Nuclear Spin Effects in Optical Lattice Clocks
Nuclear Spin Effects in Optical Lattice Clocks
Martin M. Boyd, Tanya Zelevinsky, Andrew D. Ludlow, Sebastian
Blatt, Thomas Zanon-Willette, Seth M. Foreman, and Jun Ye
JILA, National Institute of Standards and Technology and University of Colorado,
Department of Physics, University of Colorado, Boulder, CO 80309-0440
(Dated: August 28, 2018)
We present a detailed experimental and theoretical study of the effect of nuclear spin on the
performance of optical lattice clocks. With a state-mixing theory including spin-orbit and hyperfine
interactions, we describe the origin of the 1S0-
3P0 clock transition and the differential g-factor be-
tween the two clock states for alkaline-earth(-like) atoms, using 87Sr as an example. Clock frequency
shifts due to magnetic and optical fields are discussed with an emphasis on those relating to nuclear
structure. An experimental determination of the differential g-factor in 87Sr is performed and is
in good agreement with theory. The magnitude of the tensor light shift on the clock states is also
explored experimentally. State specific measurements with controlled nuclear spin polarization are
discussed as a method to reduce the nuclear spin-related systematic effects to below 10−17 in lattice
clocks.
Optical clocks [1] based on alkaline-earth atoms con-
fined in an optical lattice [2] are being intensively ex-
plored as a route to improve state of the art clock accu-
racy and precision. Pursuit of such clocks is motivated
mainly by the benefits of Lamb-Dicke confinement which
allows high spectral resolution [3, 4], and high accuracy
[5, 6, 7, 8] with the suppression of motional effects, while
the impact of the lattice potential can be eliminated using
the Stark cancelation technique [9, 10, 11, 12]. Lattice
clocks have the potential to reach the impressive accu-
racy level of trapped ion systems, such as the Hg+ opti-
cal clock [13], while having an improved stability due to
the large number of atoms involved in the measurement.
Most of the work performed thus far for lattice clocks has
been focused on the nuclear-spin induced 1S0-
3P0 tran-
sition in 87Sr. Recent experimental results are promis-
ing for development of lattice clocks as high performance
optical frequency standards. These include the confir-
mation that hyperpolarizability effects will not limit the
clock accuracy at the 10−17 level [12], observation of tran-
sition resonances as narrow as 1.8 Hz [3], and the excel-
lent agreement between high accuracy frequency mea-
surements performed by three independent laboratories
[5, 6, 7, 8] with clock systematics associated with the lat-
tice technique now controlled below 10−15 [6]. A main
effort of the recent accuracy evaluations has been to min-
imize the effect that nuclear spin (I = 9/2 for 87Sr) has
on the performance of the clock. Specifically, a linear
Zeeman shift is present due to the same hyperfine inter-
action which provides the clock transition, and magnetic
sublevel-dependent light shifts exist, which can compli-
cate the stark cancelation techniques. To reach accuracy
levels below 10−17, these effects need to be characterized
and controlled.
The long coherence time of the clock states in alkaline
earth atoms also makes the lattice clock an intriguing
system for quantum information processing. The closed
electronic shell should allow independent control of elec-
tronic and nuclear angular momenta, as well as protec-
tion of the nuclear spin from environmental perturbation,
providing a robust system for coherent manipulation[14].
Recently, protocols have been presented for entangling
nuclear spins in these systems using cold collisions [15]
and performing coherent nuclear spin operations while
cooling the system via the electronic transition [16].
Precise characterization of the effects of electronic and
nuclear angular-momentum-interactions and the resul-
tant state mixing is essential to lattice clocks and po-
tential quantum information experiments, and therefore
is the central focus of this work. The organization of
this paper is as follows. First, state mixing is discussed
in terms of the origin of the clock transition as well as
a basis for evaluating external field sensitivities on the
clock transition. In the next two sections, nuclear-spin
related shifts of the clock states due to both magnetic
fields and the lattice trapping potential are discussed.
The theoretical development is presented for a general
alkaline-earth type structure, using 87Sr only as an ex-
ample (Fig. 1), so that the results can be applied to other
species with similar level structure, such as Mg, Ca, Yb,
Hg, Zn, Cd, Al+, and In+. Following the theoretical dis-
cussion is a detailed experimental investigation of these
nuclear spin related effects in 87Sr, and a comparison to
the theory sections. Finally, the results are discussed in
the context of the performance of optical lattice clocks,
including a comparison with recent proposals to induce
the clock transition using external fields in order to elim-
inate nuclear spin effects [17, 18, 19, 20, 21, 22]. The
appendix contains additional details on the state mixing
and magnetic sensitivity calculations.
I. STATE MIXING IN THE nsnp
CONFIGURATION
To describe the two-electron system in intermediate
coupling, we follow the method of Breit and Wills [23]
and Lurio [24] and write the four real states of the ns np
configuration as expansions of pure spin-orbit (LS) cou-
http://arxiv.org/abs/0704.0912v2
)5( Ss
)55( Pps
)55( Pps
)55( Pps
)55( Pps
State A(MHz) Q(MHz)
-260              -35
-212                67
-3.4                39
State Mixing
FIG. 1: (color online) Simplified 87Sr energy level diagram
(not to scale). Relevant optical transitions discussed in the
text are shown as solid arrows, with corresponding wave-
lengths given in nanometers. Hyperfine structure sublevels
are labeled by total angular momentum F , and the magnetic
dipole (A) and electric quadrupole (Q, equivalent to the hy-
perfine B coefficient) coupling constants are listed in the inset.
State mixing of the 1P1 and
3P1 states due to the spin-orbit
interaction is shown as a dashed arrow. Dotted arrows repre-
sent the hyperfine induced state mixing of the 3P0 state with
the other F = 9/2 states in the 5s5p manifold.
pling states,
P0〉 = |
P1〉 = α|
1 〉 + β|
P2〉 = |
P1〉 = −β|
1 〉 + α|
Here the intermediate coupling coefficients α and β
(0.9996 and -0.0286 respectively for Sr) represent the
strength of the spin-orbit induced state mixing between
singlet and triplet levels, and can be determined from
experimentally measured lifetimes of 1P1 and
3P1 (see
Eq. 15 in the appendix). This mixing process results in
a weakly allowed 1S0-
3P1 transition (which would other-
wise be spin-forbidden), and has been used for a variety
of experiments spanning different fields of atomic physics.
In recent years, these intercombination transitions have
provided a unique testing ground for studies of narrow-
line cooling in Sr [25, 26, 27, 28, 29] and Ca [30, 31], as
well as the previously unexplored regime of photoassocia-
tion using long lived states [32, 33, 34]. These transitions
have also received considerable attention as potential op-
tical frequency standards [35, 36, 37], owing mainly to
the high line quality factors and insensitivity to external
fields. Fundamental symmetry measurements, relevant
to searches of physics beyond the standard model, have
also made use of this transition in Hg [38]. Furthermore,
the lack of hyperfine structure in the bosonic isotopes
(I = 0) can simplify comparison between experiment and
theory.
