring.
Cesium has a melting point of 28.4 ◦ C which is second lowest of all metals, the first being Mercury. Because of thermodynamic properties, it can easily melt within a vacuum
sealed cell by the body heat of a hand. Cesium is the second most reactive of all metals in
the atmosphere and will explode instantaneously when it comes in contact with air at room
temperature. This is due to the exothermic reaction between cesium and water molecules,
2Cs(s) + 2H2 O −→ 2CsOH(aq) + H2 (g) + Heat. The reaction is due to the production of
Hydrogen gas and the heat generated from the reaction that ignites the gas. The reaction is
very fast, thus the almost instantaneous burst of flames. Cesium should be handled carefully,
and under copious amounts of mineral oil if needed to be open to the air. However, it is quite
safe and stable with a small amount inside a vacuum sealed cell and heated to moderate
temperatures.
The electronic configuration of the
133
Cs is very simple with only one valence electron in
the ground 6s2 S1/2 state. Cesium has only one valence electron making it hydrogen-like
3
and making it easy formulate the atomic theory calculations. This valence electron moves
in spherically symmetric potential described by the central filed approximation. Thus, the
energy states of the cesium atom obey the Rydberg formula Enl = −R/n2∗ where n∗ = n − δl ,
n is the effective principle quantum number (an integer) and δl is the quantum defect.
Inherent with all alkali atoms are their D spectral lines, the lines representing the first
two excited energy states from the ground state. The D1 line represents the transition from
the 6p 2 P1/2 state to the ground state (6s 2 S1/2 ). The D2 line represents the 6p 2 P3/2 to
ground state transition. In addition to the D lines, the energy levels of cesium can experience hyperfine splitting. This experiment examines the D1 lines excited state as well as
the hyperfine structure at that level (described in Chapter 3). Some of the properties of the
Cs atoms are listed in Table 2.1. With such a low melting point of Cs, it can make cesium
Table 2.1: Properties of Naturally Occurring 133 Cs Atoms [10]
Natural Isotope Abundance
100.0 %
Nuclear Spin [S]
7/2
wavelength D1
894.6 nm
Lifetime (τ6p2 P1/2 )
34.791 (90) ns
Decay rate (Γ1/2 )
28.743(75) MHz
Fine Structure Splitting (D1-D2)
16623 GHz
Hyperfine Splitting 2 S 1/2
9193 MHz
Hyperfine Splitting 2 P
1168 MHz
1/2
Doppler Width (D1 at 300 K)
361 MHz
Melting Point
301 K
Vapor Density at 423 K
2.25 × 1014 atoms/cm3
vapor even at room temperature. The vapor pressure and atomic density depend on the
temperature of the system. The vapor density of cesium atoms ranges from 1010 atoms/cm3
to 1015 atoms/cm3 at the temperature range of 25 to 200 ◦ C. The vapor pressure rises nearly
exponentially with the rise of temperature and can be calculated with the following equations
4
[10]. The first equation calculates the vapor pressure in the solid phase:
log10 Pv = 2.881 + 4.711 −
3999
T
where Pv is the vapor pressure in torr and T is the temperature in K. The vapor pressure in
the liquid phase (used in this experiment) is similarly calculated (Fig. 2.1) with
log10 Pv = 2.881 + 4.165 −
3830
.
T
If pressure is to be calculated from atmospheres instead of torr, the 2.881 term need only be
excluded.
In addition to vapor pressure, the vapor density is important to calculate to determine
a sufficient number of atoms are present to be excited within the cells heated at various temperatures. To calculate the number density of the vapor pressure (Fig. 2.1), the following
equation can be used:
vapor density = Pv /kT
where Pv is the pressure in pascals and k is the Boltzmann constant.
1.E+00
1.00E+16
0
50
100
150
1.E-01
200
250
1.00E+15
Pressure [torr]
1.00E+14
1.E-03
1.00E+13
1.E-04
1.00E+12
1.E-05
1.00E+11
1.E-06
1.E-07
Number Density [Atoms/cm3]
1.E-02
Temperature [ºC]
1.00E+10
Figure 2.1: Calculated Vapor pressure and number density as a function of temperature for
atomic Cs.
5
Figure 2.2 is the Grotrian diagram of the transitions to various excited states in cesium. It
is the basis for the begin of the two photon excitation experiment. The excitation scheme
desired can be mapped and decisions can then be made (What dyes are needed for the transitions? What interference filters are needed to observe cascading decay? What wavelength
would be needed to excite to an energy state with no listed transition line? etc.)
6
Figure 2.2: Grotrian Diagram of Electronic States of Cs [19].
7
Chapter 3
Excitation Theory
3.1
Excitation Scheme and Selection Rules
Excitation occurs when atoms absorb energy from external sources. This can be due to a
collision between an atom and a photon or another atom. As the energy from the photon or
other atom is absorbed, a result can be the transport of an electron to a higher energy. When
an electron at the ground state absorbs energy equal to the difference in energy between the
ground state and a higher excited state, the electron jumps to the upper excited state which
ultimately decays to the ground state after some time by releasing the previous absorbed
energy in the form of radiation, known as photons. In this experiment, cesium atoms are
Figure 3.1: Electron absorption and emission between two energy levels.
excited to the 6p2 P1/2 state from the ground state. The ground state of cesium is 6s2 S1/2
and is at a neutral energy which will be referred to as zero energy. A second laser was for
the transition from 6p2 P1/2 −→ 10s2 S1/2 . To excite the transitions, dye nano-second pulsed
lasers created a two-photon, two-color pump-probe technique. The first laser was set at a
8
wavelength of 894.37 nm, matching the energy difference between 6p2 P1/2 and the ground
state. This intermediate state has an average lifetime of 34.8 ns before the electron decays
down to the ground state. The second laser is tuned to 583.88 nm matching the transition
from 6p2 P1/2 −→ 10s2 S1/2 , the final state. These two photons excite electrons to the final
state from which they will decay back to the ground state. The transition from 10s to 6s is
considered a forbidden transition, thus electrons will cascade to the nearest permissible transition more often and then decay down to the ground state. In reality, a 10s to 6s transition
does occur, but only a very small percentage of atoms decay in this fashion. The highest
probability is that atoms will cascade to the 9p2 P1/2 state and decay to ground state (Fig.
3.2).
3.509 eV
3.427 eV
10s2S1/2
9p2P1/2
583.88 nm
(probe)
1.386 eV
6p2P1/2
361.73 nm
(cascade)
894.37 nm
(pump)
0.00eV
6s2S1/2
Figure 3.2: Partial energy level diagram for the two-photon excitation of the electronic states
of atomic cesium.
The selection rules describe the change that takes place in quantum numbers during the
atomic transition. It rules narrows in on the highest probability of transition between all
possible pairs of energy levels. Those transitions which are not allowed by selection rules are
known as forbidden transitions and are of extremely small probablity. These rules determine
the most likely transitions among the quantum levels with the emission or absorption of
radiation. Table 3.1 shows the selection rules for the experiment. S refers to the total spin,
9
Table 3.1: Selection rules and Zeeman sublevel transition characterization
∆L = ±1
m=0
π Lin. Pol.
∆J = 0, ±1
m = +1
σ + RCP
∆S = 0
m = −1
σ − LCP
L represents the total orbital angular momentum, and J is the total angular momentum. As
mentioned before, a decay from 10s to 6s is not probable. Employing the selection rules, 10s
and 6s both have no orbital angular momentum, L=0. In this case, the change of L is not
equal to 1, thus violating the selection rules. In the same instance, 9p has an L equal to 1.
The transition from 9p to 6s would have a change of L (1 − 0 = 1) of an acceptable value
for to satisfy the selection rules, and a high probability of occurring [23].
3.2
Linear Polarization
→
−
Linear polarization describes the orientation of electromagnetic vectors of light. The E -Field
propagates down the z-axis, and is the combination of x and y components [20],
~ t) = E
~ x (z, t) + E
~ y (z, t)
E(z,
where,
~ x (z, t) = îE0x cos(kz − ωt)
E
and
~ y (z, t) = ĵE0y cos(kz − ωt + ).
E
represents the relative phase difference between the x and y waves. k is the vector component of propagation, z is the direction of propagation, and ω is the frequency. If > 0, Ey
is out of phase and behind Ex . In the same respect, if < 0, Ey is out of phase and leading
Ex . However, if = 0, both x and y components are in phase with each other. The resulting
→
−
vector is considered Linear, or plane polarized. The E -field oscillates in one plane and has
→
−
no angular frequency. The magnitude of the E -field grows and reduces along the sinusoidal
fashion of the two component vectors, as shown in Fig. 3.3. As described in the table 3.1,
10
Figure 3.3: Linear, or plane polarization. E-field travels down the z-axis and is the combination of vector additions of the x and y components.
linear polarized light is denoted as π and has transitions only to like m sublevels (see Fig.
3.4). A use of a half-wave plate (HWP) can be used to flip the axis of linear polarization.
