Download C. New Level Search

1
I.
INTRODUCTION ................................................................................................................................ 3
A.
B.
C.
BACKGROUND ..................................................................................................................................... 3
PURPOSE.............................................................................................................................................. 4
MOTIVATION ....................................................................................................................................... 4
1. Lifetimes and Polarizabilities ......................................................................................................... 4
2. New Level Search ........................................................................................................................... 9
D. METHOD .............................................................................................................................................10
1. Lifetimes and Polarizabilities ........................................................................................................10
2. New Level Search ..........................................................................................................................13
II. APPARATUS .......................................................................................................................................15
A.
B.
THE VACUUM SYSTEM .......................................................................................................................15
THE OVEN ..........................................................................................................................................16
1. The Current Setup ..........................................................................................................................16
2. Comments and Possible Improvements .........................................................................................18
C. THE LASER SYSTEM ...........................................................................................................................18
D. THE INTERACTION REGION.................................................................................................................19
E. THE HIGH VOLTAGE SYSTEM .............................................................................................................20
F. FLUORESCENCE DETECTION ...............................................................................................................21
G. SCATTERED LIGHT CONTROL .............................................................................................................23
III.
A.
B.
C.
IV.
A.
B.
C.
D.
E.
F.
G.
V.
DATA COLLECTION/ANALYSIS ...............................................................................................23
DATA COLLECTION ............................................................................................................................23
SIGNAL MODELING ............................................................................................................................26
DATA FITTING ....................................................................................................................................26
THEORY ..........................................................................................................................................29
OVERVIEW .........................................................................................................................................29
THE DENSITY MATRIX .......................................................................................................................30
EVOLUTION OF THE MATRIX ..............................................................................................................31
THE ABSORPTION OF RADIATION. ......................................................................................................32
THE QUADRATIC STARK EFFECT. .......................................................................................................37
THE ZEEMAN EFFECT. ........................................................................................................................41
APPLICATION OF THE DENSITY MATRIX TO THE THEORY OF RESONANCE FLUORESCENCE................42
RESULTS/DISCUSSION ...................................................................................................................48
A.
B.
PRESENTATION OF RESULTS ...............................................................................................................48
COMPARISON WITH OTHER EXPERIMENTS ..........................................................................................51
1. Lifetimes ........................................................................................................................................51
2. Polarizabilities ..............................................................................................................................52
C. COMPARISON WITH THEORY ..............................................................................................................52
1. Lifetime Calculations .....................................................................................................................52
2. Parametric Analysis ......................................................................................................................53
VI.
A.
B.
SYSTEMATICS ..............................................................................................................................54
OVERVIEW .........................................................................................................................................54
LIFETIME AND POLARIZABILITY MEASUREMENTS..............................................................................55
1. Radiation Trapping........................................................................................................................55
2. Motion of Atoms in the Beam.........................................................................................................60
3. PMT Afterpulses ............................................................................................................................62
4. PMT Linearity ...............................................................................................................................64
5. Cascade Fluorescence ...................................................................................................................65
6. Zeeman Beats Induced by the Residual Magnetic Field ................................................................66
7. Hyperfine Beats .............................................................................................................................66
8. Hyperfine Stark Beats ....................................................................................................................67
2
9. Dependence of Decay Time on the Electric Field .........................................................................67
10.
Electric Field Inhomegeneity .....................................................................................................68
C. ELECTRIC FIELD MEASUREMENT .......................................................................................................69
1. Voltage Divider Calibration ..........................................................................................................69
2. Change of Electrode Gap Due to Vacuum .....................................................................................70
D. REDUCED QUANTUM BEAT CONTRAST ..............................................................................................70
1. Error in Polarizer/Analyzer Orientation .......................................................................................70
2. Isotope Shift ...................................................................................................................................71
3. PMT Response ...............................................................................................................................72
4. AC Stark Shift and Saturation Effects ............................................................................................72
VII.
CONCLUSION ................................................................................................................................72
A.
B.
OVERVIEW .........................................................................................................................................72
IMPORTANCE FOR THE EDM SEARCH .................................................................................................73
1. Estimate of the Dipole Matrix Element..........................................................................................73
2. Estimate of the Enhancement Factor .............................................................................................74
C. NEW LEVEL SEARCH ..........................................................................................................................76
1. Application to PNC/EDM ..............................................................................................................76
2. Ratio of Polarizabilites ..................................................................................................................76
VIII. APPENDICES..................................................................................................................................78
A.
1.
2.
3.
SOFTWARE .........................................................................................................................................78
Calculation of the Signal and Determination of Experimental Geometry .....................................78
Data Collection .............................................................................................................................80
Data Analysis.................................................................................................................................80
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I. Introduction
A. Background
Atomic spectroscopy is over a hundred years old. It has played an essential role in
the development of our current view of nature, providing experimental data that led to the
formulation of quantum mechanics. Later on, spectroscopic experiments were used to
confirm the theory of quantum electrodynamics, and the existence of parity violation and
electron-quark neutral currents. Today, atomic spectroscopy remains an important way of
doing science, providing the means for searches for time-reversal violation and
extensions to the Standard Model.
For the simplest elements, the basic spectroscopic task of establishing the energy
level structure of each atom is largely complete. In the case of the more complex rare
earth elements, such as samarium, however, many of the expected energy levels are
missing from the tables, and only about a third of observed spectral lines have been
identified [1]. Furthermore, additional information relating to many of the energy levels,
such as lifetimes and tensor polarizabilities, are unknown.
The lowest energy configuration of samarium is (Xe)4f 66s2 (Fig. 1). The odd
parity levels are known relatively well because they connect to the ground term by E1
transitions. Of the even parity levels, however, many levels are “missing,” even in the
lowest terms. Lifetimes and electric polarizibilities have been measured for only a small
number of the energy levels, and typically with poor accuracy.
4
B. Purpose
In this work we have measured the lifetimes and electric polarizabilities of the
low-lying odd parity levels of the 4f 66s6p term of the samarium atom (Fig. 1).
Follow-up work will include looking for unknown even parity levels of samarium.
In particular, we would like to find levels of the 4f 66s2 5D term predicted by theory but
never seen [2]. An earlier experiment [3] found three levels of this term and we will look
for the two remaining levels.
“missing” 3 eV
levels
4f 55d6s2
4f 66s2 5DJ
J=0-4
lifetime and
4f 66s6p polarizability
measurements
4f 65d6s
1 eV
4f 66s2 7FJ
J=0-6
Even parity
Odd parity
Fig. 1. The low-lying configurations of atomic samarium. The
rectangles indicate “bands” of closely spaced energy levels. The
diagram also indicates the odd parity levels of interest for the
lifetime and polarizability measurements, and the predicted
positions of the “missing” even parity levels.
C. Motivation
1. Lifetimes and Polarizabilities
a) EDM in Metastable States
The principal motivation for this work is the prospect of an experiment to
measure the Electric Dipole Moment (EDM) of the electron, by measuring the EDM of a
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metastable state of samarium. The existence of a permanent EDM would violate both
parity- and time-reversal symmetry (Fig. 2).

d
T

s
P
Fig. 2. Time reversal changes sign of spin but not that of EDM, parity
reversal changes the sign of the EDM, but leaves the spin
unaffected.
While both P- and T-violation are known to exist (T-violation was found in 0-decay in
1964 [4]) a permanent EDM has never been observed in any system [5]. The lowest limit
placed on the electron EDM is 410-27 ecm, set by Commins et. al. in an experiment in
thallium [6].
The enhancement of P- and T-odd effects in the excited states of rare earth atoms
due to small energy separations between levels of opposite parity was pointed out in
[7,8]. The effect of an electron EDM in an atom like samarium is to induce a potentially
much larger atomic EDM. In [9], the authors estimate the enhancement factor (the ratio
of atomic EDM to electron EDM) for a metastable state of samarium and describe a
possible EDM experiment.
An atomic EDM would produce a linear Stark shift when the atom is placed in an
electric field, in addition to the usual quadratic Stark shift (Fig. 3).
6
E=0
M = -1
0
1
EDM
E0
Fig. 3. An EDM induces a linear Stark shift, in addition to quadratic.
The relative energy shift between the Zeeman sublevels when the atom is placed in an
external electric field of strength Eext is due to an atomic EDM d atom induced by the
electron EDM and the effective internal field Einteff :
EDM  datom Eext  delectron Einteff .
Here
 X d Y E ext   e 3 2 
eff
 2 Z   ,
E int


 E X ,Y   a 0
where
(1)
X d Y is the dipole matrix element between the state in question X, and its
nearest opposite-parity “partner” state Y, E X ,Y is the energy difference between these
two states, Z is the nuclear charge, and  is the fine structure constant. The first factor on
the right-hand side of this relation represents the degree of polarization of the atom, while
the second is the electric field produced by a completely polarized atom. Additional
factors depending on the configurations of the relevant states are not included here, and
would require a much more detailed analysis.
We can see the advantages of samarium for this measurement by examining this
relation. The effect scales as Z3, and samarium is a heavy atom, with Z=62. In addition,
the density of states in the energy-level structure of samarium make it possible to find
closely-lying pairs of opposite parity states with small E X ,Y .
7
From relation (1), we can see that to estimate the enhancement factor for a
particular atomic state, we need a measurement of the dipole matrix element to its
opposite parity state. In addition, knowledge of various matrix elements may aid in a
complete theoretical analysis.
The linear Stark shift causes the atom to precess in the static electric field, just as
the linear Zeeman shift causes atoms to precess in a magnetic field. Thus we can measure
the atomic EDM by polarizing an atomic beam in a metastable state and measuring the
precession angle after it has spent some time in an electric field (Fig. 4).
Photodetectors
Analyzer
(x-y or y-x)
pump laser beam
probe laser beam
Sm* beam
z
y
x
Electric field
plate
Polarizer (x or y)
Polarizer (x or y)
Fig. 4. Layout of a possible EDM experiment.
The lifetime of the metastable state needs to be sufficiently long to observe precession of
the samarium atoms. In addition, since the metastable state is mixed with its oppositeparity neighbor state, the lifetime of that state is important also, since it may dominate at
8
high electric fields. Thus, in addition to dipole matrix elements, lifetimes need to be
measured.
The preceding discussion assumes that the even-parity neighbor state of the oddparity state in question is known. Due to limited knowledge of the even-parity spectrum,
as described above, this may not always be the case, especially for the higher-lying
levels. This is a primary motivation for the new level search (see Sec. I.C.2).
Knowledge of polarizabilities may aid in the search for new levels. An odd parity
level with a particularly high polarizability and no known close partner may indicate an
unknown neighbor state nearby. In addition, if both scalar and tensor polarizabilities are
known, they give information about the angular momentum of the dominant opposite
parity state, should one exist (see Sec. VII.B).
b) Astrophysics
A large quantity of accurate atomic data, particularly state lifetimes, is required
for analysis of stellar spectra [10,11]. While rare earth elements are not prominent in the
solar spectrum, they are important for understanding the surface chemistry of upper main
sequence stars (chemically peculiar stars) [12].
c) Efficient Lighting
Rare-earth elements are increasingly used in metal-halide arc lamps to provide
high quality and efficient light sources. Spectroscopic data are needed to further
development of these lamps [13].
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d) Check of Lifetime Calculations
Atomic theory in the rare earth elements is very complex—thus experimental data
can help to improve calculational techniques by serving as a check of their accuracy. A
new calculation of the lifetimes of several levels of samarium has recently been
completed [14] and the results of this experiment will be compared to these theoretical
results.
2. New Level Search
a) Possibility for PNC Work
Identification of new even parity levels may be required to assess the possibility
of EDM work in samarium, as described in Sec. I.C.1.a). In addition, discovery of these
new levels may provide promising candidates for work in parity non-conservation (PNC).
The weak interaction between nucleon and electron (exchange of the Z0 vector
boson) leads to parity non-conservation in the atom. As with EDM, this effect depends on
the energy denominator between opposite-parity partner states, and has a rough Z3
dependence where Z is the atomic number. Thus the rare earths are attractive systems for
the measurement of this effect for the same reasons as for the EDM measurement. While
atomic theory is difficult in the rare earths, samarium's multiple stable isotopes allow
multiple PNC measurements. Comparisons of the PNC measurements on different
isotopes can circumvent some of the atomic theory.
Samarium was the first rare earth element considered for PNC experiments,
beginning with theoretical work in 1986 [8,15]. From 1987-89, work was undertaken in
Novosibirsk to find new levels suitable for parity-violating optical rotation studies [3].
10
Following this, PNC experiments examined the proposal in [15] and the PNC
enhancement of the new levels found in Novosibirsk and have determined that none of
these schemes produce an observable PNC effect [16,17,18]. However, more extensive
and precise spectroscopic knowledge, including location of new levels, may provide new
possibilities for PNC research. Spectroscopy in samarium was carried out to this end in
[19] where scalar and tensor polarizabilities were measured, and in [20,21] with
measurement of hyperfine structure, g values and tensor polarizabilities of certain
samarium states. Prior to the current work, though, no one has systematically studied the
lowest odd-parity states. A systematic survey for new even-parity states may provide
additional help.
b) Better Understanding of Sm Spectrum
In addition, a search for new levels will further understanding of the samarium
spectrum, which besides its importance for basic atomic physics, can be beneficial to
astrophysics research and the lighting industry, as described above.
D. Method
1. Lifetimes and Polarizabilities
The general approach and experimental setup of this work is essentially the same
as previous experiments in dysprosium [22,23] and ytterbium [24] carried out in this
laboratory. A block diagram of the setup is shown in Fig. 5.
11
z
ov
T = en
100
0C
Sm
y
x
la se
ye
d
lsed
pu
b eam
r
PM T
in te rfe ren ce f ilte r
in te rac tion reg ion
m ag . sh ie ld
E -f ie ld p la tes (E = 0 -40 kV /cm )
M ag . co il
(B = 0 -3 g au ss )
Fig. 5. Block diagram of experimental setup.
An atomic beam of samarium produced in an oven operating in the effusive region is
used. To measure lifetimes, time-resolved fluorescence spectroscopy is used: a pulsed
laser excites the state under investigation via an E1 transition, and then the exponentially
decaying fluorescence is detected with a photomultiplier tube and analyzed to determine
the decay time (Fig. 6).
Odd-parity excited state (J=1)
Pulse of laser light
excites E1 transition