The hyperfine interaction (HFI) in fermionic isotopes
provides an additional state mixing mechanism between
states having the same total spin F , mixing the pure 3P0
state with the 3P1,
3P2 and
1P1 states.
|3P0〉 = |
〉+ α0|
3P1〉+ β0|
1P1〉+ γ0|
〉. (2)
The HFI mixing coefficients α0, β0, and γ0 (2×10
−4, −4×
10−6, and 4 × 10−6 respectively for 87Sr) are defined in
Eq. 16 of the appendix and can be related to the hyperfine
splitting in the P states, the fine structure splitting in the
3P states, and the coupling coefficients α and β [23, 24].
The 3P0 state can also be written as a combination of
pure states using Eq. 1,
P0〉 =|
0 〉 + (α0α− β0β)|
+ (α0β + β0α)|
1 〉 + γ0|
The HFI mixing enables a non-zero electric-dipole tran-
sition via the pure 1P 01 state, with a lifetime which can
be calculated given the spin-orbit and HFI mixing coef-
ficients, the 3P1 lifetime, and the wavelengths (λ) of the
3P0 and
3P1 transitions from the ground state [39].
3P0 =
3P1−1S0
(α0β + β0α)2
3P1 . (4)
In the case of Sr, the result is a natural lifetime on the
order of 100 seconds [9, 40, 41], compared to that of a
bosonic isotope where the lifetime approaches 1000 years
[41]. Although the 100 second coherence time of the
excited state exceeds other practical limitations in cur-
rent experiments, such as laser stability or lattice life-
time, coherence times approaching one second have been
achieved [3]. The high spectral resolution has allowed a
study of nuclear-spin related effects in the lattice clock
system discussed below.
The level structure and state mixing discussed here
are summarized in a simplified energy diagram, shown
in Fig. 1, which gives the relevant atomic structure and
optical transitions for the 5s5p configuration in 87Sr.
II. THE EFFECT OF EXTERNAL MAGNETIC
FIELDS
With the obvious advantages in spectroscopic precision
of the 1S0-
3P0 transition in an optical lattice, the sensi-
tivity of the clock transition to external field shifts is a
central issue in developing the lattice clock as an atomic
frequency standard. To evaluate the magnetic sensitivity
of the clock states, we follow the treatment of Ref. [24] for
the intermediate coupling regime described by Eqns. 1-3
in the presence of a weak magnetic field. A more general
treatment for the case of intermediate fields is provided
in the appendix. The Hamiltonian for the Zeeman inter-
action in the presence of a weak magnetic field B along
the z-axis is given as
HZ = (gsSz + glLz − gIIz)µ0B. (5)
Here gs ≃ 2 and gl = 1 are the spin and orbital an-
gular momentum g-factors, and Sz, Lz, and Iz are the
z-components of the electron spin, orbital, and nuclear
spin angular momentum respectively. The nuclear g-
factor, gI , is given by gI=
µI(1−σd)
µ0|I|
, where µI is the nuclear
magnetic moment, σd is the diamagnetic correction and
. Here, µB is the Bohr magneton, and h is Planck’s
constant. For 87Sr, the nuclear magnetic momement and
diamagnetic correction are µI = −1.0924(7)µN [42] and
σd = 0.00345 [43] respectively, where µN is the nuclear
magneton. In the absence of state mixing, the 3P0 g-
factor would be identical to the 1S0 g-factor (assuming
the diamagnetic effect differs by a negligible amount for
different electronic states), equal to gI . However since
the HFI modifies the 3P0 wavefunction, a differential g-
factor, δg, exists between the two states. This can be
interpreted as a paramagnetic shift arising due to the
distortion of the electronic orbitals in the triplet state,
and hence the magnetic moment [44]. δg is given by
δg = −
〈3P0|HZ |
3P0〉 − 〈
3P 00 |HZ |
3P 00 〉
mFµ0B
= − 2 (α0α− β0β)
〈3P 00 ,mF |HZ |
3P 01 , F = I,mF 〉
mFµ0B
+ O(α
0 , γ
0 , . . .).
Using the matrix element given in the appendix for
87Sr (I = 9/2), we find 〈3P 00 ,mF |HZ |
3P 01 , F =
,mF 〉=
mFµ0B, corresponding to a modification of the
g-factor by ∼60%. Note that the sign in Eq. 6 differs
from that reported in [39, 44] due to our choice of sign
for the nuclear term in the Zeeman Hamiltonian (oppo-
site of that found in Ref. [24]). The resulting linear
Zeeman shift ∆
B = −δgmFµ0B of the
3P0 transition
is on the order of ∼110×mF Hz/G (1 G = 10
−4 Tesla).
This is an important effect for the development of lattice
clocks, as stray magnetic fields can broaden the clock
transition (deteriorate the stability) if multiple sublevels
are used. Furthermore, imbalanced population among
the sublevels or mixed probe polarizations can cause fre-
quency errors due to line shape asymmetries or shifts.
It has been demonstrated that if a narrow resonance is
achieved (10 Hz in the case of Ref. [6]), these systematics
can be controlled at 5×10−16 for stray fields of less than
5 mG. To reduce this effect, one could employ narrower
resonances or magnetic shielding.
An alternative measurement scheme is to measure
the average transition frequency between mF and −mF
states of to cancel the frequency shifts. This requires
application of a bias field to resolve the sublevels, and
therefore the second order Zeeman shift ∆
B must be
considered. The two clock states are both J = 0 so the
shift ∆
B arises from levels separated in energy by the
fine-structure splitting, as opposed to the more tradi-
tional case of alkali(-like) atoms where the second order
shift arises from nearby hyperfine levels. The shift of
the clock transition is dominated by the interaction of
0 500 1000 1500 2000 2500 3000
0 1 2 3
=-9/2C
Magnetic Field (G)
=+9/2
Magnetic Field (G)
FIG. 2: (color online) A Breit-Rabi diagram for the 1S0-
clock transition using Eq. 22 with δgµ0 = −109 Hz/G. Inset
shows the linear nature of the clock shifts at the fields relevant
for the measurement described in the text.
the 3P0 and
3P1 states since the ground state is sepa-
rated from all other energy levels by optical frequencies.
Therefore, the total Zeeman shift of the clock transition
∆B is given by
∆B = ∆
B + ∆
|〈3P0, F,mF |HZ |
3P1, F
′,mF 〉|
ν3P1,F ′ − ν3P0
The frequency difference in the denominator is mainly
due to the fine-structure splitting and is nearly indepen-
dent of F ′, and can therefore be pulled out of the sum-
mation. In terms of the pure states, and ignoring terms
of order α0, β0, β
2, and smaller, we have
B ≃− α
|〈3P 00 , F,mF |HZ |
3P 01 , F
′,mF 〉|
ν3P1 − ν3P0
2α2(gl − gs)
3(ν3P1 − ν3P0)
where we have used the matrix elements given in the
appendix for the case F = 9/2. From Eq. 8 the sec-
ond order Zeeman shift (given in Hz for a magnetic field
given in Gauss) for 87Sr is ∆
B =−0.233B
2. This is con-
sistent with the results obtained in Ref. [20] and [45] for
the bosonic isotope. Inclusion of the hyperfine splitting
into the frequency difference in the denominator of Eq. 7
yields an additional term in the second order shift pro-
portional to m2F which is more that 10
−6 times smaller
than the main effect, and therefore negligible. Notably,
the fractional frequency shift due to the second order
Zeeman effect of 5×10−16 G−2 is nearly 108 times smaller
than that of the Cs [46, 47] clock transition, and more
than an order of magnitude smaller than that present in
Hg+ [13], Sr+ [48, 49],and Yb+ [50, 51] ion optical clocks.
