Page 1

Minimizing the Required Trap Depth in

Optical Lattice Clocks

Pierre Lemonde

SYRTE, Observatoire de Paris

61 av. de l’Observatoire, 75014 Paris, France

Peter Wolf

SYRTE, Observatoire de Paris

and

Bureau International des Poids et Mesures

Pavillon de Breteuil, 92312 S` evres, Cedex, France

Abstract—We study the trap depth requirement for the

realization of an optical clock using atoms confined in a

lattice. We show that site-to-site tunnelling leads to a residual

sensitivity to the atom dynamics hence requiring large depths

(50 to 100Er for Sr) to avoid any frequency shift or line

broadening of the atomic transition at the 10−17− 10−18

level. Such large depths and the corresponding laser power

may, however, lead to difficulties (e.g. higher order light shifts,

two-photon ionization, technical difficulties) and therefore one

would like to operate the clock in much shallower traps. To

circumvent this problem we propose the use of an accelerated

lattice. Acceleration lifts the degeneracy between adjacents

potential wells which strongly inhibits tunnelling. We show

that using the Earth’s gravity, much shallower traps (down to

5Er for Sr) can be used for the same accuracy goal.

I. Introduction

The control of the external degrees of freedom of atoms,

ions and molecules and of the associated frequency shifts

and line broadenings is a long standing issue of the fields of

spectroscopy and atomic frequency standards. They have

been a strong motivation for the development of many

widely spread techniques like the use of buffer gases[1],

Ramsey spectroscopy[2], saturated spectroscopy[3], two-

photon spectroscopy[4], trapping and laser cooling[5], [6],

etc.

In the case of ions, the problem is now essentially solved

since they can be trapped in relatively low fields and

cooled to the zero point of motion of such traps[5]. In this

state, the ions are well within the Lamb-Dicke regime[1]

and experience no recoil nor first order Doppler effect[5].

The fractional inaccuracy of today’s best ion clocks lies

in the range from 3 to 10×10−15[7], [8], [9], [10], [11]

with still room for improvement. The main drawback of

trapped ion frequency standards is that only one to a few

ions can contribute to the signal due to Coulomb repulsion.

This fundamentally limits the frequency stability of these

systems and puts stringent constraints on the frequency

noise of the oscillator which probes the ions[12].

These constraints are relaxed when using a large number

of neutral atoms[13] for which, however, trapping requires

much higher fields, leading to shifts of the atomic levels.

This fact has for a long time prevented the use of

trapped atoms for the realization of atomic clocks and

today’s most accurate standards use freely falling atoms.

Microwave fountains now have an inaccuracy below 10−15

and are coming close to their foreseen ultimate limit which

lies around 10−16[14], which is essentially not related

to effects due to the atomic dynamics[15], [16]. In the

optical domain, atomic motion is a problem and even

with the use of ultra-cold atoms probed in a Ramsey-

Bord´ e interferometer[17], optical clocks with neutrals still

suffer from the first order Doppler and recoil effects[18],

[19], [20], [21]. Their state-of-the-art inaccuracy is about

10−14[20].

The situation has recently changed with the proposal

of the optical lattice clock[22]. The idea is to engineer

a lattice of optical traps in such a way that the dipole

potential is exactly identical for both states of the clock

transition, independently of the dipole laser power and

polarisation. This is achieved by tuning the trap laser to

the so-called ”magic wavelength” and by the choice of

clock levels with zero electronic angular momentum. The

original scheme was proposed for87Sr atoms using the

strongly forbidden1S0−3P0 line at 698nm as a clock

transition[23]. In principle however, it also works for all

atoms with a similar level structure like Mg, Ca, Yb, Hg,

etc. including their bosonic isotopes if one uses multi-

photon excitation of the clock transition[24], [25].

