1. Lifetimes and Polarizabilities I-4
Total Page：16
File Type：pdf, Size：1020Kb
1
I. INTRODUCTION...... I-3
A. BACKGROUND...... I-3 B. PURPOSE...... I-4 C. MOTIVATION...... I-4 1. Lifetimes and Polarizabilities...... I-4 2. New Level Search...... I-9 D. METHOD...... I-10 1. Lifetimes and Polarizabilities...... I-10 2. New Level Search...... I-13 II. APPARATUS...... II-15
A. THE VACUUM SYSTEM...... II-15 B. THE OVEN...... II-16 1. The Current Setup...... II-16 2. Comments and Possible Improvements...... II-18 C. THE LASER SYSTEM...... II-18 D. THE INTERACTION REGION...... II-19 E. THE HIGH VOLTAGE SYSTEM...... II-20 F. FLUORESCENCE DETECTION...... II-21 G. SCATTERED LIGHT CONTROL...... II-23 III. DATA COLLECTION/ANALYSIS...... III-23
A. DATA COLLECTION...... III-23 B. SIGNAL MODELING...... III-26 C. DATA FITTING...... III-26 IV. THEORY...... IV-29
A. OVERVIEW...... IV-29 B. THE DENSITY MATRIX...... IV-30 C. EVOLUTION OF THE MATRIX...... IV-31 D. THE ABSORPTION OF RADIATION...... IV-32 E. THE QUADRATIC STARK EFFECT...... IV-37 F. THE ZEEMAN EFFECT...... IV-41 G. APPLICATION OF THE DENSITY MATRIX TO THE THEORY OF RESONANCE FLUORESCENCE...... IV-42 V. RESULTS/DISCUSSION...... V-48
A. PRESENTATION OF RESULTS...... V-48 B. COMPARISON WITH OTHER EXPERIMENTS...... V-51 1. Lifetimes...... V-51 2. Polarizabilities...... V-52 C. COMPARISON WITH THEORY...... V-52 1. Lifetime Calculations...... V-52 2. Parametric Analysis...... V-53 VI. SYSTEMATICS...... VI-54
A. OVERVIEW...... VI-54 B. LIFETIME AND POLARIZABILITY MEASUREMENTS...... VI-55 1. Radiation Trapping...... VI-55 2. Motion of Atoms in the Beam...... VI-60 3. PMT Afterpulses...... VI-62 4. PMT Linearity...... VI-64 5. Cascade Fluorescence...... VI-65 6. Zeeman Beats Induced by the Residual Magnetic Field...... VI-66 7. Hyperfine Beats...... VI-66 2
8. Hyperfine Stark Beats...... VI-67 9. Dependence of Decay Time on the Electric Field...... VI-67 10. Electric Field Inhomegeneity...... VI-68 C. ELECTRIC FIELD MEASUREMENT...... VI-69 1. Voltage Divider Calibration...... VI-69 2. Change of Electrode Gap Due to Vacuum...... VI-70 D. REDUCED QUANTUM BEAT CONTRAST...... VI-70 1. Error in Polarizer/Analyzer Orientation...... VI-70 2. Isotope Shift...... VI-71 3. PMT Response...... VI-72 4. AC Stark Shift and Saturation Effects...... VI-72 VII. CONCLUSION...... VII-73
A. OVERVIEW...... VII-73 B. IMPORTANCE FOR THE EDM SEARCH...... VII-73 1. Estimate of the Dipole Matrix Element...... VII-73 2. Estimate of the Enhancement Factor...... VII-74 C. NEW LEVEL SEARCH...... VII-76 1. Application to PNC/EDM...... VII-76 2. Ratio of Polarizabilites...... VII-77 VIII. APPENDICES...... VIII-78
A. SOFTWARE...... 79 1. Calculation of the Signal and Determination of Experimental Geometry...... 79 2. Data Collection...... 81 3. Data Analysis...... 81 3
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.
6 2 The lowest energy configuration of samarium is (Xe)4f 6s (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
6 low-lying odd parity levels of the 4f 6s6p 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
6 2 5 lifetime and 4f 6s DJ 6 J=0-4 4f 6s6p polarizability measurements 4f 65d6s 1 eV
6 2 7 4f 6s FJ 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 5 metastable state of samarium. The existence of a permanent EDM would violate both parity- and time-reversal symmetry (Fig. 2).
Fig. 2. Time reversal changes sign of spin but not that of EDM, parity
T P
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 410-27 ecm, 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 datom induced by the
eff electron EDM and the effective internal field Eint :
eff EDM datomEext delectron Eint .
Here
eff X d Y Eext e 3 2 Eint 2 Z , (0) E X ,Y a0 where 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 (0), 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)
probe laser beam pump laser beam
Sm* beam
z
x Electric field y 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 opposite- parity 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 odd- parity 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. 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. 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]. 9
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. 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 [Error: Reference source not found,15]. From
1987-89, work was undertaken in Novosibirsk to find new levels suitable for parity- 10 violating optical rotation studies [Error: Reference source not found]. Following this,
PNC experiments examined the proposal in [Error: Reference source not found] 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
o v e n T = 1 0 0 y 0 x C r S a s e m e l b e a d y m s e d p u l
PMT
interference filter mag. shield interaction region E-field plates (E = 0-40 kV/cm) Mag. coil (B = 0-3 gauss)
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 Lifetime is reciprocal of excites E1 transition 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. 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 tensor- polarizability of the state.
Polarized laser light excites a coherent superposition of the split sublevels Apply E-field M sublevels are split Odd-parity excited state (J=1) J Interference of transition amplitudes causes beats at frequency of splitting
Even-parity excited state (J=1)
J=2 polarizer 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 5D 0 “missing” level
4f 66s6p 7G 1 3. Scan pulsed dye laser 2. Use resonance frequency. When transition fluorescence to to the sought-after level is monitor population 1. Tune cw laser to a excited, the ground state of the ground state. known E1 transition. population is depleted.
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.
Fig. 9. The first step of the proposed method for new level search: crude determination of the position of the "new" levels and a
3. Select previously 2. Detect resonance unidentified decay fluorescence in a variety channels and determine of decay channels. their wavelength to 5 nm Known high- lying level
“new” level 1. Excite with frequency- doubled pulsed dye laser light.
4f 66s2 7F 0-6 possible range of their angular momenta up to J=1. 15
3. Measurement of the probe laser frequency on 2. A second (probe) laser is used to resonance precisely determines the energy of the re-excite to high-lying odd parity "new" level; using odd parity states of different states; resonance fluorescence in J, allows to determine its angular momentum. known decay channels is monitored.