It works by retarding the component of polarization (x or y component) which is along the
HWPs optical axis. The retarding component will be slowed by half a wave, thereby having
→
−
the components out of phase by 180◦ , thus changing the polarization of the E -field by 90◦ .
10s2S1/2
-1/2
+1/2
# - "m=-1
LCP
# + "m=+1
RCP
6p2P1/2
-1/2
+1/2
! "m=0
! "m=0
6s2S1/2
-1/2
+1/2
Figure 3.4: Allowed transitions to Zeeman levels by m selection rules.
3.3
Circular Polarization
In the same respect with linear polarization, the two x and y components can be out of phase
as much as 90◦ , or π2 . In this case,
~ x (z, t) = îE0x cos(kz − ωt)
E
11
and
~ y (z, t) = îE0y sin(kz − ωt).
E
→
−
The resulting E -field and be written as
~ t) = E
~ x (z, t) ± E
~ y (z, t)
E(z,
which depends on which component is leading. If the x and y component are added together,
→
−
this deems that the E -field is propagating in a clockwise angular motion around the z axis.
This is considered to be Right Circularly Polarized light (RCP). Through the propagation,
→
−
~ t) =
the magnitude of the E -field remains constant, and only direction oscillates. If E(z,
Figure 3.5: Circularly polarized light propagation with time.
−
~ x (z, t) − E
~ y (z, t), the →
E
E -field propagates in a counter-clockwise motion, while remaining
with a constant magnitude. This is considered Left Circularly Polarized light (LCP). As
Figure 3.6: Vector addition of the axis’s and resulting electric field for circularly polarized
light.
described in the table 3.1, circularly polarized light is denoted as σ ± and has transitions
12
of different m sublevels satisfying the m = ±1 rule (see Fig. 3.4). A description of the
excitation scheme using circularly polarized light is shown in Fig. 3.7. The excitation
scheme uses same helicity (both photons for double excitation are the same RCP or LCP
light) and different helicity (the pump photon is LCP and the probe photon is RCP). Notice
10s2S1/2
-1/2
+1/2
LCP
RCP
LCP
RCP
6p2P1/2
-1/2
+1/2
LCP
RCP
6s2S1/2
-1/2
+1/2
Figure 3.7: Kastler Diagram displaying dual helicity circular polarization excitation.
that in the ideal case, excluding hyperfine splitting, same helicity excitation excites the atom
to a non-resonance level. A use of a quarter-wave plate (QWP) can be used to flip the axis of
linear polarization. It works by retarding the component of polarization (x or y component)
which is along the QWPs optical axis. The retarding component will be slowed by a quarter
of a wave, thereby having the components out of phase by 90◦ , thus changing the circular
→
−
polarization of the E -field by from RCP to LCP or vice versa.
3.4
Broadening Mechanisms
Solutions of the time independent Schrödinger equation yield eigenfunction and energy eigenvalues for a system described by a Hamiltonian. The eigenvalue spectrum obtained under
these conditions is infinitely discrete; the eigenvalues have a single precise value for each set
of quantum numbers and so radiative emission or absorption lines would be expected to be
infinitely narrow, in theory only, in frequency. In reality, the lines would be narrowed to
the natural line width due to interactions with the quantum vacuum. However, interactions
of the isolated system with an environment will change the eigenvalues. If the interaction
13
is time dependent or has a distribution in some parameter determining the eigenvalues, the
emission or absorption lines will no longer be minimally narrowed to the natural line width.
They will be broadened.
There are two general classes of broadening phenomena: homogeneous and inhomogeneous.
When the line broadening is a result of a statistical distribution of some external parameter
(to the atom), the line is said to reflect inhomogeneous broadening. For example, in Doppler
broadening one can assign a definite velocity to each frequency on the line profile. This
results constitutes inhomogeneous broadening. Homogeneous broadening is the result of a
process which is the same for any given atom. Examples would include broadening due to
the radiative decay of an atom or to collisions (where the duration of the collision is short
and is viewed as phase changing only).
3.4.1
Line Broadening
Natural line broadening (NLB) is very small compared to other causes of broadening. NLB
is an example of homogeneous broadening since each atom behaves identical within the
system. No excited state has an infinite lifetime; therefore no excited state has a precisely
defined energy. It follows that spectral lines always spread over a range of frequencies due
to Heisenberg’s uncertainty principle ∆E ∝
1
,
∆ν
which defines the width of the transition.
NLB is defined by:
δν =
1
τ × 2π
(3.1)
where τ is the average lifetime of the excited energy state.
3.4.2
Doppler Broadening
Atoms in gas phase will have thermal motion with a spread in their velocities that give rise
to spread in the frequencies at which atoms could absorb or emit radiation. This effect is
caused by the Doppler Effect, the result of which is to reduce the actual number of interactions an atom has in a laser field.
14
The Doppler shifted resonance frequency for an atom traveling in the same direction as
the propagating direction of the laser beam, the z direction with velocity vz is given by:
vz ω◦
' ω◦ 1 ±
.
ω=
1 ∓ vz /c
c
(3.2)
For an atom in an external radiation field, the motion of the atom in the direction of the
field wave vector produces a frequency shift. The new frequency, dependent on velocity, is
given by:
r
f (v) =
M v2 M
exp −
.
2πkB T
2kB T
(3.3)
From the Boltzmann distribution, the number of atoms with velocity between v ↔ v + dv
in the direction of the observed light is found with
r
M v2 M
exp −
dv
f (v)dv = N
2πkB T
2kB T
(3.4)
where N is the total number of atoms, and M is the atomic mass. In terms of the atomic
frequency this is given by
r
f (ω)dω = ρ(T )
M c2 (ω − ω )2 c
M
◦
exp −
dω.
2
2πkB T
2kB T ω◦
ω◦
The full width half maximum (FWHM) of this distribution results by the following:
r
2ω◦ 2kB T
ln2.
∆ω =
c
M
(3.5)
(3.6)
The effective density of atoms that interact with the laser is the fraction of atoms that fall
within the natural line width of the transition. This can be approximated as
r
Z ∆v
M v 2 (v − v )2 c
2
M
∆v
◦
ρint (T ) =
ρ(T )
exp −
dv ∼
=N
2
2πkB T
2kB T v
v◦
∆νD
− ∆v
2
(3.7)
where ρ(T ) is the density of atoms in the cesium cell, ∆νD is the Doppler width, ∆v is the
natural line width, and ρint (T ) is the density of atoms resonant with the laser.
15
3.4.3
Collisional (Pressure) Broadening
The theory of collision broadening is an extensive field of study. For the purpose of the
experiment, focus will be restricted to collision broadening due to pressure. According to
kinetic theory, the time between collisions is calculated using the following relationship:
1 mkB T 1/2
V πkB T 1/2
τc =
= 2
4σc P
3
4d N
m
(3.8)
where P is the pressure, V is the volume, m is the mass, and d is the distance between two
atoms. The collisional cross section, σc , is the effective area that determines whether two
atoms will collide or not. It is less than or equal to the size of the atom. In the case of
cesium, with a radius of 0.26 nm,
2
σc = πr2 ∼
= π × [0.26nm] ∼
= 2.12 × 10−19 m2 .
From here the determination of the collisional time can be made. The relationship between
collision broadening and pressure is
τc ∝
T 1/2
P
which means a reduction of pressure increases τc and thus reduces the line width. Typically,
an increase of pressure broadens the line width.
3.5
Alignment and Orientation
Excitation of an atom or molecule by interaction with light in the gaseous medium can
leave the atom in an anisotropic state. To observe the variations of the anisotropy, or
directional dependent population trends, of excited atoms during the light emission, Fano
and Macek [13] introduced the concept of alignment and orientation to observe variations of
the anisotropy of an atom during the light emission and introduced a general expression of
the intensity of the polarized light emitted in the right-angle geometry in terms of alignment
and orientation. Alignment and orientation measure the net angular momentum of electrons
in a given atomic level. Alignment is the electric quadrupole component of the density in
the excited state. The expectation value of alignment is to populate the larger extreme m
sub-levels and minimize smaller valued ones. Figure 3.8 shows the minimizing population
16
expectation of the ± 12 m sub-levels and the higher population expectancy in the ± 32 m sublevel. Alignment can be calculated one of two ways, using the total angular quantum number
Figure 3.8: Alignment for a transitions to a J=3/2 state (hAi = 52 ) [21, 18].
of the excited state (J 0 ) or the magnetic quantum number of the same excited state (m0 )
and can be written as follows:
h3Jz02 − J 02 i X |a(m0 )|2 3m02 − J 0 (J 0 + 1)
hA◦ i = 0 0
=
J (J + 1)
J 0 (J 0 + 1)
m0
(3.9)
where a(m0 ) is the Clebsch-Gordon coefficient, discussed in Chapter 4. In this experiment,
since J=1/2 and functions only as a electronic dipole, the alignment does not factor into the
polarization, hA◦ i = 0.
hA◦ i =
X |a(m0 )|2 3m02 − J 0 (J 0 + 1)
m0
J 0 (J 0
+ 1)
=
|1|2 3( 12 )2 −
3
4
3
4
+
|1|2 3( −1
)2 −
2
3
4
3
4
=
0
3
4
+
0
3
4
=0
Orientation is the magnetic dipole component of the density in the excited state. The expectation value of orientation is to populate higher m sub-levels and minimize lower sub-levels.
