Eur. Phys. J. Special Topics 163, 71–88 (2008)
c EDP Sciences, Springer-Verlag 2008
DOI: 10.1140/epjst/e2008-00810-0
THE EUROPEAN
PHYSICAL JOURNAL
SPECIAL TOPICS
A laboratory search for variation of the fine-structure
constant using atomic dysprosium
A. Cingöz1 , N.A. Leefer1 , S.J. Ferrell1 , A. Lapierre2 , A.-T Nguyen3 , V.V. Yashchuk4 ,
D. Budker1,4 , S.K. Lamoreaux5 , and J.R. Torgerson6
1
2
3
4
5
6
Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300, USA
TRIUMF National Laboratory, 4004 Wesbrook Mall, Vancouver, British Columbia,
V6T 2A3, Canada
Department of Neurobiology, University of Pittsburgh, Pittsburgh, Pennsylvania 15213, USA
Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Department of Physics, Yale University, New Haven, Connecticut 06520-8120, USA
Physics Division, Los Alamos National Laboratory, P-23, MS-H803, Los Alamos,
New Mexico 87545, USA
Abstract. Electric-dipole transitions between nearly degenerate, opposite parity
levels of atomic dysprosium (Dy) were monitored over an eight-month period to
search for a variation in the fine-structure constant, α. The frequencies of these
transitions are sensitive to variation of α due to large relativistic corrections of
opposite sign for the opposite-parity levels. In this unique system, in contrast to
atomic-clock comparisons, the difference of the electronic energies of the oppositeparity levels can be monitored directly utilizing a radio-frequency (rf), electricdipole transition between them. Our measurements for the frequency variation of
the 3.1-MHz transition in 163 Dy and the 235-MHz transition in 162 Dy can be analyzed for both a temporal variation and a gravitational-potential dependence of α
since, during the data acquisition period, the Earth is located at different values of
the gravitational potential of the Sun. The data provide a rate of fractional temporal variation of α of (−2.7 ± 2.6) × 10−15 yr−1 or a value of (−8.7 ± 6.6) × 10−6
for kα , the linear-variation coefficient for α in a changing gravitational potential.
These results are independent of assumptions regarding variation of other fundamental constants. The latter result can be combined with other experimental
constraints to extract the first limits on ke and kq , which characterize the variation of me /mp and mq /mp in a changing gravitational potential, where me , mp ,
and mq are electron, proton, and quark masses. All results indicate the absence
of significant variation at the present level of sensitivity.
1 Introduction
A component of Einstein’s equivalence principle (EEP) is local position invariance (LPI), which
states that the laws of physics, including the values of fundamental constants, should be
independent of space and time. Modern theories attempting to unify gravitation with the
other fundamental interactions allow, or even predict, violations of EEP [1], which has sparked
searches for violation of LPI, and hence EEP, through searches for temporal and spatial variation of fundamental constants.
In particular, various recent studies have reported results of searches for a temporal variation of the fine-structure constant, α = e2 /c. Evidence for temporal variation of α was discovered in observations of quasar absorption spectra [2,3], which probe cosmological time scales.
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The result in [3] corresponds to α̇/α = (6.40 ± 1.35) × 10−16 /yr assuming a linear shift over
1010 years. However, more recent measurements present sensitive results that are consistent
with no variation [4,5]1 . On a geological time scale of 109 years, a test for α-variation comes
from the analysis of fission products of a natural reactor in Oklo (Gabon). While this analysis
yields a stringent limit of |α| < 1 × 10−17 /yr [8–10], these observational measurements are
dependent upon models and assumptions which cannot be fully verified at present.
In contrast to observational studies involving analyses of processes that have occurred
billions of years ago, laboratory searches (see, for example, Refs. [11–16]) are sensitive to
possible present-day variation of fundamental constants. Their results are easier to interpret
since the experiments are repeatable, and systematic uncertainties can be studied by changing
experimental conditions. The best laboratory limit of |α̇/α| < 1.3 × 10−16 /yr (published after
our main result in Ref. [16]) was obtained from a comparison of a Hg+ optical clock to a Cs
frequency standard [15]. This result assumes that other constants, in particular me /mp , do
not vary in time. However, me /mp is expected to have ∼40 times larger fractional variation
than the variation in α [17], and interestingly, evidence for variation of this parameter on an
astronomical time scale was recently reported in Ref. [18]2 . The best limit that is independent
of assumptions regarding other constants was obtained by combining the Hg+ optical clock vs.
Cs frequency standard comparison with a comparison of Yb+ optical clock with a Cs frequency
standard [14] and a comparison of 1S − 2S transition frequency in H with a Cs frequency
standard [13]. This analysis gives |α̇/α| < 1.5 × 10−15 /yr [15].
Another type of search for an LPI violation is a “null” gravitational red-shift experiment
where two clocks with different composition are compared side by side in a changing gravitational potential [20]. Laboratory clock comparisons can be used for this type of test due to the
eccentricity of Earth’s orbit around the Sun, which leads to a small oscillatory component of
the gravitational potential with a period of a year. In earlier work [15,21–23], clocks of different
types were compared, and the ratios of the clock rates were analyzed for a possible correlation with the gravitational potential, which led to bounds on parameters that characterize
structure-dependent modifications to the clock frequencies. However, it is not straightforward
to interpret the results of such comparisons in terms of variation of fundamental constants in
a gravitational potential because the rate of each of the clocks depends on a different combination of such constants. These comparisons, summarized in section 5.2, constrain specific
combinations of fundamental constants as a function of gravitational potential (see Eq. (4)) at
the 10−5 −10−6 level.