A Breit-Rabi like diagram is shown in Fig. 2, giving
the shift of the 1S0-
3P0 transition frequency for different
mF sublevels (assuming ∆m = 0 for π transitions), as a
function of magnetic field. The calculation is performed
using an analytical Breit-Rabi formula (Eq. 22) provided
in the appendix. The result is indistinguishable from the
perturbative derivation in this section, even for fields as
large as 104 G.
III. THE EFFECT OF THE OPTICAL LATTICE
POTENTIAL
In this section we consider the effect of the confining
potential on the energy shifts of the nuclear sublevels. In
the presence of a lattice potential of depth UT , formed by
a laser linearly polarized along the axis of quantization
defined by an external magnetic field B, the level shift of
a clock state (h∆g/e) from its bare energy is given by
∆e = −mF (gI + δg)µ0B − κ
e ξmF
F − F (F + 1)
∆g = −mF gIµ0B − κ
g ξmF
F − F (F + 1)
Here, κS , κV , and κT are shift coefficients proportional
to the scalar, vector (or axial), and tensor polarizabil-
ities, and subscripts e and g refer to the excited (3P0)
and ground (1S0) states respectively. ER is the energy
of a lattice photon recoil and UT /ER characterizes the
lattice intensity. The vector (∝ mF ) and tensor (∝ m
light shift terms arise solely from the nuclear structure
and depend on the orientation of the light polarization
and the bias magnetic field. The tensor shift coefficient
includes a geometric scaling factor which varies with the
relative angle φ of the laser polarization axis and the axis
of quantization, as 3cos2 φ − 1. The vector shift, which
can be described as an pseudo-magnetic field along the
propagation axis of the trapping laser, depends on the
trapping geometry in two ways. First, the size of the
effect is scaled by the degree of elliptical polarization ξ,
where ξ = 0 (ξ = ±1) represents perfect linear (circular)
polarization. Second, for the situation described here,
the effect of the vector light shift is expected to be orders
of magnitude smaller than the Zeeman effect, justifying
the use of the bias magnetic field direction as the quan-
tization axis for all of the mF terms in Eq. 9. Hence
the shift coefficient depends on the relative angle be-
tween the pseudo-magnetic and the bias magnetic fields,
vanishing in the case of orthogonal orientation [52]. A
more general description of the tensor and vector effects
in alkaline-earth systems for the case of arbitrary ellipti-
cal polarization can be found in Ref. [10]. Calculations of
the scalar, vector, and tensor shift coefficients have been
performed elsewhere for Sr, Yb, and Hg [9, 10, 11, 52]
and will not be discussed here. Hyperpolarizability ef-
fects (∝ U2T ) [9, 10, 11, 12] are ignored in Eq. 9 as they
are negligible in 87Sr at the level of 10−17 for the range of
lattice intensities used in current experiments [12]. The
second order Zeeman term has been omitted but is also
present.
Using Eq. 9 we can write the frequency of a π-
transition (∆mF = 0) from a ground state mF as
νπmF = νc −
F (F + 1)
mF ξ + ∆κ
− δgmFµ0B,
where the shift coefficients due to the differential polar-
izabilities are represented as ∆κ, and νc is the bare clock
frequency. The basic principle of the lattice clock tech-
nique is to tune the lattice wavelength (and hence the
polarizabilities) such that the intensity-dependent fre-
quency shift terms are reduced to zero. Due to the mF -
dependence of the third term of Eq. 10, the Stark shifts
cannot be completely compensated for all of the sublevels
simultaneously. Or equivalently, the magic wavelength
will be different depending on the sublevel used. The
significance of this effect depends on the magnitude of
the tensor and vector terms. Fortunately, in the case of
the 1S0-
3P0 transition the clock states are nearly scalar,
and hence these effects are expected to be quite small.
While theoretical estimates for the polarizabilities have
been made, experimental measurements are unavailable
for the vector and tensor terms. The frequencies of σ±
(∆mF = ±1) transitions from a ground mF state are
similar to the π-transitions, given by
= νc −
F (F + 1)
e (mF ± 1) − κ
g mF )ξ
e 3(mF ± 1)
− (±gI + δg(mF ± 1))µ0B.
IV. EXPERIMENTAL DETERMINATION OF
FIELD SENSITIVITIES
To explore the magnitude of the variousmF -dependent
shifts in Eq. 10, a differential measurement scheme can
be used to eliminate the large shifts common to all levels.
Using resolved sublevels one can extract mF sensitivities
by measuring the splitting of neighboring states. This is
the approach taken here. A diagram of our spectroscopic
setup is shown in Fig. 3(a). 87Sr atoms are captured
from a thermal beam into a magneto-optical trap (MOT),
based on the 1S0-
1P1 cycling transition. The atoms are
then transferred to a second stage MOT for narrow line
cooling using a dual frequency technique [26]. Full de-
tails of the cooling and trapping system used in this
work are discussed elsewhere [5, 28]. During the cooling
process, a vertical one-dimensional lattice is overlapped
-30 -20 -10 0 10 20 30
Laser Detuning (Hz)
FIG. 3: (color online) (a) Schematic of the experimental ap-
paratus used here. Atoms are confined in a nearly vertical
optical lattice formed by a retro-reflected 813 nm laser. A
698 nm probe laser is co-aligned with the lattice. The probe
polarization EP can be varied by an angle θ relative to that of
the linear lattice polarization EL. A pair of Helmholtz coils
(blue) is used to apply a magnetic field along the lattice po-
larization axis. (b) Nuclear structure of the 1S0 and
3P0 clock
states. The large nuclear spin (I = 9/2) results in 28 total
transitions, and the labels π, σ+, and σ− represent transi-
tions where mF changes by 0, +1, and −1 respectively. (c)
Observation of the clock transition without a bias magnetic
field. The 3P0 population (in arbitrary units) is plotted (blue
dots) versus the probe laser frequency for θ = 0, and a fit to
a sinc-squared lineshape yields a Fourier-limited linewidth of
10.7(3) Hz. Linewidths as narrow as 5 Hz have been observed
under similar conditions and when the probe time is extended
to 500 ms.
with the atom cloud. We typically load ∼104 atoms into
the lattice at a temperature of ∼1.5µK. The lattice is
operated at the Stark cancelation wavelength [6, 12] of
813.4280(5) nm with a trap depth of U0 = 35ER. A
Helmholtz coil pair provides a field along the lattice po-
larization axis for resolved sub-level spectroscopy. Two
other coil pairs are used along the other axes to zero the
orthogonal fields. The spectroscopy sequence for the 1S0-
3P0 clock transition begins with an 80 ms Rabi pulse from
a highly stabilized diode laser [53] that is co-propagated
with the lattice laser. The polarization of the probe laser
is linear at an angle θ relative to that of the lattice. A
shelved detection scheme is used, where the ground state
population is measured using the 1S0-
1P1 transition. The
3P0 population is then measured by pumping the atoms
through intermediate states using 3P0-
3S1, and
the natural decay of 3P1 , before applying a second
-300 -200 -100 0 100 200 300
+1/2-1/2
Laser Detuning (Hz)
FIG. 4: (color online) Observation of the 1S0-
3P0 π-
transitions (θ = 0) in the presence of a 0.58 G magnetic field.