Orientation can be calculated using J 0 and m0 as well.
X |a(m0 )|2 [m0 ]
hJz i
p
=
hO◦ i = p
J 0 (J 0 + 1)
J 0 (J 0 + 1)
0
m
(3.10)
Figure 3.10 shows the populations of magnetic sublevels when alignment and orientation are
both zero (a) and when how the sublevels will populate if they are not zero.
17
Figure 3.9: Orientation populating a J=3/2 state (hOi =
3.6
√5 )
2 15
[21, 18].
Intensity and Polarization
The most common geometry of an experiment where light interacts with matter is given
in Fig. 3.11. As shown in the figure, there are two coordinate frames in interest[8]. The
first frame is the collision frame, where the photon-matter interaction takes place [16, 8].
The second is the detection frame, or the geometry of the detector from the collision frame.
The collision frame can be adapted into the symmetry of the excitation process. Setting the
excitation process to be along the z-axis (quantization axis) introduces cylindrical symmetry
about the axis.
The detector frame on the other hand is where the detection of the emitted light is observed.
It is free to rotate about the common origin of the two frames. The observation direction of
the photon is along the z 0 axis, therefore the polarization vector lies on the x0 − y 0 plane and
can be written as,
ˆ = î cos β + ĵ sin β.
(3.11)
As shown in Fig. 3.11, the orientation of the collision frame with its x, y, z coordinates and
the detection frame with its x0 , y 0 , z 0 coordinates [16]. However, this experiments uses a
different orientation to observe the fluorescence photons. In Fig. 3.12, the detector is along
the y axis. Because of this, the orientation of the collision frame is freed to be oriented in any
direction, thus θ is restricted at zero. In addition, the second photon is used as the detection
polarizer. The final state will theoretically only be populated by atoms polarized opposite to
18
Figure 3.10: Population verses magnetic sublevels for (a) a pure monopole moment, (b)
alignment, the magnetic quadrupole moment, and (c) orientation, the electric dipole moment.
the second photon. Both methods were tested, and the second, Fig. 3.12, was found to have
significantly less background noise on the detector. Using a detection of relative intensity,
one can measure the polarization degree of the excitation. Theoretically, intensity of the
fluorescence can be written in terms of the Alignment and Orientation [8]. The intensity of
the fluorescence can be constructed to be dependent on three variables. These variables are
the Euler angles of the reference frame in which excitation is employed and detected. θ and
φ are the geometry angles relating to the detecter frame with the collision frame. χ defines
the orientation of the linear polaroid in the detector frame, since the polaroid is part of the
detection. Lastly, β defines the polarization of light to be detected according to table 3.2.
The Intensity of the fluorescence radiation can is given by Greene and Zare [8] to be
I◦ 1
1 − h(2) (J, J 0 )hA◦ iP2 (cos θ)
3
2
3 (1)
+ h (J, J 0 )hO◦ i(cos θ)(sin 2β)
2
3 (2)
0
2
+ h (J, J )hA◦ i(sin θ)(cos 2χ)(cos 2β)
4
I(θ, χ, β) =
(3.12)
whereP2 (cos θ) representing the 2nd order Legendre polynomial and h(k) (Ji , Jf0 ) representing
19
ization direction of the probe laser is along z axis and I?
of state multipoles [25,21] as
a
b
Fig. 2. Light-matter interaction geometry representing (a) one-photon absorption: the detector in this case detects only linearly polarized light as may be obtain
polaroid mounted in front of a polarization insensitive detector, (b) two-photon absorption: the detector in this case detects cascade fluorescence from a final l
Figure 3.12: Experimental reference frame,
3.11: only.
Reference Frames of excitation
proportional to Figure
probe population
detection angle aligned on the y axis [12].
and detection for theoretical calculations.
the geometrical quantities. For h, the orientation is represented when k=1, and the alignment
is represented when k=2.
This calculation is significantly reduced because the alignment, the quadrupole component,
hAo i is zero for the J=1/2 case. The terms with hA◦ i terms are not considered, and only the
hO◦ i, the dipole component, is considered. The remaining components are now dependent
on two variable angles. In addition, the detection scheme was affixed to the y-axis of the
collision scheme, as seen in Fig. 3.12, for practical reasons regarding set up of the actual
experiment components.
I◦ 3 (1)
0
I(θ, β) =
1 + h (J, J )hO◦ i(cos θ)(sin 2β)
3
2
(3.13)
Polarization is extracted from the intensity simply by creating a ratio of the intensity and
differentiating between circular polarization excitation schemes. The intensities are categorized by the helicity of the excitation, where Iσ−− is an excitation scheme with both photons
are LCP and Iσ−+ is where the first photon is LCP and the second photon is RCP.
− Iσ−+ PC =
× 100
Iσ−− + Iσ−+
I
σ −−
20
(3.14)
Table 3.2: Associated Polarization for β
β=0
Linearly Polarized along x0
π
2
Linearly Polarized along y 0
β=
3.7
3.7.1
β = ± π4
Circularly Polarized
β = other
Elliptical Polarization
Hyperfine Interaction
Fine Structure
→
−
The Fine structure is a result of the coupling between the orbital angular momentum L
→
−
of the outmost electron and its spin angular momentum S [10, 24]. Classically, the total
electron angular momentum is given by
→
−
→
− →
−
J = L+S
(3.15)
→
−
were J takes on values of
|L − S| 5 J 5 L + S.
However, L and S are both quantized and their relative directions are restricted in the
→
−
quantum mechanical case. The magnitude of the the total angular momentum J is given
by [24],
p
→
−
| J | = j(j + 1)~
where
j = l + s or j = |l − s|
→
−
and the z component of J is
Jz = mj ~ where mj = −j, −j + 1, ...., j − 1, j.
21
(3.16)
3.7.2
Hyperfine Structure
→
−
→
−
Coupling of J with the total nuclear angular momentum I results in the hyperfine structure.
→
−
The total electron angular momentum F is given by [10, 24],
→
−
→
− →
−
F = J + I.
(3.17)
As before, the magnitude of F can take values of
|J − I| 5 F 5 J + I.
In addition, F follows selection rule of ∆F = 0, ±1 and a transition from F = 0 −→ F = 0
is considered forbidden. To calculate the hyperfine splitting in cesium, Arimondo’s paper,
reiterated by Steck’s paper, [14, 10] provides the following equation,
EFhpf
3 K(K + 1) − 2I(I + 1)J(J + 1) 1
= hAK + hB 2
2
2I(2I − 1)2J(2J − 1)
(3.18)
where h is Planck’s constant, A is the magnetic dipole coefficient, B is the electric quadrupole
coefficient, I is the total nuclear angular momentum, J is the fine structure, and K =
F (F + 1) − I(I + 1) − J(J + 1). F describes the hyperfine structure, or levels as they split.
In the case of the experiment, I =
7
2
and J =
1
2
for the 6p2 P1/2 state. In turn, F = 4,3. Since
the transition 6s2 S1/2 −→ 6p2 P1/2 does not have an electric quadrupole, the B coefficient is
zero [14, 10]. Thus, the hyperfine structure is:
1
EFhpf = hAK.
2
Each level of the splitting can be calculated using the appropriate F, and describes the
splitting off the natural energy state. For F=3, the hyperfine state under the natural energy
state, K = − 92 ; the state F=4, above the natural energy state, K = 72 . The energy above
the natural state is
7
1
1
= 510.825MHz(h).
EF =4 = hAK = h(291.90MHz)
2
2
2
The energy below the natural state is
9
1
1
EF =3 = hAK = h(291.90MHz) −
= −656.775MHz(h).
2
2
2
22
The total splitting from the natural line is: ∆E = (EF =4 − EF =3 ) = 1167.6 MHz (h), or
1.1676 GHz (h). In addition, ∆E = hν = 1.1676 GHz (h), and thus the frequency is the
change of energy (ν = 1.1676 GHz). The hyperfine structure is always present, but it can
be resolved over a period of time to actually see the specific levels effect the signal. The
minimum time for hyperfine to be resolved is the inverse of the frequency that was just
calculated. Thus,
τhf =
1
1
=
= 8.564 × 10−10 sec = 0.856ns.
ν
1.1676 × 109 sec−1
This hyperfine time is important in developing an excitation scheme (see Section 6.1).