Advances in direct comparison of single-ion optical or neutral optical lattice clocks through
frequency-comb metrology promises significant improvements in sensitivity by up to three
orders of magnitude, as well as simplification in interpretation of the results since such comparisons directly probe α-variation, independent of other fundamental constants (see, for
example Refs. [24,25]). In this article, we present the first results obtained with an alternative
and independent method of competitive sensitivity, which utilizes rf electric-dipole transitions
between nearly degenerate electronic levels in atomic Dy. In this system, the difference of the
optical frequencies are directly measured, eliminating the need for frequency combs or transfer
cavities. Similar to optical clock comparisons, the results are independent of other fundamental
constants.
2 Effect of α-variation in dysprosium
Tests for variation of α in atomic systems rely upon the fact that relativistic corrections, which
scale as (Zα)2 , where Z is the atomic number, have different magnitudes for different energy
levels. The total energy of a level can be written as
E = hν = E0 + q(α2 /α02 − 1),
1
(1)
The controversy concerning the validity of the analysis in Ref. [5] still continues [6, 7].
Interestingly, this result is also in disagreement with the recent work on inversion spectrum of
ammonia presented in Ref. [19].
2
Atomic Clocks and Fundamental Constants
73
12
Energy (GHz )
10
F = 12.5
F = 12.5
11.5
8
11.5
6
10.5
4
9.5
8.5
2
10.5
7.5
9.5
8.5
7.5
0
A
B
Fig. 1. Relevant levels and transitions in atomic dysprosium. Population and detection scheme: Level
B is populated in a three-step process. The atoms are excited to level f in the first two steps using
833- and 699-nm laser light (solid arrows). The third step is a spontaneous decay (labeled by a shortdashed arrow) from level f to level B with ∼30% branching ratio. The population is transferred to
level A by the rf electric field (wavy line). Atoms in level A decay to level c and then to the ground
state. The fluorescence from the second decay (564 nm) is used for detection. Inset on the right:
Hyperfine structure of levels A and B for 163 Dy. Zero energy is arbitrarily chosen to coincide with
the lowest hyperfine component.
where E0 is the present-day energy, α0 is the present-day value of the fine-structure constant,
and q is a coefficient which determines the sensitivity to variation of α. This coefficient mainly
depends upon the electronic configuration of the level. A calculation [26], utilizing relativistic
Hartree-Fock and configuration interaction methods, found values of q for two nearly degenerate opposite-parity levels in atomic dysprosium (Fig. 1, levels A and B) that are large and
of opposite sign. For the even-parity level (level A), qA /hc 6 × 103 cm−1 , while for the
odd-parity level (level B), qB /hc −24 × 103 cm−1 .
A unique aspect of the Dy system is that the directly observable quantity is the difference
νB − νA , due to the fact that an electric-dipole transition can be induced between the two levels
using a rf electric field. Since, according to Eq. (1), the change in ν is related to a change in
α by (α ≈ α0 )
2 q δα
,
(2)
δν =
h α
the time variation of the transition frequency between levels A and B is related to temporal
variation of α by
˙ = ν̇B − ν̇A = 2 qB − qA α̇ ,
(3)
∆ν
h
α
˙ where |2(qB − qA )/h| 1.8 × 1015 Hz [26]. For instance, |α̇/α| = 10−15 /yr implies |∆ν|
2 Hz/yr.
Furthermore, if the variation of α in a changing gravitational potential is parameterized as
∆U
δα
= kα 2 ,
α
c
(4)
where ∆U/c2 ∼ 10−10 is the variation in the Sun’s gravitational potential at Earth, the corresponding change in the transition frequency is given by
δ(∆ν) = 2
qB − qA ∆U
kα .
h
c2
(5)
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Numerically, a 2-Hz variation in the rf-transition frequency synchronous with Earth’s orbital
motion implies |kα | ∼ 10−5 .
There are several advantages to using the nearly degenerate levels in Dy. Because the transition frequency between levels A and B is ∼1 GHz or smaller depending on the isotope shifts and
hyperfine splitting for the odd isotopes, direct frequency-counting techniques can be used. This
allows comparison of two electronic transitions without the need for optical frequency combs or
transfer cavities. A preliminary analysis of statistical and systematic uncertainties shows that
the measurement of the transition frequency and the control of the systematics at a mHz level
is feasible, which corresponds to an ultimate sensitivity of |α̇/α| ∼ 10−18 /yr or kα ∼ 10−8 for
two measurements separated by a year’s time [27]. A mHz resolution on a transition frequency
of 1 GHz requires a relatively modest fractional stability of 10−12 for the reference frequency
standard. This also means that the results are insensitive to variation of the Cs reference frequency due to changes in the values of fundamental constants since the experimental upper
limit on this relative variation rate is ∼10−15 yr−1 (see, for example, Ref. [12]).
3 Experimental technique
As shown in Fig. 1, the population transfer from the ground state to the long-lived (τB >
200 µs [28]) odd-parity level B requires three transitions. The first two transitions are induced
via 833- and 669-nm laser light, while the last transition is a spontaneous decay with a ∼30%
branching ratio [30]. Atoms are transferred to the even-parity level A (τA = 7.9 µs [28]) with
a frequency-modulated rf electric field referenced to a commercial Cs frequency standard. The
transition frequency is determined via a lock-in technique by monitoring the 564-nm fluorescence.
3.1 RF transitions
The energy difference between the nearly degenerate levels are on the order of hyperfine splittings and isotope shifts. Since Dy has seven stable isotopes, there are many rf transitions
corresponding to the same electronic transition, including, in particular, transitions with
energy differences of opposite sign. As shown in Fig. 2, since the relativistic corrections to the
energy only depend upon electronic levels3 , the variations of these rf-transition frequencies
should have equal magnitudes but opposite signs. This correlation can be used to detect
and eliminate certain systematics. In our first experiments [16,29], we monitored two such
transitions: the 3.1 MHz (F = 10.5 → F = 10.5) transition in 163 Dy and the 235 MHz
(J = 10 → J = 10) transition in 162 Dy.