Data is shown in grey and a fit to the eight observable line-
shapes is shown as a blue curve. The peaks are labeled by
the ground state mF -sublevel of the transition. The relative
transition amplitudes for the different sublevels are strongly
influenced by the Clebsch-Gordan coefficients. Here, transi-
tion linewidths of 10 Hz are used. Spectra as narrow as 1.8
Hz can be achieved under similar conditions if the probe time
is extended to 500 ms.
1P1 pulse. The 461 nm pulse is destructive, so for each
frequency step of the probe laser the ∼800 ms loading
and cooling cycle is repeated.
When π polarization is used for spectroscopy (θ = 0),
the large nuclear spin provides ten possible transitions,
as shown schematically in Fig. 3(b). Figure 3(c) shows
a spectroscopic measurement of these states in the ab-
sence of a bias magnetic field. The suppression of mo-
tional effects provided by the lattice confinement allows
observation of extremely narrow lines [3, 4, 19], in this
case having Fourier-limited full width at half maximum
(FWHM) of ∼10 Hz (quality factor of 4 × 1013). In our
current apparatus the linewidth limitation is 5 Hz with
degenerate sublevels and 1.8 Hz when the degeneracy is
removed [3]. The high spectral resolution allows for the
study of nuclear spin effects at small bias fields, as the
ten sublevels can easily be resolved with a few hundred
mG. An example of this is shown in Fig. 4, where the ten
transitions are observed in the presence of a 0.58 G bias
field. This is important for achieving a high accuracy
measurement of δg as the contribution from magnetic-
field-induced state mixing is negligible. To extract the
desired shift coefficients we note that for the π transi-
tions we have a frequency gap between neighboring lines
fπ,mF = νπmF − νπmF −1
= −δgµ0B − ∆κ
3(2mF − 1)
From Eq. 12, we see that by measuring the differences in
-500 -250 0 250 500
-9/2 ( +)
-7/2 (
Laser Detuning (Hz)
+7/2 ( +)
+9/2 ( )
FIG. 5: (color online) Observation of the 18 σ transitions
when the probe laser polarization is orthogonal to that of the
lattice (θ = π
). Here, a field of 0.69 G is used. The spectro-
scopic data is shown in grey and a fit to the data is shown
as a blue curve. Peak labels give the ground state sublevel of
the transition, as well as the excitation polarization.
frequency of two spectroscopic features, the three terms
of interest (δg, ∆κV , and ∆κT ) can be determined inde-
pendently. The differential g factor can be determined
by varying the magnetic field. The contribution of the
last two terms can be extracted by varying the inten-
sity of the standing wave trap, and can be independently
determined since only the tensor shift depends on mF .
While the π transitions allow a simple determination
of δg, the measurement requires a careful calibration of
the magnetic field and a precise control of the probe
laser frequency over the ∼500 seconds required to pro-
duce a scan such as in Fig. 4. Any linear laser drift
will appear in the form of a smaller or larger δg, de-
pending on the laser scan direction. Furthermore, the
measurement can not be used to determine the sign of
δg as an opposite sign would yield an identical spectral
pattern. In an alternative measurement scheme, we in-
stead polarize the probe laser perpendicular to the lattice
polarization (θ = π
) to excite both σ+ and σ− tran-
sitions. In this configuration, 18 spectral features are
observed and easily identified (Fig. 5). Ignoring small
shifts due to the lattice potential, δg is given by extract-
ing the frequency splitting between adjacent transitions
of a given polarization (all σ+ or all σ− transitions) as
fσ±,mF =νσ±mF
mF −1
=−δgµ0B . If we also measure the
frequency difference between σ+ and σ− transitions from
the same sublevel, fd,mF =νσ+mF
=−2(gI + δg)µ0B,
we find that the differential g-factor can be determined
from the ratio of these frequencies as
fd,mF
σ±,mF
. (13)
-600 -300 0 300 600
Laser Detuning (Hz)
FIG. 6: (color online) Calculation of the 18 σ transition fre-
quencies in the presence of a 1 G bias field, including the influ-
ence of Clebsch-Gordan coefficients. The green (red) curves
show the σ+ (σ−) transitions. (a) Spectral pattern for g-
factors gIµ0 = −185 Hz/G and δgµ0 = −109 Hz/G. (b) Same
pattern as in (a) but with δgµ0 = +109 Hz/G. The qualita-
tive difference in the relative positions of the transitions allows
determination of the sign of δg compared to that of gI .
In this case, prior knowledge of the magnetic field is not
required for the evaluation, nor is a series of measure-
ment at different fields, as δg is instead directly deter-
mined from the line splitting and the known 1S0 g factor
gI . The field calibration and the δg measurement are in
fact done simultaneously, making the method immune to
some systematics which could mimic a false field, such as
linear laser drift during a spectroscopic scan or slow mag-
netic field variations. Using the σ transitions also elim-
inates the sign ambiguity which persists when using the
π transitions for measuring δg. While we can not extract
the absolute sign, the recovered spectrum is sensitive to
the relative sign between gI and δg. This is shown explic-
itly in Fig. 6 where the positions of the transitions have
been calculated in the presence of a ∼1 G magnetic field.
Figure 6(a) shows the spectrum when the signs of gI and
δg are the same while in Fig. 6(b) the signs are oppo-
site. The two plots show a qualitative difference between
the two possible cases. Comparing Fig. 5 and Fig. 6 it is
obvious that the hyperfine interaction increases the mag-
nitude of the 3P0 g-factor (δg has the same sign as gI).
We state this point explicitly because of recent inconsis-
tencies in theoretical estimates of the relative sign of δg
and gI in the
87Sr literature [7, 8].
To extract the magnitude of δg, data such as in Fig. 5
are fit with eighteen Lorentzian lines, and the relevant
splitting frequencies fd,mF and fσ± are extracted. Due
to the large number of spectral features, each experimen-
tal spectrum yields 16 measurements of δg. A total of
31 full spectra was taken, resulting in an average value
of δgµ0 = −108.4(4) Hz/G where the uncertainty is the
0.0 0.5 1.0 1.5 2.0
Lattice Depth (U/U
FIG. 7: (color online) Summary of δg-measurements for dif-
ferent lattice intensities. Each data point (and uncertainty)
represents the δg value extracted from a full σ± spectrum
such as in Fig. 5. Linear extrapolation (red line) to zero lat-
tice intensity yields a value −108.4(1) Hz/G.
standard deviation of the measured value. To check for
sources of systematic error, the magnetic field was varied
to confirm the field independence of the measurement.