1. A "new" level is populated as “new” level in Step I by laser excitation followed by spontaneous decay.
4f 66s2 7F 0-6 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. 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. 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. 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.
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 24 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
6 2 5 6 4f 6s DJ 4f 6s6p J=0-4 4f 65d6s 1 eV
6 2 7 4f 6s FJ 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.
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 25 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. 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. 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. 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.
B. Signal Modeling
t Lifetime data are modeled with an exponential decay, s ae 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 26 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. 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 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. Dbelow).
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 27 square root of this number. This procedure accounts for inaccuracies in the estimation of the statistical error—see again [Error: Reference source not found]). This fitting is carried out in a program written in the Labview environment (see Sec. 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. 1) and a correction is made if necessary. 28
0.3 time of excitation pulse 0.1
-0.1
-0.3 ) data V fit
m -0.5 (
l a
n -0.7 g i = 1.07(2) s S -0.9
-1.1
-1.3
-1.5 0 1 2 3 4 5 6 7 8 9 10 Time (s)
6 7 6 2 7 Fig. 18. Decay fluorescence data from 4f 6s6p F2 to 4f 6s F3 transition, with fit used to extract lifetime.
2 time of excitation pulse 0
-2
) data
V -4 fit m ( l a -6 n g i
S E-field = 1.433(4) kV/cm -8
-10
-12
0 1 2 3 4 5 6 7 8 9 10 Time (s)
6 7 6 2 7 Fig. 19. Stark beat data from 4f 6s6p G1 to 4f 6s F0 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 tensor- polarizability 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. 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
th their coherence. Consider a sample of N atoms. The wavefunction i (t) of the i atom in the sample can be written as a linear superposition of the elements m of a complete orthonormal basis of the wavefunction space:
(t) a (i) (t) m . i m (0)
(i) Here the am (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 n M m (0) m,n where
(i) (i) (i) mn am (t)an (t) . (0)
(i) (i) Thus the matrix mn , called the atomic density matrix, contains all the information about the atom. For example, from equation (0), the expectation value of the operator M can be written:
(i) i M i Tr( M) , (0) 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. 31
N N 1 1 (i) M N i M i N mn n M m . (0) i1 i1 m,n Thus if we use the averages of the elements of the density matrix to define the elements of the density matrix of the ensemble
N (0) 1 (t ) mn m n N mn , i1 then the mean value of the observable M is given by
(0) M m n n M m TrM . 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 (0) and (0) the rate of change of the matrix element mn is given by
N mn 1 (i) (i)* (i) (i)* N a˙ m an am a˙ n . (0) t i1 (i) In the adiabatic approximation, the time dependence of probability amplitude am (t) is
(iEmt / ) given bye , where Em is the energy of state m and is Plank’s constant. In this 32 situation, the off-diagonal elements of the density matrix are periodic in time with
characteristic angular frequencies given byEm En . Given the Hamiltonian operator, the evolution of the atomic wavefunction is determined by the Schrödinger equation
i H . (0) t Substituting equation (0) into this equation gives
(i) 1 (i) (0) a˙ m mH k ak . i k Using this result in equation (0) together with the definition of the density matrix, equation (0), and the Hermiticity ofH leads to the required differential equation:
mn 1 (0) H mk kn mkH kn t i k or in operator notation,
1 H , (0) t i where the square bracket denotes the commutator product H H . 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 33
approximation: the atoms will be described quantum mechanically while the
electromagnetic field will be treated classically. This is appropriate as long as the
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
i(krt ) E(t) ReE e . (0)
d e ri This wave will interact with a stationary atom with dipole moment i . For
transitions involving the absorption of a single photon, we can use the classical
interaction energy dE(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) (0) i(krt ) d Re[E e ].
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 (0), with Hamiltonian
H H 0 H 1 . We seek solutions of the form
(iEmt / ) (r,t) am (t) m (r)e , (0) m where the spatial wavefunctions are eigenstates of the time-independent equation
(0)
H 0 m (r) Em m (r) . 34
For the moment, we will consider these wavefunctions to represent non-degenerate eigenstates.
Say the atom is in state i at time t0 0. Then ai (0) 1 andani (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 (0) into equation (0) gives
(iEnt ) (iEnt ) (i a˙ m Emam ) me (H 0 H 1 ) am me . (0) m m
Since m (r) is an eigenfunction of H 0 [equation (0)] equation (0) reduces to
(iEnt ) (iEnt ) i a˙ m me amH 1 me . (0) m m
Now we will assume that the radiation field is sufficiently weak that the amplitudes
am (t) do not change significantly with time [i.e. a˙ m (t)t am (t) ]. Thus, as an
approximation, we use the initial values am (t0 ) on the right-hand side of equation (0), giving
(iEnt ) (iEit ) i a˙ m me H 1 i e . (0) m
To obtain a˙ k we utilize the orthonormality of the m by multiplying equation (0) by
iEk t/ k e and integrating over the spatial coordinates, with the result:
i( Ek Ei )t i a˙ k = kH 1i dve . (0)
Substituting the formula for the Hamiltonian H 1 into this equation gives
i(krt ) i ki t i a˙ k = - k d E Re[e ] i e (0)
1 ikr i( ki )t 1 ikr i( ki )t 2 k d E e i e 2 k d E e i e ,
whereki Ek Ei . We can now integrate equation (0) to obtain the probability
amplitude ak , using the initial condition ak (t0 ) 0. 35
i(ki )t 1 ikr 1 e (0) ak 2 k dE e i . (ki )
The contribution of the second term to the integral is negligible, becauseki 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 (0). The probability of finding the atom in the state k at time t is given by
2 2 1 2 sin ki t 2 (0) a (t) k d E eikr i k 4 2 2 . ki 2 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 (0) as a transition probability per
unit frequency interval and integrating over frequency holding E gives
2 2 1 2 sin ki t 2 a (t) k d E eikr i d k 4 2 2 0 ki 2 (0)
2 k d E eikr i t 2 2 where we have used the standard integral
2 sin x dx . x 2
The absorption probability per unit time is therefore
2 P k dE eikr i . (0) ik 2 2 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
eikr 1 ik r 36 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:
2 P k dE i (0) ik 2 2
Introducing the polarization vector e by defining E eE and using the expression for the spectral energy density of an electromagnetic wave,
2 E (0) U , 8 the transition probability becomes
2 4 2 (0) P k e d i U . ik 2 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.
2 4 1 2 (0) Pik 2 U mk e d mi , gi mi ,mk where gi is the statistical weight of the initial states.