3.2 The first-generation apparatus
A detailed description of the atomic-beam source is given in Ref. [30]. An oven, operating at
∼1500 K, heats the Dy atoms which effuse through a multislit nozzle-array. The oven consists of
a molybdenum tube containing dysprosium metal, surrounded by resistive heaters made from
tantalum wire inside alumina ceramic tubes. The radiation shielding is provided by five layers
of tantalum sheets, which surround the oven and the heaters. In addition to the multislit nozzle,
two external collimators are used to collimate the atomic beam and the oven light. The latter is
necessary to minimize the background due to scattered oven light. The resulting atomic beam
has a mean velocity of ∼5 × 104 cm/s with a full-angle divergence of ∼0.2 rad (1/e2 level) in
both transverse directions. The measured atomic density in the population and detection region
is 5 × 109 atoms/cm3 [30].
3
There are additional corrections due to hyperfine interactions and isotope shifts that depend on α,
as well as other fundamental constants, such as me /mp , but these corrections are smaller by several
orders of magnitude.
Atomic Clocks and Fundamental Constants
A
Energy
75
B
Fig. 2. Energy-level diagram for levels A and B for two different isotopes of Dy with energy differences
of opposite sign. Solid lines indicate energies for the present value of α, while dashed lines indicate shifts
in the energy levels due to a possible change in α (not to scale). Notice that shifts for the entire A
manifold is identical since these levels correspond to the same electronic states in different isotopes.
The same is also true for the B manifold. As a result δν1 = ν1 − ν1 = −(ν2 − ν2 ) = − δν2 . The
same result holds also for different hyperfine-structure components of the same nonzero-spin isotope
(161 Dy or 163 Dy).
t
ligh
nm
3
3
8
OV
EN
a
d
b
e
c
f
d
t
ligh
nm
669
T
PM
g
i
h
Fig. 3. Experimental setup (not shown to scale): a) Atomic beam produced by effusive oven source
at ∼1500 K; b) atomic-beam collimator; c) oven light collimator; d) cylindrical lenses to diverge laser
beams; e) interaction region of atoms with the electric field enclosed in a magnetic shield (not shown);
f) cylindrical mirror to collect fluorescent light; g) lucite light pipe; h) interference filter; and i) shortpass filter.
The atoms enter the interaction region (Fig. 3), where they are first transferred to level B
by two laser-induced transitions and a spontaneous decay. Approximately 600 mW of 833 nm
light is produced by a Ti:Sapphire ring laser and ∼300 mW of 669 nm light is produced by a
ring dye laser with DCM dye. After passing through several beam-splitters, which split portions
of the light necessary to monitor both the absolute frequency and the frequency stability of the
laser outputs, ∼200 mW of 833 nm light and ∼100 mW of 669 nm light enter the interaction
region.
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Fig. 4. Schematic diagram of the rf-generation, control, and pressure-measurement systems. The
rf synthesizer is referenced to a Cs standard, which is compared to a CDMA (Code Division Multiple
Access) primary reference source used in commercial cellular networks. The output of the synthesizer
is amplified and then sent to the electric-field plates and a 50-Ω power-resistor termination in series
with the larger impedance of the plates. The output of the photomultiplier tube is sent to the lock-in
amplifier for detection and acquisition.
Since the population transfer of a weakly collimated atomic beam due to narrow-band cw
lasers is inefficient, an adiabatic-passage technique is utilized to transfer atoms to level B.
A detailed description of this technique and references to earlier work are given in Ref. [31].
Briefly, cylindrical lenses are used to diverge the two laser beams in the interaction region so
that the divergences of the light beams match the atomic-beam divergence. Due to the Doppler
effect, the atoms “see” a frequency chirp in laser detuning which adiabatically transfers the
population to the excited state.
The rf electric field is formed between two parallel, 10 cm by 5 cm, wire grids made of
0.002-in Be-Cu wire. The separation between adjacent wires is 2.5 mm which minimizes surface
area and hence surface-charge accumulation. The separation between the grids is ∼ 2.5 cm.
The entire assembly is enclosed inside a high-permeability magnetic shield which reduces the
magnetic field in the interaction region to ∼1 mG. The 564-nm fluorescence is collected by a
lucite light pipe and detected with a photomultiplier tube (PMT) with bandpass-interference
and short-pass filters at its input window.
The rf-generation and detection system is shown in Fig. 4. The frequency-modulated rf field
is generated by a synthesizer (Hewlett Packard 8647A) and sent through a 45-dB amplifier (ENI
604L). The rf power at the output is ∼0.5 W. The synthesizer is referenced to a Cs frequency
standard (Hewlett Packard 5061A) with a long term stability (> 10 minutes) of 10−12 . This
frequency standard is compared (via a frequency counter) to a second clock (Symmetricom
TS2700) to monitor its stability4 . A Signal Recovery 7265 lock-in amplifier provides a 10-kHz
modulation signal to the rf synthesizer and demodulates the signal from the PMT. The signal
from the lock-in amplifier, as well as the clock, pressure, and temperature data, are sent to a
computer for data acquisition.
3.3 Procedure
To minimize fluctuations such as those due to laser-power drifts or density fluctuations of the
atomic beam, the rf voltage is frequency modulated at 10 kHz with a peak modulation depth of
4
The long-term stability of this clock was also determined to be 10−12 through comparisons with a
Loran-C frequency standard.
Atomic Clocks and Fundamental Constants
77
Fig. 5. First- and second-harmonic signals for 10-kHz frequency modulation. Solid line is a fit to the
function derived from first-order time-dependent perturbation theory.