Lifetime is reciprocal of
observed exponential
decay rate
J=2
1
t
0
Ground Term
Fig. 6. Time-resolved spectroscopy.
12
To measure tensor polarizabilities and thus estimate the dipole matrix elements
the method of Stark-induced quantum beats is employed (Fig. 7, see Sec. IV.E for
theory). An electric field is applied to the interaction region to break the degeneracy
between Zeeman sublevels. The laser beam is prepared in a particular polarization by a
linear polarizer. The laser light pulse excites a partially coherent superposition of the
sublevels, and the collection of transition dipole moments interfere so that the intensity of
fluorescence with a particular polarization exhibits temporal oscillations superimposed on
the exponential decay. These oscillations are called quantum beats. The frequency of the
beating is proportional to the energy separation between the Zeeman sublevels of the
upper state in the presence of the electric field and thus is a measure of the tensorpolarizability of the state.
Apply E-field
MJ sublevels are split
Polarized laser light excites
a coherent superposition of
the split sublevels
Odd-parity excited state (J=1)
Interference of transition
amplitudes causes beats
at frequency of splitting
Even-parity excited state
(J=1)

polarizer
J=2
t
1
0
Ground term
Fig. 7. Stark-induced quantum beats.
For general reviews of time-resolved spectroscopy and quantum beats see, e.g.,
[25] and [26].
13
2. New Level Search
The search for new even-parity levels can be accomplished with essentially the
same apparatus as for the lifetime and polarizability measurments. An outline of the
methods to be employed will be given here—a detailed discussion will be contained in a
separate paper.
A direct technique can be used to search for the missing levels of the 4f 66s2 5D
term (J=0, J=4) utilizing the large predicted M1 coupling to the ground state of these
levels. Two lasers will be employed, one pulsed, one continuous wave. The cw laser will
be used to monitor the population of a particular level of the ground term by tuning it to a
known E1 transition and detecting the resonance fluorescence with a photomultiplier tube
(Fig. 8). The pulsed laser will be scanned through frequency in the expected range of the
transition from the ground state to the missing level. If the M1 transition is excited the
ground state population will be depleted, and a dip will be seen in the monitored
fluorescence signal. This technique is known as pulsed-cw saturation spectroscopy.
To perform a general search for all missing low-lying even parity levels, an
alternate technique can be used. The problem with the traditional spectroscopic methods
employed to map out atomic spectra for a complicated atom like samarium is that there is
no a priori information about the atomic levels from which a given spectral line
originates, making identification difficult. To improve this situation, laser excitation can
be used. A high-lying odd parity state can be excited, and its decay spectrum observed
(Fig. 9). Any unclassified transitions will provide the approximate position and angular
momentum of a new even parity level. The new level can then be re-excited to another
high-lying level to exactly determine its position and angular momentum (Fig. 10).
14
4f 66s2 5D0
“missing” level
4f 66s6p 7G1
3. Scan pulsed dye laser
frequency. When transition
to the sought-after level is
excited, the ground state
population is depleted.
2. Use resonance
fluorescence
to
monitor population
of the ground state.
1. Tune cw laser to a
known E1 transition.
J=1
4f 66s2 7F
J=0
Fig. 8. Pulsed-cw saturation spectroscopy utilizing strong M1
amplitudes to search for missing levels of the 4f 66s2 5D term.
3.
Select
previously
unidentified
decay
channels and determine
their wavelength to 5 nm
“new” level
2. Detect resonance
fluorescence in a variety
of decay channels.
Known highlying level
1. Excite with frequencydoubled pulsed dye laser light.
4f 66s2 7F0-6
Fig. 9. The first step of the proposed method for new level search:
crude determination of the position of the "new" levels and a
possible range of their angular momenta up to J=1.
15
3. Measurement of the probe laser frequency on
resonance precisely determines the energy of the
"new" level; using odd parity states of different
J, allows to determine its angular momentum.
2. A second (probe) laser is used to
re-excite to high-lying odd parity
states; resonance fluorescence in
known decay channels is monitored.
“new” level
1. A "new" level is populated as
in Step I by laser excitation
followed by spontaneous decay.
4f 66s2 7F0-6
Fig. 10. The second step of the proposed method for new level search:
precise determination of the position of the "new" levels and
their angular momenta using a two-laser pump-probe technique.
II. Apparatus
A. The Vacuum System
The various elements of the apparatus are described in the following sections. The
apparatus includes a vacuum chamber pumped with a diffusion pump and mechanical
roughing pump and maintained at a pressure of about 1x10-6 Torr. The chamber is
cylindrical, with eight ports on the circumference where the flanges holding the oven,
photomultiplier tube and other pieces of the apparatus are attached. The high voltage is
supplied through the top of the chamber, and the diffusion pump port is at the bottom.
Chamber pressure is monitored with an ion gauge; thermocouple pressure gauges are
used to monitor pump-down and foreline pressure.
16
B. The Oven
1. The Current Setup
The samarium beam is produced in an oven constructed out of tantalum and
molybdenum—materials chosen because they do not react strongly with samarium. The
oven is a cylinder, 6.9 cm long by 2.5 cm diameter (Fig. 11). The exit nozzle is
rectangular, 0.5 cm x 1.5 cm and is composed of slits to collimate the atomic beam.
These slits are cut with a wire electric discharge machining (EDM) machine and are
about 0.25 mm wide. The slits primarily collimate in the horizontal direction. They
produce an angular spread in the atomic beam of 0.15-0.25 rad. The samarium, cut into
small chunks, is loaded through a hole in the back of the oven, which is then closed with
a tantalum plug. The hole is made small enough so that if the samarium melts (melting
point for Sm is 1345 K) it will not leak out of the oven. Ten to 20 grams of Sm are loaded
into the oven at a time, giving 15 to 30 hours running time.
Fig. 11. Cross-section and end view of the molybdenum oven for
producing samarium beam. The tapered plug is made out of
tantalum.
The oven is resistively heated to approximately 1200 K—corresponding to a
samarium vapor pressure of about 0.2 Torr. Four tantalum-wire heating-elements are
17
used, electrically insulated with ceramic tubing (Fig. 12). Two are long heaters, running
the entire length of the oven, which are run in parallel from the same voltage supply.
These are each bent into six sections that lie lengthwise along the oven in an s-curve. The
other two are shorter heaters that are used to keep the front of the oven at a higher
temperature than the body of the oven, in order to reduce clogging of the nozzle. These
two heaters are each bent into four sections and are run in series from a separate voltage
supply. Under this arrangement, both power supplies are driving approximately the same
impedance. The total power supplied to the oven is about 100 Watts. To reduce heat loss,
five layers of tantalum foil heat shields surround the oven. The assembly is mounted on a
water-cooled brass flange. To monitor temperature, two sets of Pt/Pt-Rh (type R)
thermocouples are used—one for the body of the oven, and one to measure the
temperature at the front of the oven. The wires are fed out through a hole in the back of
the heat shielding to feedthroughs near the back port of the brass flange (Fig. 12).
Fig. 12. Cross-section of the oven flange assembly.
18
2. Comments and Possible Improvements
The main problem with the current design arises when reloading the oven. The
plug used to close the hole in the back of the oven has invariably become welded to the
body of the oven during operation, and it must be drilled out in order to be removed. This
involves removing the oven entirely from the apparatus, and subjecting it to possible
damage while machining. The oven is very fragile due to repeated heating.
A possible alternate design has been suggested which may be implemented in the
future. This involves making a removable oven nozzle out of tantalum foil that has been
textured using a gear crinkler system. For reloading, the nozzle can be pulled or cut out of
the oven and the samarium can be loaded through the front of the oven. The nozzle is
then replaced. Construction of replacement nozzles is simple and cheap, as opposed to
the previous wire EDM method. This simplifies the design of the oven, as only one hole
is required, not two, and the potential for leakage of the molten material—which can be
disastrous—is reduced.
C. The Laser System
The laser used is a tunable dye laser (Quanta Ray PDL-2) pumped by a pulsed
Nd-YAG laser (Quanta Ray DCR-2). The YAG laser operates at a 10 Hz repetition rate,
with pulses ~8 ns long. Two dyes were employed, LDS 751 and DCM, giving two
different tuning ranges of the dye laser. Using LDS 751 the dye laser could be tuned from
about 720 to 763 nm on the fourth order of the laser diffraction grating, with a typical
output of ~2 mJ per pulse. With DCM dye, the laser could be tuned from 610 nm to at
19
least 673 nm on the fourth order, producing typical output of ~10 mJ at the peak
wavelength.
D. The Interaction Region
The atomic beam passes between 6.4 cm diameter high-voltage electrodes, spaced
about 1 cm apart, where it is illuminated with the laser beam. The laser beam intersects
the atomic beam at right angles, and its polarization is selected to maximize signal as
described in Sec. VIII.A.1. One layer of CO-NETIC AA high permeability alloy
surrounds the interaction region, keeping the background magnetic field below 10 mGs.
Holes are cut in the shield to allow the passage of the atomic and laser beams, insertion of
the high-voltage cable, and detection of the fluorescence light. Inside the magnetic shield,
there is a magnetic coil so that a z-directed magnetic field of up to few Gauss can be
applied (Fig. 13).
Fig. 13. Side cross-section of the chamber, showing the interaction region.
20
Proper alignment of the two electrodes relative to each other is important in order
to produce the most homogenous electric field possible. The grounded electrode is fixed
to the magnetic shield—alignment is performed by adjusting the height of the locking
nuts holding the magnetic shield to the support posts attached to the lid of the chamber.
By measuring the plate separation at various points, the electrodes can be aligned with
each other to within 7x10-4 rad. The high voltage system is discussed in more detail in the
next section.
E. The High Voltage System
High voltage of up to 40 kV is applied to the top electrode using a cable running
through a ceramic stand-off, eliminating the need for a high-voltage feed-through (Fig.
14). In conventional designs, high voltage is exposed to air in order to feed it through the
chamber wall, creating dangerous discharges. In the arrangement used in this work,
however, the outer, grounded conductor of the co-axial high-voltage cable is attached
directly to the chamber, and the high-voltage conductor is sent through an insulating
ceramic tube. The high-voltage electrode makes a vacuum seal with the other end of the
tube, and the high-voltage cable makes contact with the electrode at that point. In this
way, the high-voltage is inside a grounded enclosure at all times, and insulated
everywhere but at the electrodes. Very few discharges are observed, all occurring inside
the chamber. Applied voltage is measured from a voltage-divided output of the power
supply using a precision digital voltmeter.
21
Fig. 14. Cross-section of chamber showing high-voltage assembly.
F. Fluorescence Detection
Fluorescence is detected with a 2” diameter photomultiplier tube (EMI 9658 with
a S-20 Prismatic photocathode) at 45 to both laser and atomic beams. This arrangement,
besides being convenient given our choice of vacuum chamber and experimental setup, is
chosen to maximize the Stark-beat signal (see Sec. VIII.A.1). The phototube is held in a
flange that projects into the vacuum chamber. A glass window makes a vacuum seal with
the flange and allows light to reach the photocathode (Fig. 15). The gain of the PMT was
~6.3x105 and the typical quantum efficiency for the wavelengths used was about 5%.
Interference filters are used to select the decay channel of interest, and colored glass
filters are used to further reduce scattered light from the laser and oven. Typically, one or
22
two interference filters with bandwidth 10 nm and one or two colored glass filters are
used. In addition, for the Stark-beat measurement, a particular polarization needs to be
detected in order to see the oscillation in fluorescence. Two polarizations were used:
linear polarization at 45 to the electric field direction (vertical) and circular polarization.
Polaroid linear and circular analyzers were used to produce these polarizations. The PMT
signal is displayed on a Tektronix TDS 410A digitizing oscilloscope, across 50. The
oscilloscope is triggered by the same signal that activates the Q switch of the Nd:YAG
laser. Typically 1000 points at 100 Msamples/sec are taken, and the signal is averaged
over ~300 laser pulses using the oscilloscope’s internal averaging function. Time jitter in
the oscilloscope trigger was small compared to the oscilloscope’s time resolution. The
averaged signal is read-out over a GPIB interface to a PC running Labview (Sec.
VIII.A.2).
Fig. 15. Horizontal cross-section of the chamber, showing the
interaction region and the detection geometry.
23
G. Scattered Light Control
A major source, both direct and indirect, of time-correlated noise and systematics
in the signal is scattered light from the laser pulse. To reduce scattered light from the
laser, 38 cm-long collimating arms with multiple knife-edge diaphragms are used for the
entry and exit of the laser beam (Fig. 16). In each arm, the diaphragms are pointed away
from the chamber. Thus, for the entry arm, the diaphragms block the “halo” around the
incoming laser beam, and for the exit arm, the diaphragms block any off-axis reflections
from the exit window. To reduce scattered light from the oven, the area around the
interaction region was painted black with a permanent marker and an anodized tube was
inserted to contain the atomic beam on the way to the interaction region.
Fig. 16. Cross-section of one of the laser beam entry/exit tubes,
showing the knife-edge diaphragms.
III.
Data Collection/Analysis
A. Data Collection
Data collection for the lifetime measurement and the tensor-polarizability
measurement is similar.
24
Samarium atoms begin thermally distributed among the various levels of the
ground term. At 1200 K, there is significant population in each of them (Fig. 22). We
tune the dye laser to excite atoms from a particular J-state to an upper state of interest.
The excited state decays to lower lying even-parity states. Because of the dependence of
the branching ratios on the cube of the transition energy, the decays to the ground term
are usually dominant (Fig. 17). Fluorescence from a particular decay channel is selected
using an interference filter in front of the photomultiplier tube. Because of the E1
selection rules for angular momentum, the initial state must have a value of J the same or
one greater or less than the excited state, as must the final state. We want to avoid, where
possible, detecting light at the same frequency as the excitation pulse because of the large
amount of scattered light from the laser.
3 eV
4f 55d6s2
4f 66s2 5DJ
J=0-4
4f 66s6p
4f 65d6s
1 eV
4f 66s2 7FJ
J=0-6
Even parity
Odd parity
Fig. 17. Samarium level diagram showing an excitation and possible decay
channels. Decays to higher even-parity terms are usually less probable.
25
Therefore, we normally choose to detect a decay channel leading to some other ground
level besides the initial one. However, if the upper state has J  0 , the lower initial and
final states must have J  1 , since J  0 to J  0 transitions are forbidden. Whenever
possible, multiple transition schemes are used for each upper state as a crosscheck for
results. For lifetime data, we record the decay fluorescence on the scope, averaging over
several hundred pulses (30-60 sec. integration time) to improve signal to noise. This file
is then read-out to a PC running Labview (Sec. VIII.A.2) for storage and analysis. For
Stark beats, we turn on the electric field at the interaction region and put polarizers in the
laser beam and in front of the PMT. We then take averaged data files at several values of
the electric field to verify the quadratic dependence of the Stark beat frequency on the
electric field. Depending on the decay branching ratios, the wavelength-dependant PMT
efficiency, filter transmission, etc., the amplitude of detected fluorescence signals ranged
from below 1 mV to 200 mV. The largest signals were attenuated by putting a aperture or
glass filter in the laser beam path or putting more filters in front of the PMT to reduce
various systematic effects (Sec. VI.B.3-4).
For each resonance at which data is taken, an off-resonance file is taken,
recording the background level, the scattered laser light pulse, and phototube afterpulses
and electrical noise (see Sec. VI.B.3 for details) associated with the laser pulse. During
analysis, the off-resonance file is subtracted from the resonance file to remove these
effects from the data.
26
B. Signal Modeling
Lifetime data are modeled with an exponential decay, s  ae
 t
 b , where a is the
signal amplitude, t is time,  is the state lifetime, and b is the background signal level. To
develop a model for the signal in the presence of electric and magnetic fields, density
matrix formalism is used as described in Sec. IV. We first determine which partially
coherent superposition of Zeeman sublevels of the upper state results from interaction
with the polarized laser light. A relative energy shift for each sublevel is calculated,
which is proportional to the square of the electric field. The energy separation between
the sublevels determines the frequencies of the beat patterns seen in the signal. This
calculation, done in Mathematica (see Sec. VIII.A.1), also helps determine the
optimum experimental geometry, as well as the polarization of the laser beam and the
polarization of the light that should be detected in order to maximize the effect. Data
fitting is then carried out with the objective of measuring the fundamental beat frequency.
The constant of proportionality between this frequency and the square of the electric field
is 2, the tensor polarizability. State lifetime data is also extracted from the Stark beat
files and combined with the lifetime data from field-free files.
C. Data fitting
To fit the data, a Levenberg-Marquardt algorithm is used (see, e.g., [27]). For
lifetime data, a three-parameter fit using the model given above is performed (Fig. 18).
For Stark-beat data a six-parameter fit is done, fitting background, amplitude, decay time,
the beat start time (i.e. phase), beat frequency, and beat contrast (Fig. 19). The beat start
time should be the same as the laser pulse time—deviation from this is an indication that
27
there is something wrong with the fit. While beat contrast is predicted by the density
matrix calculation, there are several effects that can reduce contrast—see Sec. VI.D
below). A statistical error is assigned to each data point using an estimate of background
and shot noise. The reduced 2 is calculated for the fit, and the fitting errors are
multiplied by the square root of this number. This procedure accounts for inaccuracies in
the estimation of the statistical error—see again [27]). This fitting is carried out in a
program written in the Labview environment (see Sec. VIII.A.3).
To obtain final values for lifetimes, a weighted average is performed of the results
of all the lifetime data files for each state. For the weighting, systematic errors due to
incomplete cancellation of the off-resonance file and low-frequency Zeeman beats are
calculated (Sec. VI). These errors are added to the statistical errors in quadrature. The
reduced 2 is calculated for the average—if it is greater than unity, the error for the
average is multiplied by the square root of the reduced 2, producing a statistically
consistent error on the mean as above. If the reduced 2 is less than unity, this is assumed
to be because of the systematic error, and the final error is not modified. The errors due to
imperfect cancellation of the off-resonance files should average out over many
measurements, but the Zeeman beat error will not. Thus, the final error must be at least as
large as the smallest Zeeman beat error. The data files are also analyzed for the presence
of radiation trapping (see Sec. VI.B.1) and a correction is made if necessary.
28
time of
excitation pulse
0.3
0.1
Signal (mV)
-0.1
-0.3
data
fit
-0.5
-0.7
 = 1.07(2) s
-0.9
-1.1
-1.3
-1.5
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Fig. 18. Decay fluorescence data from 4f 66s6p 7F2 to 4f 66s2 7F3
transition, with fit used to extract lifetime.
2
time of
excitation pulse
0
Signal (mV)
-2
data
fit
-4
-6
E-field = 1.433(4) kV/cm
-8
-10
-12
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Fig. 19. Stark beat data from 4f 66s6p 7G1 to 4f 66s2 7F0 transition, with
fit used to extract polarizability.
29
For the polarizability measurements, a series of data files are taken at different
values of the electric field for each level during any given experimental run. Each data
file is fit in the same manner as the lifetime data files, and a value for the tensorpolarizability is extracted from each by dividing by the square of the electric field. The
values from each series are then compared to each other by performing the weighted
average and normalizing the error on the mean as for the lifetime data. The systematic
errors are then calculated for each series (see Sec. VI) and the results and errors from
each series are combined to produce the quoted result.
With a measurement of the tensor-polarizability of a given state, we can make an
estimate of the dipole reduced matrix element to the nearest opposite-parity “neighbor”
state, as described in Sec. IV.G.
IV.
Theory
A. Overview
The theoretical basis for the data analysis is discussed in this chapter. First, the
formalism of the density matrix is introduced, which allows the description of an
ensemble of atoms. Then the Liouville equation is derived, which determines the time
evolution of the density matrix when given the Hamiltonian of the system. The effects of
optical and static electromagnetic fields are calculated, and then these are applied to the
density matrix describing the experimental system, allowing calculation of the fitting
formulae.
This description of the theory follows and extends the treatments in [28,29,30].
30
B. The Density Matrix
The density matrix allows the description of an ensemble of atoms, and quantifies
their coherence. Consider a sample of N atoms. The wavefunction i ( t ) of the ith atom in
the sample can be written as a linear superposition of the elements m of a complete
orthonormal basis of the wavefunction space:
i (t )   a m(i ) (t ) m .
(2)
Here the a m(i ) (t ) are the time-dependent amplitudes of each element of the decomposition
of i ( t ) . Then the expectation value of any physical observable, represented by an
operator M, is given by
(i )
i M i   mn
nMm
(3)
m ,n
where
(i )
mn
 am(i ) (t )an(i ) (t ) .
(4)
(i )
Thus the matrix  (i )   mn
 , called the atomic density matrix, contains all the
information about the atom. For example, from equation (3), the expectation value of the
operator M can be written:
i M i  Tr(  (i ) M ) ,
(5)
where M is understood to mean the matrix representation of the operator, and Tr means to
take the trace of the product matrix.
Now, the measurable properties of a sample of N atoms are the averages over all
atoms of the observables of a single atom, i.e.
M 
N
1
N