In Sec. G we will want to apply this theory to the density matrix. Equation (0) as it stands applies to the diagonal elements of the excited state density matrix, through the relation
dmm (0) Pm , dt 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 37 ground state density matrix are zero since the ground state atoms are not coherent (their phases are not correlated).
The general element of the density matrix is [equation (0)] mm am am . Going back to equation (0) and following the development analogously gives the generalization of equations (0) and (0) for any element of the excited state density matrix:
2 dmm 4 (0) 2 U m e d e d m . dt
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
(0) fit the experimental data quite well. They may have some significance, however (see
Sec. 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: 38
(0)
H elec E d . First order perturbation theory gives the energy shift for a state k under this Hamiltonian:
(0) (1) Ek k H elec k k E d k . 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 stateJM (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 JM J M (2) H elec EJM J M EJ E J (0) 2 JM d z J M E 2 , J EJ E J 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 (0) we use the Wigner-Eckart theorem [28]:
J J (0) JM T JM (1) J M J T J . q M q M
th Here Tq is the q 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 d0 , so 39
JM d z J M JM d0 J M J 1 J (1) J M J d J M 0 M J 2 M 2 d for J J 1, M M J2J 12J 1 (0) M d for J J, M M J(J 1)(2J 1) 2 2 (J 1) M d for J J 1, M M (J 1)(2J 1)(2J 3) 0 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
2 2 (0) EJM E AJ BJ M ,
where AJ and BJ are constants not depending on M. Thus the levelJ is split into 2J 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 and2 , by
0 2 (J 1) AJ 2 2(2J 1) (0) 3 B 2 J 2J2J 1 then we have
2 3M 2 J(J 1) E (0) EJM 0 2 . 2 J2J 1
The advantage of 0 and2 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-polarizability2 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
(0) dominates. From equations (0) and (0) we have for the contribution to the energy shift due to only one level J :
d 2E 2 J 2 M 2 for J J 1 (EJ E J ) J2J 12J 1 2 d E 2 M 2 (0) E for J J JM (EJ E J ) J J 1 2J 1 d 2E 2 (J 1) 2 M 2 for J J 1. (EJ E J ) (J 1) 2J 1 2J 3 Thus, in this case, the scalar polarizability becomes
2 d 2 (0) 0 3(2J 1) EJ E J and the tensor polarizability becomes
2 J(2J 1) d 2 1 for J J 1 3 J(2J 1)(2J 1) EJ E J 2 2 J(2J 1) d 1 (0) for J J 2 3 J(J 1)(2J 1) EJ E J 2 J(2J 1) d 2 1 for J J 1, 3 E E (J 1)(2J 1)(2J 3) J J so the reduced matrix element is given by 41
3 EJ E J 2 (2J 1) for J J 1 2 3 (J 1)(2J 1) (0) J d J EJ E J 2 for J J 2 (2J 1) 3 (J 1)(2J 1)(2J 3) EJ E J 2 for J J 1. 2 J(2J 1)
Given this result, we are also interested in the maximum value that dz can take for any
Zeeman sub-level. From equation (0) this is given by
J d for J J 1 2J 12J 1 J (0) JM d J M d for J J z max (J 1)(2J 1) (J 1) d for J J 1. (2J 1)(2J 3)
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. 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 H mag , (0) where is the magnetic moment of the atom, and is given by 42
gJ . 0 (0) e Here is the Bohr magneton, J is the total electronic angular momentum, and g 0 2mc is the gyromagnetic ratio. If H points in the z-direction, we have
E g HM , H mag 0 (0) so that the level J is split into 2J 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 d (2) H 0, H 1 , dt t i i (0)
1 (1) (2) H , d d . i 0 dt dt
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 Je described by basis functions m , and a final state
f J f , to which the atoms make spontaneous transitions, described by basis functions 43
. In the ground state, the phases of the atoms’ wavefunctions are not correlated with
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
(0)]
2 d (1) 4 (0) dt mm 2 U Fmm ,
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 , (0)
where is the rate of decay, the reciprocal of the state lifetime.
Substituting these results into equation (0), we see that the elements of the
excited-state density matrix are solutions of the equation
1 4 2 mm (0) m H ext , m 2 U() Fmm mm . t i
For Stark beats, we have a z-directed electric field and no magnetic field. In this case,
H ext assumes the value EJm derived in equation (0):
E 2 3m2 J (J 1) (0) H m E m e e m . ext e Jem 2 0 2 Je 2Je 1 Now we can evaluate the first term on the right-hand side of equation (0), using the
Hermiticity of H ext : 44
m H ext , m m H ext H ext m
mH ext m m H ext m m H ext m m H ext m
m H ext m m H ext m (0) EJm m m EJm m m
EJm m m EJm m m
(EJm EJm )mm E 2 3m2 3m 2 2 mm 2 Je 2Je 1 so that we have
2 mm 4 (0) imm mm 2 U() Fmm , t where
3 E 2 m2 m 2 (0) 2 mm . 2 Je 2Je 1 For Zeeman beats, everything is the same except that
g Hm m . mm 0 (0) In our pulsed experiment, U() is a function of time. Multiplying (0) by the integrating
i t factor e mm gives
4 2 d imm t imm t (0) dt mm e 2 Fmm U(,t)e .
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 t0 t t0 t U,t 0 for all other values of t so integrating equation (0) we have fort t0 t 45
2 imm t 4 i (t t ) i t e (0) mm 0 mm 0 , mm 2 U() Fmm e e imm 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 fort t0 . 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
2 i t 4 e mm 1 imm (t t0 ) mm 2 U() Fmm e imm 2 4 1 imm t 1 (0) imm (t t0 ) 2 U() Fmm e imm 4 2 imm (t t0 ) 2 t U() Fmm e .
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
[Error: Reference source not found]
3 2 W s d m e d d . (0) m 2 c3
We can thus form the fluorescent light monitoring operator LF defined by
3 (0) LF 3 e d e d . 2 c
Then the observed intensity of fluorescent light can be obtained from the Liouville equation: 46
dI Tr(L ) d F (0) 3 imm t t0 2 U()t Fmm G mm e , c mm
whereG mm m e d e d m is called the emission matrix.
To calculate the matrix elements in equation (0), we expand the polarization vector e and the electric dipole moment operator d in terms of the spherical unit vectors, for instance,
1 e ∓ (e ie ) 1 2 x y (0)
e0 ez .