10 kHz (modulation index of 1). Since the rf-transition linewidth is ∼20 kHz, this modulation
provides a fast sweep across the absorption lineshape, minimizing lineshape distortions due to
fluctuations during a lineshape scan. The outputs from the lock-in amplifier at the first and
second harmonic of the modulation frequency are shown in Fig. 5. The first-harmonic signal is
an odd function of detuning while the second-harmonic signal is an even function. For small
modulation amplitudes, the output signal at the harmonics of the modulation frequency are
proportional to the corresponding derivatives of the absorption line shape [32]. In our system,
since the modulation amplitude is comparable to the rf-transition linewidth, the harmonic
signals contain higher-derivative contributions. Therefore, a fitting function was derived from
first-order time-dependent perturbation theory. This model does not include any effects due to
power broadening or beam velocity distribution and is used to confirm qualitative behavior of
the line shape.
The transition frequencies are obtained from the ratio of the first- and second-harmonic
signals. The ratio, which is computed by the digital lock-in amplifier, is used in order to
reduce the effect of fluctuations due to laser-power drifts or atomic-beam density changes since
the signals at both harmonics have the same dependence on these parameters. The transition frequency is extracted from the ratio by a two-step process. First, the amplitudes at the
first and second harmonics and their ratio, as derived from the lock-in amplifier, are recorded
simultaneously as the carrier frequency is scanned 4 kHz around the resonance. For such a small
scan compared to the linewidth, the first harmonic is linear while the second harmonic is nearly
a constant (Fig. 5). By fitting a linear function to the ratio, a conversion parameter from the
ratio to frequency is obtained. Once the calibration is established, the rf carrier frequency is
locked at a fixed value near resonance (which is 3,074,000 Hz and 234,661,000 Hz for each of the
two transitions), and the ratio is measured repeatedly. The average values of ratios from these
measurements are converted to frequency using the calibration.
Using this √technique, statistical sensitivity achieved for the 235-MHz transition is
≈0.65 τ −1/2 Hz s, in good agreement with what one would expect for a counting rate of 109 s−1
and a 20 kHz linewidth [27]. Since the isotopic abundances of 162 Dy and 163 Dy are nearly identical, the statistical sensitivity of the 3.1-MHz transition is determined
by the dilution of the
√
atomic population due to the hyperfine splittings and is ∼ 6 times worse than that of the
235-MHz transition.
4 Systematic effects
There are several imperfections that affect the stability of our transition-frequency measurements. A detailed analysis of systematic effects and a discussion of feasible levels of stability
are given in Ref. [27]. Estimated and experimentally measured values for various systematic
effects for the data presented in section 5 are listed in the second column of Table 1. All
systematic effects, excluding the first three, lead to estimated shifts of less than 1 Hz. Many of
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Table 1. A summary of measured and estimated size of systematic effects for the first-generation
experiment and the new apparatus. The first three effects on column 2 are experimental measurements, while the rest are estimates. Systematic effects for the new apparatus are estimated and under
investigation. For detailed discussion and calculations see Ref. [27].
Systematic shifts
First-generation apparatus (Hz)
New apparatus (Hz)
Stray B-field
Collisional effects
Detuning effect
ac Starka
Doppler
Room temp. black-body radiation
Oven temp. black-body radiation
dc Starka
Millman effect
Quadrupole moment
2–5
1–2
< 0.4
(10−3 −1)
< 0.2
0.1
0.02
∼ (10−4 −10−2 )
5 × 10−4
10−5
1 × 10−3
(1−10) × 10−4
< 0.04
∼ (0.02−5)
< 0.02
1 × 10−3
5 × 10−3
∼ (10−4−10−2 )
2 × 10−3
10−4
a
transition dependent.
these systematics, such as ac Stark shifts and oven black-body radiation shifts, were studied
experimentally by exaggerating certain imperfections and were found to be negligible at the
present level of sensitivity.
In the remainder of this section, we concentrate on the dominant systematic effects that
limit the sensitivity of the experiment to α variation at the 10−15 yr−1 level.
4.1 Magnetic field and polarization imperfections
The largest systematic effect is due to a combination of the residual magnetic field with the
laser-light polarization imperfections. For a zero-spin isotope, individual magnetic sublevels in
the presence of a residual magnetic field shift by an amount gJ µ0 mJ B, where µ0 is the Bohr
magneton, gJ is the Landé g factor, and mJ is the quantum number corresponding to the
projection of total angular momentum onto the quantization axis. The corresponding shift in
the transition frequency for a ∆m = 0 transition (induced by an rf field linearly polarized
along the quantization axis) can be as large as (gJA − gJB )mJ µ0 B ∼ 2 kHz for gJA = 1.21,
gJB = 1.367 [33], mJ = 10, and a residual field of 1 mG. However, if all sublevels are populated
equally, the net effect is a symmetric broadening of the rf lineshape, which is not accompanied
by an overall frequency shift.
In the experimental geometry of the apparatus (Fig. 3), an imbalance in the sublevel population may be caused by residual circular polarization of the laser light coupled with a misalignment in the propagation direction of the laser beams. This imbalance in the sublevels leads to
asymmetric broadening of the line in the presence of stray magnetic fields and causes apparent
shifts [27]. In systematic studies where we deliberately used circularly polarized light to create
a population imbalance, shifts in the transition frequencies as large as ±160 Hz were observed.
During normal operation, however, the polarization of both laser beams are set to linear
to minimize this effect. The linear polarizations of the laser beams are determined by calcite
polarizers with a 10−5 extinction ratio outside the vacuum chamber. A residual systematic
effect arises from stress-induced birefringence on the optical windows and lenses, utilized for
adiabatic passage population technique (see section 3.2), that the laser beams travel through
after the polarizers before reaching the interaction region. The amount of induced residual
ellipticity depends upon where the laser beams sample these optical elements. As a result,
the stability of the transition frequencies is dependent on the repeatability of our laser-beam
alignment procedures. The residual systematic uncertainties due to this effect range from 2 to
5 Hz on different experimental runs with the smaller uncertainty corresponding to later runs in
which the lens mounts were modified to relieve the stress. The apparent (anti)correlation seen
Atomic Clocks and Fundamental Constants
79
in Fig. 6 between the transition frequencies for the two isotopes is due to the stray magnetic
field effect. Since the difference in the Landé g factors of level A and B for the two isotopes are
within 5 percent of each other (due to the hyperfine splitting, gF values are different than gJ
values, see Ref. [27]), and because the energy separation for the two isotopes are of opposite
sign (see Fig. 2), the two transition frequencies shift in opposite direction with nearly identical
magnitudes.