For example, if the smallest hyperfine time is 0.856 ns and the overlap time of the two
pulses is much smaller than 0.856 ns, the atoms will be excited to a final state before the
hyperfine structure has resolved so that the specific level being dealt with is identified. In
this instance, the signal will not be affected by the hyperfine structure. One setback to this
approach is the percentage of excited atoms that can reach the 10s2 S1/2 state. In using a
small percentage of total pulse time, in addition with a small percentage of overlap time
for the two pulses, the number of atoms excited is greatly reduced resulting in a significant
drop in intensity. The drop of intensity effects the signal detection, and allows for systematic
noise to interfere with the detection.
However, in this experiment, the overlaps of the two pulses is 2.53 ns. Therefore, hpf
structure does affect the signal. The effects of the hpf structure can be taken into account
in the data analysis and their effects explained, as shown in section 4.4 and 6.1.
23
E
10s2S1/2
110.6 MHz
171268 cm-1
252.8 MHz
142.2 MHz
τhf =3.95 ns
583.88 nm
F=4
6p 2P1/2
510 MHz
11178 cm-1
1.167 GHz
τ =34.8 ns
656 MHz
τhf =0.856 ns
F=3
894.34 nm
D1 Line
F=4
6s 2S1/2
0 cm-1
4.021 GHz
9.192 GHz
Ground State
5.171 GHz
F=3
Figure 3.13: Hyperfine Structure of Ground and Excited States.
24
Chapter 4
Polarization Consideration
Two methods of calculating intensity will be demonstrated here. The intensity calculated will
be used to calculate the theoretical value of polarization. The first is by applying the WignerEckart theorem which utilizes Clebsch-Gordon coefficients. Alternatively, polarization can
be calculated using the fluorescence radiation theory utilizing Euler angles and Hyper-fine
splitting.
4.1
Clebsch-Gordon Coefficients
Clebsch-Gordon coefficients (CGC) are referred to as vector addition coefficients. They arise
from adding the electron and core spins to obtain the spin of the atom and the electron and
core orbital momenta to obtain the atomic orbital momentum. CGCs are represented in
adding together the angular momentum of vectors of two particles, or angular momenta of
different origin (for example, spin and orbital angular momenta) for one particle, to obtain
the total angular momentum of a system. The polarization degree of any atomic transition
can be calculated by using transition matrix elements. The general form of the transition
matrix element is given as
hj 0 m0 |Tqk |jmi = C(jj 0 k; mm0 q)hj 0 ||T k ||ji
(4.1)
where C(jkj 0 ; mqm0 ) is the Clebsch-Gordon coefficient. It follow the conservation of angular
momentum and vanishes unless m0 = q + m. The second term hj 0 ||T k ||ji is called a reduced
25
matrix element of tensor operator Tqk and is independent of quantum numbers m and m0 .
The primes denote the final states and no prime signify initial states for both j and m.
The tensor rank (k) and the total quantum number (q) of the transition both describe the
tensor operator and affect the matrix elements. From the reduction of the m’s, dependency
of the state is now only on j. From here there are several tables to apply to the reduced
matrix elements to calculate the CGCs that go with a corresponding state. However, be
fore warned. Many tables use different labels, orders, or representation and trying to use
many can become confusing. It is extremely important to pay attention to excited state
representation and initial state representation, as well as ranks and orders. If the matrix
elements are understood, any table can be used, with a slight adjustment of the order in
the C-term. The following table shows the equations used to determine the CGCs for this
experiment.
Table 4.1: Clebsch-Gordon Coefficients Equations use for Experiment [26]
j=
j1 + 1/2
4.2
m=1/2
q
1
j1 +m+ 2
2j1 +1
m=-1/2
q
1
j1 −m+ 2
2j1 +1
Polarization Calculation using Wigner-Eckart
Theorem
The Wigner-Eckart Theorem, as described with Clebsch-Gordon coefficients, describes the
addition of angular momentum and separates the dependence of the matrix elements on
spatial orientation (m0 , m and q) from the rest [22]. The spatial orientation is expressed
entirely in terms of CGCs [18, 19]. The transitions A and B from the ground state to the
intermediate state can be represented by the following density matrices (see Fig. 4.1).
Transition A
hj 0 m0 |rq |jmi =
D 1 −1
26
2 2
|r1 |
1 1E
22
10s2S1/2
-1/2
+1/2
LCP
RCP
LCP
RCP
6p2P1/2
-1/2
+1/2
LCP
RCP
A
6s2S1/2
-1/2
+1/2
B
Figure 4.1: Transitions for Clebsch-Gordon calculations.
1 −1 − D 1 k 1 E
=C
1;
1
||T ||
22 2 2
2
2
1 1
(4.2)
Transition B
D1 1
1 −1 E
|r1 |
hj 0 m0 |rq |jmi =
22
2 2
1 1 −1 1 D 1
1E
1;
1
||T k ||
=C
22
2 2
2
2
(4.3)
The CGC for transition A is calculated using the following equation [26]:
s
j1 − m + 21
0
aA (m = −1/2) =
= 1.
2j1 + 1
The CGC for transition B uses a slightly different equation [26]:
s
j1 + m + 12
aB (m0 = 1/2) =
= 1.
2j1 + 1
For both A and B transitions, the m’s in the equation represents q, the total quantum
number of the tensor operator, and j1 is the initial state j. Since there is no alignment for
the j=1/2 states, there will be no alignment consideration in the polarization calculation.
From this point, the figures found can be applied to an alternate intensity equation.
I = |hj20 m02 |rq,2 |j2 m2 ihj 0 1 m01 |rq,1 |j1 m1 i|2
The intensity of the same helicity excitation process is as follows.
D 1 3
1 1 ED 1 1
1 1 E2
Iσ−− = − |r−1 | −
− |r−1 |
=0
2 2
2 2 2 2
22
27
(4.4)
Same helicity excitation theoretically makes a undefined transition, as shown in bold, thus
there should be no double excitation. This can be seen by the diagram for the dashed arrow,
and also by the rule that m0 = q + m. Since q =-1 and m=-1/2, the excited m should be
-3/2, however, there is not an m= ±3/2 for the j=1/2 state. The intensity for the opposite
helicity case does exist and is written as:
I
σ −+
D 1 1
1 1 ED 1 1
1 1 E2
=
|r1 | −
− |r1 |
.
22
2 2 2 2
22
Expounding with the CGCs calculated before, intensity is now:
D 1 1 E D 1 1 E2
Iσ−+ = 1 |T | 1 |T | .
2 2
2 2
Since polarization is the ratio of intensities, and the CGCs for these transitions happen to be
identity matrices, the reduced matrix elements, which are only dependent on j, will cancel
in the ratio. Thus, circular polarization degree is calculated as
PC =
Iσ−− − Iσ−+
Iσ−− + Iσ−+
D
E D
E2
1 1
1 1
−1 2 |T | 2 1 2 |T | 2 −1
× 100 = −100%.
× 100 = D
E D
E2 × 100 =
1
1 1
1
1 |T
|
1
|T
|
1
2 2
2
2 Depending on the starting polarization of light, the calculated polarization is positive or
negative. This experiment used LCP for the transition A, and it would be expected that
negative polarization (0 −→ −1).
4.3
Polarization Calculation Using Fluorescence
Radiation Theory
From chapter 3, the intensity calculation from Greene and Zare [8] is dependent of factors
of the detection frame, collisional frame, and atomic properties as well.
I◦ 1
I(θ, χ, β) =
1 − h(2) (J, J 0 )hA◦ iP2 (cos θ)
3
2
3 (1)
+ h (J, J 0 )hO◦ i(cos θ)(sin 2β)
2
3 (2)
0
2
+ h (J, J )hA◦ i(sin θ)(cos 2χ)(cos 2β)
4
28
(4.5)
For the J=1/2 case, both LCP and RCP light will have no alignment component since
alignment is based on the quadrapole. Thus, since hA◦ i = 0, the second and last expression
in the parentheses, both dependent on alignment, are excluded from the calculation. The
resulting equation is:
I(θ, β) =
3
I◦ 1 + h(1) (J, J 0 )hO◦ i(cos θ)(sin2β) .
3
2
(4.6)
In addition, the orientation for circularly polarized light is
hJz i
±1
hO◦ i = p
=√
3
J(J + 1)
respectfully for σ ± . The ratio of Racah coefficients (Racah coefficients will be discussed
in the next section), which are dependent on the total angular momentum of the states
(h(1) (J, J 0 )) was solved to be:
h(1) (J, J 0 ) = p
1
J(J + 1)
=q
2
=√ .
1 1
3
( + 1)
2 2
1
In combining the orientation and Racah coefficients into the intensity equation,
I(θ, β) =
I◦ 3 2 −1
1 + ( √ )( √ ) cos θ sin 2β
3
2 3
3
which reduces to
I◦ I(θ, β) =
1 − cos θ sin 2β .