i 1
i M i 
N
1
N
  mn(i )
i 1 m ,n
nM m .
(6)
31
Thus if we use the averages of the elements of the density matrix to define the elements
of the density matrix  of the ensemble
mn  m  n 
N
1
N
 mn( t ) ,
(7)
i 1
then the mean value of the observable M is given by
M   m  n n M m  Tr  M  .
(8)
n,m
The probability that an atom of the sample is in state m is given by mm ; the off-diagonal
elements mn of the density matrix describe the degree of coherence between the states
m and n .
C. Evolution of the Matrix
We now want to determine the differential equation that controls the time
development of the density matrix. This equation, along with the Hamiltonians for optical
excitation, interaction with static electric and magnetic fields, and raditative decay, allow
us to determine the expectation value of any observable of the system as a function of
time. We will use the adiabatic condition, discussed below, as the total Hamiltonian of
the system is dominated by the time-independent part.
From equations (4) and (7) the rate of change of the matrix element mn is given by
mn

t
  a m(i ) an(i )*  am(i ) a n(i )*  .
N
1
N
(9)
i 1
In the adiabatic approximation, the time dependence of probability amplitude a m(i ) (t ) is
given by e ( iEmt / ) , where E m is the energy of state m and  is Plank’s constant. In this
situation, the off-diagonal elements of the density matrix are periodic in time with
32
characteristic angular frequencies given by  Em  En   . Given the Hamiltonian operator,
the evolution of the atomic wavefunction is determined by the Schrödinger equation

H  .
t
Substituting equation (2) into this equation gives
(10)
i
a m(i ) 
1
 mH k a k(i ) .
i k
(11)
Using this result in equation (9) together with the definition of the density matrix,
equation (6), and the Hermiticity ofH leads to the required differential equation:
mn 1
  H mk kn  mkH kn 
t
i k
(12)
or in operator notation,
 1
 H,  
t i
where the square bracket denotes the commutator product H  H
(13)
 . This result is
known as the Liouville equation.
D. The Absorption of Radiation.
In the next three sections, we will calculate the effects of external electromagnetic
fields—both time-varying (optical) and static electric and magnetic fields. This section
deals with atomic transitions induced by the absorption of electromagnetic radiation.
When laser light is incident on an atom, there is some probability that light will be
absorbed and the atom will make a transition to a higher energy state. We will describe
the interaction of our atomic sample with this radiation using a semi-classical
approximation: the atoms will be described quantum mechanically while the
electromagnetic field will be treated classically. This is appropriate as long as the
33
electromagnetic field is sufficiently strong that the absorption or emission of a single
photon does not affect the field as a whole. We can thus consider the laser beam to be a
superposition of linearly polarized plane waves. A linearly polarized electromagnetic
plane wave of a single frequency  and wavevector k can be represented by
E(t )  ReE e i ( kr t )  .
(14)
This wave will interact with a stationary atom with dipole moment d  e ri . For
i
transitions involving the absorption of a single photon, we can use the classical
interaction energy  d  E(t ) as the perturbation operator. The contribution to the
Hamiltonian of the system due to this effect is then given by
H 1  d  E(t )
 d  Re[E e
(15)
i ( k r t )
].
This contribution is small compared to the Hamiltonian of the unperturbed atom, H 0 . We
are neglecting upper state relaxation, since the laser pulse is much shorter than any
relaxation rates involved. The precession of the upper state due to external fields is also
neglected. The wavefunction (r , t ) describing the atom perturbed by the incident
radiation, is a solution of the time-dependant Schrödinger equation (10), with
Hamiltonian H  H 0  H 1 . We seek solutions of the form
(r, t )   am (t ) m (r)e ( iEmt / ) ,
(16)
m
where the spatial wavefunctions are eigenstates of the time-independent equation
H 0 m (r )  E m m (r ) .
(17)
For the moment, we will consider these wavefunctions to represent non-degenerate
eigenstates.
34
Say the atom is in state i at time t 0  0 . Then ai (0)  1 and ani (0)  0 . We now
want to find the probability that after a time t, the atom has made a transition to the state
k . Substituting equation (16) into equation (10) gives
 (ia m
m
 Em am ) m e ( iEnt )  (H 0 H 1 ) am m e ( iEnt ) .
(18)
m
Since  m (r ) is an eigenfunction of H 0 [equation (17)] equation (18) reduces to
 ia m m e ( iEnt
)
m
 amH 1 m e ( iEnt ) .
(19)
m
Now we will assume that the radiation field is sufficiently weak that the amplitudes
a m ( t ) do not change significantly with time [i.e. a m (t ) t  a m (t ) ]. Thus, as an
approximation, we use the initial values a m ( t 0 ) on the right-hand side of equation (19),
giving
 ia m m e ( iEnt
m
)
 H 1 i e ( iEi t ) .
(20)
To obtain a k we utilize the orthonormality of the  m by multiplying equation (20) by
 k eiE t / and integrating over the spatial coordinates, with the result:
k
ia k =   kH 1 i dve i ( Ek  Ei ) t  .
(21)
Substituting the formula for the Hamiltonian H
1
into this equation gives
ia k =- k d  E Re[e i ( kr t ) ] i e i ki t

1
2
k d  E e
ik r
ie
i ( ki  ) t

(22)
1
2
k d  E e
ik r
ie
i ( ki  ) t
,
where ki   Ek  Ei   . We can now integrate equation (22) to obtain the probability
amplitude a k , using the initial condition a k (t 0 )  0 .
1  ei (ki  ) t 
ak   21 k d  E eikr i 
.
 (ki   ) 
(23)
35
The contribution of the second term to the integral is negligible, because  ki   for
absorption, so that the ki    in the denominator of the first term causes it to dominate,
thus only the resonant term was retained in (23). The probability of finding the atom in
the state k at time t is given by
1
ak (t )   2 k d  E eik r i
4
2
2

.
sin2 ki   t 2

ki

  2
2
(24)
In this work, we used a pulsed laser with ~8 ns pulse length, broadband compared to the
width of the atomic resonance. Thus some fraction of the laser light will be in resonance
and can be absorbed by the atom. Interpreting equation (24) as a transition probability per
unit frequency interval and integrating over frequency holding E gives
1
a k (t )  2 k d  E e ikr i
4
2


2
2
k d  E e ikr i

2

0
2

 d
sin 2  ki   t 2

ki

  2
2
(25)
t
where we have used the standard integral
sin 2 x
 x 2 dx   .

The absorption probability per unit time is therefore
Pik 

2
2
k d  E e ikr i
2
.
(26)
Since r is of the order of 10-8 cm and k is of the order of 105 cm-1, k  r is small
compared to unity and we can expand the exponential as
eikr  1  ik  r 
and retain only the first term of this expansion. Then the transition probability is given in
terms of the matrix elements of the electric dipole operator:
36
Pik 

2
2
k d  E i
2
(27)
Introducing the polarization vector e by defining E  eE and using the expression for
the spectral energy density of an electromagnetic wave,
U  
E2
,
8
(28)
the transition probability becomes
4 2
2
(29)
Pik  2 k e  d i U  .