In addition to the residual magnetic field, the magnetic shielding was inadvertently magnetized in the runs of April 2006 during studies utilizing external magnetic-field coils. Since the
amount of magnetization and the resultant shifts in the transition frequencies were unknown
at the time, systematic uncertainties have been assigned by driving the magnetization to saturation and noting the maximum shift induced for both field polarities. The uncertainty due to
this effect is 7.5 Hz. To guard against such systematic effects the shields were demagnetized at
the beginning of each subsequent run.
4.2 Collisional shifts
Another important systematic uncertainty is due to the collisions of Dy atoms with the background gas in the vacuum chamber. In the low-pressure regime, the shift in the transition
frequency is given by ∆ν = (1/2π)nσv, where σ is the collision cross-section, v is the velocity,
and n is the density of the perturbing gas, which is directly proportional to the pressure. Linear
collisional shift rates for the rf transitions due to various gases were determined in Ref. [34] and
were found to be on the order of Hz/µTorr level, consistent with our estimates in Ref. [27].
The pressure in the vacuum chamber is ∼5 × 10−7 Torr when the Dy oven is off. When the
oven is turned on, the pressure rises to ∼2 × 10−6 Torr, limited by H2 outgassing from the oven.
In Ref. [34], the observed shift rates for H2 were found to be consistent with zero. However,
the pressure of other gases, such as N2 , also increases at the level of ∼4 × 10−7 Torr when the
oven is turned on, and continually decreases and stabilizes after ∼6 hours of continuous oven
operation. The partial pressures are monitored with a calibrated residual gas analyzer, and the
transition frequencies are corrected for the presence of N2 , O2 , H2 , and Ar using the shift rates
measured for each gas. The variation in the pressure of other gases such as H2 O and CO2 are
∼10−7 Torr or smaller, and no significant correlation between these pressure variations and the
transition frequency is observed. The total correction is <2 Hz at any time during each run.
In addition to collisional shifts with background gas, effects of self-collisions within the
atomic beam were investigated. The beam density can be controlled by varying the oven temperature which changes the Dy saturated-vapor pressure. In our studies, presented in Ref. [34],
the oven temperature was varied from 1077◦ C to 1170◦ C, while we were monitoring both transition frequencies. The measured shift rates were −5(9) × 10−3 Hz/◦ C and −3(3) × 10−2 Hz/◦ C
for the 235- and 3.1-MHz transitions, respectively. The results presented in section 6 include
uncertainties of <0.6 Hz due to this effect.
4.3 Detuning effect
There are also systematic effects associated with rf electric-field inhomogeneities. The rf electrodes are fed by twisted-pair wires at one corner. Numerical simulations show that with this
feed, at rf wavelengths comparable to the size of the electrodes (∼10 cm), spatially varying
amplitude and phase inhomogeneities become substantial. These spatial phase inhomogeneities
in the lab frame are converted into time-varying phase inhomogeneities in the atoms’ reference
frame and can lead to drifts in the transition frequency when they are combined with changes
in the atomic-beam velocity distribution.
This effect was studied experimentally by making changes to the transverse velocity distribution. Deliberate detuning of the laser frequencies from the optical resonances leads to changes
in the transverse velocity distribution of the excited atoms because the adiabatic passage technique used in the population is not 100 percent efficient. Since the transverse velocity group
resonant with the laser frequency is populated more efficiently than other velocity groups that
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Fig. 6. Measured transition frequencies for (a) the 235-MHz and (b) 3.1-MHz transitions over an
eight-month period, as well as (c) the sum- and (d) difference-frequency plots. The data have been
corrected for collisional shifts. The solid lines are the least-squares linear fits to the data. The dashed
lines are the least-squares fits to a constant function. The apparent correlation between the transition
frequencies for the two isotopes are due to the stray magnetic field effect (see section 4.1).
rely on adiabatic passage, detuning of the laser frequencies from resonance moves the peak of
the population from the center velocity group to the groups traveling in different directions.
This leads to sampling of a different region in the rf electrodes with different phase gradients and thus different shifts. While this effect is large (∼200 Hz for half-width detuning) for
high-frequency transitions of ∼1 GHz, for the two transitions considered in this study, shifts of
<0.5 Hz for 1 MHz detuning in the laser frequencies were observed. To keep this effect at the
1-Hz level during the runs, the laser frequencies were periodically retuned (∼every 2 minutes)
to the center of their respective transitions.
To check for slow variations in the beam velocity distribution due to other changes such as
collimator clogging, this study was repeated several times over a two-year period, during which
the oven was reloaded and collimators cleaned several times. No significant change in the size
or dependence of the effect on laser detuning was observed.
5 Results and analysis
5.1 Temporal variation
We have measured the two rf transition frequencies in the course of seven runs over eight
months. Figure 6 shows the results of these measurements corrected for collisional shifts. Error
bars are due both to statistical and aforementioned systematic uncertainties which have been
added in quadrature. Least-squares linear fit to the data points gives slopes of −0.6 ± 6.5 Hz/yr
and 9.0±6.7 Hz/yr for the 235-MHz and 3.1-MHz transitions, respectively. According to Eq. (3),
these results correspond to α̇/α = (−0.3 ± 3.6) × 10−15 yr−1 for the 235-MHz transition and
(−5.0 ± 3.7) × 10−15 yr−1 for the 3.1-MHz transition.