We also varied the clock laser intensity by an order of
magnitude to check for Stark and line pulling effects. It
is also necessary to consider potential measurement er-
rors due to the optical lattice since in general the splitting
frequencies fd,mF and fσ± will depend on the vector and
tensor light shifts. For fixed fields, the vector shift is in-
distinguishable from the linear Zeeman shift (see Eqs. 10-
12) and can lead to errors in calibrating the field for a δg
measurement. In this work, a high quality linear polar-
izer (10−4) is used which would in principle eliminate the
vector shift. The nearly orthogonal orientation should
further reduce the shift. However, any birefringence of
the vacuum windows or misalignment between the lattice
polarization axis and the magnetic field axis can lead to
a non-zero value of the vector shift. To measure this ef-
fect in our system, we varied the trapping depth over a
range of ∼ (0.6 − 1.7)U0 and extrapolated δg to zero in-
tensity, as shown in Fig. 7. Note that this measurement
also checks for possible errors due to scalar and tensor
polarizabilites as their effects also scale linearly with the
trap intensity. We found that the δg-measurement was
affected by the lattice potential by less then 0.1%, well
below the uncertainty quoted above.
Unlike the vector shift, the tensor contribution to the
sublevel splitting is distinguishable from the magnetic
contribution even for fixed fields. Adjacent σ transitions
can be used to measure ∆κT and κTe due to the m
F de-
pendence of the tensor shift. An appropriate choice of
transition comparisons results in a measurement of the
tensor shift without any contributions from magnetic or
vector terms. To enhance the sensitivity of our measure-
0 5 10 15 20 25
Measurement
UT=1.7U0 UT=0.85U0 UT=1.3U0
FIG. 8: (color online) Measurement of the tensor shift coef-
ficients ∆κT (blue triangles), and κTe (green circles), using σ
spectra and Eq. 14. The measured coefficients show no sta-
tistically significant trap depth dependence while varying the
depth from 0.85–1.7 U0.
ment we focus mainly on the transitions originating from
states with large mF ; for example, we find that
fσ+,mF=7/2 − fσ+,mF =−7/2
e = −
fd,mF =7/2 − fd,mF =−7/2
while similar combinations can be used to isolate the dif-
ferential tensor shift from the σ− data as well as the
tensor shift coefficient of the 1S0 state. From the σ split-
ting data we find ∆κT = 0.03(8) Hz/U0 and |κ
e |=0.02(4)
Hz/U0. The data for these measurements is shown in
Fig. 8. Similarly, we extracted the tensor shift coeffi-
cient from π spectra, exploiting the mF -dependent term
in Eq. 12, yielding ∆κT = 0.02(7) Hz/U0. The measure-
ments here are consistent with zero and were not found
to depend on the trapping depth used for a range of 0.85–
1.7 U0, and hence are interpreted as conservative upper
limits to the shift coefficients. The error bars represent
the standard deviation of many measurements, with the
scatter in the data due mainly to laser frequency noise
and slight under sampling of the peaks. It is worth noting
that the tensor shift of the clock transition is expected to
be dominated by the 3P0 shift, and therefore, the limit
on κTe can be used as an additional estimate for the up-
per limit on ∆κT . Improvements on these limits can be
made by going to larger trap intensities to enhance sen-
sitivity, as well as by directly stabilizing the clock laser
to components of interest for improved averaging.
Table I summarizes the measured sensitivities to mag-
netic fields and the lattice potential. The Stark shift
coefficients for linear polarization at 813.4280(5) nm are
given in units of Hz/(UT /ER). For completeness, a recent
measurement of the second order Zeeman shift using 88Sr
has been included [45], as well as the measured shift coef-
ficient ∆γ for the hyperpolarizability [12] and the upper
TABLE I: Measured Field Sensitivities for 87Sr
Sensitivity Value Units Ref.
B /mFB -108.4(4) Hz/G This work
2 -0.233(5) Hz/G2 [45]a
∆κT 6(20) ×10−4 Hz/(UT /ER) This work
∆κT 9(23)×10−4 Hz/(UT /ER) This work
κTe 5(10)×10
−4 Hz/(UT /ER) This work
κ -3(7)×10−3 Hz/(UT /ER) [6]
∆γ 7(6)×10−6 Hz/(UT /ER)
2 [12]d
a Measured for 88Sr
b Measured with π spectra
c Measured with σ± spectra
d Measured with degenerate sublevels
limit for the overall linear lattice shift coefficient κ from
our recent clock measurement [6]. While we were able
to confirm that the vector shift effect is small and con-
sistent with zero in our system, we do not report a limit
for the vector shift coefficient ∆κV due to uncertainty
in the lattice polarization purity and orientation relative
to the quantization axis. In future measurements, use of
circular trap polarization can enhance the measurement
precision of ∆κV by at least two orders of magnitude.
Although only upper limits are reported here, the re-
sult can be used to estimate accuracy and linewidth lim-
itations for lattice clocks. For example, in the absence
of magnetic fields, the tensor shift can cause line broad-
ening of the transition for unpolarized samples. Given
the transition amplitudes in Fig. 4, the upper limit for
line broadening, derived from the tensor shift coefficients
discussed above, is 5 Hz at U0. The tensor shift also
results in a different magic wavelength for different mF
sublevels, which is constrained here to the few picometer
level.
V. COMPARISON OF THE δg MEASUREMENT
WITH THEORY AND
3P0 LIFETIME ESTIMATE
The precise measurement of δg provides an opportu-
nity to compare various atomic hyperfine interaction the-
ories to the experiment. To calculate the mixing param-
eters α0 and β0 (defined in Eq. 16 of the Appendix),
we first try the simplest approach using the standard
Breit-Wills (BW) theory [23, 24] to relate the mixing
parameters to the measured triplet hyperfine splitting
(hfs). The parameters α (0.9996) and β (−0.0286(3))
are calculated from recent determinations of the 3P1 [32]
and 1P1 [54] lifetimes. The relevant singlet and triplet
single-electron hyperfine coefficients are taken from Ref.
[55]. From this calculation we find α0 = 2.37(1) × 10
β0 = −4.12(1) × 10
−6, and γ0 = 4.72(1) × 10
−6, resulting
in δgµ0 = −109.1(1) Hz/G . Using the mixing values in
conjunction with Eq. 4 we find that the 3P0 lifetime is
152(2) s. The agreement with the measured g-factor is
excellent, however the BW-theory is known to have prob-
lems predicting the 1P1 characteristics based on those of
the triplet states. In this case, the BW-theory frame-
work predicts a magnetic dipole A coefficient for the 1P1
state of -32.7(2) MHz, whereas the experimental value is
-3.4(4) MHz [55]. Since δg is determined mainly by the
properties of the 3P1 state, it is not surprising that the
theoretical and experimental values are in good agree-
ment. Conversely, the lifetime of the 3P0 state depends
nearly equally on the 1P1 and
3P1 characteristics, so the
lifetime prediction deserves further investigation.