This result is true for transitions between nondegenerate states induced by the absorption
of radiation from a linearly polarized beam. For degenerate levels we need to sum over
the degenerate final states mk and average over the degenerate initial states mi , i.e.
Pik 
4 2
1
 
2 U 
gi



mk e  d mi
2
,
(30)
mi ,mk
where gi is the statistical weight of the initial states.
In Sec. IV.G we will want to apply this theory to the density matrix. Equation (29)
as it stands applies to the diagonal elements of the excited state density matrix, through
the relation
dmm
  Pm  ,
dt

(31)
where mm are the diagonal elements of the excited state density matrix, and  are the
diagonal elements of the ground state density matrix. The off-diagonal elements of the
ground state density matrix are zero since the ground state atoms are not coherent (their
phases are not correlated).
37
The general element of the density matrix is [equation (4)] mm  a m a m . Going
back to equation (23) and following the development analogously gives the
generalization of equations (30) and (31) for any element of the excited state density
matrix:
dmm 4 2


 2 U  m e  d   e  d m  .
dt


(32)
This formula will be the one applied in Sec. G.
We have made a few assumptions that are not necessarily valid in our case. We
have assumed weak pumping by neglecting stimulated emission, although in our
experimental conditions we have laser saturation. In addition, we have neglected the
Stark shift due to the optical electromagnetic field. These two effects do not greatly
influence the form of our experimental data, since the formulae calculated using equation
(32) fit the experimental data quite well. They may have some significance, however (see
Sec. VI.D.4).
E. The Quadratic Stark effect.
In the previous section we discussed atoms’ interaction with light. We now turn to
the consideration of the static fields. In this section, we determine the effect of an
external static electric field. Interaction with an external electric field shifts and splits the
atomic levels. The perturbation HamiltonianH
elec
is the same asH 1 , in the previous
section, except that now the electric field is static and homogenous:
H
elec
 E  d .
(33)
First order perturbation theory gives the energy shift for a state k under this Hamiltonian:
38
E k(1)  k H elec k  k E  d k .
(34)
Since d is an odd operator, this integral is zero. Thus, we need to use the second-order
corrections. Let the electric field point along the z-axis. ThenH
elec
 E d z , and for the
energy correction to a state JM (where J is the angular momentum quantum number, M
is the z-projection of angular momentum, and  represents the remaining quantum
numbers) due to all other states  J M  we have
2)
E (JM


 J M 
E
2

 J 
JM H
elec
2
 J M 
E J  E  J 
JM d z  J M
E J  E  J 
(35)
2
,
since, as we see below, the dz operator only mixes states with the same z-projection of
total angular momentum. To calculate the M-dependence of the integral in equation (35)
we use the Wigner-Eckart theorem [29]:

  M
JM Tq  J M   ( 1) J  M J T  J  
J

q
J 
.
M 
(36)
Here Tq is the qth spherical component of an irreducible tensor operator of rank , and
J T


 J  is called the reduced matrix element—it does not depend on M, M  , or q.
The matrix represents the 3-J symbol. The operator d is a vector, so it has tensor rank 1,
and dz expressed in spherical tensor components is just d 0 , so
39
JM d z  J M   JM d 0  J M 
 J
 ( 1) J  M J d  J 
 M

d


d


d

0

J2  M2
J 2 J  12 J  1
M
J ( J  1)(2 J  1)
1
0
J 

M 
for J   J  1, M   M
(37)
for J   J , M   M
( J  1) 2  M 2
for J   J  1, M   M
( J  1)(2 J  1)(2 J  3)
for J   J  1 or M   M or J  J   0
where d is shorthand for the reduced matrix element. From this, we can see that


EJM  E 2 AJ  BJ M 2 ,
(38)
where AJ and BJ are constants not depending on M. Thus the level J is split into 2 J  1
sub-levels, each two-fold degenerate with respect to the sign of M (except for the M  0
level).
If we define the scalar and tensor-polarizability,  0 and  2 , by
AJ  
BJ  
0
2

 2 ( J  1)
2(2 J  1)
(39)
3 2
2 J 2 J  1
then we have
EJM  
E2
2

3 M 2  J ( J  1) 



 0
.
2
J 2 J  1 

(40)
The advantage of  0 and  2 is that they describe the Stark shift of the level as a whole and
the splitting between different M sublevels, respectively.
40
In this work, we first measure the value of the tensor-polarizability  2 for certain
odd-parity levels of samarium. We then use this value for a given odd-parity state to
estimate the value of the reduced matrix element J d  J  to the nearest even-parity
“partner” state, assuming that this nearest partner’s contribution to the sum in equation
(35) dominates. From equations (35) and (37) we have for the contribution to the energy
shift due to only one level  J  :
EJM
 d 2E 2
J2  M2
for J   J  1

 ( E J  E J  ) J  2 J  1 2 J  1
 d 2E 2
M2

for J   J
 ( E J  E J  ) J  J  1 2 J  1
 d 2E 2
( J  1) 2  M 2
for J   J  1.

 ( E J  E J  ) ( J  1) 2 J  1 2 J  3
(41)
Thus, in this case, the scalar polarizability becomes
2
2
d
0  
3(2 J  1) EJ  E J 


(42)
and the tensor polarizability becomes
 2 J (2 J  1) d 2
1

for J   J  1
 3 EJ  E J  J (2 J  1)(2 J  1)

2
1
 2 J (2 J  1) d
 2  
for J   J
 3 EJ  E J  J ( J  1)(2 J  1)
 2 J (2 J  1) d 2
1

for J   J  1,
 3 EJ  E J  ( J  1)(2 J  1)(2 J  3)







so the reduced matrix element is given by
(43)
41



J d  J   






3
E  E J  2 (2 J  1)
for J   J  1
2 J
3
( J  1)(2 J  1)
 EJ  E J  2
for J   J
2
(2 J  1)
3
( J  1)(2 J  1)(2 J  3)
EJ  E J  2
for J   J  1.
2
J (2 J  1)



(44)

Given this result, we are also interested in the maximum value that dz can take for any
Zeeman sub-level. From equation (37) this is given by
JM d z  J M
max

d


d


d

J
2 J  12 J  1
J
( J  1)(2 J  1)
( J  1)
(2 J  1)(2 J  3)
for J   J  1
for J   J
(45)
for J   J  1.
F. The Zeeman Effect.
Having considered static electric fields, we now turn to the Zeeman effect—the
splitting of atomic levels under the action of a magnetic field. The Zeeman effect is
mainly of interest as a systematic (Sec. VI.B.6). In addition, the frequency of the Zeeman
effect quantum beats (Zeeman beats) produced by a given magnetic field on a particular
samarium transition is known. Thus, a measurement of Zeeman beats is a partial check
that both the experimental and data analysis procedures are working correctly.
The interaction of an atom with a magnetic field H has the form
H mag     H ,
where  is the magnetic moment of the atom, and is given by
(46)
42
   0 gJ .
Here 0 
(47)
e
is the Bohr magneton, J is the total electronic angular momentum, and g
2mc
is the gyromagnetic ratio. If H points in the z-direction, we have
E  H
mag
 g0 HM ,
(48)
so that the level J is split into  2 J  1 components.
G. Application of the Density Matrix to the Theory of Resonance Fluorescence
We now apply the results of the preceding sections to calculate the observed
signal for a pulsed-excitation quantum beat experiment of any geometry. We assume that
the Hamiltonian of the system,H , is the sum of a time-independent part, H 0  , and a
perturbation H
1
which represents the effect of optical excitation. We will include the
effect of radiative decay by hand as another term. Thus, the Liouville equation becomes
 1
1
(2)
 H 0 ,    H 1 ,    dtd 
t i
i

(49)
1
(1)
(2)
H 0 ,    dtd   dtd .

i
The time-independent HamiltonianH 0  is actually the sum of an operatorH 0 , which
determines the unperturbed states of the atomic electrons, and the operatorH
ext
, which
describes the interaction of the atom with static external magnetic and electric fields. We
are interested in three electronic levels of the atom: a ground state  g J g described by the
basis functions  , an excited state  e J e described by basis functions m , and a final state
 f J f , to which the atoms make spontaneous transitions, described by basis functions
  . In the ground state, the phases of the atoms’ wavefunctions are not correlated with
43
each other, so the density matrix g is totally incoherent, and the off-diagonal
elements   are identically zero. The effect of H 1 is derived in Sec. D—we have [from
equation (32)]
d (1)
dt
mm
4 2
 2 U  Fmm  ,


(50)
where Fmm  m e  d   e  d m is often called the excitation matrix.
d ( 2)
dt
 describes the effect of spontaneous emission. Any given atom has a
constant probability to spontaneously decay, so that
d ( 2)
dt
mm  mm ,
(51)
where  is the rate of decay, the reciprocal of the state lifetime.
Substituting these results into equation (49), we see that the elements of the
excited-state density matrix are solutions of the equation
mm 1
4 2

m H ext ,  m  2 U( ) Fmm   mm .
t
i


(52)
For Stark beats, we have a z-directed electric field and no magnetic field. In this
case,H
ext
H
assumes the value EJm derived in equation (40):
ext
E2
m  E e Jem m  
2

3m2  J e ( J e  1) 





 0
m .
2
J
2
J

1




e
e


(53)
Now we can evaluate the first term on the right-hand side of equation (52), using the
Hermiticity of H
ext
:
44
m H
ext
,   m   m H
 mH
  H
ext
ext
 m  H
ext
 m
 m   m H
ext
  m  m H
m

ext
 m  H
ext
m  m H
ext
ext
m
m
(54)
 E Jm m   m  E Jm m  m 
 E Jm m  m   E Jm m  m 
 ( E Jm  E Jm ) mm

 2E 2  3m 2  3m  2 
2

 mm
 J e  2 J e  1 
so that we have
mm
4 2
    imm mm  2 U( ) Fmm  ,
t


(55)
where
 mm
3 E
 2
2
2
 m 2  m 2 

 .
J
2
J

1


 e

e
(56)
For Zeeman beats, everything is the same except that
mm  g0 Hm  m .
(57)
In our pulsed experiment, U( ) is a function of time. Multiplying (54) by the integrating
factor e  imm t gives
d
dt


4 2
mm e   i t  2
mm 

 Fmm  U( , t )e   i
mm 
t
.
(58)

We now assume that the pulse of incident radiation has a rectangular shape (we are going
assume that the pulse is very short so the shape is not important),
U  for t 0  t  t 0  t
U , t   
0 for all other values of t
so integrating equation (58) we have for t  t 0  t
45
mm 
   i mm   t
4 2
  i mm  ( t0  t )
  i mm  t0 e
U(

)
F

e

e
,
 mm 
  i mm
2



(59)
where we have assumed that all the atoms are in the ground state prior to the arrival of
the excitation pulse, i.e. mm (t )  0 for t  t 0 . If we now assume that the length of the
pulse, t , is much shorter than the lifetime of the excited state or the period of the
induced Stark modulation, i.e. t ,  mm t  1 , then we can write
mm 
  i mm  t
4 2
1
   i mm   ( t  t 0 ) e
U(

)
F

e

mm 
2
  i mm





1     i mm  t  1
4 2
   i mm   ( t  t 0 )
2 U( )  Fmm  e
  i mm



4 2
t U( )  Fmm  e   imm  ( t  t0 ) .
2



(60)
We see here clearly that the off-diagonal elements of the density matrix oscillate with
frequencies given by the Stark splitting of the excited state, mm .
From quantum electrodynamics, we have that the transition rate for spontaneous
emission of a photon into the solid angle d is given in the dipole approximation by [30]
Wms  d 
2
3

d .
3 m e  d  
2c
(61)
We can thus form the fluorescent light monitoring operator LF defined by
LF 
3
 e   d     e   d .
2c 3  
(62)
Then the observed intensity of fluorescent light can be obtained from the Liouville
equation:
46
dI
 Tr( LF )
d

 2 U( ) t  
 c 
3
(63)
 Fmm G mm  e   i
mm 
 t  t0 
mm
 
,
where G mm  m e   d     e   d m is called the emission matrix.
To calculate the matrix elements in equation (63), we expand the polarization
vector e and the electric dipole moment operator d in terms of the spherical unit vectors,
for instance,
e1  
1
(e x  ie y )
2
(64)
e0  e z .
With spherical components, we have
e  d  e1d 1  e1d 1   e0 d 0 .
(65)
The components of the dipole operator can then be calculated using the Wigner-Eckart
theorem, equation (36), so that
m e  d    ( 1) q e q m d q 
q
  ( 1) e q ( 1)
q
Je m
q
 Je
 e Je d  g J g 
 m


1
q
Jg 
.
 