In this paper we study the effect of the atom dynamics

in the lattice on the clock performances. In ref.[22], it is

implicitly assumed that each microtrap can be treated

separately as a quadratic potential in which case the

situation is very similar to the trapped ion case and

then fully understood[5]. With an inaccuracy goal in the

10−17− 10−18range in mind (corresponding to the mHz

level in the optical domain), we shall see later on, that

this is correct at very high trap depths only. The natural

energy unit for the trap dynamics is the recoil energy

associated with the absorption or emission of a photon of

the lattice laser, Er=

lattice laser and mathe atomic mass. For Sr and for the

above accuracy goal the trap depth U0 corresponding to

the independent trap limit is typically U0= 100Er, which

corresponds to a peak laser intensity of 25kW/cm2.

For a number of reasons however, one would like to work

¯ h2k2

2mawith kLthe wave vector of the

L

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0-7803-9052-0/05/$20.00 © 2005 IEEE.

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with traps as shallow as possible. First, the residual shift

by the trapping light of the clock transition is smaller

and smaller at a decreasing trap depth. The first order

perturbation is intrinsically cancelled by tuning to the

magic wavelength except for a small eventual tensorial

effect which depends on the hyperfine structure of the

atom under consideration. Higher order terms may be

much more problematic depending on possible coinci-

dences between two photon resonances and the magic

wavelength[22], [26]. The associated shift scales as U2

The shifts would then be minimized by a reduction of U0

and its evaluation would be greatly improved if one can

vary this parameter over a broader range. Second, for some

of the possible candidate atoms, such as Hg for which the

magic wavelength is about 340 nm, two-photon ionization

can occur which may limit the achievable resonance width

and lead to a frequency shift. Finally, technical aspects like

the required laser power at the magic wavelength can be

greatly relaxed if one can use shallow traps. This can make

the experiment feasible or not if the magic wavelength is

in a region of the spectrum where no readily available

high power laser exists, such as in the case of Hg. For this

atom, a trap depth of 100Er would necessitate a peak

intensity of 500kW/cm2at 340nm.

When considering shallow traps, the independent trap

limit no longer holds, and one cannot neglect tunnelling of

the atoms from one site of the lattice to another. This leads

to a delocalization of the atoms and to a band structure in

their energy spectrum and associated dynamics. In section

III we investigate the ultimate performance of the clock

taking this effect into account. We show that depending

on the initial state of the atoms in the lattice, one faces

a broadening and/or a shift of the atomic transition of

the order of the width of the lowest energy band of the

system. For Sr, this requires U0of the order of 100Erto

ensure a fractional inaccuracy lower than 10−17.

The deep reason for such a large required value of U0

is that site-to-site tunnelling is a resonant process in a

lattice. We show in section IV that a much lower U0

can be used provided the tunnelling process is made non-

resonant by lifting the degeneracy between adjacent sites.

This can be done by adding a constant acceleration to the

lattice, leading to the well-known Wannier-Stark ladder

of states[27], [28]. More specifically, we study the case

where this acceleration is simply the Earth’s gravity. The

experimental realization of the scheme in this case is then

extremely simple: the atoms have to be probed with a laser

beam which propagates vertically. In this configuration,

trap depths down to U0∼ 5Er can be sufficient for the

above accuracy goal.

0

1.

1note that this effect cannot be quantified without an accurate

knowledge of the magic wavelength and of the strength of transitions

involving highly excited states.

II. Confined atoms coupled to a light field

In this section we describe the theoretical frame used

to investigate the residual effects of the motion of atoms

in an external potential. The internal atomic structure

is approximated by a two-level system |g? and |e? with

energy difference ¯ hωeg. The internal Hamiltonian is:

ˆHi= ¯ hωeg|e??e|.

(1)

We introduce the coupling between |e? and |g? by a laser

of frequency ω and wavevector ks propagating along the

x direction:

ˆHs= ¯ hΩcos(ωt − ksˆ x)|e??g| + h.c.,

with Ω the Rabi frequency.