As mentioned earlier, because the energy differences for these two transitions are of opposite
sign, these two data sets can be combined to construct sum- and difference-frequency plots.
Atomic Clocks and Fundamental Constants
81
For the 235-MHz transition, the energy of level A is higher than level B, and the change in the
transition frequency is defined as
˙ 235 = ν˙A − ν˙B = 2 qA − qB α̇ .
∆ν
h
α
(6)
For the 3.1-MHz transition, the opposite is true and the variation is defined as
˙ 3.1 = ν˙B − ν˙A = 2 qB − qA α̇ .
∆ν
h
α
(7)
We note that qA and qB only depend on relativistic effects and are essentially identical for
various rf transitions in different isotopes. Given Eq. (6) and Eq. (7), it is clear that the sumfrequency slope is
qB − qA α̇
qA − qB
˙
˙
+2
= 0.
(8)
Σ̇ = ∆ν 235 + ∆ν 3.1 = 2
h
h
α
Any nonzero slope in measured data is an indication of systematic effects that shift different
isotopes and hyperfine components by different amounts. As a result, the sum frequency is
used as a check to make sure systematic uncertainties are estimated properly. Similarly, the
difference frequency slope is given by
qB − qA α̇
qA − qB α̇
qA − qB
˙
˙
˙
−2
=4
.
(9)
∆ = ∆ν 235 − ∆ν 3.1 = 2
h
h
α
h
α
It is twice as sensitive to α variation as the individual frequencies and eliminates any commonmode systematic effects that would shift the transition frequencies in the same direction and
possibly mask the effects of α variation.
As shown in Fig. 6, the least-squares fit to the sum frequency gives a slope of 8.1±9.4 Hz/yr,
which is consistent with zero. The fit to the difference frequency gives the final result of
α̇/α = (−2.7 ± 2.6) × 10−15 yr−1 ,
(10)
consistent (1σ) with no variation at the present level of sensitivity [16].
5.2 Gravitational-potential dependence
The gravitational potential at the Earth due to the Sun is given by
U=
−GMS
,
r
(11)
where G is the gravitational constant and MS is the mass of the Sun5 . For an elliptical orbit,
the distance r between the Sun and Earth is
1 − 2
,
(12)
1 + cos φ
where a is the semi-major axis of the Earth’s orbit, = 1 − b2 /a2 ≈ 0.0167 is the eccentricity,
b is the semi-minor axis, and φ is the true anomaly (see Fig. 7). Substituting Eq. (12) into
Eq. (11), the gravitational potential becomes
r=a
U=
GMS
−GMS
−
cos φ + O(2 ).
a
a
(13)
5
The variation of the gravitational potential due to Jupiter and the other planets was ignored since
they are at least two orders of magnitude smaller than that due to the Sun.
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The European Physical Journal Special Topics
Fig. 7. (a) The true anomaly, φ, is the angle subtended from perihelion, the point of closest approach.
The eccentric anomaly, ε, is the angle between perihelion and the position of the Earth in its orbit
projected onto the auxiliary circle of the ellipse (the eccentricity of the ellipse is exaggerated for clarity).
(b) The change in gravitational potential of the Sun at the Earth due to the eccentricity of the orbit.
The points indicate the dates on which data were taken.
The first term in Eq. (13) is the constant part of the gravitational potential at Earth, while
the second term is a change in gravitational potential, ∆U , which arises due to the eccentricity
of the Earth’s orbit. This term is dependent on time through the time dependence of φ as
discussed in Ref. [29].
The data were taken between October 2005 and June 2006; the true anomaly is zero at the
2005 perihelion, January 2 [35]. The calculated values for cos φ are substituted into ∆U to find
the gravitational potential for each data point. Then, the measured frequencies for each isotope
are fitted to the gravitational potential using a two-parameter least-squares fit given by
δ(∆ν) − ν ∗ = x0
∆U (t)
+ x1 ,
c2
(14)
where ν ∗ is an arbitrary reference frequency, and x0 and x1 are the fit parameters. The parameter x1 accounts for the offset due to choice of the reference frequency while the parameter
x0 determines the correlation between the change in gravitational potential and the transition
frequency. The data plots and the fits of the measured frequencies for each isotope as well as
the sum and difference frequencies are shown in Fig. 8.
The parameter x0 obtained by the least-squares fit is (−0.2 ± 1.7) × 1010 Hz for the
235-MHz transition, (2.9 ± 1.7) × 1010 Hz for the 3.1-MHz transition, (2.7 ± 2.4) × 1010 Hz
for the sum frequency, and (−3.1 ± 2.4) × 1010 Hz for the difference frequency. The sum frequency is consistent with zero at a 1.2σ level, indicating the uncertainties due to systematic
effects are adequately addressed.
Gravitational-potential dependence of α can be deduced from the parameter x0 using Eq. (5)
and yields kα = (−1.1 ± 9.2) × 10−6 for the 235-MHz transition, kα = (−16.3 ± 9.4) × 10−6 for
the 3.1-MHz transition. The final result of
kα = (−8.7 ± 6.6) × 10−6
(15)
is obtained from the difference frequency, which is consistent with zero at 1.3σ level, indicating
no significant variation at the present level of sensitivity [29].
Experimental limits on the gravitational-potential dependence of fundamental constants
are summarized in Table 2. The results in the first three rows are extracted in Ref. [36] using
least-squares fits presented in the experimental papers similar to the one used in this paper.