A modified BW (MBW) theory [44, 55, 56] was at-
tempted to incorporate the singlet data and eliminate
such discrepancies. In this case 1P1,
3P1, and
3P2 hfs are
all used in the calculation, and two scaling factors are
introduced to account for differences between singlet and
triplet radial wavefunctions when determining the HFI
mixing coefficients (note that γ0 is not affected by this
modification). This method has been shown to be suc-
cessful in the case of heavier systems such as neutral Hg
[44]. We find α0 = 2.56(1)× 10
−4 and β0 = −5.5(1)× 10
resulting in δgµ0 = −117.9(5) Hz/G and τ
3P0 = 110(1) s.
Here, the agreement with experiment is fair, but the un-
certainties in experimental parameters used for the the-
ory are too small to explain the discrepancy.
Alternatively, we note that in Eq. 6, δg depends
strongly on α0α and only weakly (< 1%) on β0β, such
that our measurement can be used to tightly constrain
α0 = 2.35(1)×10
−4, and then use only the triplet hfs data
to calculate β0 in the MBW theory framework. In this
way we find β0 = −3.2(1) × 10
−6, yielding τ
3P0 = 182(5)s.
The resulting 1P1 hfs A coefficient is −15.9(5) MHz,
which is an improvement compared to the standard BW
calculation. The inability of the BW and MBW theory to
simultaneously predict the singlet and triplet properties
seems to suggest that the theory is inadequate for 87Sr.
A second possibility is a measurement error of some of
the hfs coefficients, or the ground state g-factor. The
triplet hfs is well resolved and has been confirmed with
high accuracy in a number of measurements. An error in
the ground state g-factor measurement at the 10% level
is unlikely, but it can be tested in future measurements
TABLE II: Theoretical estimates of δg and τ
3P0 for 87Sr
Values used in Calculation
α = 0.9996 β = −0.0286(3)
Calc. α0 β0 τ
3P0 δgµ0 A
×104 ×106 (s) mF (Hz/G) (MHz)
BW 2.37(1) -4.12(1) 152(2) -109.1(1) -32.7(2)
MBW I 2.56(1) -5.5(1) 110(1) -117.9(5) -3.4(4)a
MBW II 2.35(1) -3.2(1) 182(5) -108.4(4)b -15.9(5)
Ref [40] — — 132 — —
Ref [41, 59] 2.9(3) -4.7(7) 110(30) -130(15) c —
Ref [8, 9] — — 159 106d —
a Experimental value [55]
b Experimental value from this work
c Calculated using Eq. 6
d Sign inferred from Figure 1 in Ref. [8]
by calibrating the field in an independent way so that
both gI and δg can be measured. On the other hand,
the 1P1 hfs measurement has only been performed once
using level crossing techniques, and is complicated by the
fact that the structure is not resolved, and that the 88Sr
transition dominates the spectrum for naturally abun-
dant samples. Present 87Sr cooling experiments could be
used to provide an improved measurement of the 1P1 data
to check whether this is the origin of the discrepancy.
Although one can presumably predict the lifetime with
a few percent accuracy (based on uncertainties in the
experimental data), the large model-dependent spread
in values introduces significant additional uncertainty.
Based on the calculations above (and many other similar
ones) and our experimental data, the predicted lifetime is
145(40) s. A direct measurement of the natural lifetime
would be ideal, as has been done in similar studies with
trapped ion systems such as In+ [39] and Al+ [57] or neu-
tral atoms where the lifetime is shorter, but for Sr this
type of experiment is difficult due to trap lifetime limi-
tations, and the measurement accuracy would be limited
by blackbody quenching of the 3P0 state [58].
Table II summarizes the calculations of δg and τ
discussed here including the HFI mixing parameters α0
and β0. Other recent calculations based on the BW the-
ory [8, 9], ab initio relativistic many body calculations
[40], and an effective core calculation [41] are given for
comparison, with error bars shown when available.
VI. IMPLICATIONS FOR THE
SR LATTICE
CLOCK
In the previous sections, the magnitude of relevant
magnetic and Stark shifts has been discussed. Briefly, we
will discuss straightforward methods to reduce or elim-
inate the effects of the field sensitivities. To eliminate
linear Zeeman and vector light shifts the obvious path is
to use resolved sublevels and average out the effects by al-
ternating between measurements of levels with the same
|mF |. Figure 9 shows an example of a spin-polarized mea-
surement using the mF = ±9/2 states for cancelation of
the Zeeman and vector shifts. To polarize the sample,
we optically pump the atoms using a weak beam reso-
nant with the 1S0-
3P1 (F = 7/2) transition. The beam
is co-aligned with the lattice and clock laser and linearly
polarized along the lattice polarization axis (θ = 0), re-
sulting in optical pumping to the stretched (mF = 9/2)
states. Spectroscopy with (blue) and without (red) the
polarizing step shows the efficiency of the optical pump-
ing as the population in the stretched states is dramati-
cally increased while excitations from other sublevels are
not visible. Alternate schemes have been demonstrated
elsewhere [8, 26] where the population is pumped into a
single mF = ±9/2 state using the
3P1 (F = 9/2)
transition. In our system, we have found the method
shown here to be more efficient in terms of atom number
in the final state and state purity. The highly efficient
-150 -100 -50 0 50 100 150
Laser Detuning (Hz)
FIG. 9: (color online) The effect of optical pumping via the
3P1 (F = 7/2) state is shown via direct spectroscopy with θ =
0. The red data shows the spectrum without the polarizing
light for a field of 0.27 G. With the polarizing step added
to the spectroscopy sequence the blue spectum is observed.
Even with the loss of ∼ 15% of the total atom number due to
the polarizing laser, the signal size of the mF = ±9/2 states
is increased by more than a factor of 4.
optical pumping and high spectral resolution should al-
low clock operation with a bias field of less than 300 mG
for a 10 Hz feature while keeping line pulling effects due
to the presence of the other sublevels below 10−17. The
corresponding second order Zeeman shift for such a field
is only ∼21 mHz, and hence knowledge of the magnetic
field at the 10% level is sufficient to control the effect
below 10−17. With the high accuracy δg-measurement
reported here, real time magnetic field calibration at the
level of a few percent is trivial. For spin-polarized sam-
ples, a new magic wavelength can be determined for the
mF -pair, and the effect of the tensor shift will only be to
modify the cancelation wavelength by at most a few pi-
cometers if a different set of sublevels are employed. With
spin-polarized samples, the sensitivity to both magnetic
and optical fields (including hyperpolarizability effects)
should not prevent the clock accuracy from reaching be-
low 10−17.