(66)
Calculating the complex conjugate of the polarization vector, we see that
 e 
q
 ( 1)q eq ,
so that the excitation matrix is given by
(67)
47
Fmm   m e  d   e  d m


 Je
   ( 1) q e q ( 1) Je  m  e J e d  g J g 
 m
 
q
1
 Jg
 eq  ( 1) J e    g J g d  e J e    q

q




1
q
Jg 

 
(68)
Je  
 .
m  
We also have [29]
 J  d J   (1) J d  J  ,
JJ
(69)
so that equation (68) can be written as

Fmm   e J e d  g J g

2

 Je
   ( 1) q  Je  J g m  eq eq    m

 q q
J g  J g

   
1
q
1
q
Je 
.
m 
(70)
The formula for G mm is identical, except that the ground state Jg is replaced by the final
state, Jf. Now when the direction and polarization of the excitation pulse and detection
direction and polarization are specified—as well as Jg, Je, and Jf—then the functional
form of the signal can be calculated in terms of the product  2E 2 , the decay rate , and
the pulse time t 0 . This was done using Mathematica (see section VIII.A.1.) For
example, for the geometry in our experiment, laser and detection polarization at 45 to
vertical, and J g  2 , J e  2 , J g  1 , we have
 2 2
 (t  t 0 ) w  12 2
 3(t  t 0 ) w  
I  ae  ( t t0 ) 1 
cos
cos

 ,




51
4
51
4

where a is an amplitude coefficient, and w 
 2E

2
.
48
V. Results/Discussion
A. Presentation of Results
We have measured lifetimes and tensor polarizabilities of the lowest-lying levels
of the 4f 66s6p configuration. The lifetimes are given in Table 1 along with all other
known lifetimes for samarium levels in the range that we examined (below 18,504 cm-1).
Tensor polarizabilities are given in Table 2 along with all other known tensor
polarizabilities in samarium.
Level
(cm-1)
13796.36
13999.50
14380.50
14863.85
14915.83
15039.59
15507.35
15567.32
15579.12
15586.30
15650.55
16112.33
16116.42
16131.53
16211.12
16344.77
16681.74
16690.76
16748.30
16859.31
17190.20
17243.55
17462.37
17504.63
17587.46
17769.71
17810.85
17830.80
18075.67
18225.13
18350.40
18475.28
18503.49
Configuration
Term
4f6(7F)6s6p(3Po)
9Go
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f5(6Ho)5d6s2
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f5(6Ho)5d6s2
4f6(7F)6s6p(3Po)
9Go
9Go
9Fo
9Go
9Do
9Do
9Go
5
Do
7Go
5Do
7Go
9Do
9Fo
9Go
7Do
7Go
9Do
7Fo
5Go
7Go
9Fo
7Fo
7Fo
7Ho
7Go
7Fo
5Go
J
0
1
2
1
3
2
3
2
4
0
1
1
2
4
3
5
2
1
3
5
2
3
2
4
5
1
0
3
2
1
5
1
4
Lifetime
(This work)
[31]
3.043(29)
2.464(34)
2.087(42)
0.954(41)
1.828(20)
1.817(73)
2.227(21)
2.74(11)
1.716(39)
2.466(44)
2.626(17)
1.955(40)
2.71(22)
2.657(49)
4.38(12)
1.569(58)
1.974(18)
2.594(96)
2.86(22)
1.086(16)
1.589(10)
Lifetime
(Theory)
Lifetime (Previous Measurements)
[32]
[33]
[34]
Other
3.2
2.9
2.6
1.2
2.4
3.4
2.4
3.3
1.46(13)
1.70(10)
1.71(10)
1.45(20)
1.02(10)
1.20(13) 1.50(10)
2.22(10)
0.122(9) 2.42(10)
1.10(10) [35]
1.80 [35]
2.420(34)
2.66(14)
1.265(11)
[14]
0.157(5) 0.159(10) 0.165(10)
0.342(10)
1.10(10)
1.58(10)
0.450(50) 0.440(40) 0.158(5)
0.146(6) 0.146(8)
0.038 [36]
0.480(20) [35]
2.558(79)
0.071(2)
0.061(5)
0.069(4) [37]
1.471(10)
Table 1. All determinations of lifetimes of samarium levels below 18,504 cm-1,
including present work. Lifetimes in s. Most of the previous results can be
found in review [38], where results for higher-lying levels can also be found. 
denotes an estimate obtained from measurements in [16,39,18] (see text).
49
Tensor Polarizability
Level
(cm-1)
Estimate of
Matrix
Element
Closest Even Parity Neighbors
Configuration Term J
This work
Other work
6 7
13999.50
4f6(7F)6s6p(3Po)
9
Go
1
38.89(13)*
14380.50
4f6(7F)6s6p(3Po)
9
Go
2
27.7(12)*
14863.85
4f6(7F)6s6p(3Po)
9 o
F
1
4.326(9)
14915.83
4f6(7F)6s6p(3Po)
9
Go
3
27.94(19)*
15039.59
4f6(7F)6s6p(3Po)
2
33.179(83)*
15507.35
4f6(7F)6s6p(3Po)
9
3
77.3(26)*
15567.32
4f6(7F)6s6p(3Po)
9
2
24.90(82)
15579.12
4f6(7F)6s6p(3Po)
9
4
9.57(21)*
Do
Do
Go
15650.55
4f6(7F)6s6p(1Po)
7
Go
1
-561.7(11)
16112.33
4f6(7F)6s6p(3Po)
5
1
70.94(31)
16116.42
4f6(7F)6s6p(1Po)
7
2
-115.23(79)
16131.53
4f6(7F)6s6p(3Po)
9
Do
4
76.26(12)*
16211.12
4f6(7F)6s6p(3Po)
9 o
3
3.785(49)
16681.74
4f6(7F)6s6p(3Po)
2
260.09(67)*
Do
Go
F
16690.76
4f6(7F)6s6p(1Po)
7
Do
1
16748.30
4f6(7F)6s6p(1Po)
7
3
124.3(26)
16859.31
4f6(7F)6s6p(3Po)
9
Do
5
129.00(42)*
17190.20
4f6(7F)6s6p(1Po)
7 o
2
-25.30(15)
17243.55
4f6(7F)6s6p(1Po)
3
95.18(31)
17462.37
4f6(7F)6s6p(3Po)
Go
F
5
Go
2
4(1) [19]
-556(12) [20]
-548(12) [21]
-563(34) [19]
-410(50) [40]
-112.5(24)[21]
-103(10) [19]
-13.95(65)[33]
119.8(26) [21]
127(18) [19]
-20.8(15) [33]
-23(2) [19]
1315(120) [19]
Level
(cm-1)
E
1
1
2
2
2
3
2
2
2
3
2
3
3
2
3
3
2
3
3
1
3
3
3
4
14026.45
13732.53
13687.75
14365.50
14550.50
14612.44
14783.51
14550.5
14365.5
14920.45
14783.51
14612.44
14920.45
14783.51
14612.44
15524.56
15955.24
14920.45
15524.56
15639.8
15834.6
15524.56
14920.45
16354.6
26.95
-266.97
-311.75
-15.00
170.00
231.94
-80.34
-313.35
-498.35
4.62
-132.32
-303.39
-119.14
-256.08
-427.15
17.21
447.89
-586.90
-42.76
72.48
267.28
-54.56
-658.67
775.48
Configuration Term J
8
4f ( F)5d( F)6s
4f6(7F)5d(8G)6s
“
4f6(7F)5d(8F)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8F)6s
4f6(7F)5d(8F)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8P)6s
4f6(7F)5d(8F)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8G)6s
4f6(7F)5d(8F)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8G)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8F)6s
4f6(7F)5d(8D)6s
9
F
G
“
9
F
7
P
9
P
7
D
7
P
9
F
9
F
7
D
9
P
9
F
7
D
9
P
7
D
7
G
9
F
7
D
7
G
9
7
D
F
7
D
9
4f6(7F)5d(8G)6s
4f6(7F)5d(8F)6s
4f66s2
7
G
F
5
D
1
0
1
15639.8
15793.68
15914.55
-10.75
143.13
264.00
4f6(7F)5d(8G)6s
4f66s2
4f6(7F)5d(8F)6s
4f6(7F)5d(8G)6s
4f66s2
4f6(7F)5d(6P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8F)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8G)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8G)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8G)6s
4f6(7F)5d(8G)6s
4f66s2
4f6(7F)5d(8F)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8G)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8H)6s
4f6(7F)5d(8D)6s
4f66s2
4f6(7F)5d(6H)6s
4f6(7F)5d(8G)6s
4f66s2
6 7
4f ( F)5d(8D)6s
4f6(7F)5d(6H)6s
4f66s2
4f6(7F)5d(6H)6s
4f6(7F)5d(8G)6s
7
2
1
0
2
1
3
4
3
3
4
2
3
2
3
1
2
1
0
4
2
3
4
6
6
2
2
2
2
4
2
2
2
2
15955.24
15914.55
15793.68
15955.24
15914.55
15834.6
16354.6
15524.56
14920.45
16354.6
15955.24
15834.6
15955.24
15834.6
15639.8
15955.24
15914.55
15793.68
16354.6
15955.24
15834.6
16354.6
15617.45
15082.94
17864.29
18176.17
15955.24
17864.29
16354.6
18176.17
17864.29
18176.17
15955.24
-157.09
-197.78
-318.65
-161.18
-201.87
-281.82
223.07
-606.97
-1211.08
143.48
-255.88
-376.52
-726.50
-847.14
-1041.94
-735.52
-776.21
-897.08
-393.70
-793.06
-913.70
-504.71
-1241.86
-1776.37
674.09
985.97
-1234.96
620.74
-888.95
932.62
401.92
713.80
-1507.13
7
G
D
7
F
7
G
5
D
5
7
D
D
9
F
7
D
7
G
7
7
G
7
G
7
G
5
D
7
F
7
D
7
G
7
D
H
9
D
5
D
7
H
7
G
5
D
7
D
7
H
5
D
7
H
7
G
7
||d||
|dz|
max
0.42 0.17
0.24 0.09
0.54 0.20
0.14 0.05
1.4
0.40
0.45 0.15
0.72 0.21
0.36 0.09
1.0
0.41
3.0
1.1
1.6
0.58
1.7
0.52
N/A N/A
5.1
1.9
N/A N/A
4.8
1.2
4.4
1.0
N/A N/A
N/A N/A
8.5
3.1
50
17504.63
4f6(7F)6s6p(1Po)
17587.46
4f6(7F)6s6p(3Po)
17769.71
4f6(7F)6s6p(1Po)
17830.80
4f6(7F)6s6p(1Po)
18075.67
7
Go
9 o
F
4
5
-5.84(10)
36.20(32)
13.13(57)[33]
13.55(6) [41]
1
7 o
F
3
4f5(6Ho)5d6s2
7
Ho
2
9.15(35) [33]
10(1) [19]
18209.04
4f6(7F)6s6p(3Po)
5
3
-150(9) [19]
18225.13
4f6(7F)6s6p(1Po)
1
-6.08(31) [33]
-6(1) [19]
18416.62
4f6(7F)6s6p(1Po)
2
9.12(50) [33]
9(4) [19]
18503.49
4f6(7F)6s6p(3Po)
5
Go
4
18788.08
4f6(7F)6s6p(1Po)
7 o
F
2
-7.69(54)[33]
-7.3(5) [19]
18985.70
4f6(7F)6s6p(3Po)
5 o
1
473(35) [19]
19009.52
4f5(6Ho)5d6s2
2
9(6) [19]
19501.27
4f6(7F)6s6p(1Po)
7 o
F
3
16(1) [19]
19990.25
4f5(6Ho)5d6s2
7
Ho
4
52(6) [19]
20153.47
4f6(7F)6s6p(1Po)
7 o
F
5
809(36) [19]
20163.00
4f6(7F)6s6p(1Po)
7 o
F
4
-6(1) [19]
21055.76
4f6(7F)6s6p(1Po)
7 o
F
6
313(20) [19]
21458.89
4f6(7F)6s6p(1Po)
7 o
F
5
38(6) [19]
22914.07
4f6(7F)6s6p(3Po)
7
Go
1
4(1) [19]
Go
F
-202.74(94)
-717.2(15)
-660(46) [19]
4f6(7F)5d(8D)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8H)6s
4f6(7F)5d(8D)6s
4f66s2
4f6(7F)5d(6H)6s
4f6(7F)5d(8G)6s
4f66s2
4f6(7F)5d(6H)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(6H)6s
4f66s2
4f66s2
4f6(7F)5d(6H)6s
4f66s2
4f6(7F)5d(8D)6s
4f6(7F)5d(6H)6s
4f66s2
4f66s2
4f6(7F)5d(6H)6s
4f66s2
4f66s2
4f66s2
4f6(7F)5d(8D)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(6H)6s
4f66s2
4f66s2
4f66s2
4f6(7F)5d(8G)6s
4f66s2
4f6(7F)5d(6H)6s
4f66s2
4f66s2
4f66s2
4f6(7F)5d(6H)6s
4f66s2
4f66s2
4f6(7F)5d(8D)6s
4f6(7F)5d(6P)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8H)6s
4f6(7F)5d(8D)6s
4f66s2
4f6(7F)5d(6H)6s
4f66s2
4f6(7F)5d(8H)6s
4f6(7F)5d(8H)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(8H)6s
4f6(7F)5d(8D)6s
4f6(7F)5d(6H)6s
4f66s2
4f6(7F)5d(8G)6s
7
D
7
D
D
7
H
9
D
5
D
7
H
7
G
5
D
7
H
7
D
7
H
5
D
5
D
7
H
5
D
7
D
7
H
5
D
5
D
7
H
5
D
5
D
5
D
7
D
7
7
H
D
5
D
5
D
7
G
5
D
7
H
5
D
5
D
5
D
7
H
5
D
5
D
7
D
5
7
D
H
9
D
5
D
7
H
5
D
7
H
7
H
9
D
7
D
7
H
9
D
7
H
5
D
7
G
7
4
3
3
4
6
6
2
2
2
2
2
4
2
2
3
2
2
4
2
2
3
2
2
3
3
4
3
2
2
3
2
2
1
2
2
3
3
2
2
3
4
3
4
6
6
3
2
2
7
6
6
4
6
6
2
2
2
16354.6 -1150.03
15834.6 -1670.03
15524.56 -1980.07
16354.6 -1232.86
15617.45 -1970.01
15082.94 -2504.52
17864.29
94.58
18176.17 406.46
15955.24 -1814.47
17864.29
33.49
18176.17 345.37
16354.6 -1476.20
18176.17 100.50
17864.29 -211.38
20195.76 2120.09
18176.17
-32.87
17864.29 -344.75
16354.6 -1854.44
18176.17
-48.96
17864.29 -360.84
20195.76 1970.63
18176.17 -240.45
17864.29 -552.33
20195.76 1779.14
20195.76 1692.27
16354.6 -2148.89
15834.6 -2668.89
18176.17 -611.91
17864.29 -923.79
20195.76 1407.68
17864.29 -1121.41
15955.24 -3030.46
15914.55 -3071.15
18176.17 -833.35
17864.29 -1145.23
20195.76 1186.24
20195.76 694.49
18176.17 -1325.10
17864.29 -1636.98
20195.76 205.51
16354.6 -3635.65
15834.6 -4155.65
16354.6 -3798.87
15617.45 -4536.02
15082.94 -5070.53
20195.76
32.76
18176.17 -1986.83
17864.29 -2298.71
16392.93 -4662.83
15617.45 -5438.31
15082.94 -5972.82
16354.6 -5104.29
15617.45 -5841.44
15082.94 -6375.95
18176.17 -4737.90
17864.29 -5049.78
15955.24 -6958.83
Table 2. All experimental results for tensor polarizabilities of states of neutral samarium,
including present work. Polarizabilities in (kHz/(kV/cm) 2), matrix elements in
units of (ea0). Results marked with an asterisk are absolute values—sign of
polarizability was not experimentally determined. Closest Even Parity Neighbors
lists the three closest known even parity states for each odd parity state (taken
from [2,3]). The estimates of ||d|| and |dz| max are based on only the nearest known
partner state (see Sec. IV.E). If the nearest partner state is unknown, (as is often
the case) the matrix element estimate will be incorrect. N/A means that the closest
known state could not account for the sign of the polarizability and thus was
clearly not the dominantly mixed state.
1.1
0.32
3.7
0.83
N/A N/A
1.1
0.33
0.36 0.13
N/A N/A
N/A N/A
N/A N/A
17
4.4
0.80 0.29
21
7.6
N/A N/A
1.3
0.43
N/A N/A
30
6.9
0.22 0.06
29
5.5
7.7
1.7
4.0
1.4
51
B. Comparison with other Experiments
1. Lifetimes
Table 1 compares the lifetime results from this work to all other lifetime
measurements performed in the range considered in this work. A variety of techniques
were employed in these experiments: [31,33,35] used fluorescence decay with laser
excitation, [34] used luminescence decay with laser excitation, [32,37] used the delayedcoincidence method, and [36] used the level-crossing method in zero magnetic field
(Hanle method).  denotes an estimate obtained from measurements of the dipole matrix
element to the 7F0 ground state in [16] and relative oscillator strengths to the 7F0-2 ground
term levels in [39] and [18]. Consideration of 3 and spin-flip suppression indicates that
ignoring infrared decay channels should not greatly affect this estimate.
There is evidently some disagreement between the results of the various
experiments. Comparing the current work to previous experiments, we see that the
present work disagrees with the result of [33] for the 5D1 level beyond 2. However, [33]
also disagrees with the consensus of three other experiments for the lifetime of the 7H2
level, so it is possible that there is a problem with its result for the 5D1 level, as well. The
present work also disagrees with [34] beyond the 2 level for the lifetimes of four states:
the level at 16681.74 cm-1 with J=2, the 7F2 at 17190.20 cm-1, the level at 17243.55 cm-1
with J=3, and the 7F3 level at 17830.80 cm-1. However, in the case of the 7F2 level, [34]
contradicts the consensus of four other experiments, including the present work, and in
the case of the 7F3 level, the present work agrees at the 2 level with [31]. Thus the
disagreement with [34] does not seem to be a serious problem. There are no other cases
52
of disagreement between the present and previous work, and in the one case where the
lifetime of a level that already had a consensus between more than one experiment was
measured in the current work (7F2) the new result agrees with the previous ones within
the stated error.
2. Polarizabilities
The current work in polarizabilities is presented along with all other
measurements of tensor polarizabilities in samarium. [20,21,41] used laser-atomic beam
spectroscopy, [19] used laser-atomic beam spectroscopy and the nonlinear level-crossing
technique, [33] used quantum-beat spectroscopy, and [40] used electric-field-induced
pseudorotation.
There is good agreement between the present work and previous experiments.
Most results agree within the stated errors—the rest within twice the error bars, except
for two levels (7G1, 7F2) but in these cases there is agreement between the present work
and at least one other experiment. There is also good agreement among the previous
experiments for measurements for which there are no new results.
C. Comparison with Theory
1. Lifetime Calculations
An Ab initio calculation of the lifetimes of the lowest-lying odd-parity levels of
samarium has recently been completed using the configuration interaction method [14].
The results are included in Table 1. There is good correspondence between theory and
53
experiment—the theoretical values are generally a factor of ~1.2 higher than those from
experiment (Fig. 20).
1.9
1.8
theory/expt
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
13600 13800 14000 14200 14400 14600 14800 15000 15200 15400 15600
Level (cm-1)
Fig. 20. Ratio of theoretical to experimental values for lifetimes
showing trend with energy.
2. Parametric Analysis
Knowledge of both lifetimes and tensor polarizabilities and the angular parts of
the atomic wavefunctions allows calculation of the radial parts of the wavefunctions from
a parametric analysis [33]. Lifetimes and tensor polarizabilities calculated from these
integrals are compared to experiment in [33]. Scalar and tensor polarizabilities calculated
from these integrals are compared to experiment in [19]. For some levels the agreement is
quite good. However, lack of knowledge of the even-parity spectrum causes this theory to
fail in many cases, especially for the higher-lying levels, where many even-parity levels
are unknown.
54
VI.
Systematics
A. Overview
We now turn to a discussion of the various effects which could produce
systematic errors in the results. In Sec. VI.B effects which alter the signal and could
affect the measured values for lifetimes and Stark beat frequencies are considered. In
general, the Stark beat measurements, especially for higher frequencies, are more robust
and are affected much less by distortions of the decay lineshape. The first two effects,
radiation trapping and motion of atoms in the beam, have to do with conditions in the
atomic beam and primarily affect the lifetime measurements. The next two effects
discussed are introduced by the photomultiplier tube. Then cascade fluorescence caused
by “accidental” energy coincidence is discussed. Zeeman beats caused by residual
magnetic field are then considered, and then two effects due to the non-zero nuclear spin
isotopes. Finally, two effects due to the applied electric field are discussed: the
dependence of decay time on the electric field and the washing out of beats due to electric
field inhomegeneity.
In Sec. VI.C issues having to do with the measurement of the applied electric field
are discussed. Lastly, in Sec. VI.D factors that affect the modulation depth of the
observed quantum beats are considered. Beat contrast varied widely, from 20% to 100%
of expected—the reasons for this are discussed in this section.
55
B. Lifetime and Polarizability Measurements
1. Radiation Trapping
An optically thick atomic beam would increase the apparent radiative lifetime of
an excited state by absorbing and re-emitting decay fluorescence. To gain an idea of
whether or not this effect is important in our experiment, we estimate the absorption
length for resonant light in the atomic beam. First, we need to find the density of the
beam at the interaction region.
About 0.5 g of samarium is emitted from the oven per hour. Taking into account
the collimating channels in the oven nozzle, which reduce emission in the off-axis
directions, this corresponds to an oven vapor pressure of about 0.2 Torr. (This indicates
an oven temperature of ~1150 K, reasonably consistent with thermocouple readings of
~1250 K, as the thermocouple is located on the outside of the oven, right next to the
heaters themselves.) Thus the density in the oven is ~2x1015 atoms/cm3, and as the
interaction region is 20 cm from the oven, the density at the interaction region is
~5x1011 atoms/cm3. Any given level of the ground term will have a population of not
greater than 1/3, so the density of atoms of interest is ~2x1011.
Now we need the cross-section for resonant absorption. At the peak, this is given
by [42]
  2 J e  1
2  partial
.
2 Doppler
Assuming that this is the dominant decay channel, we overestimate by substituting
(71)
1
for