The results in the first row of Table 2 depend on the variation of α as well as the variation of
me /mp , characterized by ke . The results in the second and third rows in Table 2 depend on the
variation of α and the variation of mq /mp , characterized by kq . The last three rows are separate
limits on kα , ke , and kq from this work. Since our experimental system is independent of other
Atomic Clocks and Fundamental Constants
83
Fig. 8. Same data as in Fig. 6. Solid lines are least-squares fit to a function given by Eq. (14) where
∆U is the oscillatory component of the gravitational potential of the Sun at Earth with a period
of a year.
Table 2. A summary of results for changing fundamental constants in a varying gravitational potential.
The first three results are extracted in Ref. [36]. Note: Edited in proof: since the submission of the
manuscript, a sign error in the result of Ref. [15] has been discovered, and additional results have been
published [40]. The results in the table and the text have been updated in light of this information.
Parameter
kα + 0.17ke
|kα + 0.13kq |
kα + 0.13kq
kα
ke
kq
Constraint
(3.5 ± 6) × 10−7
<2.5 × 10−5
(1 ± 17) × 10−7
(−8.7 ± 6.6) × 10−6
(2.9 ± 4.4) × 10−5
(6.8 ± 5.2) × 10−5
Experimental Ref.
[15]
[22]
[23]
this work ([29])
this work ([29])
this work ([29])
fundamental constants, our deduced value of kα can be combined with the results for kα + 0.17ke
and kα + 0.13kq in the first three lines of Table 2 to extract a value of (2.9 ± 4.4) × 10−5 for
ke and (6.8 ± 5.2) × 10−5 for kq . In addition, our results may possibly constrain some specific
theories, such as those in Refs. [37–39], which use a parametrization that is different than the
one in terms of ki presented here.
6 New apparatus
In the past year, we have been designing and constructing a new apparatus, which will reduce
the influence of the systematic effects due to the background pressure, residual magnetic fields,
inhomogeneities in the rf electric field, as well as other systematic effects expected to be important below the 1-Hz per year level. A schematic drawing of the new experimental apparatus is
shown in Fig. 9.
The new rf apparatus is composed of three differentially pumped vacuum chambers pumped
by various turbo-molecular and oil-diffusion vacuum pumps. The first chamber houses the
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The European Physical Journal Special Topics
c
i
j
b
a
k
h
m
100 c
d
g
f
e
Fig. 9. Section view of the cryogenic ultra-high vacuum Dy atomic-beam rf spectrometer (vacuum
pumps not shown). The spectrometer incorporates three differentially pumped vacuum chambers:
a) first vacuum chamber housing the Dy oven (∼10−6 Torr); b) gate valve used as a second intermediate
chamber (∼10−8 Torr); c) third chamber housing the laser and rf interaction regions (∼10−10 Torr);
d) Dy oven; e) atomic beam/photon collimators; f) laser access port and in-vacuum polarizer;
g) rf electrodes; h) optical light collection mirrors; i) light pipe; j) magnetic field coils; k) two-layer
magnetic shield.
Dy oven and has a residual pressure on the order of 10−6 Torr. This residual pressure is limited
by the outgassing of the effusive oven. The second chamber, a modified gate valve at ∼10−8 Torr,
isolates the third chamber from the first. The third chamber, which houses the laser- and
rf-interaction regions, is kept under an ultra-high vacuum of ∼10−10 Torr, which is necessary
to reduce the effects of collisional shifts to 1 mHz.
The laser- and rf-interaction regions are enclosed inside a two-layer magnetic shield in order
to reduce stray magnetic field to ∼100µG. In addition, the rf electrodes are surrounded by
3 orthogonal pairs of magnetic coils, which will be used to compensate for the remaining stray
field. The stray fields orthogonal to the rf electric-field polarization are of secondary importance
and may couple through gradients. For the longitudinal stray field, the Dy atoms themselves
can be used as a magnetometer. The atoms can be made sensitive to linear Zeeman shifts by
inducing atomic orientation through use of circularly polarized laser light. The stray field can
be eliminated by noting the difference in frequency shifts for opposite-helicity polarizations
and reducing this difference to zero with the application of a compensation field. Once this
procedure is over, linear polarization can be used for measurements which should suppress
sensitivity to magnetic fields by several orders of magnitude with good polarization control.
Laser-light polarization has been improved by in-vacuum, high quality, dichroic glass polarizers
(colorPol VISIR and colorPol VIS700) as the last optical elements to avoid stress-induced
birefringence that occurs on optical vacuum ports and adiabatic passage lenses. In addition,
fiber optic launch stages for the laser beams are used in order to stabilize beam pointing through
the ports and the lenses.
We have also redesigned the new rf electrodes using finite-element numerical analysis software. Since we would like to work with several transitions with frequencies ranging from 3 MHz
to 1 GHz simultaneously, cylindrical cavities used in microwave clocks that utilize uniform
transverse electric and magnetic (TE and TM) modes can not be used due to the fact that
the frequency range spans three orders of magnitude. Instead, similar to the parallel plate
Atomic Clocks and Fundamental Constants
85
electrodes of the original apparatus, the new rf electrodes utilize a standing-wave TEM mode
of a terminated rectangular coaxial transmission line, which has no cutoff frequency.
The new electrode design is shown in Fig. 10. It consists of a 9 cm by 7 cm wire-grid suspended in the middle of 10 cm by 8 cm by 4.2 cm grounded wire-grid box made of gold-coated
0.002-in Be-Cu wire. On either side, the separation between the middle electrode and the sides
of the grounded box parallel to the plane of the middle electrode is 2 cm. Since most of the
phase inhomogeneities of the first-generation electrodes are due to the exponentially decaying
higher-order TE and TM modes near the rf feed, which must be present to match boundary
conditions, the dimensions of the transmission line near the rf feed are smaller than the rest of
the electrode in order to keep the phase gradients away from the interaction region. This also
improves the rf coupling into the electrodes. An impedance-matched conical “transition region”
links the feed location to the rf-atom interaction region. Moreover, the electrode geometry has
been modified to make sure that the atomic beam propagation direction is perpendicular to
any residual traveling rf wave components in order to suppress residual Doppler shifts. These
two directions are parallel for the first-generation apparatus. Finally, the new electrodes contain two opposing feed points which allow the introduction of and compensation for various
imperfections such as phase gradients or traveling waves by modifying the amplitude, phase,
or termination at the two inputs.