Initial concerns that nuclear spin effects would limit
the obtainable accuracy of a lattice clock have prompted
a number of recent proposals to use bosonic isotopes
in combination with external field induced state mixing
[17, 18, 20, 21, 22] to replace the mixing provided natu-
rally by the nuclear spin. In these schemes, however, the
simplicity of a hyperfine-free system comes at the cost
of additional accuracy concerns as the mixing fields also
shift the clock states. The magnitudes of the shifts de-
pend on the species, mixing mechanism, and achievable
spectral resolution in a given system. As an example,
we discuss the magnetic field induced mixing scheme [20]
which was the first to be experimentally demonstrated
for Yb [19] and Sr [45]. For a 10 Hz 88Sr resonance (i.e.
the linewidth used in this work), the required magnetic
and optical fields (set to minimize the total frequency
shift) result in a second order Zeeman shift of −30 Hz
and an ac Stark shift from the probe laser of −36 Hz.
For the same transition width, using spin-polarized 87Sr,
the second order Zeeman shift is less than −20 mHz for
the situation in Fig. 9, and the ac Stark shift is less than
1 mHz. Although the nuclear-spin-induced case requires
a short spin-polarizing stage and averaging between two
sublevels, this is preferable to the bosonic isotope, where
the mixing fields must be calibrated and monitored at the
10−5 level to reach below 10−17. Other practical concerns
may make the external mixing schemes favorable, if for
example isotopes with nuclear spin are not readily avail-
able for the species of interest. In a lattice clock with
atom-shot noise limited performance, the stability could
be improved, at the cost of accuracy, by switching to a
bosonic isotope with larger natural abundance.
In conclusion we have presented a detailed experimen-
tal and theoretical study of the nuclear spin effects in op-
tical lattice clocks. A perturbative approach for describ-
ing the state mixing and magnetic sensitivity of the clock
states was given for a general alkaline-earth(-like) system,
with 87Sr used as an example. Relevant Stark shifts from
the optical lattice were also discussed. We described in
detail our sign-sensitive measurement of the differential
g-factor of the 1S0-
3P0 clock transition in
87Sr, yield-
ing µ0δg = −108.4(4)mF Hz/G, as well as upper limit for
the differential and exited state tensor shift coefficients
∆κT = 0.02 Hz/(UT /ER) and κ
e = 0.01 Hz/(UT /ER).
We have demonstrated a polarizing scheme which should
allow control of the nuclear spin related effects in the 87Sr
lattice clock to well below 10−17.
We thank T. Ido for help during the early stages of
the g-factor measurement, and G. K. Campbell and A.
Pe’er for careful reading of the manuscript. This work
was supported by ONR, NIST, and NSF. Andrew Lud-
low acknowledges support from NSF-IGERT through the
OSEP program at the University of Colorado.
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VII. APPENDIX
The appendix is organized as follows, in the first sec-
tion we briefly describe calculation of the mixing coeffi-
cients needed to estimate the effects discussed in the main
text. We also include relevant Zeeman matrix elements.
In the second section we describe a perturbative treat-
ment of the magnetic field on the hyperfine-mixed 3P0
state, resulting in a Breit-Rabi like formula for the clock
transition. In the final section we solve the more general
case and treat the magnetic field and hyperfine interac-
tion simultaneously, which is necessary to calculate the
sensitivity of the 1P1,
3P1 and
3P2 states.
A. State mixing coefficients and Zeeman elements
The intermediate coupling coefficients α and β are typ-
ically calculated from measured lifetimes and transition
frequencies of the 1P1 and
3P1 states and a normalization
constraint, resulting in
= 1. (15)
The HFI mixing coefficients α0, β0, and γ0 are due to
the interaction between the pure 3P0 state and the spin-
orbit mixed states in Eq. 1 having the same total angular
momentum F . They are defined as
〈3P1, F = I |HA|
3P 00 , F = I〉
ν3P0 − ν3P1
〈1P1, F = I |HA|
3P 00 , F = I〉
ν3P0 − ν1P1
〈3P2, F = I |HQ|
3P 00 , F = I〉
ν3P0 − ν3P2
Where HA and HQ are the magnetic dipole and electric
quadrupole contributions of the hyperfine Hamiltonian.
A standard technique for calculating the matrix elements
is to relate unknown radial contributions of the wavefunc-
tions to the measured hyperfine magnetic dipole (A) and
electric quadrupole (Q) coefficients. Calculation of the
matrix elements using BW theory [23, 24, 39, 44, 55] can
be performed using the measured hyperfine splitting of
the triplet state along with matrix elements provided in
[24]. Inclusion of the 1P1 data (and an accurate predic-
tion of β0) requires a modified BW theory [44, 55, 56]
where the relation between the measured hyperfine split-
ting and the radial components is more complex but man-
ageable if the splitting data for all of the states in the
nsnp manifold are available. A thorough discussion of
the two theories is provided in Refs. [44, 55].
Zeeman matrix elements for singlet and triplet states in
the nsnp configuration have been calculated in Ref. [24].
Table III summarizes those elements relevant to the work
here, where the results have been simplified by using the
electronic quantum numbers for the alkaline-earth case,
but leaving the nuclear spin quantum number general
for simple application to different species. Note that the
results include the application of our sign convention in
Eq. 5 which differs from that in Ref. [24].
B. Magnetic field as a perturbation
To determine the magnetic sensitivity of the 3P0 state
due to the hyperfine interaction with the 3P1 and
states, we first use a perturbative approach to add the
Zeeman interaction as a correction to the |3P0〉 state in
Eq. 3. The resulting matrix elements depend on spin-
orbit and hyperfine mixing coefficients α, β, α0, β0, and
γ0. For the
3P0 state, diagonal elements to first order in
α0 and β0 are relevant, while for
1P1 and
3P1, the contri-
bution of the hyperfine mixing to the diagonal elements
can be ignored. All off-diagonal terms of order β2, α0α,
α0β, α
, and smaller can be neglected. Due to the selec-
tion rules for pure (LS) states, the only contributions of
the 3P2 hyperfine mixing are of order α0γ0, γ
, and β0γ0.
Thus the state can be ignored and the Zeeman interac-
tion matrixMz between atomic P states can be described
in the
|1P1, F,mF 〉, |
3P0, F,mF 〉, |
3P1, F,mF 〉
basis as
ν1P1 M
ν3P0 M
, (17)
where we define diagonal elements as
ν3P0 = ν
0 |HZ |
+ 2(αα0 − ββ0)〈
1 , F = I |HZ |
ν3P1 = ν
1 , F
|HZ |
1 , F
1 , F
|HZ |
1 , F
ν1P1 = ν
1 , F
|HZ |
1 , F
1 , F
|HZ |
1 , F
Off diagonal elements are given by
|〈3P 01 , F
′|HZ |3P 00 , F 〉|
|〈3P 00 , F |HZ |
3P 01 , F
′〉|2.