In the following we consider external potentials induced

by trap lasers tuned at the magic wavelength and/or by

gravity. The external potentialˆHext is then identical for

both |g? and |e? with eigenstates |m? obeyingˆHext|m? =

¯ hωm|m? (Note that |m? can be a continuous variable in

which case the discrete sums in the following are replaced

by integrals). If we restrict ourselves to experiments much

shorter than the lifetime of state |e? (for87Sr, the lifetime

of the lowest3P0state is 100 s) spontaneous emission can

be neglected and the evolution of the general atomic state

?

is driven by

(2)

|ψat? =

m

ag

me−iωmt|m,g? + ae

me−i(ωeg+ωm)t|m,e? (3)

i¯ h∂

∂t|ψat? = (ˆHext+ˆHi+ˆHs)|ψat?,

(4)

leading to the following set of coupled equations

i ˙ ag

m

=

?

?

m?

Ω∗

2ei∆m?,mt?m|e−iksˆ x|m??ae

Ω

2e−i∆m,m?t?m|eiksˆ x|m??ag

m?

(5)

i ˙ ae

m

=

m?

m?.

To derive eq. (5) we have made the usual rotating wave

approximation (assuming ω − ωeg << ωeg) and defined

∆m?,m= ω − ωeg+ ωm− ωm?.

In the case of free atoms,

the atomic momentum and ma the atomic mass. The

eigenstates are then plane waves: |g,? κ? is coupled to

|e,? κ +?ks? with ∆? κ,? κ+?ks= ω − ωeg+¯ h? κ.?ks

recovers the first order Doppler and recoil frequency shifts.

Conversely in a tightly confining trap ?m|eiksˆ x|m??=

m? << ?m|eiksˆ x|m?, and the spectrum of the system

consists of a set of unshifted resonances corresponding

to each state of the external hamiltonian. Motional effects

then reduce to the line pulling of these resonances by small

(detuned) sidebands[5].

ˆHext =

¯ h2ˆ κ2

2ma

with ¯ hˆ? κ

ma

+

¯ hk2

2ma. One

s

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III. Periodic potential

A. Eigenstates and coupling by the probe laser

We now consider the case of atoms trapped in an optical

lattice. As is clear from eq. (5), only the motion of the

atoms along the probe laser propagation axis plays a role

in the problem and we restrict the analysis to 1D2. We

assume that the lattice is formed by a standing wave

leading to the following external hamiltonian:

ˆHI

ext=¯ h2ˆ κ2

2ma

+U0

2(1 − cos(2klˆ x)),

(6)

where kl is the wave vector of the trap laser. The

eigenstates |n,q? and eigenenergies ¯ hωn,q of

derived from the Bloch theorem[29]. They are labelled by

two quantum numbers: the band index n and the quasi-

momentum q. Furthermore they are periodic functions of

q with period 2kl and the usual convention is to restrict

oneself to the first Brillouin zone q ∈] − kl,kl].

Following a procedure given in Ref.[30] a numerical

solution to this eigenvalue problem can be easily found

in the momentum representation. The atomic plane wave

with wave vector κ obeys

?¯ h2κ2

For each value of q, the problem then reduces to the

diagonalization of a real tridiagonal matrix giving the

eigenenergies and eigenvectors as a linear superposition

of plane waves:

ˆHI

ext|n,q?

=

ˆHI

extare

ˆHI

ext|κ? =

2ma

+U0

2

?

|κ? −U0

4(|κ + 2kl? + |κ − 2kl?).

(7)

=¯ hωI

?

n,q|n,q?

∞

Cn,κi,q|κi,q?,

|n,q?

i=−∞

(8)

with κi,q = q + 2ikl. For each value of q one obtains

a discrete set of energies ¯ hωI

which are real and normalized such that?

values of U0. Except when explicitly stated, all numerical

values throughout the paper are given for87Sr at a lattice

laser wavelength 813nm which corresponds to the magic

wavelength reported in Ref.[31]. In frequency units Er

then corresponds to 3.58kHz. In figure3 is shown the

width (|ωI

a function of U0in units of Er and in frequency units.