3
Due to the orientation of the detection angle, as discussed in chapter 3 and 5, the value of
θ is 90◦ . In addition, because we are using CPL, the value of β will be ± π4 , as seen in table
3.2. The resulting intensity will be:
I=
I◦ π
1 ± sin
3
2
(4.7)
Here the plus refers to the excitation of CPL with different helicities (σ +− or σ −+ ). The
minus refers to CPL with same helicities, or both Laser 1 and 2 with the same polarization
with respect to the collisional frame (σ ++ or σ −− ). If the excitation is of opposite helicities,
2
I = I◦
3
29
wherein if excitation is of the same helicity, I = 0. Now if this is applied to the polarization
calculation,
− Iσ−+ PC =
× 100
Iσ−− + Iσ−+
I
σ −−
(4.8)
it is clear that the theoretical polarization is 100%:
PC =
0 − (− 2 I ) −I ◦
3 ◦
× 100 = −100%.
=
2
I
0 + (− 3 I◦ )
◦
Polarization can be left in terms of orientation. The same intensity equations, with orientation left in, will result in a polarization ratio of:
3
PC = − √ hO◦ i
3
(4.9)
Table 4.2: Calculated values of orientation, alignment, and polarization for the D line excited
states
Excitation to J 0 with σ + Light
4.4
J
hAi
1/2
0
3/2
2/5
hOi
√
1/ 3
√
5/2 15
Excitation to J 0 with σ − Light
PC
hAi
1
0
1
2/5
hOi
√
-1/ 3
√
-5/2 15
PC
-1
-1
Hyperfine Consideration in Polarization
The depolarization coefficients generally describe the influence of an initially unpolarized and
unobserved angular momentum I on an average tensor multiple hTqk i described in the state
angular momentum J [15]. The orientation and alignment are then axially symmetric tensors
with rank 1 and 2 respectively. If a multipole of initial value hTqk (0)i becomes depolarized via
an interaction between I and J, then hTqk i = g (k) hTqk (0)i, and the hyperfine depolarization
coefficient (g (k) ) is
g (k) =
X (2F 0 + 1)(2F + 1) W 2 (F F 0 JJ; KI)
[15].
2
2I
+
1
1
+
(ω
FF0τ )
0
FF
30
(4.10)
F and F0 are the set quantum numbers formed by coupling I and J, W(...) is the Racah
coefficient (similar to Clebsch-Gordon coefficients, but dealing with three sources of angular
momentum and not two), and ωF F 0 is the angular frequency splitting produced by the interaction of O and J. The average duration of the interaction is τ . Hyperfine splitting must
be considered due to the interaction time. It actually oscillates over time, and in addition
to that, orientation, and if applicable, alignment oscillate. To resolve this, a depolarization
coefficient (g (1) ) is employed to the theoretical calculation. According to Havey and Vahala
[15], g (1) for Cs 6p 2 P1/2 is 0.344. Applied to the theoretical calculation, an expectation of
polarization for a transition to 6p 2 P1/2 in pure cesium should be an average of -34.4%. Polarization can be written in terms of orientation and the hyperfine depolarization coefficient
which looks like:
3
PC = − √ hO◦ ig (1) .
3
31
(4.11)
Chapter 5
Lasers
The following describes the lasers and laser apparatus scheme used in the experiment. Information about general principles as well as experiment specific information is presented.
5.1
The Laser
The invention of lasers fifty years ago revolutionized the science world. Light Amplified by
Stimulated Emission of Radiation (LASER) creates coherent photons that are in-phase with,
has the polarization of, and propagates in the same direction as the stimulating radiation
[20]. The feature of stimulated emission is crucial to lasers so that each emitted photon is
coherent with all other emitted photons. From here on, flash lamp pumped crystal lasers
will be highlighted.
To achieve stimulated emission, a state of population inversion is key, because when the
laser is turned on, the atoms are most always in the ground state. As the flash lamp pulses,
the energy from the light excites the atoms from the ground state to an excited state. To
achieve a population inversion, a substantial percentage of the atoms must be in the excited
state leaving the ground state all but empty.
32
Figure 5.1: The Stages of stimulated emission due to Population Inversion [4].
5.2
Nd:YAG Laser
The primary laser used in the experiment is a (Continuum Surelight I-20) neodymium-doped
yttrium aluminium garnet crystal flash pulsed laser, also known as an Nd:YAG laser. The
crystal was flash pulsed at a rate of 20 Hz. The fundamental wavelength of the Nd:YAG is
1064 nm. Using different harmonic add-ons, we can double or triple the frequency of the
laser exiting the non-linear crystal 2nd and 3rd harmonic generators and generate higher
wavelengths. Two wavelengths can be emitted from the laser at any given time based upon
which are selected (i.e. 1st and 2nd harmonics, 1st and 3rd harmonics, or 2nd and 3rd
harmonics). In addition, the laser can be set to only emit one wavelength. This will give
the highest power output for a single beam, because the initial laser is not being split into
two different harmonic channels. The wavelength of the laser, generated from the second
harmonic resonator, is 532 nm. Due to the age of the laser, power has been reduced slightly
from previous experiments to 0.50 watts. This laser is used to pump the two dye lasers for
the experiment. Each dye was specifically chosen around the efficiency of obtaining a specific
wavelength from excitation of the Nd:YAG laser.
33
5.3
Dye Lasers
Dye lasers consist of pumping a solution of an organic dye with a laser, causing the dye
to fluoresce. Based upon the concentration of the dye in solution, the pumping laser, and
type of dye, it is possible to achieve specific wavelengths of fluorescent light. The fluorescent
can be harnessed and made coherent by using a Littman-Metcalf cavity configuration. The
Littman-Metcalf gives the dye laser the ability to be tuned, anywhere between 2nm and
20nm depending on the dye chemical and source laser. The Nd:YAG laser excited the dye to
Figure 5.2: Littman-Metcalf Configuration for tunable dye laser.
fluoresce, and perpendicular to the laser excitation in the Lab x-axis is the highest probability
of fluorescence. Behind the cuvette is a diffraction grating (minimum 1200 division per mm)
that allows the high probability of fluorescence to cascade across the entire grating. A mirror
is set next to the grating to reflect the first order excited line back to the grating and into
the cuvette again. In front of the cuvette is a reflective lens (output coupler) that, after
some threshold is achieved, allows light to pass through. What does not pass is cycled back
to excite more dye molecules, creating the laser.
5.3.1
Laser 1
What will from now on be designated as Laser 1 is a 894.37 nm dye laser for excitation
from 6s 2 S1/2 →6p 2 P1/2 energy level. The dye chosen for this particular wavelength was
34
LDS 867 [Exciton 08670] which is dissolved into 99% Methanol at a solution concentration
of 2.95 × 10 −4 Molar. The cell is set up using the Littman-Metcalf design. For a dye cuvette,
a special AR coated flow cell(NSG Precision Cells Inc. T-524) is used with the addition of
a dye circulator (Spectra Physics Model 376) because the dye used degrades quickly and
circulation helps to stabilize the power output. A gold-coated grating (Edmund Optics Y55261) is used for the Laser 1 cavity because gold is highly reflective for wavelengths in the
IR range. The grating has 1200 grooves/mm. In addition to the grating, an output coupler
(CVI LW-2-1037-C) is fixed at the front of the cell. To excite the dye, the Nd:YAG laser
passes through a cylindrical lens with a 5.08 cm focal length. To complete the oscillator, an
infrared coated mirror (Thorlabs BB1-E03) with a reflectivity of 99% is used in a precision
mount for tuning Laser 1 [18]. While not as great as Laser 2, Laser 1 has a tunability of
about 5 nm. Since the wavelength is stationary during data collection, a small tunable range
satisfies the needs of the laser.
5.3.2
Laser 2
The designated Laser 2 is a 585 nm dye laser for electron excitation of the 6p 2 P1/2 →10s 2 S1/2
transition. The dye used for this transition is Rhodamine 610 [Exciton 06100] which is
dissolved into 99% Methanol at a solution concentration of 2.1 × 10 −4 Molar. In addition, to
boost power output, we also used a amplifier cell made from the same dye at a concentration
of 3.1×10 −5 Molar. The amplifier is not part of the Littman-Metcalf cavity but was recorded
as increasing the power output 2 fold. Particular to Laser 2 is a Kinematic mirror mount
(Newport 610 Series Ultra-Resolution Kinematic Mirror Mount) that provided tunability.
The mirror mount was fitted with a motor mike (Ardei Kinematic Motor Mike: Linear
Motorized Actuator) that took the place of the fine adjustment of the vertical rotational
axis. The motor mike was able to adjust from analog signals from the computer, allowing
for automated wavelength adjustment during data collection. In addition to the mirror, as
in the Littman-Metcalf set up, a holigraphic grating with 1200 grooves/mm is used to unveil
the fundamental line of the excited dye (Edmund Industrial Optics, Y43-215). The efficiency
of this grating is 55%. The quartz cuvettes for the cell and amplifier used were AR coated
cuvettes (T-509). Also, a cylindrical lens and output coupler similar to Laser 1 is used to
35
excite the dye cell. Laser 2 has a tunable range of about 20 nm (see Fig. 5.3) and remains
linear with each corresponding adjustment. The wide range is needed for spectral scanning
and data collection using LabVIEW programs. The computer programs are discussed in
Appendix A.
