Frequency shifts in an optical lattice clock due to magnetic-dipole and electric-quadrupole transitions.

Abstract

We report a hitherto undiscovered frequency shift for forbidden J = 0-->J = 0 clock transitions excited in atoms confined to an optical lattice. These shifts result from magnetic-dipole and electric-quadrupole transitions, which have a spatial dependence in an optical lattice that differs from that of the stronger electric-dipole transitions. In combination with the residual translational motion of atoms in an optical lattice, this spatial mismatch leads to a frequency shift via differential energy level spacing in the lattice wells for ground state and excited state atoms. We estimate that this effect could lead to fractional frequency shifts as large as 10(-16), which might prevent lattice-based optical clocks from reaching their predicted performance levels. Moreover, these effects could shift the magic wavelength in lattice clocks in three dimensions by as much as 100 MHz, depending on the lattice configuration.

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Frequency Shifts in an Optical Lattice Clock Due to Magnetic-Dipole
and Electric-Quadrupole Transitions
A.V. Taichenachev*and V.I. Yudin*
Institute of Laser Physics SB RAS, Novosibirsk 630090, Russia
Novosibirsk State University, Novosibirsk 630090, Russia
Novosibirsk State Technical University, Novosibirsk 630092, Russia
V.D. Ovsiannikov
Physics Department, Voronezh State University, Voronezh 394006, Russia
V.G. Pal’chikov
Institute of Metrology for Time and Space at National Research Institute for Physical-Technical and Radiotechnical Measurements,
Mendeleevo, Moscow Region, 141579 Russia
C.W. Oates
National Institute of Standards and Technology, Boulder, Colorado 80305, USA
(Received 10 March 2008; revised manuscript received 22 August 2008; published 4 November 2008)
We report a hitherto undiscovered frequency shift for forbidden J ¼ 0 ! J ¼ 0 clock transitions
excited in atoms confined to an optical lattice. These shifts result from magnetic-dipole and electric-
quadrupole transitions, which have a spatial dependence in an optical lattice that differs from that of the
stronger electric-dipole transitions. In combination with the residual translational motion of atoms in an
optical lattice, this spatial mismatch leads to a frequency shift via differential energy level spacing in the
lattice wells for ground state and excited state atoms. We estimate that this effect could lead to fractional
frequency shifts as large as 10?16, which might prevent lattice-based optical clocks from reaching their
predicted performance levels. Moreover, these effects could shift the magic wavelength in lattice clocks in
three dimensions by as much as 100 MHz, depending on the lattice configuration.
DOI: 10.1103/PhysRevLett.101.193601PACS numbers: 42.50.Gy, 42.62.Fi, 42.62.Eh
The last few years have been marked by theoretical [1]
and experimental [2–6] breakthroughs in the field of fun-
damental laser frequency metrology that has demonstrated
the feasibility of exciting strongly forbidden optical tran-
sitions in a large number of neutral atoms that are confined
to an optical lattice. Tight confinement of the atoms to the
Lamb-Dicke regime has enabled spectroscopy of atomic
transitions with Hertz level linewidths [4,6] and raises the
prospect of neutral atom-based optical frequency standards
with a fractional frequency uncertainty at a level below
10?17[3].
Critical to reaching this level of performance is the
suppression of the shifts due to the lattice light itself.
Indeed, the red-detuned optical lattices used in these ex-
periments rely on these Stark shifts to confine the atoms.
These shifts result primarily from electric dipole (E1)
transitions, have a linear dependence on the lattice inten-
sity, and can be as large as 1 MHz (10?9fractionally).
Thus, common-mode rejection of the Stark shifts at the
10?8level or below is required. This challenge is accom-
plished largely by tuning the wavelength of the lattice to a
value (the so-called magicwavelength) that producesequal
shifts for the ground and excited states of the largely
forbidden (i.e., extremely narrow) clock transition [1].
