Minimizing the required trap depth in optical lattice clocks
ABSTRACT We study the trap depth requirement for the realization of an optical clock using atoms confined in a lattice. We show that sitetosite tunnelling leads to a residual sensitivity to the atom dynamics hence requiring large depths (50 to 100 E_{r} 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, twophoton 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 5 E_{r} for Sr) can be used for the same accuracy goal
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Article: Molecular Beams
M. A. Yaffee, E. F. Taylor, S. G. Kukolich, R. D. Posner, C. L. Searle, R. Golub, R. S. Badessa, Thornburg, C. O, G. L. Guttrich, J. F. Brenner, Johnston, W. D, K. W. Billman, J. R. Zacharias, J. G. King[Show abstract] [Hide abstract]
ABSTRACT: Contains research objectives and reports on six research projects.01/1957;  SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: A sharp resonance line that appears in threephoton transitions between the 1S0 and 3P0 states of alkaline earth and Yb atoms is proposed as an optical frequency standard. This proposal permits the use of the even isotopes, in which the clock transition is narrower than in proposed clocks using the odd isotopes and the energy interval is not affected by external magnetic fields or the polarization of trapping light. With this method, the width and the rate of the clock transition can, in principle, be continuously adjusted from the MHz level to submHz without loss of signal amplitude by varying the intensities of the three optical beams. Doppler and recoil effects can be eliminated by proper alignment of the three optical beams or by point confinement in a lattice trap. Lightshift effects on the clock accuracy can be limited to below a part in 10(18).Physical Review Letters 03/2005; 94(5):050801. · 7.73 Impact Factor  SourceAvailable from: christian.j.borde.free.fr
Article: Atomic clocks and inertial sensors
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ABSTRACT: We show that the language of atom interferometry provides a unified picture for microwave and optical atomic clocks as well as for gravitoinertial sensors. The sensitivity and accuracy of these devices is now such that a new theoretical framework common to all these interferometers is required that includes: (a) a fully quantum mechanical treatment of the atomic motion in free space and in the presence of a gravitational field (most coldatom interferometric devices use atoms in ``free fall'' in a fountain geometry); (b) an account of simultaneous actions of gravitational and electromagnetic fields in the interaction zones; (c) a second quantization of the matter fields to take into account their fermionic or bosonic character in order to discuss the role of coherent sources and their noise properties; (d) a covariant treatment including spin to evaluate general relativistic effects. A theoretical description of atomic clocks revisited along these lines is presented, using both an exact propagator of atom waves in gravitoinertial fields and a covariant Dirac equation in the presence of weak gravitational fields. Using this framework, recoil effects, spinrelated effects, beam curvature effects, the sensitivity to gravitoinertial fields and the influence of the coherence of the atom source are discussed in the context of present and future atomic clocks and gravitoinertial sensors.Metrologia 02/2003; 39(5):435. · 1.90 Impact Factor
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 sitetosite 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,
twophoton 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 LambDicke 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 ultracold 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 stateoftheart 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 socalled ”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
947
0780390520/05/$20.00 © 2005 IEEE.
Page 2
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, twophoton 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 sitetosite 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 wellknown WannierStark 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 twolevel system g? and e? with
energy difference ¯ hωeg. The internal Hamiltonian is:
ˆHi= ¯ hωege??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? obeyingˆHextm? =
¯ hωmm? (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−iωmtm,g? + ae
me−i(ωeg+ωm)tm,e? (3)
i¯ h∂
∂tψat? = (ˆHext+ˆHi+ˆHs)ψat?,
(4)
leading to the following set of coupled equations
i ˙ ag
m
=
?
?
m?
Ω∗
2ei∆m?,mt?me−iksˆ xm??ae
Ω
2e−i∆m,m?t?meiksˆ xm??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 ?meiksˆ xm??=
m? << ?meiksˆ xm?, 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
948
Page 3
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
extn,q?
=
ˆHI
extare
ˆHI
extκ? =
2ma
+U0
2
?
κ? −U0
4(κ + 2kl? + κ − 2kl?).
(7)
=¯ hωI
?
n,qn,q?
∞
Cn,κi,qκi,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:?dqn,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
950
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 endup in the continuum by tunnel effect. For U0= 5Er
the lifetime of the quasibound 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. WannierStark states and coupling by the probe laser
The shift and broadening encountered in the previous
section are both due to sitetosite tunnelling and to the
corresponding complete delocalization of the eigenstates
of the lattice. As is wellknown from solidstate 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 LandauZener
tunnelling and can be seen as nonadiabatic 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
subspace 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
extWm?
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
mWm?
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 LandauZener tunnelling between energy
bands is negligible. Otherwise, terms characterising the
contribution of the other bands must be added and the
description of the quasibound 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 WannierStark 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