5.4
Free Spectral Range
The free spectral range (FSR) of an optical resonator cavity is the frequency spacing of its
axial resonator mode [18]. It basically determines the minimum spacing of the lines in the
optical spectrum of the laser output. The FSR of a laser cavity can be calculated by the
following equation:
F SR =
6
580.93
4.5
583.62
3
586.32
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
589.01
588.11
587.22
586.32
585.87
584.52
584.07
583.17
582.27
581.37
580.03
579.58
c
2nd
581.82 d is the cavity length and n is the refractive index of the
where c is the speed 5.5
of
light,
5
582.72
4 of 584.52
medium. With the use
a Fabry-Perot as the resonant cavity, the index of refraction for
3.5
585.42
2.5
587.22
the air in between is 1.0.
Thus,
2
588.11
F SR =
c
.
2d
Using this calculation, the free spectral range for both Laser 1 and Laser 2 is about 1.1 GHz.
590
Wavelength [nm]
588
y = -1.7387x + 591.59
586
584
582
580
578
0
1
2
3
4
5
6
7
8
Micrometer Mount [um]
Figure 5.3: Calibration of Laser 2 with adjustment to the micrometer mount.
36
Chapter 6
Experimental Apparatus
Herein is an extensive description of the setup for the experiment. Described are the optical
system and detection scheme for the circular polarization spectroscopy.
6.1
Beam Path
The path of each of the lasers is important to ensure excitation. Because the laser used is
a pulse laser and not a continuous wavelength (CW) laser, the interactions of the photons
from each of the lasers needs to be coordinated. This is done by the manipulation of the
beam path length. Light travels at 2.99×10 8 m/s, which translates to about 1 foot every
nanosecond. In addition to this, the Nd:YAG pulses at a frequency of 20 Hz, which when
adding in dead time between pulses produces packets of photons lasting about 5-6 ns each.
Ensuring photon interaction from the two lasers requires coordinating two 6ns packets from
two different laser cavities to overlap in a two inch space by one inch space.
The overlap of the lasers was performed by having the pulse from Laser 2 reach the cell
first. The goal is to allow the second excitation photon to arrive at the cell initially, and
then have the first excitation photon come to the cell and excite the electron to the 10s level
in one fail swoop. The path of Laser 2 from the Littman-Metcalf cavity to the cesium cell is
about 5.328 ft. Laser 1 travels through an optical delay, and has a beam path length of about
8.798 ft, creating an interaction overlap of 2.53ns (see Fig. 6.1). There are two cautions in
37
regards to overlap that limit the detection ability. If the overlap time is too short, there is a
reduction in the resolution of hyperfine splitting interaction (refer to Section 3.7.2), however
that interaction oscillates over time, and gauging that oscillation adds tremendous difficulty
in the accuracy of applying collected data to the theoretical calculations. This difficulty is
caused by the reduction of signal strength due to the minimizing of overlap time.
Figure 6.1: Beam overlap time scale for two-photon excitation
6.2
Polarization Scheme
Polarization of Lasers 1 and 2 is critical for accuracy in the results of this experiment. The
slightest misalignment of the polarization can produce elliptical polarization and therefore
give false data readings.
The polarization scheme for this experiment first requires making the Laser linearly polarized. This is achieved by using Glan-Thompson Polarizers (GTP) (ThorLabs GTH10M).
GTPs have a high extinction ratio of 100,000:1. Using a GTP for each of the lasers, the next
step is to make both lasers in the same polarization with each other. The GTPs were set
to be linearly polarized along the z-axis. This was done by determining the vertical linear
polarity of one GTP and determining the vertical polarity of the second GTP from the first
one using the minimum transmittance at 90 degrees off vertical. After both GTPs were set
vertically polarized and fixed from movement, they were placed in the path of the laser.
38
The next objective is to achieve circularly polarized light. Quarter-wave plates (QWP)
were used to transform the linear light into circularly polarized light. QWPs have restrictive
characteristics based upon wavelengths of light passing through them. Basically, the QWP
effect can be optimized for certain wavelengths but not others based upon the material used
in the product. An IR QWP, with an optimum range of 830nm [ThorLabs WPQ05M-830]
was used for Laser 1. With Laser 2, a broader band QWP was used, ranging from 450nm
to 800nm [ThorLabs AQWP05M-630]. Lastly, a liquid crystal variable retarder (LCVR)
was used for switching the polarization of Laser 2 from RCP to LCP. This was done by
programing the LCVR to act as a half-wave plate (HWP). HWPs alter the propagation of
→
−
the E -field of light by 90◦ . If vertically linear polarized light entered a HWP, the exiting
light would be horizontally polarized and would continue propagating as such. In that same
→
−
respect, if RCP light enters a HWP, the shift of the E -field would result in an exiting LCP
light. The LCVR provided this function. As with the QWPs, LCVRs react differently with
light of different wavelengths. The potential voltage difference on the LCVR, controlled by
the computer, adjusts a polarized crystal medium, and adjusting the voltage will change the
orientations of the crystals. The proper voltage can be tuned to find HWP properties of the
LCVR for any wavelength of light from near UV to near IR. In addition to the LCVR acting
as a HWP, testing was done so the LCVR has no effect on the entering light (i.e. RCP
enters, RCP exits). Both voltages were programmed into the LabVIEW program, discussed
in the next chapter, for automative data collection. The entire set up is shown in Figure 6.4.
6.3
Cesium Oven Chamber
The cesium oven is designed to perform several functions. First is to house the cesium cell
(Fig. 6.2) for testing. The cell slides into position and is held so that the ends of the cell are
parallel and centered with ends of the chamber. In addition to position, the ends of the cell
are unobstructed to the windows in the ends of the chamber. Next, the cell holder positions
a portion of the cell to be centered and unobstructed to the PMT (see Chapter 7) window.
Moreover, a mirror is fixed behind the cell in line with the PMT window to reflect photon
scattering axially in line with the photomultiplier tube (PMT) window. Third, the cell holder
39
Figure 6.2: The scale of one of the cesium cells used in the experiment.
has a cold sink attached to the obstructed portion for the purpose of gathering cesium atoms
during cool down, keeping the window sections of the cell clean and unobstructed. Fourth,
a thermocouple is connected to the holder of the cesium cell, so the cell is in contact and
accurately read by the Temperature controller (see Chapter 8). Last, the cell holder is a
separate piece from the cesium oven that can be removed easily, allowing for cell switching
with minimum invasion to the total oven system. The oven does not move during the
switching of samples, and the holder sits flush with the oven to ensure exacting placement
from one sample to another. In other words, all the samples are positioned in the exact same
arrangement and location in the oven.
Figure 6.3: The actual oven chamber used in the experiment wrapped in aluminum.
40
Figure 6.4: Experiment Optics Table Diagram.
41
Chapter 7
Overview of Measurement
In this chapter, the detection scheme and signal generation for data taking is described.
7.1
Detecter
The Photomultiplier Tube Detector (PMT) is a standard detecter for fluorescence measurements. The signal detected from the PMT is based upon the photoelectric effect [25].
Photons pass into the PMT and strike a photocathode, it releases an electron. The electron
is then sent through a series of anodes (”dynodes”- dynamic anodes) by the potential difference from the cathode. As the electron hits the first dynode, two electrons are ejected.
Those two travel to the next dynode and strike it resulting in four electrons being released,
and so on. Typically, PMTs contain a set of nine or more dynodes. Each dynode is held
at a voltage more positive than the preceding one so the electrons will be accelerated from
one to the next. Supply voltages to achieve this dynamic for the PMT range from 600V
to 3000V. The current resulting from this multiplied photoelectric effect is directly related
to the number of photons striking the initial cathode. The stronger the current, the more
initial photons are detected on the cathode and vice versa.
PMTs are very susceptible to UV, visible and infrared light. Because of the sensitivity
of the device, limitations to the amount and type of light entering the PMT must be made.
To achieve this limit, interference filters were used. A 900nm wavelength 10nm FWHM inter42
ference filter (Ealing 35-4589-000) is used to identify single-photon excitation of the cesium
atoms by Laser 1. Once maximum excitation has been achieved and observed, that filter is
switch to the detection of two-photon excitation. The combination of a 365nm wavelength
11nm FWHM interference filter (Coherent 35-3045-000) and a colored glass filter (UG-11)
is used with the PMT for double excitation.
7.2
Boxcar
The Boxcar Averager is a device that integrates the applied input signal. Once real signal
has been established, the boxcar can take that signal and perform a series of functions to the
signal. These include signal multiplication, signal averaging, and signal delay. In addition
to signal manipulation, the boxcar can also adjust for maximum detection. The boxcar can
adjust its sensitivity to the size of the signal potential, and adjust the size of the gate width
for signal detection by the computer.