This approach has been extremely effective, and clock
performance with a 1:5 ? 10?16fractional uncertainty
has been demonstrated [3]. However, to reach such levels
and below, it is necessary to consider the effects of the
lattice light in more detail. Higher order effects due to two
photon transitions (with a quadratic dependence on inten-
sity) have been carefully evaluated in several works and
have been shown to impose no serious barriers to reaching
fractional uncertainties below 10?17[5,7,8].
In the present Letter, we evaluate the contributions due
to magnetic-dipole (M1) and electric-quadrupole (E2)
transitions and demonstrate a previously unknown fre-
quency shift. Such contributions were first considered in
this context in Ref. [1], where it was concluded that they
would affect only the value of the magic wavelength, and
even then in only a negligible way due to their much
weaker line strengths (e.g., 10?7times that of E1 transi-
tions for Sr atoms [1]). While this is indeed the case for
traveling light waves, for the standing waves that are
characteristic of an optical lattice, the situation is different.
In this case, there is an inhomogeneous spatial distribution
of the electric and magnetic fields that modifies the spacing
of the energy levels in the potential wells formed by lattice
light. This leads to a shift associated with the quantization
of atomic translational motion in an optical lattice for a
forbidden optical transition J ¼ 0 ! J ¼ 0 (for instance,
1S0!3P0in alkaline-earth-like atoms). This shift is pro-
portional to the square root of the lattice field intensity (in
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the Lamb-Dicke regime), and it does not vanish at the
magic wavelength?m, at which the first-order (in intensity)
light shift cancels. Estimates show that this shift has con-
siderable significance for lattice-based atomic clocks as we
strive for fractional uncertainties at the 10?16level and
below.
Consider an atom confined to an optical lattice that is
produced by a one-dimensional elliptically polarized
standing wave (with the frequency !). The electric field
vector has the form
Eðr;tÞ ¼ E0ecosðk ? rÞe?i!tþ c:c:;
where E0is the scalar amplitude, k is thewavevector (k ¼
jkj ¼ !=c), and e is the complex unit polarization vector
ðe ? e?Þ ¼ 1. The condition ðe ? kÞ ¼ 0 is satisfied due to
the transverse nature of the electromagnetic field.
First we consider the frequency shift of a transition Jg¼
0 ! Je¼ 0 in a potential produced only by the contribu-
tions of E1 transitions Jj¼ 0 ! J ¼ 1 (j ¼ g, e). For the
standing-wave field (1) the light shift (potential) of a jth
level is spatially modulated and has the following form(the
negative sign results from the red detuning):
(1)
UE
jðrÞ ¼ ?Wjcos2ðkzÞ;
Here, for convenience, we choose the z axis to lie along the
wave vector k. The potential amplitude Wjdepends on
the frequency ! and is proportional to the field intensity I
at the lattice antinode, which can be written as I ¼
cjE0j2=2?.
Now we quantize the translational motion of atoms.
Here we assume the atoms are localized around the field
antinodes kz ¼ l? (l ¼ 0;?1;?2...) to within much less
than the lattice wavelength (the Lamb-Dicke regime). In
this case we can describe the atomic motion with a har-
monic oscillator approximation around the point z ¼ 0.
For the condition jkzj ? 1 we can use the approximation
cos2ðkzÞ ? 1 ? k2z2, which allows us to rewrite the shift
(2) as a constant plus a harmonic oscillator potential:
Wj>0
ðj ¼ g;eÞ:
(2)
UE
jðrÞ ? ?WjþMð2??jÞ2z2
where M is the atomic mass, and the oscillator frequency
?jfor the jth level has the form
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
2
;
ðj ¼ g;eÞ;
(3)
?j¼
1
2?