TABLE III: Zeeman Matrix Elements for Pure (2S+1L0J ) States
Relevant Elements for the
3P0 State:
〈3P 00 , F = I |HZ|
3P 00 , F = I〉= −gImFµ0B
〈3P 00 , F = I |HZ|
3P 01 , F
′ = I〉 =(gs − gl)mFµ0B
3I(I+1)
〈3P 00 , F = I |HZ|
3P 01 , F
′ = I + 1〉 =(gs − gl)µ0B
((I+1)2−m2
)(4I+6)
3(I+1)(4(I2+1)−1)
〈3P 00 , F = I |HZ|
3P 01 , F
′ = I − 1〉 =(gs − gl)µ0B
(I2−m2
)(4I−2)
3I(4I2−1)
Relevant Diagonal Elements within
3P1 Manifold:
〈3P 01 , F = I |HZ|
3P 01 , F = I〉=
gl+gs−gI (2I(I+1)−2)
2I(I+1)
mFµ0B
〈3P 01 , F = I + 1|HZ |
3P 01 , F = I + 1〉=
gl+gs−2gII
2(I+1)
mFµ0B
〈3P 01 , F = I − 1|HZ |
3P 01 , F = I − 1〉=
gl+gs+2gI (I+1)
mFµ0B
Relevant Diagonal Elements within
1P1 Manifold:
〈1P 01 , F = I |HZ|
1P 01 , F = I〉=
gl−gI (I(I+1)−1)
I(I+1)
mFµ0B
〈1P 01 , F = I + 1|HZ |
1P 01 , F = I + 1〉=
gl−gII
(I+1)
mFµ0B
〈1P 01 , F = I − 1|HZ |
1P 01 , F = I − 1〉=
gl+gI(I+1)
mFµ0B
The eigenvalues of Eq. 17 can be written analytically as
three distinct cubic roots
ν20 + 3ν
arccos
2ν30 + 9ν0ν
1 + 27ν
2(ν20 + 3ν
νmF ≡ν3P0,mF =
ν20 + 3ν
arccos
2ν30 + 9ν0ν
1 + 27ν
2(ν20 + 3ν
where we have
ν0 =ν3P0 + ν3P1 + ν1P1
−ν3P0ν3P1 − ν3P1ν1P1 − ν3P0ν1P1 + (M
ν3P0ν3P1ν1P1 − ν3P1(M
− ν1P1(M
Since the main goal is a description of the 3P0 state sen-
sitivity, the solution can be simplified when one considers
the relative energy spacing of the three states, and that
elements having terms β, αβ, and smaller are negligible
compared to those proportional to only α. Therefore we
can ignore M
terms and find simplified eigenvalues
arising only from the interaction between 3P1 and
that can be expressed as a Breit-Rabi like expression for
the 3P0 state given by
ν3P0,mF =
ν3P0 + ν3P1
ν3P0 − ν3P1
1 + 4
α2|〈3P 00 , F |HZ|
3P 01 , F
(ν3P0 − ν3P1)
For magnetic fields where the Zeeman effect is small com-
pared to the fine-structure splitting, the result is identi-
cal to that from Eq. 8 of the main text. The magnetic
0 10 20 30 40 50 60 70 80
F=9/2
F=11/2
F=7/2
Magnetic Field(G)
FIG. 10: (color online) Magnetic sensitivity of the 1P1 state
calculated with the expression in Eq. 24 using A = −3.4 MHz
and Q = 39 MHz [55]. Note the inverted level structure.
sensitivity of the clock transition (plotted in Fig. 2) is de-
termined by simply subtracting the 〈3P 00 |HZ |
3P 00 〉 term
which is common to both states.
C. Full treatment of the HFI and magnetic field
For a more complete treatment of the Zeeman effect
we can relax the constraint of small fields and treat the
hyperfine and Zeeman interactions simultaneously using
the spin-orbit mixed states in Eq. 1 as a basis. The total
Hamiltonian is written Htotal = HZ+HA+HQ including
hyperfine HA and quadrupole HQ effects in addition to
the Zeeman interaction HZ defined in Eq. 5 of the main
0 500 1000 1500 2000 2500 3000
 Magnetic Field (G)
F=11/2
F=9/2
F=7/2
FIG. 11: (color online) Magnetic sensitivity of the 3P1 state
calculated with the expression in Eq. 24 using A = −260 MHz
and Q = −35 MHz [63].
text. The Hamiltonian Htotal can be written as
Htotal =HZ + A~I · ~J
~I · ~J(2~I · ~J + 1) − IJ(I + 1)(J + 1)
2IJ(2I − 1)(2J − 1)
Diagonalization of the full space using Eq. 23 does not
change the 3P0 result discussed above, even for fields
as large as 104 G. This is not surprising since the 3P0
state has only one F level, and is therefore only af-
fected by the hyperfine interaction through state mix-
ing which was already accounted for in the previous cal-
culation. Alternatively, for an accurate description of
the 1P1,
3P1 and
3P2 states, Eq. 23 must be used. For
an alkaline-earth 2S+1L1 state in the |I, J, F,mF 〉 basis
we find an analytical expression for the field dependence
of the F = I, I ± 1 states and sublevels. The solution
is identical to Eq. 20 except we replace the frequencies
in Eq. 21 with those in Eq. 24. We define the relative
strengths of magnetic, hyperfine, and quadrupole inter-
actions with respect to an effective hyperfine-quadrupole
coupling constant WAQ = A +
4I(1−2I)
as XBR =
, and XQ =
I(1−2I)WAQ
, respectively. The so-
lution is a generalization of the Breit-Rabi formula [61]
for the 2S+1L1 state in the two electron system with nu-
clear spin I. The frequencies are expanded in powers of
XBR as
ν0 = −2WAQ
mFXBR
ν1 = WAQ
2(geff − gI)XA + 3geffXQ
mFXBR +
(geff + gI)
3m2F g
(geff+gI)
ν2 = WAQ
I(I + 1)X
I(I+1)
3(1−2I)(3+2I)
I(I+1)
−XAXQ
geff(2 −
2I(I+1)
) + gI
mFXBR
2gIgeff
I(I+1)
(geff+gI )
3m2F g
I(I+1)(geff+gI)
gI((geff+gI )
2−(gImF )
I(I+1)
with abbreviations
=I(I + 1)
− I(I + 1)XQ(XA − 1)
=Xeff
XQXeff +
X2A −X
3(3 + 2I)(1 − 2I)
Xeff =XA +XQ
(3 + 2I)(1 − 2I)
geff =
− gs)
(L(L+ 1)− S(S + 1)) .
The resulting Zeeman splitting of the 5s5p1P1 and
5s5p3P1 hyperfine states in
87Sr is shown in Fig. 10 and
Fig. 11. For the more complex structure of 3P2, we
have solved Eq. 23 numerically, with the results shown in
Fig. 12. The solution for the 1P1 state depends strongly
on the quadrupole (Q) term in the Hamiltonian, while
for the 3P1 and
3P2 states the magnetic dipole (A) term
is dominant.
0 500 1000 1500 2000 2500 3000
10 F=5/2
F=7/2
F=11/2
F=13/2
F=9/2
Magnetic Field  (G)
FIG. 12: (color online) Magnetic sensitivity of the 3P1 state
calculated numerically with Eq. 23 using A=-212 MHz and
Q=67 MHz [62].