With spherical components, we have
e d e1d 1 e1d 1 e0d0 . (0)
The components of the dipole operator can then be calculated using the Wigner-Eckart theorem, equation (0), so that
q m e d (1) eq m dq q (0) Je 1 J g q Je m (1) eq (1) e Je d g J g . q m q
Calculating the complex conjugate of the polarization vector, we see that
e (1)q e , q q (0) so that the excitation matrix is given by 47
Fmm m e d e d m
Je 1 J g (1) q e (1) Je m J d J (0) q e e g g m q q
J g 1 J e e (1) J e J d J . q g g e e q m q
We also have [28]
J d J (1) J J J d J , (0) so that equation (0) can be written as
2 Fmm e Je d g J g (0) J 1 J J 1 J q Je Jg m e g g e (1) eq eq . q q m q q m
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
2 form of the signal can be calculated in terms of the product 2E , the decay rate , and
the pulse time t0 . This was done using Mathematica (see section 1.) For example, for the geometry in our experiment, laser and detection polarization at 45 to vertical, and
J g 2 , Je 2 , J g 1, we have
2 2 (t t0 )w 12 2 3(t t0 )w I ae (t t0 ) 1 cos cos , 51 4 51 4
E 2 where a is an amplitude coefficient, and w 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.
Lifetime Lifetime (Previous Measurements) (Theory) [Error: Level Lifetime Referen Configuration Term J (cm-1) (This work) ce [31] [32] [33] [34] Other source not found] 13796.36 4f6(7F)6s6p(3Po) 9Go 0 3.043(29) 3.2 13999.50 4f6(7F)6s6p(3Po) 9Go 1 2.464(34) 2.9 14380.50 4f6(7F)6s6p(3Po) 9Go 2 2.087(42) 2.6 14863.85 4f6(7F)6s6p(3Po) 9Fo 1 0.954(41) 1.2 14915.83 4f6(7F)6s6p(3Po) 9Go 3 1.828(20) 2.4 15039.59 4f6(7F)6s6p(3Po) 2 1.817(73) 3.4 15507.35 4f6(7F)6s6p(3Po) 9Do 3 2.227(21) 15567.32 4f6(7F)6s6p(3Po) 9Do 2 2.74(11) 15579.12 4f6(7F)6s6p(3Po) 9Go 4 1.716(39) 2.4 15586.30 4f6(7F)6s6p(3Po) 5Do 0 2.466(44) 15650.55 4f6(7F)6s6p(1Po) 7Go 1 2.626(17) 3.3 16112.33 4f6(7F)6s6p(3Po) 5Do 1 1.955(40) 1.46(13) 16116.42 4f6(7F)6s6p(1Po) 7Go 2 2.71(22) 16131.53 4f6(7F)6s6p(3Po) 9Do 4 2.657(49) 16211.12 4f6(7F)6s6p(3Po) 9Fo 3 4.38(12) 16344.77 4f6(7F)6s6p(3Po) 9Go 5 1.569(58) 16681.74 4f6(7F)6s6p(3Po) 2 1.974(18) 1.70(10) 16690.76 4f6(7F)6s6p(1Po) 7Do 1 1.71(10) 1.45(20) 16748.30 4f6(7F)6s6p(1Po) 7Go 3 2.594(96) 16859.31 4f6(7F)6s6p(3Po) 9Do 5 2.86(22) 17190.20 4f6(7F)6s6p(1Po) 7Fo 2 1.086(16) 1.02(10) 1.20(13) 1.50(10) 1.10(10) [35] 17243.55 4f6(7F)6s6p(1Po) 3 1.589(10) 2.22(10) 1.80 [Error: 17462.37 4f6(7F)6s6p(3Po) 5Go 2 0.122(9) 2.42(10) Reference source not found] 17504.63 4f6(7F)6s6p(1Po) 7Go 4 2.420(34) 17587.46 4f6(7F)6s6p(3Po) 9Fo 5 2.66(14) 17769.71 4f6(7F)6s6p(1Po) 1 0.157(5) 0.159(10) 0.165(10) 0.038 [36] 17810.85 4f6(7F)6s6p(1Po) 7Fo 0 0.342(10) 17830.80 4f6(7F)6s6p(1Po) 7Fo 3 1.265(11) 1.10(10) 1.58(10) 0.480(20) [Error: 18075.67 4f5(6Ho)5d6s2 7Ho 2 0.450(50) 0.440(40) 0.158(5) Reference source not found] 18225.13 4f6(7F)6s6p(1Po) 1 0.146(6) 0.146(8) 18350.40 4f6(7F)6s6p(1Po) 7Go 5 2.558(79) 49
18475.28 4f5(6Ho)5d6s2 7Fo 1 0.071(2) 0.061(5) 0.069(4) [37] 18503.49 4f6(7F)6s6p(3Po) 5Go 4 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 [Error: Reference source not found,39,Error: Reference source not found] (see text). Estimate of Tensor Polarizability Closest Even Parity Neighbors Matrix Level Configuration Term J Element (cm-1) Level |dz| This work Other work Configuration Term J -1 E ||d|| (cm ) max 4f6(7F)5d(8F)6s 9F 1 14026.45 26.95 13999.50 4f6(7F)6s6p(3Po) 9Go 1 38.89(13)* 4f6(7F)5d(8G)6s 9G 1 13732.53 -266.97 0.42 0.17 “ “ 2 13687.75 -311.75 4f6(7F)5d(8F)6s 9F 2 14365.50 -15.00 14380.50 4f6(7F)6s6p(3Po) 9Go 2 27.7(12)* 4f6(7F)5d(6P)6s 7P 2 14550.50 170.00 0.24 0.09 4f6(7F)5d(8P)6s 9P 3 14612.44 231.94 6 7 8 7 [Error: Reference 4f ( F)5d( D)6s D 2 14783.51 -80.34 6 7 3 o 9 o 4(1) 