2Wjk2
s
;
ðj ¼ g;eÞ:
(4)
For the potential (3), standard quantum theory for the
harmonic oscillator yields energies of the upper and lower
levels that contain a vibrational structure:
EjðnÞ ¼ Eð0Þ
where h ¼ 2?@, Eð0Þ
state in a free space, and n ¼ 0;1;2;... is the vibrational
quantum number (see Fig. 1).
j? Wjþ h?jðn þ 1=2Þ;
ðj ¼ g;eÞ;
(5)
j
is the energy of the unperturbed jth
Consider the frequency for an optical transition between
vibrational levels with the same quantum numbers n (the
usual case for precision spectroscopy):
?nn¼EeðnÞ ? EgðnÞ
¼ ?ð0Þ? ðWe? WgÞ=h þ ð?e? ?gÞðn þ 1=2Þ;
h
(6)
where ?ð0Þ¼ ðEð0Þ
turbed Jg¼ 0 ! Je¼ 0 transition. Based on the relation-
ships between I, W, and ?, the frequency shift can be
written as
e ? Eð0Þ
g Þ=h is the frequency of the unper-
??nn? ?nn? ?ð0Þ¼ ?ð!ÞI þ ðn þ 1=2Þ?ð!Þ
where the coefficients ?ð!Þ and ?ð!Þ depend upon the
given atomic element. They are defined as follows:
ffiffiffi
I
p
;
(7)
?ð!ÞI ¼ ?ðWe? WgÞ=h;?ð!Þ
ffiffiffi
I
p
¼ ?e? ?g:
(8)
Thus, despite the fact that we started with a potential (2)
that is proportional to I, the effects of the quantization of
atomic motion lead to the appearance of an additional
square-root dependence ( /
(7). Beyond the Lamb-Dicke regime the intensity depen-
dence of the frequency shift is more complicated due to the
anharmonicity of the potential.
For the case of potential (2), which is induced only by
E1 transitions, ?ð!mÞ and ?ð!mÞ are simultaneously equal
to zero at the magic frequency ?m, because Weð!mÞ ¼
Wgð!mÞ, and the shifts cancel in the usual way. However,
as we now show, if we take into account contributions due
ffiffiffi
I
p
) for the frequency shift
FIG. 1.
tional levels for atoms confined to an optical lattice.
An illustration of optical transitions between vibra-
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to M1 and E2 transitions, we can no longer null ?ð!mÞ and
?ð!mÞ simultaneously, which has important implications
for precision metrology.
In accordance with Maxwell’s equations for a standing
wave, the magnetic field vector B has a sinðk ? rÞ spatial
dependence rather than the cosðk ? rÞ dependence of the
corresponding electric field (1):
Bðr;tÞ ¼ B0eBsinðk ? rÞe?i!tþ c:c:;
where B0¼ iE0is the scalar amplitude of the magnetic
field, and eB¼ ½k ? e?=k is the unit polarization vector of
the magnetic field. As a result, the contribution to the
potential of jth level ( / jBj2) due to M1 transitions Jj¼
0 ! J ¼ 1 has a spatial dependence different from that of
the potential (2):
(9)
UB
jðrÞ ¼ Bjsin2ðkzÞ;
ðj ¼ g;eÞ;
(10)
where the potential amplitude Bjð!Þ is proportional to the
intensity I, but has a frequency dependence that differs
from that of Wj. It can be shown that the contribution due
to E2 transitions Jj¼ 0 ! J ¼ 2 also has a sin2ðkzÞ spa-
tial dependence (for a 1D standing wave):
UQ
jðrÞ ¼ Qjsin2ðkzÞ;
ðj ¼ g;eÞ:
(11)
Hence, for the jth level the total potential proportional to
the intensity I has the form
UjðrÞ¼UE
jðrÞþUB
¼?Wjcos2ðkzÞþfBjþQjgsin2ðkzÞ;
jðrÞþUQ
jðrÞ
ðj¼g;eÞ:
(12)
In contrast to the traveling wave case, where the spatial
dependence for the E1, M1, and E2 transitions is the same,
here we find that the M1 and E2 potential wells are
spatially shifted relative to the E1 potential wells. Thus,
for complete cancellation of the Stark shifts, we would
need to find a lattice frequency that simultaneously nulls
the difference in contributions between the E1 ground and
excited states (i.e., the usual magic wavelength) and this
difference for the sum of the M1 and E2 contributions.