The Boxcar is from Stanford Research Systems. Included are a Fast Preamp with A v =
5Ω per Channel, 50 resistance per channel and functioning at DC 300 MHz, and the Boxcar
Signal system. This system has trigger functions from external to 3K kHz internal, a delay
function from 1ns -10 µs with a 1-10x multiplier, a gate width adjuster from 1 ns to 15 µs.
7.3
Computer Programs
For data recording, a series of LabVIEW programs were utilized to limit the amount of
human interpretation (as opposed to recording data from an oscilloscope display). Each program serves a specific function in either the process of running the experiment or recording
data from the PMT. In addition, each data point collected by various LabVIEW programs
represents an average of between 1000 and 1500 data points collected by the PMT. The programs were written by Jacob Hinkle (’06) and Morgan Welsh (’06). Specific applications for
this experiment were reprogrammed. A description of the programs is located in Appendix
A.
43
Figure 7.1: The Boxcar averager and integrator with oscilloscope for signal visual check.
44
Chapter 8
Systematic Effects
In categorizing the data received from the experiment, it is prudent to examine the system
and what effects are produced by different system elements. Two elements that pertain
to the system are the equilibrium temperature of the oven in which the cesium cells are
mounted and the power of the two lasers used for two photon excitation. To isolate these
two system effects and how their changes influence the data results, a cell of cesium with a
minimum argon pressure (10 −5 torr) is mounted in the oven. With the purest cell achievable
due to fiscal restrictions, adjustments to temperature of the cell and power of the lasers are
performed in isolated cases.
8.1
Temperature Dependency Runs
Temperature adjustments consist of programming a temperature controller (Digi-Sense) to
specific temperature values. The temperature controller connects to heating tape, which is
wrapped around two thirds of the cesium oven. Each temperature change is given ample
time to come to equilibrium. A thermal coupling is placed inside the oven directly touching
the cesium sample cell to ensure that the cesium cell temperature is being observed by the
thermometer reading on the temperature controller. Once equilibrium is achieved, data
taking begins. After the data taking is complete at that temperature, the cell and oven are
left to equilibrate when a new temperature is set. 15-20 minutes is an average equilibrium
wait time.
45
8.2
Power Dependency Runs
The adjustment of power for the lasers was achieved by using optical density filters in the
path of the lasers. Optical density (O.D.) filters reduce the intensity of all wavelengths
from UV to infrared without distorting the wavelength of the light. The effect of the O.D.
filters decreases the power of the laser in half for every 0.3 N.D. added. Each laser was
tested individually, leaving the other at full strength during the data taking set. The optical
density lenses are increased for each trial by 0.15 N.D. The trials of adjustment of laser
power were done in concordance with the temperature readings. A temperature was set
and allowed to achieve equilibrium. Then data was taken for each increase of OD for laser
1. Afterward, the OD filters were cleared from laser one and data was taken for the laser 2
power reduction. Finally, the temperature was set to a different temperature and the process
was repeated. What follows is the data collected for the minimum argon pressure cesium
cell for the determination of the ideal testing conditions for the experiment.
0
0
0.05
0.1
0.15
0.2
0.25
-0.2
-0.4
Polarization
50 deg C
60 deg C
75 deg C
-0.6
90 deg C
105 deg C
120 deg C
150 deg C
-0.8
-1
-1.2
Power fo Laser1 [mW]
Figure 8.1: Polarization and the effects of Temperature and Power of L1
46
8.3
Data Analysis
The results in Figure 8.1 show a wide range of temperature tests. The different temperatures seem to effect the polarization signal towards higher polarization, particularly the right
most data points. At 60 and 75 ◦ C, there is minimal fluctuation between .22 mW and .15
mW, right around the desired polarization from the calculation. Splitting the difference, a
temperature of 70 ◦ C is used for the pressure runs. Laser 1, from day to day, varies in power
between .17 and .22 mW, so it was important to choose a temperature that is fairly stable
in producing consistent polarization.
One other interesting facet to these results is that as temperature increases, and the power of
the laser decreases, the results trend toward a high degree of polarization. This phenomenon
is known as Hole-Burning (HB). As temperature increase, the number density of cesium
increases as well. With more atoms present, the lasers begins exciting atoms closest to the
side wall of the cell where the photons enter. If the laser is weak enough, a gap beings to
grow in the center of the laser profile. As the laser propagates through the cell, that gap
grows larger, as if a hole is burning the center. The separation in the gap can become wide
enough to cover the hyperfine structure, thus the effect of the hyperfine structure would not
affect the signal. In essence, when temperature increase, and the power of laser one decrease,
the polarization will increase. This is an interesting trend to observe.
Laser 2’s power was also adjusted and the results are shown in Figure 8.2. What this shows
0
-0.1 0
0.5
1
1.5
2
2.5
Polarization
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
Power of L2 [mW]
Figure 8.2: Polarization dependency on power of Laser 2.
47
is that as the power of Laser 2 varies, on average the polarization remains fairly consistent.
This makes sense because Laser 2 can only excite atoms that have already been excited by
Laser 1. Laser 1’s power is more crucial to the data because it excites to the 6p2 P1/2 state.
So as long as Laser 1 remains consistent in power, L2’s ability to create a double excitation
should be consistent, and thus polarization should not vary.
48
Chapter 9
Results
9.1
Polarization Dependency on Pressure
Based upon previous data taken in this lab and other research [16, 18, 3, 5], it is expected
that at a moderate temperature (70 ◦ C), the polarization would decrease as the pressure of
the argon buffer gas increases. Figure 9.1 suggests otherwise. The first two data points are in
0
0
20
40
60
80
100
120
-0.2
Polarization
-0.4
-0.6
-0.8
-1
-1.2
Pressure [torr]
Figure 9.1: Polarization for Cs cells under various pressures of Ar at 70 ◦ C.
agreement with the theory that an increase of pressure will result in decreasing polarization.
However the last tree points do something interesting. They trend toward an increase of
polarization with an increase of pressure.
One observation during testing was that the size of the signal was significantly reduced
49
as the pressure in the cells increased. It could be that the increase of pressure affects the
observed power of the laser, similar to Hole Burning. Another observation is that power of
the Nd:YAG is lower than what was recorded previous in house experiments (from about
4 W to .9W) due to age, and subsequently Laser’s 1 & 2 power is overall lower. With a
reduction of power, the Power Effects showed an increase in negative polarization, fitting
with the data we obtained. If this is the case, a retest using a more powerful source laser
should affect the data by a reduction of the polarization for the higher pressures. Another
investigation is needed to confirm this especially with the error involved with the last three
points.
Table 9.1: Polarization degree at various pressures of Ar
9.2
Pressure
Pol.Degreemeas.
10−4 torr
-32.3%
5 torr
-24.4%
30 torr
-28.4%
60 torr
-44.2%
100 torr
-71.3%
Cross Section Analysis
Rate equations are used to analyze the circular polarization data and extract the disorientation cross-section. The population variations among the Zeeman sublevels of the 6p2 P1/2
level due to subsequent collisoins with the buffer-gas can be expressed by a simple theoretical
model using rate equation analysis. A description of the rates is given in Fig. 9.2. Γp is the
rate of atoms being excited by the pump laser, or pump rate. Γ1&2 are the repopulating rates
due to collisions with argon atoms. γ is the radiative decay rate. Each of these rates, when
considered together, help to categorize the collisional cross section between cesium atoms
and argon atoms. The populations of the states when considering rates can be written as;
50
"1
"2
-1/2
+1/2
"p
!
!
Figure 9.2: A picture of rates and the effects of collisions.
d
N−1/2 = −(γ + Γ1 )N−1/2 + Γ2 N1/2 + Γp
dt
(9.1)
d
N1/2 = −(γ + Γ2 )N1/2 + Γ1 N−1/2
dt
(9.2)
N, the total population, is the sum of the populations of the Zeeman sublevels
N=
X
d
N = −γN + Γp .
dt
Nmj ;
mj
where γ, Γp , and Γ1,2 are the radiative decay, pump rate populating the m sublevel, and
collision induced transition rate. The net rate of change of the total population is
d
N (t) = −γN + Γp
dt
(9.3)
At time t=0, there is no population. The time dependent total population density and
orientation in the excited level is
N (t) = Γp
h1
γ
(1 − e−γt )
i
(9.4)
Orientation decay rate can be written in terms of the populations as well.
hO◦ i = p
1
j(j + 1)
X
[
Nmj mj ]
1
hO◦ i = √ [N1/2 − N−1/2 ]
3
51
(9.5)
mj
(9.6)
The derivative of the orientation has the population derivatives in its definition.