Substituting ?m| → ?n,q| and |m?? → |n?,q?? in eq. (5),

the action of the probe laser is described by the coupled

equations

n,qand coefficients Cn,κi,q,

iC2

n,κi,q= 1.

In figures 1 and 2 are shown ¯ hωI

n,qand C0,κi,qfor various

n,q=kl− ωI

n,q=0|) of the lowest energy bands as

i ˙ ag

n,q

=

?

?

n?

Ωn?,n∗

q

2

ei∆n?,n

q

tae

n?,q+ks

(9)

i ˙ ae

n,q+ks

=

n?

Ωn,n?

q

2

e−i∆n,n?

q

tag

n?,q

,

2See section V for a brief discussion of the 3D problem.

Fig. 1.

(left) and U0= 10Er (right). Each state |n,q0? is coupled to all the

states |n?,q0+ ks? by the probe laser.

Band structure for two different lattice depth: U0 = 2Er

Fig. 2.

U0 = 10Er (right). The bold vertical lines illustrate the case q =

−kl/2. The dotted lines delimit the Brillouin zones. For a state |n =

0,q = akl? with a ∈]−1,1] the solid envelope gives the contribution

of the plane waves |κi,akl= akl+ 2ikl?.

C0,κi,qfor two different lattice depth: U0= 2Er (left) and

with Ωn,n?

ωI

Bloch vectors in (8), the translation in momentum space

eiksˆ xdue to the probe laser leads to the coupling of a

given state |n,q? to the whole set |n?,q + ks? (see figure

1) with a coupling strength Ωn?,n

to the atomic resonance ωI

depend on n, n?and q and to go further we have to make

assumptions on the initial state of the atoms in the lattice.

q

= Ω?

iCn?,κi,qCn,κi,q+ksand ∆n,n?

n,q+ks. As expected from the structure of the

q

= ω−ωeg+

n?,q− ωI

q

and a shift with respect

n?,q+ks− ωI

n,q. Both quantities

B. Discussion

We first consider the case where the initial state is a

pure |n,q? state. The strengths of the resonances Ωn,n?

shown in figure 4 for the case n = 0 and various values of

q. At a growing lattice depth Ωn,n?

of q and the strength of all ”sidebands” (n?− n ?= 0)

q

are

q

become independent

Fig. 3.

U0 in units of Er/¯ h (left scale) and in frequency units (right scale).

Lowest four band widths as a function of the lattice depth

949

Page 4

Fig. 4.

(n = 0 → n?) for an atom prepared in state |n = 0,q = −kl? (bold

lines), |n = 0,q = −kl/2? and |n = 0,q = kl/2? (thin lines). Right:

detuning of the first two sidebands for an atom prepared in state

|n = 0,q = −kl? (bold lines) and |n = 0,q = 0? (thin lines) in units

of Er/¯ h (left scale) and in frequency units (right scale).

Left: Relative strength of the transitions to different bands

Fig. 5.

lattice depth U0 = 10Er. Left scale: in units of Er/¯ h. Right scale:

in frequency units.

Shift of the ”carrier” resonance in the first band for a

asymptotically decreases as U−|n?−n|/4

of the ”carrier” (n?= n). The frequency separation of

the resonances rapidly increases with U0 (Fig. 4). For

U0 as low as 5Er, this separation is of the order of

10kHz. For narrow resonances (which are required for an

accurate clock) they can be treated separately and the

effect of the sidebands on the carrier is negligible. If for

example one reaches a carrier width of 10Hz, the sideband

pulling is of the order of 10−5Hz. On the other hand, the

”carrier” frequency is shifted from the atomic frequency

by ωI

the order of the width of the nthband (Fig. 5 and 3). It can

be seen as a residual Doppler and recoil effect for atoms

trapped in a lattice and is a consequence of the complete

delocalisation of the eigenstates of the system over the

lattice. The ”carrier” shift is plotted in figure 5 for the

case n = 0 and U0= 10Er. For this shift to be as small

as 5mHz over the whole lowest band, which corresponds

in fractional units to 10−17for Sr atoms probed on the

1S0−3P0transition, the lattice depth should be at least

90Er(Fig. 3).