Since such a value is prohibitively unlikely, the concept of
the ideal magic frequency is really only valid for a single
traveling wave. In this case, however, the confining optical
lattice potential is absent and therefore it is not useful for
lattice-based atomic clocks. For the standing-wave case,
we can estimate the size of this effect by realizing that
the contribution due to E1 transitions dominates, so the
other contributions can be considered as very small
perturbations.
Expanding the expression (12) in powers of (kz) and
using the harmonic approximation [i.e., cos2ðkzÞ ?
1 ? k2z2and sin2ðkzÞ ? k2z2], we obtain an expression
for the potential analogous to (3):
UjðrÞ ? ?WjþMð2?~?jÞ2z2
2
;
ðj ¼ g;eÞ:
(13)
However, we now have to use a modified expression for the
vibrational frequency~?jthat accounts for the M1 and E2
contributions:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
~?j¼
1
2?
2fWjþ Bjþ Qjgk2
s
;
ðj ¼ g;eÞ: (14)
From Eqs. (13) and (14) it follows that for a 1D standing
wave the M1 and E2 transitions affect only the coefficient
?ð!Þ in the formula for the shifts (7), while the coefficient
?ð!Þ is governed solely by the E1 transitions as before.
The magic frequency of lattice field !mcan still be
defined from the conditionthat nulls the linear shift (/ I) in
(7) [i.e., ?ð!mÞ ¼ 0]. In this case Weð!mÞ ¼ Wgð!mÞ ¼
W. However, the remaining part (/
differs from zero:
ffiffiffi
ffiffiffi
I
p
) in Eq. (7) now
??nn¼ ðn þ 1=2Þ?ð!mÞ
Expanding the expression (14) in the small parameter
jðBjþ QjÞ=Wj ? 1 and leaving only the first-order
term, we obtain
ffiffiffi
Here the frequency ?ð0Þis equal to
I
p
? 0:
(15)
?ð!mÞ
I
p
¼ ð~?e?~?gÞ ? ?ð0Þ?:
(16)
?ð0Þ¼
1
2?
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
2Wk2
s
;
(17)
and its value coincides with the vibrational frequency,
which is nearly (but not exactly) the same (at !m) for the
upper and lower levels of the clock transition Jg¼ 0 !
Je¼ 0. The dimensionless small coefficient ? in (16) is
defined as
? ¼ ?M1þ ?E2¼Be? Bg
j?M1;?E2j ? 1;
and it does not depend on the intensity I and polarization e
(for the odd isotopes, which have nonzero nuclear spin,
there is a weak polarization dependence that is negligibly
small). The terms ?M1and ?E2in (18) are governed by M1
transitions and E2 transitions, respectively.
We now estimate the metrological significance of the
square-root-dependent shift (16). Based on general consid-
erations, we expect the coefficient ? to have a value in the
range 10?7–10?6for elements currently being used in
optical lattice clock development. For typical experimental
lattice intensities the vibrational frequency ?ð0Þ
?50 kHz. Then the shift ??nn could be as large as
50 mHz (10?16, fractionally), large enough to be of con-
siderable concern for standards with projected uncertain-
ties below 10?17. Of even more concern is that this shift
appears to be unavoidable and cannot be substantially
2W
þQe? Qg
2W
;
(18)
is
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reduced by decreasing the field intensity, due to its weak
square-root dependence,
I
. This is in contrast to the case
of the hyperpolarizability shift, which is proportional to I2.
This shift ??nnmay also need to be taken into considera-
tion for high precision measurements of the magic fre-
quency !m.