6 7 6 7 14863.85 4f ( F)6s6p( P ) F 1 4.326(9) source not found] 4f ( F)5d( P)6s P 2 14550.5 -313.35 0.54 0.20 4f6(7F)5d(8F)6s 9F 2 14365.5 -498.35 4f6(7F)5d(8F)6s 9F 3 14920.45 4.62 14915.83 4f6(7F)6s6p(3Po) 9Go 3 27.94(19)* 4f6(7F)5d(8D)6s 7D 2 14783.51 -132.32 0.14 0.05 4f6(7F)5d(8P)6s 9P 3 14612.44 -303.39 4f6(7F)5d(8F)6s 9F 3 14920.45 -119.14 15039.59 4f6(7F)6s6p(3Po) 2 33.179(83)* 4f6(7F)5d(8D)6s 7D 2 14783.51 -256.08 1.4 0.40 4f6(7F)5d(8P)6s 9P 3 14612.44 -427.15 4f6(7F)5d(8D)6s 7D 3 15524.56 17.21 15507.35 4f6(7F)6s6p(3Po) 9Do 3 77.3(26)* 4f6(7F)5d(8G)6s 7G 2 15955.24 447.89 0.45 0.15 4f6(7F)5d(8F)6s 9F 3 14920.45 -586.90 4f6(7F)5d(8D)6s 7D 3 15524.56 -42.76 15567.32 4f6(7F)6s6p(3Po) 9Do 2 24.90(82) 4f6(7F)5d(8G)6s 7G 1 15639.8 72.48 0.72 0.21 4f6(7F)5d(6P)6s 3 15834.6 267.28 4f6(7F)5d(8D)6s 7D 3 15524.56 -54.56 15579.12 4f6(7F)6s6p(3Po) 9Go 4 9.57(21)* 4f6(7F)5d(8F)6s 9F 3 14920.45 -658.67 0.36 0.09 4f6(7F)5d(8D)6s 7D 4 16354.6 775.48 -556(12) [Error: Reference source not
found] 6 7 8 7 [20] 4f ( F)5d( G)6s G 1 15639.8 -10.75 6 7 1 o 7 o -548(12) 6 7 8 7 15650.55 4f ( F)6s6p( P ) G 1 -561.7(11) [Error: 4f ( F)5d( F)6s F 0 15793.68 143.13 1.0 0.41 -563(34) 6 2 5 Reference source not 4f 6s D 1 15914.55 264.00
found] -410(50) [40] 4f6(7F)5d(8G)6s 7G 2 15955.24 -157.09 16112.33 4f6(7F)6s6p(3Po) 5Do 1 70.94(31) 4f66s2 5D 1 15914.55 -197.78 3.0 1.1 4f6(7F)5d(8F)6s 7F 0 15793.68 -318.65 [20] -112.5(24) 6 7 8 7 [Error: 4f ( F)5d( G)6s G 2 15955.24 -161.18 6 7 1 o 7 o -103(10) 6 2 5 16116.42 4f ( F)6s6p( P ) G 2 -115.23(79) Reference source not 4f 6s D 1 15914.55 -201.87 1.6 0.58 6 7 6 found] 4f ( F)5d( P)6s 3 15834.6 -281.82 4f6(7F)5d(8D)6s 7D 4 16354.6 223.07 16131.53 4f6(7F)6s6p(3Po) 9Do 4 76.26(12)* 4f6(7F)5d(8D)6s 7D 3 15524.56 -606.97 1.7 0.52 4f6(7F)5d(8F)6s 9F 3 14920.45 -1211.08 4f6(7F)5d(8D)6s 7D 4 16354.6 143.48 16211.12 4f6(7F)6s6p(3Po) 9Fo 3 3.785(49) 4f6(7F)5d(8G)6s 7G 2 15955.24 -255.88 N/A N/A 4f6(7F)5d(6P)6s 3 15834.6 -376.52 4f6(7F)5d(8G)6s 7G 2 15955.24 -726.50 16681.74 4f6(7F)6s6p(3Po) 2 260.09(67)* 4f6(7F)5d(6P)6s 3 15834.6 -847.14 5.1 1.9 4f6(7F)5d(8G)6s 7G 1 15639.8 -1041.94 -13.95(65)[Error: 4f6(7F)5d(8G)6s 7G 2 15955.24 -735.52 16690.76 4f6(7F)6s6p(1Po) 7Do 1 Reference source not 4f66s2 5D 1 15914.55 -776.21 N/A N/A found] 4f6(7F)5d(8F)6s 7F 0 15793.68 -897.08 16748.30 4f6(7F)6s6p(1Po) 7Go 3 124.3(26) 119.8(26) [20] 4f6(7F)5d(8D)6s 7D 4 16354.6 -393.70 4.8 1.2 127(18) [Error: 4f6(7F)5d(8G)6s 7G 2 15955.24 -793.06 50
Reference source not 6 7 6 found] 4f ( F)5d( P)6s 3 15834.6 -913.70 4f6(7F)5d(8D)6s 7D 4 16354.6 -504.71 16859.31 4f6(7F)6s6p(3Po) 9Do 5 129.00(42)* 4f6(7F)5d(8H)6s 7H 6 15617.45 -1241.86 4.4 1.0 4f6(7F)5d(8D)6s 9D 6 15082.94 -1776.37 -20.8(15) [Error: Reference source not 6 2 5 found] 4f 6s D 2 17864.29 674.09 6 7 1 o 7 o 6 7 6 7 17190.20 4f ( F)6s6p( P ) F 2 -25.30(15) [Error: 4f ( F)5d( H)6s H 2 18176.17 985.97 N/A N/A -23(2) 6 7 8 7 Reference source not 4f ( F)5d( G)6s G 2 15955.24 -1234.96
found] 4f66s2 5D 2 17864.29 620.74 17243.55 4f6(7F)6s6p(1Po) 3 95.18(31) 4f6(7F)5d(8D)6s 7D 4 16354.6 -888.95 N/A N/A 4f6(7F)5d(6H)6s 7H 2 18176.17 932.62 1315(120) [Error: 4f66s2 5D 2 17864.29 401.92 17462.37 4f6(7F)6s6p(3Po) 5Go 2 Reference source not 4f6(7F)5d(6H)6s 7H 2 18176.17 713.80 8.5 3.1 found] 4f6(7F)5d(8G)6s 7G 2 15955.24 -1507.13 4f6(7F)5d(8D)6s 7D 4 16354.6 -1150.03 17504.63 4f6(7F)6s6p(1Po) 7Go 4 -5.84(10) 4f6(7F)5d(6P)6s 3 15834.6 -1670.03 1.1 0.32 4f6(7F)5d(8D)6s 7D 3 15524.56 -1980.07 4f6(7F)5d(8D)6s 7D 4 16354.6 -1232.86 17587.46 4f6(7F)6s6p(3Po) 9Fo 5 36.20(32) 4f6(7F)5d(8H)6s 7H 6 15617.45 -1970.01 3.7 0.83 4f6(7F)5d(8D)6s 9D 6 15082.94 -2504.52 [Error: 13.13(57) 6 2 5 Reference source not 4f 