Another extreme situation is the case where one band is

uniformly populated. In this case the ”carrier” shift aver-

aged over q cancels and one can hope to operate the clock

at a much lower U0than in the previous case. The problem

is then the ultimate linewidth that can be achieved in the

system, which is of the order of the width of the band and

is reminiscent of Doppler broadening. This is illustrated in

0

for the benefit

n,q+ks−ωI

n,qdue to the band structure. This shift is of

Fig. 6.

uniformely populated for Ω = 10Hz and U0 = 20Er, 30Er and

40Er. The duration of the interaction is such that the transition

probability Pe is maximized at resonance.

Expected resonances in the case where the first band is

figure 6 for which we have computed the expected ”carrier”

resonances in the case where the lowest band is uniformly

populated, by numerically solving equations (5). This was

done for a Rabi frequency Ω = 10Hz and an interaction

duration which is adjusted for each trapping depth so as

to maximize the transition probability at zero detuning.

We have checked that all resonances plotted in figure 6

are not shifted to within the numerical accuracy (less

than 10−5Hz). However, at decreasing U0 the contrast

of the resonance starts to drop for U0 < 40Er and

the resonance broadens progressively, becoming unusable

for precise spectroscopy when the width of the energy

band reaches the Rabi frequency. To get more physical

insight into this phenomenon, let’s consider the particular

example of this uniform band population where one well

of the lattice is initially populated. This corresponds to

a given relative phase of the Bloch states such that the

interference of the Bloch vectors is destructive everywhere

except in one well of the lattice. The time scale for

the evolution of this relative phase is the inverse of

the width of the populated energy band which then

corresponds to the tunnelling time towards delocalization

(once the relative phases have evolved significantly, the

destructive/constructive interferences of the initial state

no longer hold). The broadening and loss of contrast shown

in figure 6 can be seen as the Doppler effect associated

with this tunnelling motion.

The two cases discussed above (pure |n,q? state and uni-

form superposition of all states inside a band:?dq|n,q?)

populating only the bottom band. They illustrate the

dilemma one has to face: either the resonance is affected

by a frequency shift of the order of the width of the bottom

band (pure state), or by a braoadening of the same order

(superposition state), or by a combination of both (general

case). In either case the solution is to increase the trap

depth in order to decrease the energy width of the bottom

band.

In the experimental setup described in [31] about 90% of

the atoms are in the lowest band and can be selected by an

adequate sequence of laser pulses. The residual population

correspond to the two extremes one can obtain when

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Page 5

Fig. 7. External potential seen by the atoms in the case of a vertical

lattice (U0 = 5Er). An atom initially trapped in one well of the

lattice will end-up in the continuum by tunnel effect. For U0= 5Er

the lifetime of the quasi-bound state of each well is about 1010s.

of excited bands can then be made negligible (< 10−3).

On the other hand, knowing and controlling with accuracy

the population of the various |q? states in the ground

band is a difficult task. The actual initial distribution of

atomic states will lie somewhere between a pure state in

the bottom band and a uniform superposition of all states

in the bottom band. If we assume that the population

of the |q? states in the ground band can be controlled

so that the frequency shift averages to within one tenth

of the band width, then a fractional inaccuracy goal of

10−17implies U0= 70Er or more. Note that due to the

exponential dependence of the width of the ground band

on U0 (see figure 3) the required lattice depth is largely

insensitive to an improvement in the control of the initial

state. If for example the averaging effect is improved down

to 1% the depth requirement drops from 70Erto 50Er.