 partial . The transverse Doppler width is ~100 MHz, a factor of 10 less than the forward
Doppler width due to the 10:1 collimation. Using the transverse Doppler width, and
56
  2 s ,   700 nm as typical values, equation (71) gives   10 11 cm2 . Thus the
absorption length for this radiation is

1
1

cm  0.5 cm ,
11
n 2  10 10 11 
of the order of magnitude of the actual diameter of the beam. Thus radiation trapping is a
potentially significant systematic effect.
If this effect is present, then the apparent state lifetimes should increase with
higher oven temperature, since samarium vapor pressure and thus beam density strongly
depends on oven temperature. Therefore, obtaining measurements at different
temperatures for a given upper state is an effective check for radiation trapping. This was
done for three states chosen for their strong coupling to the ground state. The density in
the beam should be proportional to the signal amplitude if no other experimental
parameters are changing. Plotting the apparent lifetime versus the signal amplitude, we
see the effect of radiation trapping at the highest beam densities, but a much less
significant effect at lower temperatures (Fig. 21).
57
3.0
lifetime (s)
2.9
2.8
2.7
2.6
2.5
0
10
20
30
40
50
60
70
signal amplitude (mV)
Fig. 21. Lifetime dependence on density for 9D2 state.
Utilizing the signal amplitude, we can make an estimate of the absorption coefficient
(inverse of the absorption length) for any given data file in order to assess which files are
most likely to be affected by radiation trapping. The value of this estimate for any
particular file may not be accurate enough to determine whether there is a likelihood for
radiation trapping in that file. However, there should be some correlation between any
radiation trapping effect and the absorption coefficient, so that we can look for trends in
the fitted lifetime with the absorption coefficient. From equation (71) we have
58
 
 
 1  N P  f   N P  f 2 J e  1
2  partial
,
2 Doppler
(72)
where N is the density of atoms in the interaction region and P f  is the fractional
population of atoms in the final state, given by thermal equilibrium in the oven (Fig. 22).
0.5
J=1
Population
0.4
J=0
0.3
J=2
0.2
0.1
J=3
J=4
J=5
J=6
0.0
500
700
900
1100
1300
1500
1700
1900
Temperature (K)
Fig. 22. Population of the levels of the ground term of samarium,
calculated on the basis of thermal equilibrium.
Now we want to estimate the charge delivered to the anode of the photomultiplier
tube in response to decay fluorescence produced by one laser light pulse. The number of
atoms in the initial state in the interaction region is NP i V , where V is the volume of
the interaction region. The laser light saturates the transition, so half of the atoms are
transferred to the excited state. Multiplying by the branching ratio, B.R., gives the number
of photons emitted at the frequency of interest. Multiplying by the solid angle O,
59
transmission of the filters and polarizers T, quantum efficiency  and gain G of the
photomultiplier, and the charge of a single electron, e, we have the result:
Q  21 N P i VB. R. OTGe .
(73)
In terms of the amplitude of the signal and the lifetime, we have
Q
a
,
R
(74)
where a is the signal amplitude and R is the input impedance of the oscilloscope. Letting
F  21 VOGeR , we have
N partial  N
B. R.


a
FT P i 
(75)
,
so from equation (72)