6s D 2 17864.29 94.58 6 7 1 o 6 7 6 7 17769.71 4f ( F)6s6p( P ) 1 found] 4f ( F)5d( H)6s H 2 18176.17 406.46 N/A N/A 4f6(7F)5d(8G)6s 7G 2 15955.24 -1814.47 13.55(6) [41] 4f6(7F)6s6p(1Po) 7Fo 3 -202.74(94) 4f66s2 5D 2 17864.29 33.49 1.1 0.33 17830.80 4f6(7F)5d(6H)6s 7H 2 18176.17 345.37 4f6(7F)5d(8D)6s 7D 4 16354.6 -1476.20 9.15(35) [Error: Reference source not 6 7 6 7 found] 4f ( F)5d( H)6s H 2 18176.17 100.50 5 6 o 2 7 o 6 2 5 18075.67 4f ( H )5d6s H 2 [Error: 4f 6s D 2 17864.29 -211.38 0.36 0.13 10(1) 6 2 5 Reference source not 4f 6s D 3 20195.76 2120.09
found] -150(9) [Error: 4f6(7F)5d(6H)6s 7H 2 18176.17 -32.87 18209.04 4f6(7F)6s6p(3Po) 5Go 3 Reference source not 4f66s2 5D 2 17864.29 -344.75 N/A N/A found] 4f6(7F)5d(8D)6s 7D 4 16354.6 -1854.44 -6.08(31) [Error: Reference source not 6 7 6 7 found] 4f ( F)5d( H)6s H 2 18176.17 -48.96 6 7 1 o 6 2 5 18225.13 4f ( F)6s6p( P ) 1 [Error: 4f 6s D 2 17864.29 -360.84 N/A N/A -6(1) 6 2 5 Reference source not 4f 6s D 3 20195.76 1970.63
found] 9.12(50) [Error: Reference source not 4f6(7F)5d(6H)6s 7H 2 18176.17 -240.45 18416.62 4f6(7F)6s6p(1Po) 2 found] 4f66s2 5D 2 17864.29 -552.33 N/A N/A 9(4) [Error: Reference 4f66s2 5D 3 20195.76 1779.14 source not found] -660(46) [Error: 4f66s2 5D 3 20195.76 1692.27 18503.49 4f6(7F)6s6p(3Po) 5Go 4 -717.2(15) Reference source not 4f6(7F)5d(8D)6s 7D 4 16354.6 -2148.89 17 4.4 found] 4f6(7F)5d(6P)6s 3 15834.6 -2668.89 -7.69(54)[Error: Reference source not 6 7 6 7 found] 4f ( F)5d( H)6s H 2 18176.17 -611.91 6 7 1 o 7 o 6 2 5 18788.08 4f ( F)6s6p( P ) F 2 [Error: 4f 6s D 2 17864.29 -923.79 0.80 0.29 -7.3(5) 6 2 5 Reference source not 4f 6s D 3 20195.76 1407.68
found] 473(35) [Error: 4f66s2 5D 2 17864.29 -1121.41 18985.70 4f6(7F)6s6p(3Po) 5Fo 1 Reference source not 4f6(7F)5d(8G)6s 7G 2 15955.24 -3030.46 21 7.6 found] 4f66s2 5D 1 15914.55 -3071.15 6 7 6 7 [Error: Reference 4f ( F)5d( H)6s H 2 18176.17 -833.35 5 6 o 2 9(6) 6 2 5 19009.52 4f ( H )5d6s 2 source not found] 4f 6s D 2 17864.29 -1145.23 N/A N/A 4f66s2 5D 3 20195.76 1186.24 16(1) [Error: 4f66s2 5D 3 20195.76 694.49 19501.27 4f6(7F)6s6p(1Po) 7Fo 3 Reference source not 4f6(7F)5d(6H)6s 7H 2 18176.17 -1325.10 1.3 0.43 found] 4f66s2 5D 2 17864.29 -1636.98 19990.25 4f5(6Ho)5d6s2 7Ho 4 52(6) [Error: 4f66s2 5D 3 20195.76 205.51 N/A N/A Reference source not 4f6(7F)5d(8D)6s 7D 4 16354.6 -3635.65 51
found] 4f6(7F)5d(6P)6s 3 15834.6 -4155.65 809(36) [Error: 4f6(7F)5d(8D)6s 7D 4 16354.6 -3798.87 20153.47 4f6(7F)6s6p(1Po) 7Fo 5 Reference source not 4f6(7F)5d(8H)6s 7H 6 15617.45 -4536.02 30 6.9 found] 4f6(7F)5d(8D)6s 9D 6 15082.94 -5070.53 -6(1) [Error: 4f66s2 5D 3 20195.76 32.76 20163.00 4f6(7F)6s6p(1Po) 7Fo 4 Reference source not 4f6(7F)5d(6H)6s 7H 2 18176.17 -1986.83 0.22 0.06 found] 4f66s2 5D 2 17864.29 -2298.71 313(20) [Error: 4f6(7F)5d(8H)6s 7H 7 16392.93 -4662.83 21055.76 4f6(7F)6s6p(1Po) 7Fo 6 Reference source not 4f6(7F)5d(8H)6s 7H 6 15617.45 -5438.31 29 5.5 found] 4f6(7F)5d(8D)6s 9D 6 15082.94 -5972.82 38(6) [Error: 4f6(7F)5d(8D)6s 7D 4 16354.6 -5104.29 21458.89 4f6(7F)6s6p(1Po) 7Fo 5 Reference source not 4f6(7F)5d(8H)6s 7H 6 15617.45 -5841.44 7.7 1.7 found] 4f6(7F)5d(8D)6s 9D 6 15082.94 -6375.95 6 7 6 7 [Error: Reference 4f ( F)5d( H)6s H 2 18176.17 -4737.90 6 7 3 o 7 o 4(1) 6 2 5 22914.07 4f ( F)6s6p( P ) G 1 source not found] 4f 6s D 2 17864.29 -5049.78 4.0 1.4 4f6(7F)5d(8G)6s 7G 2 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 (ea0). 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 [Error: Reference source not found,Error: Reference source not found]). The
estimates of ||d|| and |dz| max are based on only the nearest known partner state (see Sec. 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.