In addition to these effects, the largest systematic effect in Table 1 is the ac Stark shifts
that vary due to the rf-field amplitude (E) fluctuations. The second-order ac Stark shift for a
two-level system is given by
1
d2 E 2
1
Re
+
,
(16)
δνac =
2
∆ − ν + iγ
∆ + ν + iγ
where ν is the applied rf frequency, d is the dipole matrix element, and ∆ is the frequency
separation in the absence of rf electric fields. Because the first term is an odd function of
detuning, only the second so-called Bloch-Siegert term contributes to an actual shift of the
central frequency of the resonance line shape. Near saturation, d2 E 2 /(γγT ) ∼ 1, where 1/γT is
the relaxation rate due to transit time through the electrodes. For ν = ∆ γ and a transition
near saturation, the corresponding shift is δνac ∼ γγT /(4ν). Hence for a mean velocity of
5 × 104 cm/s, a travel distance of 2 cm, and transition frequencies ∼3− 1000 MHz, the shift
varies from ∼0.02 to 5 Hz. For the 3.1-MHz transition, for which the shift is largest, in order to
achieve a sensitivity to frequency shifts of a few mHz, the amplitude stability must be ∼10−3 .
However, this requirement becomes much less stringent for higher-frequency transitions. For
ν ∼ 1 GHz, only a modest control at a level of 10 percent is required. As discussed in Ref. [27],
taking into account shifts due to other levels does not lead to more stringent requirements on the
amplitude stability of the rf electric field beyond those obtained in the two-level approximation.
Variations in the Dy oven and vacuum chamber temperatures can also induce frequency
shifts due to the ac Stark effect [27]:
δνBB ∼
d2 E 2 ,
4(∆ − νBB )
(17)
where ∆ is a characteristic atomic energy scale, νBB is a characteristic frequency of the
black-body radiation, d = 1 ea0 is a typical optical transition dipole moment, and E 2 =
(8.3 V/cm)2 [T (K)/300 K]4 is the rms value of the black-body radiation electric field. In order to
control the fluctuations in the ambient temperature, the rf interaction region will be thermally
stabilized at near boiling point of liquid nitrogen. To achieve this, the light collection mirrors
surrounding the interaction region will be cooled down to liquid-nitrogen temperature by a
pair of cold fingers. Numerical simulations show that the final temperature of the interaction
chamber should be around 100 K. For ∆ ∼ 1014 Hz, T = 100 K should be sufficient to reduce
the shifts to ∼1 mHz.
The interaction region is also illuminated by much hotter black-body radiation from the
atomic oven. However, its effect is much smaller due to the decreased solid angle. The distance
between the oven and the interaction region for the new apparatus reduces this effect to an
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The European Physical Journal Special Topics
Fig. 10. Section view of the new rf electrodes. The wire grid on the grounded box surrounding the
middle electrode is not shown for clarity. Atoms travel through the electrodes from left to right and
transition from level B to level A between the first surface of the box and the middle electrode. The
black rectangle marks half the approximate size of the interaction region. In this geometry, residual
rf traveling waves propagate along the wires, perpendicular to the nominal atomic-beam propagation
direction.
estimated size of ∼5 mHz for a 1500 K black body. Thus, the oven temperature must be stabilized
to ∼140◦ C, which is feasible. To guard against changes in the surface emissivity, we plan to
monitor the temperature of the oven with both a thermocouple and an infrared thermometer.
All other systematic effects presented in column 3 of Table 1 are expected to be at or
below the ∼ 1 mHz level. Certain systematic effects, such as Millman effect and quadrupole
moment shifts, are expected to be larger in the new apparatus due to compromises that had
to be made to balance numerous, and often conflicting, requirements. However, these effects
should be unimportant at the desired level of sensitivity. When possible, our philosophy in
dealing with all of the systematic effects will be to amplify the systematic by a few orders of
magnitude by introducing a specific imperfection, study the systematic effect as a function of
secondary imperfections and take data at a stable point on this curve with the main imperfection
compensated or at least restored to its original level.
7 Conclusion
We have presented the first result of a direct measurement of α-variation both in time and
a changing gravitational potential using atomic dysprosium. These results are of comparable
uncertainty to the best published laboratory results that are independent of other fundamental
Atomic Clocks and Fundamental Constants
87
constants. However, in our case the interpretation does not require comparison with different measurements to eliminate dependence on other constants, and provides an alternative to
measurements that utilize state-of-the-art atomic optical frequency clocks. Using our limit for
the gravitational variation of α and earlier work sensitive to combinations of fundamental constants, we have also extracted the first limits on the possible gravitational variation of me /mp
and mq /mp .
The present uncertainty is dominated by systematic effects primarily due to polarization
imperfections coupled to the residual magnetic field, collisional shifts, and rf electric-field inhomogeneities. A significant improvement is expected from the new apparatus that has recently
become operational, which provides better control over these, as well as other, systematic
effects expected to be important to achieve better than 1-Hz sensitivity. Ultimately, mHz-level
sensitivity may be achievable with this method [27].
We would like to thank V.V. Flambaum for intellectual inspiration and collaboration on the
analysis of the gravitational potential dependence of α. We are grateful to D.F. Jackson Kimball,
and D. English for valuable discussions. This work was supported in part by the University of
California – Los Alamos National Laboratory CLC program, NIST Precision Measurement Grant,
Los Alamos National Laboratory LDRD, and by grant RFP1-06-15 from the Foundational Questions
Institute (fqxi.org).
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