The results obtained above can be generalized to the
case of an arbitrary field configuration (including 2D and
3D optical lattices), when the electric field vector has the
general form
ffiffiffi
p
Eðr;tÞ¼EðrÞe?i!tþc:c:;EðrÞ¼X
a
EaeiðkarÞ;
(19)
where Eais the vector amplitude of the ath running wave
with the wave vector ka(jkaj ¼ k ¼ !=c). The spatial
dependence of potential induced by E1 transitions is gov-
erned by the expression
UE
jðrÞ ¼ wjð!ÞjEðrÞj2;
ðj ¼ g;eÞ;
(20)
where the frequency dependence wjð!Þ depends on the
particular atomic element.
The potential induced by M1 transitions has the form
UB
jðrÞ ¼ bjð!ÞjBðrÞj2;
BðrÞ ¼
a
ðj ¼ g;eÞ;
ðnka¼ ka=kÞ:
X
eiðkarÞ½nka? Ea?;
(21)
The contribution due to E2 transitions can be presented in
the form of the following scalar product [9]:
UQ
jðrÞ ¼ rjð!ÞðQ2? Q?
where the covariant components of irreducible tensor of
the second rank Q2are written as [9]
Q2q¼X
In these more general (i.e., 2D–3D) cases all the spatial
dependencies (21)–(23) differ from one another, so the
M1 and E2 transitions can now also affect the linear term
in (7), i.e., the coefficient ?ð!Þ. This will occur, if at the
minima points frming of the E1 potential UE
UB
of the ideal 1D standing wave (1)]. Depending on the field
configuration,weestimatethattheshiftinmagicfrequency
could be as large as j?j!m, or about 100 MHz. Moreover,
in a similar way, even in the 1D case, if the counterpropa-
gating waves are unbalanced (e.g., due to imperfect retro-
reflection), the M1 and E2 contributions can appreciably
modify the value of the coefficient ?ð!Þ in (7), thereby
shifting the magic frequency !m.
From an experimental standpoint these results have
several important implications. First, measurement of this
effect will become important as the clocks are pushed to
higher performance levels. However, this may be challeng-
ing as normal leveraging techniques (e.g., temporarily
increasing the lattice intensity above its usual operational
value in order to enhance the size of the shift) will be
2Þ;
ðj ¼ g;eÞ;
(22)
a
eiðkarÞfnka?Eag2q;
ðq¼0;?1;?2Þ:
(23)
jðrÞ we have
jðrminÞ ? 0 and/or UQ
jðrminÞ ? 0 [in contrast to the case
hampered by the presence of the hyperpolarizability shifts
proportional to I2. Second, in 1D experiments, the size of
residual traveling waves must be considered when report-
ing magic wavelengths (the use of optical cavities could be
used to suppress the traveling wave [5]). Third, in multi-
dimensional lattices, the magic wavelength will be con-
figuration dependent. Finally, since ??nnis proportional to
(n þ 1=2), the frequency shift will increasewith increasing
temperatureof the atoms in the lattice. Thus, it may well be
advisable to cool atoms to the lowest vibrational level (n ¼
0) (as demonstrated in Ref. [10]) before performing the
precision spectroscopy.
In conclusion, we have found that when we take into
account the effect of M1 and E2 transitions on the fre-
quency of a strongly forbidden optical transition, the opti-
cal lattice appears not to be as benign as perhaps first
thought. When the spatial inhomogeneity of the fields in
a 1D or multidimensional lattice is considered, there arises
a previously unconsidered frequency shift that results
from residual atomic translational motion in the lattice.
We find that this shift has a square-root dependence on the
lattice intensity for atoms in the Lamb-Dicke regime, and it
does not vanish at the magic wavelength. Our order-of-
magnitude estimate for the size of this shift suggests that it
might be significant for state-of-the-art optical lattice
clocks.If so,thiseffectcould haveimportantconsequences
for the design and operation of future versions of such
clocks.
A.V.T. and V.I.Yu. were supported by RFBR (07-02-
01230, 07-02-01028, 08-02-01108), INTAS-SBRAS (06-
1000013-9427), and Presidium of SB RAS. V.D.O. was
supported by RFBR (07-02-00279), CRDF, and MinES RF
(ANNEX-BP2M10).
*llf@laser.nsc.ru
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