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: [Error: Reference source not found,Error:
Reference source not found,Error: Reference source not found] used fluorescence decay
with laser excitation, [Error: Reference source not found] used luminescence decay with
laser excitation, [Error: Reference source not found,Error: Reference source not found]
used the delayed-coincidence method, and [Error: Reference source not found] used the
level-crossing method in zero magnetic field (Hanle method). denotes an estimate
7 obtained from measurements of the dipole matrix element to the F0 ground state in
7 [Error: Reference source not found] and relative oscillator strengths to the F0-2 ground 52 term levels in [Error: Reference source not found] and [Error: Reference source not found]. 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
5 present work disagrees with the result of [Error: Reference source not found] for the D1 level beyond 2. However, [Error: Reference source not found] also disagrees with the
7 consensus of three other experiments for the lifetime of the H2 level, so it is possible that
5 there is a problem with its result for the D1 level, as well. The present work also disagrees with [Error: Reference source not found] beyond the 2 level for the lifetimes
-1 7 -1 of four states: the level at 16681.74 cm with J=2, the F2 at 17190.20 cm , the level at
-1 7 -1 17243.55 cm with J=3, and the F3 level at 17830.80 cm . However, in the case of the
7 F2 level, [Error: Reference source not found] contradicts the consensus of four other
7 experiments, including the present work, and in the case of the F3 level, the present work agrees at the 2 level with [Error: Reference source not found]. Thus the disagreement with [Error: Reference source not found] does not seem to be a serious problem. There are no other cases 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
7 experiment was measured in the current work ( F2) the new result agrees with the previous ones within the stated error. 53
2. Polarizabilities
The current work in polarizabilities is presented along with all other measurements of tensor polarizabilities in samarium. [Error: Reference source not found,Error: Reference source not found,Error: Reference source not found] used laser- atomic beam spectroscopy, [Error: Reference source not found] used laser-atomic beam spectroscopy and the nonlinear level-crossing technique, [Error: Reference source not found] used quantum-beat spectroscopy, and [Error: Reference source not found] 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
7 7 for two levels ( G1, F2) 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 [Error:
Reference source not found]. The results are included in Table 1. There is good correspondence between theory and experiment—the theoretical values are generally a factor of ~1.2 higher than those from experiment (Fig. 20). 54
1.9
1.8
1.7
t 1.6 p x e
1.5 / y r o
e 1.4 h t 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 [Error: Reference source not found]. Lifetimes and tensor polarizabilities calculated from these integrals are compared to experiment in [Error:
Reference source not found]. Scalar and tensor polarizabilities calculated from these integrals are compared to experiment in [Error: Reference source not found]. 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. 55
VI. Systematics
A. Overview
We now turn to a discussion of the various effects which could produce systematic errors in the results. In Sec. 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. C issues having to do with the measurement of the applied electric field are discussed. Lastly, in Sec. 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. 56
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 partial (0) 2Je 1 . 2 Doppler
1 Assuming that this is the dominant decay channel, we overestimate by substituting 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 57
2 s , 700 nm as typical values, equation (0) gives 1011 cm2 . Thus the absorption length for this radiation is
1 1 ℓ cm 0.5 cm , n 2 1011 1011 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). 58
3.0
2.9 )
s 2.8 (
e m i t e
f 2.7 i l
2.6
2.5 0 10 20 30 40 50 60 70 signal amplitude (mV)
9 Fig. 21. Lifetime dependence on density for D2 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 (0) we have 59
2 1 partial (0) ℓ N P f N P f 2Je 1 , 2 Doppler
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 0.4
J=0 n 0.3 o i t a l u
p J=2 o 0.2 P
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, 60 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:
1 Q 2 N P i VB.R.OTGe . (0)
In terms of the amplitude of the signal and the lifetime, we have
a Q , (0) R where a is the signal amplitude and R is the input impedance of the oscilloscope. Letting
1 F 2 VOGeR , we have
B.R. a N N (0) partial , FT P i so from equation (0)
P 2 1 a f (0) ℓ 2Je 1 . TFDoppler P i 2
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 2.4 1.02 2.3 ) ) s s 2.2 1.00 ( (
e 2.1 e 0.98 m m i i t t e e
f 2.0 f i i 0.96 L 1.9 L 0.94 1.8 0.92 1.7 1.6 0.90 0.88 0 1 2 3 4 5 6 7 8 9 0 100 200 300 400 500 600 700 800 900 Absorption coefficient (arb. units) Absorption coefficient (arb. units) (a) (b) Fig. 23. Lifetime plotted against absorption coefficient for decays from 9 9 (a) a G2 level, where no correlation is seen and (b) a F1 level where a 0.05 s effect is seen. 61
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 n0 2 transmission 0 1 sin , 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). hr saptnilysgiiatefc ic o for since effect significant potentially a is there 10
s, our data record length, the fluorescing atoms move on average about 3 about average on move atoms fluorescing the length, record data our s, In this experiment, the exposed diameter of the interference filters was 4 was filters interference the of diameter exposed the experiment, this In i.Fig. i.Fig. Transmitted flux 0.8 0.6 24 25 Transmitted flux (arb. units) 0.4 fluorescence at 645 fluorescence 4 of diameter exposed an with circular, actually is IF experiment. 652 at fluorescence 0.2 1 0.8
0.6 rnmsin poie fr 650 for profile Transmission . rnmsin poie fr 650 for profile Transmission .
(arb. units) cm. 0.4 0.2 1 0 0 -2 -2 -1 -1 (cm) x x (cm) x
nm.
m acltd fr te goer n our in geometry the for calculated nm, 0 0 1 1
m itreec filter, interference nm m itreec filter, interference nm 2 2 = = 0 - -2 -2 f = 5 = -1 -1 0
0 nm, for example, the example, for nm, y(cm) y(cm) 1 1 2 2
mm. Thus mm.
cm. In cm. 62 63 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.