1
 
Pf
a
2

2 J e  1 2 .
TFDoppler P i 
(76)
For most states, we see no correlation between the extracted lifetime and the
absorption coefficient. However, in some cases an effect is seen, and a correction and
associated error is assigned to the lifetime result (Fig. 23).
1.08
2.6
1.06
2.5
1.04
1.02
2.3
Lifetime (s)
Lifetime (s)
2.4
1.00
2.2
2.1
0.98
2.0
0.96
1.9
0.94
1.8
0.92
1.7
0.90
1.6
0
1
2
3
4
5
6
7
8
9
0.88
Absorption coefficient (arb. units)
(a)
0
100
200
300
400
500
600
700
Absorption coefficient (arb. units)
(b)
Fig. 23. Lifetime plotted against absorption coefficient for decays from
(a) a 9G2 level, where no correlation is seen and (b) a 9F1 level
where a 0.05 s effect is seen.
800
900
60
2. Motion of Atoms in the Beam
The peak transmission wavelength of the interference filters change as a function
of the angle of incident light. This effect, coupled with the motion of the atomic beam as
it fluoresces, could alter the apparent lifetimes. The peak transmission wavelength of
interference filter for an angle of incidence  is given by
2
transmission  0
n 
1   0  sin 2  ,
 ne 
where n0 is the index of refraction of air, ne is the index of refraction of the interference
filter supporting medium (~2) and 0 is the peak transmission wavelength at normal
incidence. Thus for fluorescence of wavelengths f longer than 0 there is a circle of a
certain radius r on the interference filter that transmits fluorescence—outside this radius
the peak transmission wavelength has been shifted far enough so that little fluorescent
light is transmitted (Fig. 24). For wavelengths shorter than 0, transmission is in a ring
(Fig. 25).
Transmitted flux
(arb. units)
61
1
0.8
0.6
0.4
0.2
0
2
1
0
-2
y (cm)
-1
-1
0
x (cm)
1
-2
2
Transmitted flux
(arb. units)
Fig. 24. Transmission profile for 650 nm interference filter,
fluorescence at 652 nm, calculated for the geometry in our
experiment. IF is actually circular, with an exposed diameter of
4 cm.
1
0.8
0.6
0.4
0.2
0
2
1
0
-2
y (cm)
-1
-1
0
x (cm)
1
-2
2
Fig. 25. Transmission profile for 650 nm interference filter,
fluorescence at 645 nm.
In this experiment, the exposed diameter of the interference filters was 4 cm. In
10 s, our data record length, the fluorescing atoms move on average about 3 mm. Thus
there is a potentially significant effect since for  = 0 - f = 5 nm, for example, the
62
radius of transmission is just about 2 cm, and as the ring moves relative to the
interference filter, some of the detected fluorescence will be lost.
Checking individual cases shows that for most data files, there is less than a 0.1%
effect on lifetimes. For two of the worst cases, however, the effect up to 1%, and is
included in the error for these files. This has no significant effect on the final values or
errors for the lifetimes.
3. PMT Afterpulses
The photomultiplier tube itself introduces certain systematic errors. For many of
the transitions studied in our experiment, we were unable to entirely remove the dye laser
scattered light pulse from the signal. The scattered light pulse, which could be up to 1 V
in the most extreme case, can produce afterpulses, which add time-correlated noise to the
data. Afterpulses are secondary pulses that follow a main anode-current pulse (Fig. 26).
The current produced in the photomultiplier tube in response to a light pulse ionizes
residual gas in the region between the cathode and the first dynode. The ions are then
accelerated back to the cathode, where each ion can produce several photoelectrons.
Several gases, including N2+, He+, CH4+, and H2+, are known to produce afterpulses,
[43, 44] each with its own characteristic delay time. As the phototube that we used for the
bulk of the measurements was fairly old, an appreciable amount of helium may have
migrated through the glass wall of the tube, producing the observed afterpulses.
63
Phototube response (mV)
0
-10
afterpulses
-20
scattered laser light
-30
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Fig. 26. Afterpulses produced by the photomultiplier tube following an
initial scattered-light pulse.
The afterpulses can interfere with the fitting of lifetime and quantum-beat data.
To remove them from the signal, off-resonance data files are subtracted from the signal
files as described above in Sec. III.A (Fig. 27). This cancellation scheme is not perfect,
however, since the laser output power can drift between the time the on- and offresonance files are taken. Thus, some time-correlated noise due to afterpulsing may
remain in the data files. To estimate the effect of a mis-cancellation, 10% of the offresonance file is added to fake lifetime data. The difference between the fitted and “true”
lifetimes is used as the error due to this effect.
1
1
0
0
Phototube response (mV)
Phototube response (mV)
64
-1
on-resonance signal
off-resonance signal
-2
-3
-4
-5
-6
-7
-8
-9
-1
-2
-3
-4
-5
-6
-7
-8
0
1
2
3
4
5
Time (s)
6
7
8
9
10
-9
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Fig. 27. On the left, on- and off-resonance fluorescence data for a
transition from the 7G3 excited state. On the right, the difference
between the two data sets.
4. PMT Linearity
At high signal levels the response of the photomultiplier tube can deviate from
linearity due to space charge effects [43]. We tested the response of the photomultiplier
by measuring scattered-light pulses of different amplitudes from the laser and comparing
the peak voltage with the total charge recorded (integral of the PMT signal) (Fig. 28).
Deviation from linearity was seen above 100 mV. Signal levels were kept below 100 mV
in almost all cases, so PMT nonlinearity effects should not significantly affect the data.
65
Peak voltage (mV)
500
400
300
200
100
0
0
200
400
600
800
1000
1200
1400
1600
1800
Total charge (arb. units)
Fig. 28. Investigation of phototube linearity. Solid line is second-degree
polynomial fit. Dotted line is the linear term of the fit.
5. Cascade Fluorescence
It is possible that due to an “accidental” energy coincidence a two-photon
transition could be excited by the laser light pulse to a level that decays back to the state
of interest or to the ground term via some cascade. In this case, the observed fluorescence
signal will be the difference of two exponential decays [24], possibly distorting the
lifetime fit. We compared the transition frequencies to all known levels with double the
transition frequencies employed in this experiment, and found no matches that were
allowed by angular momentum selection rules. Many levels are unknown, so this is not
totally conclusive. Therefore, each decay lineshape was examined; no anomalies were
seen. Thus cascade fluorescence was probably not a factor in any of the lifetime
measurements.
66
6. Zeeman Beats Induced by the Residual Magnetic Field
Slow magnetically induced quantum beats (Zeeman beats) can distort the decay
lineshape, causing an error in the lifetime or polarizability measurements. A residual
magnetic field of ~10 mGs, approximately in the z-direction, was measured by a Hallprobe inside the mu-metal shield at the interaction region, enough to cause a few percent
systematic in fitting lifetimes. To estimate this effect for the lifetime measurements, we
calculate the beats due to a 10 mGs z-directed field for each excitation scheme (initial,
excited, and final state J) and input and output polarization as described in Sec. IV. We
generate fake data with this signal and attempt to fit it with a pure exponential decay. The
deviation of the fitted lifetime from the “true” lifetime is used as the estimate of the error
due to this systematic effect. A similar procedure is used for the polarizability
measurements.
7.
Hyperfine Beats
The samarium sample used has natural isotopic abundance (154Sm: 22.6%, 152Sm:
26.6%, 150Sm: 7.4%, 149Sm: 13.9%, 148Sm: 11.3%, 147Sm: 15.1%, 144Sm: 3.1%). Thus the
~30% of the sample composed of odd isotopes can produce beats due to the hyperfine
splitting, when a coherently excited state decays. The hyperfine structure of some of the
levels studied in this work were measured in [45]. The minimum splitting they found was
~100 MHz, which is equal to the sampling rate used in this experiment. Assuming the
hyperfine splitting is fairly uniform throughout the 4f 66s6p configuration, hyperfine beats
will be too fast to have an effect on the results of our experiment. However, the odd
isotopes can have another effect, discussed in the next section.
67
8. Hyperfine Stark Beats
The Stark beats produced by the odd isotopes in the samarium sample are not the
same as those produced by the even isotopes, due to the hyperfine structure. To calculate
the signal due to these beats, we write each hyperfine state as a superposition of Zeeman
states, and then use the same density matrix calculation as was used to calculate the evenisotope Stark beats. A sum is performed over all possible hyperfine transitions that can
occur during a particular excitation/decay scheme. A Mathematica program written to
do this calculation (see Sec. VIII.A.1) showed that in the worst case, the hyperfine Stark
beats would still have sub-1% expected contrast—not enough to affect fitting of the evenisotope Stark beats.
9. Dependence of Decay Time on the Electric Field
Lifetimes of the odd-parity excited states will be changed when an electric field is
applied as Stark mixing introduces an admixture of the even-parity metastable partner
states. From second-order perturbation theory, the weight of the partner state admixed is
2
d E2
~
. Thus the decay rate is given (for small mixing) by
E 2
 d 2E 2 
,
   X  Y   X 
2
 E 
(77)
where  X and Y are the decay rates of the odd- and even-parity states, respectively.
Assuming an infinitely long lifetime for state Y and small mixing, we have
68

   X 1 

d E 2
.
E 2 
2
(78)
For all states measured except for 7G1 this estimate predicts an effect considerably less
than 1% even at the highest fields used (~40 kV/cm). In the case of the 7G1 level decay
times from the Stark beat data were plotted as a function of electric field. The expected
dependence was not seen, however the accuracy of the measurement may not have been
sufficient to resolve the effect—the scatter in points was at least as large as the expected
effect. Further investigation in this area may prove interesting. For example, a
measurement of this effect may provide an estimate of the lifetime of the opposite parity
partner state.
10. Electric Field Inhomegeneity
An inhomogeneous electric field will cause atoms to beat at slightly different
frequencies, causing damping of the beats as the atoms go more and more out of phase.
This can affect the lifetime parameter when fitting Stark beats, as a smaller than the true
value for the lifetime will mimic the damping of the quantum beats. We observed this
effect initially when the high-voltage electrodes were not aligned properly. The
electrodes were mis-aligned so that over the width of the atomic beam there was a 1%
variation of the electric field. The beat frequency depends on the square of the electric
field, so atoms will have a range of frequencies 2% around the central frequency.
Modeling this as two groups of atoms with average frequencies differing by 2% we see
that after 25 cycles the two groups will be 180 out of phase and will cancel. This is what
we see in Fig. 29.
69
0
Phototube response (mV)
-2
-4
-6
-8
-10
-12
-14
-16
-18
-20
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Fig. 29. Washing-out of beats due to electric field inhomogeneity.
Upper state 7G1, electric field 8.9 kV/cm.
Subsequently, the electrodes were aligned approximately 10 times better,
eliminating the problem. The earlier data was not used for the lifetime results. Since the
washing-out affects the amplitude of the beats but not the frequency, it should not have
an effect on the polarizability results.
C. Electric Field Measurement
1. Voltage Divider Calibration
The high-voltage supplied to the electrodes was measured by recording the
voltage divided output from the power supply. We calibrated this output using a voltage
divider probe, calibrated using a precision voltage source to 0.1%. We determined that
the power supply voltage divider factor was 7.55(1)x103. The uncertainty in this number
was one of the dominant errors in the tensor polarizability measurements.
70
2. Change of Electrode Gap Due to Vacuum
The high-voltage electrode spacing shifts slightly when the chamber is pumped
down and the chamber lid bends under atmospheric pressure. The top electrode is
attached to the center of the chamber lid, and the bottom electrode is attached 5.1 cm
from the center (Fig. 13.) The deflection at the center is given by [46]
3PR 4 1  2 
,

16Et 3
where P = 105 Pa is atmospheric pressure, R = 12.7 cm is the lid radius,  = .34, E =
7x1010 Pa is Young’s Modulus and t = 1.3 cm is the lid thickness. Thus  = 0.0030 cm.
The difference between the top and bottom electrode deflection is actually less than this,
because both electrodes are attached to the lid, at different distances from the center.
Thus the difference in the deflection is about 2/5 of this, or 0.0012 cm. This adjustment is
included when calculating the electric field.
D. Reduced Quantum Beat Contrast
1. Error in Polarizer/Analyzer Orientation
The ideal polarizer and analyzer orientation in order to produce maximum Stark
beat contrast is calculated according to theory outlined in section IV.G using a program
written in Mathematica (Sec. VIII.A.1). However, in the experiment, the orientation of
the polarizer is set rather crudely, conceivably with a 5-10 error. This could lead to up to
a 10% reduction in beat contrast.
71
2. Isotope Shift
Stark beat contrast resulting from a given laser light pulse depends on the fraction
of nuclear spin zero atoms excited. This fraction will vary depending on laser tuning due
to the isotope shift. The isotope shift is measured for some transitions from the ground
Decay Fluorescence Amplitude (mV)
state to the 4f 66s6p term in e.g. [45].
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
7295.3
7295.4
7295.5
7295.6
7295.7
7295.8
Stark beat contrast (fraction of expected)
Laser Tuning (Angstroms)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
7295.3
7295.4
7295.5
7295.6
7295.7
7295.8
Laser Tuning (Angstroms)
Fig. 30. (top) Fluorescence amplitude as a function of laser tuning for
4f 66s2 7F14f 66s6p 9G1 transition. The observed line shape is
much broader than the isotope-shift induced width because of
the laser bandwidth. (bottom) Stark beat contrast as a function of
laser tuning. Higher contrast at longer wavelengths is probably
due to excitation of a larger fraction of even isotopes.
72
The spectrum of a given transition is a few GHz wide (   01
. Å) with the odd isotope
resonances towards higher frequencies (shorter wavelengths). We measured the contrast
of Stark beats as a function of laser tuning near a resonance, showing higher contrast at
longer wavelengths as expected (Fig. 30).
3. PMT Response
We measure the response time of our photomultiplier tube by examining the PMT
response to the short (~8 ns) scattered light pulse from our dye laser. Our PMT has a
measured response time of about 30 ns full-width half-maximum. This washes out the
fastest frequency beats, leaving the lower frequency beats unaffected. Beat contrast will
be reduced, but there is no significant effect on the beat frequency or state lifetime
measurements, as determined by simulations with fake data.
4. AC Stark Shift and Saturation Effects
Saturation has no effect unless there are uncoupled subsystems. Does this apply to
us?
Light shift has no effect if ground state is isotropic--possibly not true for highangular momentum states? Also, what about uncoupled subsystems? [47]
VII. Conclusion
A. Overview
We have performed a precision measurement of the lifetimes of 26 of the lowest
lying odd parity levels of samarium and the tensor polarizabilities of 22 of these levels.
73
Many of these values had not been previously measured; agreement with those that had
been measured was satisfactory (see Sec. V.B).
In the following sections the importance of these results are discussed. First we
describe how the levels were evaluated as possible candidates for an EDM search, and
then estimate the EDM enhancement factor for the most promising case. Then
information that can be extracted from the polarizability measurement pertaining to a
search for new even-parity levels is discussed.
B. Importance for the EDM search
1. Estimate of the Dipole Matrix Element
We now discuss the implications of the results of this work. For EDM
experiments, one desires to find pairs of closely-lying opposite parity levels with large
dipole coupling (indicated by high tensor polarizability) between them.
The tensor polarizability for a particular state is given by a sum of contributions
from the opposite parity states (see Sec. IV.E). Since the contribution is inversely
proportional to the energy difference between the odd and even parity levels, neglecting
all but the closest-lying level provides an estimate of the dipole matrix element to that
level by the formulae given in equations (44) and (45). The three closest known evenparity levels are given for each odd parity state listed in Table 2, along with an estimate
of the dipole matrix element based on only the closest known level. Due to the limited
knowledge of the samarium even-parity states, however, for many of the odd parity states
(especially the higher-lying ones) the actual closest-lying levels are likely to be unknown,
making the dipole matrix element estimate incorrect. This situation is discussed in Sec.
74
VII.C. If the closest neighbor is known, on the other hand, we can assess the benefit of
using the pair of states in an EDM experiment.
2. Estimate of the Enhancement Factor
If there is a known close partner for an odd parity state, an estimate of the
enhancement factor (ratio of atomic to electron EDM) for an EDM experiment utilizing
the two states can be made. As described in Sec. I.C.1.a) the important experimental
parameter for this estimate is
d
. Of the states for which polarizabilites have been
E
measured (Table 2), the most promising seems to be the 4f66s6p 7G1 state at 15650.55
cm-1 (designated Y) with its metastable partner 4f65d6s 7G1 state at 15639.80 cm-1
(designated X). Discovery of new even-parity states may provide other interesting
candidates, however (see Sec. VII.C).
In order to estimate the EDM of the state X induced by an electron EDM de, we
use the semi-empirical formulae given in [48,49] and assume that the main contribution
comes from the mixing with the state Y. For the dominant configurations of X (4f 65d6s)
and Y (4f 66s6p), this is a single-electron 5d-6p mixing. Thus,
128
Ry
X zY
d X ~ de 
9 E (Y )  E ( X ) a0
Z 3 2

  4  1 5d 6 p
2

3
2
 4  103  d e
(79)
Here Z=62 is the nuclear charge,  is the fine structure constant;
5d  6p
Ry
1.9 ,
I .P.  E (5d ,6 p)
(80)
75
are the effective principle quantum numbers, I.P.=45519 cm-1 is the ionization potential,
2
1