Consequently, operation of an optical lattice clock requires

relatively deep wells and correspondingly high laser power,

which, in turn, is likely to lead to other difficulties as

described in the introduction.

Fortunately, the requirement of deep wells can be

significantly relaxed by adding a constant acceleration to

the lattice, as described in the next section.

IV. Periodic potential in an accelerated frame

A. Wannier-Stark states and coupling by the probe laser

The shift and broadening encountered in the previous

section are both due to site-to-site tunnelling and to the

corresponding complete delocalization of the eigenstates

of the lattice. As is well-known from solid-state physics,

one way to localize the atoms is to add a linear component

to the Hamiltonian[27], [28]: adjacent wells are then

shifted in energy, which strongly inhibits tunnelling. In

this section we study the case where the lattice and probe

laser are oriented vertically so that gravity plays the role of

this linear component. The external hamiltonian is then:

ˆHII

ext=¯ h2ˆ κ2

2ma

+U0

2(1 − cos(2klˆ x)) + magˆ x,

(10)

with g the acceleration of the Earth’s gravity. This

hamiltonian supports no true bound states, as an atom

initially confined in one well of the lattice will end up

in the continuum due to tunnelling under the influence

of gravity (Fig. 7). This effect is known as Landau-Zener

tunnelling and can be seen as non-adiabatic transitions

between bands induced by the linear potential in the Bloch

representation[32], [33], [34], [35], [28]. The timescale for

this effect however increases exponentially with the depth

of the lattice and for the cases considered here is orders of

magnitude longer than the duration of the experiment3.

In the case of Sr in an optical lattice, and for U0as low as

5Er, the lifetime of the ground state of each well is about

1010s! The coupling due to gravity between the ground

and excited bands can therefore be neglected here. In the

frame of this approximation the problem of finding the

”eigenstates” ofˆHII

sub-space restricted to the ground band[36], [37] (we drop

the band index in the following to keep notations as simple

as possible). We are looking for solutions to the eigenvalue

equation, of the form:

ˆHII

ext|Wm?

|Wm?

In eq. (11) the |q? are the Bloch eigenstates ofˆHI

section III) for the bottom energy band (n = 0), m is a

new quantum number, and the bm(q) are periodic: bm(q+

2ikl) = bm(q). After some algebra, eq. (11) reduce to the

differential equation

extreduces to its diagonalization in a

=¯ hωII

?kl

m|Wm?

dq bm(q)|q?.

(11)

=

−kl

ext(c.f.

¯ h(ωI

q− ωII

m)bm(q) + imag∂qbm(q) = 0 (12)

where ωI

section III. Note that equations (11) and (12) only hold in

the limit where Landau-Zener tunnelling between energy

bands is negligible. Otherwise, terms characterising the

contribution of the other bands must be added and the

description of the quasi-bound states is more complex[38],

[30], [28]. In our case the periodicity of bm(q) and a

normalization condition lead to a simple solution of the

form

qis the eigenvalue of the Bloch state |n = 0,q? of

ωII

m

=

ωII

0+ m∆g

1

√2kl

?kl

(13)

bm(q)=

e−

i¯ h

mag(qωII

m−γq)

with the definitions ωII

and ∂qγq= ωI

called Wannier-Stark states and their wave functions are

plotted in figure 8 for various trap depths. In the position

representation |Wm? exhibits a main peak in the mth

well of the lattice and small revivals in adjacent wells.

These revivals decrease exponentially at increasing lattice

depth. At U0= 10Erthe first revival is already a hundred

times smaller than the main peak. Conversely, in the

0 =

1

2kl

−kldq ωI

q, ¯ h∆g= magλl/2,

qwith γ0= 0. The |Wm? states are usually

3This exponential increase is true on average only and can be

modified for specific values of U0 by a resonant coupling between

states in distant wells[38], [35], [28].

951