√
3hȮ◦ i =
d
d
N1/2 − N−1/2
dt
dt
By plugging in the populations, orientation can now be written as
√
3hȮ◦ i = (γ + 2Γ1 )(N−1/2 − N1/2 ) − Γp .
i
h1
1
−γ◦ t
√
(1 − e
)
hO◦ (t)i =
Γp
γ◦
3
(9.7)
The orientation decay rateγ◦ is defined as
γ◦ = γ + Γ1 + Γ2 = γ + 2Γ1 .
(9.8)
This decay rate is due to collisions. The rate the collision induced transition are equal to
one another (Γ1 = Γ2 ). Population in each zeeman sublevels can be written in terms of total
population and orientation as
√
N−1/2 = 1/2[N (t) −
3
hO◦ (t)i]
2
(9.9)
√
N1/2 = 1/2[N (t) +
3
hO◦ (t)i]
2
(9.10)
and the total density N as
N = N−1/2 + N1/2 .
±
The measured signals will be labelled as S σ and represent the signal taken from experimental
data. Relating this to the intensity ratio,
+
PC =
−
Sσ − Sσ
S σ+ + S σ−
S integrated over the pulse width can be written as
Z T
σ+
S =1
N−1/2 dt
(9.11)
0
S
σ−
T
Z
=1
N1/2 dt
0
52
(9.12)
The constants in front of the integrals are the Clebsch-Gordon coefficients. since it is one in
this instance, it does not have an effect. Now let IN define the integral of the total population
and IO is the integral of the orientation.
Z T
T
1 − e−γT
1 − e−γt
dt = +
IN =
γ
γ
γ2
0
T
Z
IO =
0
1 − e−γ◦ t
T
1 − e−γ◦ T
dt =
+
γ◦
γ◦
γ◦2
(9.13)
(9.14)
Each of the signals will be calculated in terms of population and orientation and the rates.
Z Th
i
√
1
σ+
N (t) − 3hO◦ (t)i
S =
2
0
Z T
√ Z T
1
N (t)dt − 3
hO◦ (t)idt
=
2 0
0
√
Z T
Z T
1
(1 − e−γt )
3
1 − e−γ◦ t
dt − √ Γp
dt
(9.15)
= Γp
2
γ
γ◦
3
0
0
S
σ−
Z
=
0
T
1
N1/2 dt = Γp
2
Z
T
0
√
Z T
1 − e−γt
3
1 − e−γ◦ t
dt + √ Γp
dt
γ
γ◦
3
0
(9.16)
The signals can now be written in the same I terms for convenience.
1
+
S σ = Γp (IN ) − Γp (IO )
2
(9.17)
1
−
S σ = Γp (IN ) + Γp (IO )
2
(9.18)
Now the signals are ready to be added and subtracted by plugging in the signal solutions
into the polarization equation.
+
−
S σ − S σ = −2Γp (IO )
&
−
+
S σ + S σ = Γp (IN )
(9.19)
The polarization now is a ratio of the signal equations. By putting in the definitions of the
I terms,
−γ◦ T
√ γT◦ + 1−eγ◦2
−2Γp (IO )
PC =
=− 3 T
g (1) .
e−γT −1
Γp (IN )
+
γ
γ2
53
(9.20)
The orientation decay is
γ◦ = γ + ΓO
where ΓO is the disorientation decay rate. In addition ΓO = 2Γ1 . The pressure dependency
in the disorientation decay rate (ΓO ) can be written as
ΓO = ρAr k◦
= ρAr σ◦ v̄Cs−Ar
p
σ◦ v̄Cs−Ar
=
kT
(9.21)
where p is the buffer gas pressure, kT is the thermal energy constant, and σO is the disorientation cross section. kO can be written as hσO vi and the cross section can be pulled out
as a constant (ko = σO hvi). hvi = v̄Cs−Ar is the average velocities of the colliding cesium
and argon atoms over the Maxwell-Boltzmann distribution of relative velocities at the cell
temperature. The cross-section is term is in the disorientation rate. By substituting Eq.
9.21 into the disorientation rate, it is possible to determine the collisional cross-section by
plotting the polarization data obtained in the experiment and using a weighted non-linear
least square fit. The weighted non-linear least square fit is used because it minimizes the
error to give the best value of the cross-section. If the scatter of data is uniform, the least
P
square regression minimizes (Ydata − Ycurve )2 and finds the best value of the parameter.
If the average amound of data is not uniform, the least square tends to give undue weight
to the points with large y-values and ignores points with low y-values. To prevent this, the
following weight is added.
X Ydata − Ycurve
(
)2 .
Ydata
Unfortunately, the computer program used to extract the collisional cross-section was not
functional at the time of this thesis. The following scheme will be applied to the data
collected and once the program is functioning, those results will be published in a future
paper. This scheme comes from a student who had success with it in a similar experiment
[18].
54
Chapter 10
Conclusions and Future Work
In this experiment, a two-photon, two-color, pump-probe technique was used with circularly
polarized light to excite cesium atoms from 6s 2 S1/2 →6p 2 P1/2 → 10s 2 S1/2 and test the depolarization of the transitions at various pressures of a buffer gas. Because the transition from
10s2 S1/2 → 6s2 S1/2 is considered ”forbidden”, the fluorescence from the 9p2 P1/2 → 6s2 S1/2 is
observed with an interference filter to obtain the experimental signal. The hyperfine structure was taken into consideration into data analysis in how it affects the depolarization of
transitions to 6p 2 P1/2 state.
The hole burning effect was observed in the experiment at constant pressure when the power
of the pump laser was significantly reduced and the temperature of the oven was increased.
In addition, the same trend was observed in the polarization results for a fixed temperature
but varying buffer gas pressure. The pure cesium cell polarization data was 6.1% off the
excepted value, and the depolarization with increase of pressure theory described in Guiry
and Krause’s paper [3] was observed over their test range from 10−6 torr to 5 torr. However,
an increase of polarization with higher pressures was also observed. Further investigation is
required to confirm the observation of an increase of polarization with the increase of argon
pressure. With further investigation, a more descriptive hypothesis of the phenomenon may
be formed. The collisional cross-section will be extracted from the data in preparation for
publication once the computer program is obtained again. Those results will be reported in
a journal publication.
55
Appendix A
LabView
A.1
Meadowlark USB Set Voltage.VI
The LCVR from Meadowlark (Fig. A.1) has a program, provided by the company and
integrated into LabVIEW, for setting the voltage potential to alter the liquid crystals. The
program can deliver up to four separate voltage potentials to four separate LCVRs at once.
This experiment only is concerned with one signal. The signal can be programmed to be
invariant, sinusoidal fluctuation, a sawtooth flux, and a stair alteration. The single constant
potential was used to perform the HWP characteristic for the LCVR. This program was
used determine the two potentials that created both the zero effect and HWP effect of the
→
−
LCVR. The zero effect acts as a clear window imposing no E field change. The HWP will
then switch the circular polarization from right to left or left to right.
A.2
takedata2 sub.vi
This program is the main source of data collection and calibration (Fig. A.2). It is first
used to calibrated the data collection. By assigning the Baseline to 0.000, and running the
program cyclicly, an adjustment to the Boxcar signal so as to minimize the background to
0.003. After a baseline has been established, it can be applied to other programs or inputed
to this program itself. If the wavelength of Laser 2 is set to maximum peak excitation of
6p 2 P1/2 −→ 10s 2 S1/2 , this program can be used to extract peak intensity data for each
56
helicity. Helicity changes can be performed using the QWP and no LCVR, however multiple
data sets should be collected of each helicity to perform averages and ratio comparisons, as
described in the appendix.
A.3
mircometer subVI2.vi
The micrometer program is a very interesting program for a few reasons (Fig. A.3). It is
the program that creates a digital potential difference that, when connected through the
DAQ board to the motor mike, signals the motor mike to move, thus tuning the laser.
This program is also connected to several programs giving the program the function of self
testing, or tuning the laser at specific increments and taking data at those increments in a
fluid process. One setback to the micrometer program is the fact that based on the potential
difference set, and the time the potential is run, the micrometer movements can vary from
program to program that are linked to the micrometer. A check should be performed using
a spectrometer to calibrate the spectral movement of a given program.
A.4
Modified-IntensityLCVR4.vi
The Modified Intensity (MI) program was used to take data on complete spectrum scans
with the use of the micrometer program (Fig. A.4). In addition, the LCVR program is also
linked with the MI to adjust between a zero effect and a HWP effect during each stop on
the micrometer. Data collected is displayed graphically for a visual check and can be saved
to the computer after each run. In addition to the saved data, this program also calculates
the polarization degree between to points on the same micrometer stop and displays this
information in the saved file. The MI program can take a baseline, or have a baseline
preassigned at the beginning of each run. This program will be used to investigate pressure
broadening as well as polarization.
57
Figure A.1: Meadowlark program for testing LCVR functionability.
Figure A.2: Data taking and baseline setting program (takedata2.vi.)
58
Figure A.3: Micrometer adjustment program for Laser 2 tuning.
Figure A.4: Spectrum data collection program (ModifedIntensity.vi.)
59
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