Ry is the Rydberg constant;    j    Z 2 2 , where j=3/2 is the only possible
2

common value of the single-electron total angular momentum for a p and a d state.
Equation (79) substitutes 8 3 X z Y for the radial integral  found in [49] and neglects
the angular coefficient required by the samarium configuration. We do not attempt to
evaluate this angular coefficient because the configurations are not pure [2] and there
have been problems with theoretical determinations of matrix elements involving states X
and Y in the past [16]. A more sophisticated theoretical analysis is required.
In addition to the contribution of the dominant configurations, one can also expect
a significant effect due to the admixture of the 4f 65d6p configuration to the state Y. Even
a relatively small admixture of this configuration is important, since the 6s-6p EDM
mixing is much stronger than 5d-6p EDM mixing; this is due to the fact that the main
contribution to the EDM matrix element is from the region close to the nucleus [49]. In
fact, the amplitude of the 4f n5d6p-4f n6s6p mixing is known to be ~0.1 throughout the
rare earth elements [50]; a similar value was also estimated for Sm [14]. Using this value
of the mixing amplitude and performing an estimate similar to the one above, one finds
that this contribution is in fact likely to be dominant, and gives the overall magnitude of
the EDM:
dX ~ 104de .
(81)
Thus, the estimated EDM enhancement factor for the state X in samarium (i.e. the ratio of
the atomic EDM to that of the electron) exceeds the enhancement factor of the ground
state of thallium (600, giving the current best limit on the value of the electron EDM
76
[6]) by over an order of magnitude. We would like to note that within the framework of
the simple estimates presented here it is impossible to determine the relative sign of the
two contributions to the EDM. Since these contributions are of comparable magnitude, in
principle, they can cancel each other; therefore, a more refined theoretical calculation is
necessary to confirm the existence of large EDM enhancement. Of course, a reliable
value of the enhancement factor will also allow interpretation of the experimental results
on atomic EDM in terms of the limit on or value of the electron EDM.
C. New Level Search
1. Application to PNC/EDM
As mentioned above, there are many cases of odd-parity levels with high
polarizability for which there are no known close-lying even-parity partners. The fact that
that the level has high polarizability indicates that there is probably an even-parity level
close by, and if it is found it is a possible candidate for an EDM search. For PNC
experiments, one also hopes to find pairs of “partner” states; even though strong dipole
coupling is not required, an odd parity state with high tensor polarizability is still a clue
to the location of a pair of close partner states. In the next section we see how more
information about missing even-parity states can be extracted from a measurement of the
polarizabilities of odd-parity levels.
2. Ratio of Polarizabilites
If both the scalar and tensor polarizabilies of a particular state are known, some
information can be obtained about whether one closest state is dominantly mixed and if
77
so, what the angular momentum of that state is. In the "close partner" approximation,
where all opposite parity states are neglected except the nearest, the ratio  2  0 of the
tensor to the scalar polarizabilities is specified by the J-values of the two states. From
eqs. (42) and (43):

1

2  2 J  1

0  J  1
  J  2 J  1
  J  1 2 J  3
for J   J  1
for J   J
(82)
for J   J  1,
where J and J  are the angular momenta of the odd- and even-parity states, respectively.
Thus if the measured ratio  2  0 is close to one of these possible values then the mixing
is likely dominated by a state of that angular momentum. If not, then the "close partner"
approximation is invalid—multiple states are significantly mixed. Both scalar and tensor
polarizabilities for some states of samarium have been measured in [19]. Ratios of these
measurements are given in Table 3, along with the three possible theoretical predictions
of the "close partner" approximation given an even parity level with J   J  1, J , J  1 .
One level in this table known to have a close opposite parity partner (4f 66s6p 7Go1, see
Table 2) has a measured ratio that agrees with the theoretical prediction for that J quite
well. Other levels with known close-by opposite parity levels have varying degrees of
agreement with the theoretical prediction. Certain levels without a known partner state
match one of the theoretical predictions quite well (for example 4f 66s6p 5Go2, which in
addition has a very high polarizability) and this indicates that there may be an unknown
even parity state of that particular J nearby. This information will be used in a search for
new levels suitable for EDM or PNC work.
78
Level
(cm-1)
Configuration
14863.85
15650.55
16116.42
16748.30
17190.20
17462.37
18075.67
18209.04
18225.13
18416.62
18503.49
18788.08
18985.70
19009.52
19501.27
19990.25
20153.47
20163.00
21055.76
21458.89
22914.07
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f5(6Ho)5d6s2
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
4f5(6Ho)5d6s2
4f6(7F)6s6p(1Po)
4f5(6Ho)5d6s2
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(1Po)
4f6(7F)6s6p(3Po)
Term
9Fo
7Go
7Go
7Go
7Fo
5Go
7Ho
5Go
5Go
7Fo
5Fo
7Fo
7Ho
7Fo
7Fo
7Fo
7Fo
7Go
J
1
1
2
3
2
2
2
3
1
2
4
2
1
2
3
4
5
4
6
5
1
2 0
-0.042(15)
0.564(57)
-2.10(59)
-0.86(19)
-0.42(13)
-0.308(33)
0.143(36)
-5.0(22)
-0.118(32)
0.089(40)
-0.476(56)
-0.111(78)
0.353(39)
0.110(76)
0.47(16)
0.77(17)
-0.640(62)
-0.088(25)
-0.775(73)
0.339(66)
0.053(15)
Closest Known
State
Dom.
E
J
J
(cm-1)
-80.34
2
1
-10.75
1
-161.2
2
2
-393.70
4
3
674.09
2
3
401.92
2
100.50
2
-32.87
2
2
-48.96
2
-240.45
2
5
1692.27
3
-611.91
2
1
-1121.41
2
-833.35
2
694.49
3
205.51
3
6
-3798.87
4
32.76
3
7
-4662.83
7
-5104.29
4
-4737.90
2
Theoretical Prediction
J-1
J
J+1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
0.5
0.5
1
1.25
1
1
1
1.25
0.5
1
1.40
1
0.5
1
1.25
1.40
1.5
1.40
1.57
1.5
0.5
-0.1
-0.10
-0.29
-0.42
-0.29
-0.29
-0.29
-0.42
-0.10
-0.29
-0.51
-0.29
-0.10
-0.29
-0.42
-0.51
-0.58
-0.51
-0.63
-0.58
-0.10
Table 3. Ratio of tensor to scalar polarizability measured in [19] along
with possible theoretical values given the "close partner"
approximation. Dom. J gives the J-values of the theoretical
predictions that match the measured ratio, indicating a closelying even parity state with the given angular momentum.
Closest Known State gives the distance to and angular
momentum of the closest known even parity state (taken from
Table 2). In many cases, this is not the actual closest level.
VIII. Appendices
A. Software
1. Calculation of the Signal and Determination of Experimental Geometry
The calculation to determine the observed signal, described in Sec. IV.G, is carried out in
carried out in a Mathematica program. Given the initial state, upper state, and final
state J-values and the laser beam and detection directions and polarizations, the program
calculates the Stark beat signal in terms of the lifetime and tensor polarizability of the
upper state. Alternatively, the program can calculate the signal in terms of free
79
parameters, say laser beam and analyzer polarizations, and then plot the signal as a
function of these in order to determine the optimum experimental geometry. For example,
the signal for various possible transitions was calculated as a function of the input and
output polarization, and then a 3-dimensional plot was made of the signal as a function of
these variables (
Fig. 31). The plot was then animated to show the variations of the signal in time.
This allowed easy selection of the laser and detection polarizations that would produce
both a large signal and maximum contrast of the beats.
j g  3, j e  2, j f  2
Intensity
of Polarized
Light
Total
0.0065
Intensity
0.0064
0.0034
0.0032
0.003
0.0028
0.0026
0
2
3
2


2


3
2
2
2 0
0.0063
0.0062
0.0061

2

3
2
2
Fig. 31. Calculated intensity of light received at the photomultiplier
tube at a particular time for Stark beats from a J=322
transition. (left) Intensity as a function of polarizer and analyzer
orientation. (right) Intensity as a function of polarizer
orientation, with no analyzer. Plot can be animated as a function
of time to determine optimum experimental geometry.
Variations of this program were written to calculate the effect of a magnetic field,
in order to analyze the systematic effects due to Zeeman beats, and also to calculate
signal due to Stark-induced hyperfine beats.
80
2. Data Collection
Fluorescence data were transferred from the digital oscilloscope to the PC using
interface software written in the Labview programming environment. Routines supplied
for the Tektronix TDS 410A by National Instruments were employed.
3. Data Analysis
Data analysis was performed using a program written in the Labview
programing environment. The Levenberg-Marquardt nonlinear fitting algorithm included
with the Labview package was modified to suit the current purpose. In addition,
routines were written to perform data set subtraction, Fourier transform of data,
convolution of the PMT response, and analysis of various systematic effects.
[1] V. A. Komarovskii and Yu. M. Smirnov, Opt. Spectrosc. 80, 313 (1996).
[2] W. C. Martin, R. Zalubas, and L. Hagan, Atomic Energy Levels—The Rare Earth Elements (National
Bureau of Standards, 1978).
[3] L. M. Barkov, M. S. Zolotorev, and D. A. Melik-Pashaev, Opt. Spektrosk. 66, 495 (1989) [Opt.
Spectrosc. (USSR) 66, 288 (1989)].
[4] J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13, 138 (1964).
[5] I. B. Khriplovich and S. K. Lamoreaux, CP Violation without Strangeness, (Springer-Verlag, 1996).
[6] E. D. Commins, S. B. Ross, D. DeMille, and B.C. Regan, Phys. Rev. A 50, 2960 (1994).
[7] O.P. Sushkov, V. V. Flambaum, and I. B. Khriplovich, Sov. Phys. JETP 60, 873 (1984).
[8] V. A. Dzuba, V. V. Flambaum, and I. B. Khriplovich, Z. Phys. D 1, 243 (1986).
[9] D. Budker and D. DeMille, “Permanent Electric Dipole Moment in Metastable States of Atoms and
Molecules”, unpublished (1995).
[10] B. Baschek, Physica Scripta. T8, 21 (1984).
[11] E. Biémont, N. Grevesse, P. Hannaford, and R. M. Lowe, Astron. Astrophys. 222, 307 (1989).
[12] C. R. Cowley, Physica Scripta. T8, 28 (1984).
[13] M. E. Wicklifee and J. E. Lawler, J. Opt. Soc. Am. B 14, 737 (1997).
[14] S. G. Porsev, Phys. Rev. A 56, 3535 (1997).
[15] A. Gongora and P. G. H. Sandars, J. Phys. B 19, 291 (1986).
[16] I. O. G. Davies, P. E. G. Baird, P. G. H. Sandars, and T. D. Wolfenden, J. Phys. B. 22, 741 (1989).
[17] T. D. Wolfenden and P. E. G. Baird, J. Phys. B 26, 1379 (1993).
[18] D. M. Lucas, D. N. Stacey, C. D. Thompson, and R. B. Warrington, Phys. Scripta T70, 145 (1997).
[19] D. Burrow and R.-H. Rinkleff, The Arabian Journal for Science and Engineering 17, 287 (1992).
[20] T. Kobayashi, I. Endo, et. al., Z. Phys. D 39, 209 (1997).
[21] A. Fukumi, I. Endo, et. al., Fur Physik D 42, 237 (1997).
[22] D. Budker, E. D. Commins, D. DeMille, and M. S. Zolotorev, Opt. Lett 16, 1514 (1991).
[23] D. Budker, E. D. Commins, D. DeMille, and M. S. Zolotorev, Phys. Rev. A 50, 132 (1994).
81
[24] C. J. Bowers, D. Budker, E. D. Commins, D. DeMille, S. J. Freedman, A.-T. Nguyen, S.-Q. Shang and
M. Zolotorev, Phys. Rev. A 53, 3103 (1996).
[25] Demtroder
[26] S. Haroche
[27] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C
(Cambridge University Press, 1988).
[28] A. Corney, Atomic and Laser Spectroscopy (Oxford University Press, 1977).
[29] I. I. Sobelman, Atomic Spectra and Radiative Transitions, Second Edition (Springer-Verlag, 1992).
[30] B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Addison Wesley Longman Ltd.,
1983).
[31] P. Hannaford and R. M. Lowe, J.Phys. B 18, 2365 (1985).
[32] V. N. Gorshkov, V. A. Komarovskii, and N. P. Penkin, Opt. Spectrosc. (USSR) 59, 268 (1985).
[33] P. Kulina and R.-H. Rinkleff, Z Phys. A 321, 15 (1985).
[34] B. B. Kronetckii and V. A. Mishin, Preprint Gen. Phys. Inst. (AN USSR) Rep. 24, 59 (1990).
[35] P. Hannaford and R. M. Lowe, Austr. J. Phys. 39, 829 (1986).
[36] H. Brand, K. H. Drake, W. Lange, and J. Mlynek, Phys. Lett. A 75, 345 (1980).
[37] K. B. Blagoev, V. A. Komarovskii, and N. P. Penkin, Opt. Spectrosc. (USSR) 42, 238 (1977).
[38] K. B. Blagoev and V. A. Komarovskii, Atomic Data and Nuclear Data Tables 56, 1 (1994).
[39] N. P. Penkin and V. A. Komarovskii, J. Quant. Spectrosc. Radiat. Transfer 16, 217 (1976).
[40] I. O. G. Davies, P. E. G. Baird, and J. L. Nicol, J. Phys. B. 21, 3857(1988).
[41] C. Neureiter, R.-H. Rinkleff, and L. Windholz, J. Phys. B 19, 2227 (1986).
[42] N. V. Karlov, Lectures on Quantum Electronics, (CRC, 1993).
[43] Philips Photonics, Photomultiplier Tubes, Principles and Applications (Philips Photonics).
[44] Burle, Photomultiplier Handbook, Theory, Design, Application (Burle Industries, Inc., 1980).
[45] H. Brand, B. Nottbeck, H. H. Schulz, and A. Steudel, J. Phys. B 11, L99 (1978).
[46] H. L. Anderson, Editor, A Physicist’s Desk Reference, The Second Edition of Physics Vade Mecum
(American Institute of Physics, New York, 1989).
[47] M. P. Silverman, S. Haroche, and M. Gross, Phys. Rev. A 18, 1507 (1978).
[48] V. V. Flambaum, Yad. Fiz. (Sov. Nuclear Phys) 24, 383, (1976).
[49] I. B. Khriplovich, Parity Nonconservation in Atomic Phenomena (Gordan and Breach, 1991).
[50] J.-F. Wyart and P. Camus, Phys. Scr. 20, 43 (1979).