+ + + + 43 44 Several gases, including N2 , He , CH4 , and H2 , are known to produce afterpulses, [ , ] 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. 64
0 ) V m (
e
s -10 n o p s e r
e
b afterpulses u t
o -20 t o h P
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. A (Fig. 27). This cancellation scheme is not perfect, however, since the laser output power can drift between the time the on- and off- resonance 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 off- resonance file is added to fake lifetime data. The difference between the fitted and “true” lifetimes is used as the error due to this effect. 65
1 1
0 0 ) -1 ) -1 V V m m ( on-resonance signal (
-2 -2 off-resonance signal e e s s n -3 n -3 o o p p s -4 s -4 e e r r
e e
b -5 -5 b u u t t o o t -6 t -6 o o h h P -7 P -7
-8 -8
-9 -9 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Time (s) Time (s)
Fig. 27. On the left, on- and off-resonance fluorescence data for a 7 transition from the G3 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 [Error: Reference source not found]. 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. 66
500
) 400 V m (
e
g 300 a t l o v
k
a 200 e P
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 [Error: Reference source not found], 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. 67
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 Hall- probe 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. 68
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 even- isotope 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. 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 even- isotope 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
d 2E 2 ~ . Thus the decay rate is given (for small mixing) by E 2
2 2 d E (0) X Y X 2 , E
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 69
d 2E 2 1 . (0) X E 2
7 For all states measured except for G1 this estimate predicts an effect considerably less
7 than 1% even at the highest fields used (~40 kV/cm). In the case of the G1 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. 70
0
-2
) -4 V m (
-6 e s
n -8 o p
s -10 e r
e
b -12 u t o
t -14 o h
P -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. 7 Upper state G1, 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. 71
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 , 16Et3 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 G using a program written in Mathematica (Sec. 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. 72
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 state to the 4f 66s6p term in e.g. [Error: Reference source not found]. )
V 1.6 m (
e 1.4 d u t i l 1.2 p m A
1.0 e c
n 0.8 e c s e
r 0.6 o u l F
0.4 y a c
e 0.2 D 0.0 7295.3 7295.4 7295.5 7295.6 7295.7 7295.8 Laser Tuning (Angstroms)
) 1.0 d e t
c 0.9 e p
x 0.8 e
f o
0.7 n o i t 0.6 c a r f
( 0.5
t s
a 0.4 r t n
o 0.3 c
t a
e 0.2 b
k
r 0.1 a t
S 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 6 2 7 6 9 4f 6s F14f 6s6p G1 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. 73
The spectrum of a given transition is a few GHz wide ( 0.1Å) 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 high- angular 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. 74
Many of these values had not been previously measured; agreement with those that had been measured was satisfactory (see Sec. 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. 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 (0) and (0). The three closest known even-parity 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. C. 75
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. a the important experimental parameter
d for this estimate is . Of the states for which polarizabilites have been measured (Table E
6 7 -1 2), the most promising seems to be the 4f 6s6p G1 state at 15650.55 cm (designated Y)
6 7 -1 with its metastable partner 4f 5d6s G1 state at 15639.80 cm (designated X). Discovery of new even-parity states may provide other interesting candidates, however (see Sec. 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,
3 2 128 Ry X z Y Z 3 (0) d X ~ de 3 4 10 de 9 E(Y) E( X ) a0 2 2 4 15d6 p
Here Z=62 is the nuclear charge, is the fine structure constant;
Ry (0) 1.9 , 5d 6p I.P. E(5d,6 p) 76 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 (0) substitutes 8 3 X z Y for the radial integral found in [Error: Reference source not found] 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 [Error: Reference source not found] and there have been problems with theoretical determinations of matrix elements involving states X and Y in the past [Error: Reference source not found]. 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 [Error:
Reference source not found]. 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 [Error: Reference source not found]. 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:
4 dX ~ 10 de . (0) 77
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
[Error: Reference source not found]) 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. 78
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 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 ratio2 0 of the tensor to the scalar polarizabilities is specified by the J-values of the two states. From eqs. (0) and (0):
1 for J J 1 2J 1 (0) 2 for J J 0 J 1 J2J 1 for J J 1, J 12J 3 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 [Error: Reference source not found]. 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
6 7 o opposite parity partner (4f 6s6p G 1, 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 79 levels without a known partner state match one of the theoretical predictions quite well
6 5 o (for example 4f 6s6p G 2, 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.
Closest Known Theoretical Prediction Level State Configuration Term J (cm-1) 2 0 Dom. E J-1 J J+1 J J (cm-1) 14863.85 4f6(7F)6s6p(3Po) 9Fo 1 -0.042(15) -1 0.5 -0.1 - -80.34 2 15650.55 4f6(7F)6s6p(1Po) 7Go 1 0.564(57) -1 0.5 -0.10 1 -10.75 1 16116.42 4f6(7F)6s6p(1Po) 7Go 2 -2.10(59) -1 1 -0.29 -161.2 2 16748.30 4f6(7F)6s6p(1Po) 7Go 3 -0.86(19) -1 1.25 -0.42 2 -393.70 4 17190.20 4f6(7F)6s6p(1Po) 7Fo 2 -0.42(13) -1 1 -0.29 3 674.09 2 17462.37 4f6(7F)6s6p(3Po) 5Go 2 -0.308(33) -1 1 -0.29 3 401.92 2 18075.67 4f5(6Ho)5d6s2 7Ho 2 0.143(36) -1 1 -0.29 100.50 2 18209.04 4f6(7F)6s6p(3Po) 5Go 3 -5.0(22) -1 1.25 -0.42 -32.87 2 18225.13 4f6(7F)6s6p(1Po) 1 -0.118(32) -1 0.5 -0.10 2 -48.96 2 18416.62 4f6(7F)6s6p(1Po) 2 0.089(40) -1 1 -0.29 -240.45 2 18503.49 4f6(7F)6s6p(3Po) 5Go 4 -0.476(56) -1 1.40 -0.51 5 1692.27 3 18788.08 4f6(7F)6s6p(1Po) 7Fo 2 -0.111(78) -1 1 -0.29 -611.91 2 18985.70 4f6(7F)6s6p(3Po) 5Fo 1 0.353(39) -1 0.5 -0.10 1 -1121.41 2 19009.52 4f5(6Ho)5d6s2 2 0.110(76) -1 1 -0.29 -833.35 2 19501.27 4f6(7F)6s6p(1Po) 7Fo 3 0.47(16) -1 1.25 -0.42 694.49 3 19990.25 4f5(6Ho)5d6s2 7Ho 4 0.77(17) -1 1.40 -0.51 205.51 3 20153.47 4f6(7F)6s6p(1Po) 7Fo 5 -0.640(62) -1 1.5 -0.58 6 -3798.87 4 20163.00 4f6(7F)6s6p(1Po) 7Fo 4 -0.088(25) -1 1.40 -0.51 32.76 3 21055.76 4f6(7F)6s6p(1Po) 7Fo 6 -0.775(73) -1 1.57 -0.63 7 -4662.83 7 21458.89 4f6(7F)6s6p(1Po) 7Fo 5 0.339(66) -1 1.5 -0.58 -5104.29 4 22914.07 4f6(7F)6s6p(3Po) 7Go 1 0.053(15) -1 0.5 -0.10 -4737.90 2 Table 3. Ratio of tensor to scalar polarizability measured in [Error: Reference source not found] 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 close-lying 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. 80
VIII. Appendices
A. Software
1. Calculation of the Signal and Determination of Experimental Geometry
The calculation to determine the observed signal, described in Sec. G, is 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 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 (). 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.
jg 3, j e 2, jf 2
Intensity of Polarized Light Total Intensity 0.0065
0.0064
0.0034 0.0063 0.0032 2 0.003 0.0028 3 0.0062 0.0026 2 0 0.0061 2 2 3 3 2 2 2 2 2 0 81
Fig. 31. Calculated intensity of light received at the photomultiplier tube at a particular time for Stark beats from a J=322 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.
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). 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).