cond-mat/somewhere
‘Stückelberg interferometry’ with ultracold molecules
M. Mark,1 T. Kraemer,1 P. Waldburger,1 J. Herbig,1 C. Chin,1,3 H.-C. Nägerl,1 R. Grimm1,2
Institut für Experimentalphysik und Forschungszentrum für Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
Physics Department and James Franck Institute, University of Chicago, Chicago, IL 60637, USA
(Dated: August 18, 2021)
We report on the realization of a time-domain ‘Stückelberg interferometer’, which is based on
the internal state structure of ultracold Feshbach molecules. Two subsequent passages through a
weak avoided crossing between two different orbital angular momentum states in combination with
a variable hold time lead to high-contrast population oscillations. This allows for a precise determi-
nation of the energy difference between the two molecular states. We demonstrate a high degree of
control over the interferometer dynamics. The interferometric scheme provides new possibilities for
precision measurements with ultracold molecules.
PACS numbers: 34.50.-s, 05.30.Jp, 32.80.Pj, 67.40.Hf
The creation of molecules on Feshbach resonances in
atomic quantum gases has opened up a new chapter in
the field of ultracold matter [1]. Molecular quantum gases
are now readily available in the lab for various applica-
tions. Prominent examples are given by the creation of
strongly interacting many-body systems based on molec-
ular Bose-Einstein condensates [2], experiments on few-
body collision physics [3], the realization of molecular
matter-wave optics [4], and by the demonstration of ex-
otic pairs in optical lattices [5]. Recent experimental
progress has shown that full control over all degress of
freedom can be expected for such molecules [6, 7, 8]. Ul-
tracold molecular samples with very low thermal spread
and long interaction times could greatly increase the sen-
sitivity in measurements of fundamental physical prop-
erties such as the existence of an electron dipole moment
[9] and a possible time-variation of the fine-structure con-
stant [10, 11].
Most of today’s most accurate and precise measure-
ments rely on interferometric techniques applied to ultra-
cold atomic systems. For example, long coherence times
in atomic fountains or in optical lattices allow ultrapre-
cise frequency metrology [12, 13]. Molecules, given their
rich internal structure, greatly extend the scope of possi-
ble precision measurements. Molecular clocks, for exam-
ple, may provide novel access to fundamental constants
and interaction effects, different from atomic clocks. The
fast progress in preparing cold molecular samples thus
opens up fascinating perspectives for precision interfer-
ometry. Recently, the technique of Stark deceleration has
allowed a demonstration of Ramsey interferometry with a
cold and slowed molecular beam [10]. Ultracold trapped
molecular ensembles are expected to further enhance the
range of possible measurements.
In this Letter, we report on the realization of an inter-
nal-state interferometer with ultracold Cs2 molecules. A
weak avoided crossing is used as a ‘beam splitter’ for
molecular states as a result of partial Landau-Zener tun-
10 12 14 16 18 20
magnetic field (G)
FIG. 1: (color online). (a) Molecular energy structure below
the dissociation threshold showing all molecular states up ℓ=
8. The relevant states for the present experiment (solid lines)
are labeled |g〉, |g′〉, and |l〉. Molecules in state |g〉 or |l〉
are detected upon dissociation as shown in (b) and (c). The
crossing used for the interferometer is the one between |g′〉 and
|l〉 near 11.4G. Initially, ultracold molecules are generated in
state |g〉 on the Feshbach resonance at 19.8G.
neling when it is traversed by means of an appropriately
chosen magnetic field ramp. Using the avoided crossing
twice, first for splitting, and then for recombination of
molecular states, leads to the well-known ‘Stückelberg
oscillations’ [14]. We thus call our scheme a ‘Stückelberg
interferometer’. Our realization of this interferometer al-
lows full control over the interferometer dynamics. In
particular, the hold time between splitting and recombi-
nation can be freely chosen. In analogy to the well-known
Ramsey interferometer [15] the acquired interferometer
phase is mapped onto the relative populations of the two
output states that can be well discriminated upon molec-
ular dissociation. To demonstrate the performance of the
Stückelberg interferometer we use it for precision molec-
ular spectroscopy to determine the position and coupling
strength of the avoided crossing.
http://arxiv.org/abs/0704.0653v1
hold time 
FIG. 2: (a) Scheme of the ‘Stückelberg interferometer’. By
ramping the magnetic field over the avoided crossing at Bc
at a rate near the critical ramp rate Rc the population in the
initial molecular state is coherently split. ∆E is the bind-
ing energy difference at the given hold field B0. After the
hold time τ a reverse ramp coherently recombines the two
populations. The populations in the two ‘output ports’ are
then determined as a function of acquired phase difference
φ ∝ ∆E × τ . (b) Corresponding magnetic field ramp.
The energy structure of weakly bound Cs2 dimers in
the relevant range of low magnetic field strength is shown
in Fig. 1 [16]. Zero binding energy corresponds to the
threshold of dissociation into two free Cs atoms in the
lowest hyperfine sublevel |F = 3,mF = 3〉 and thus to
the zero-energy collision limit of two atoms. The states
relevant for this work are labelled by |g〉, |g′〉, and |l〉 [17].
While |g〉 and |g′〉 are g-wave states with orbital angular
momentum ℓ=4, the state |l〉 is an l-wave state with a
high orbital angular momentum of ℓ= 8 [18]. Coupling
with ∆ℓ=4 between s-wave atoms and g-wave molecules
and between g- and l-wave states is a result of the strong
indirect spin-spin interaction between two Cs atoms [16].
The starting point for our experiments is a Bose-
Einstein condensate (BEC) with ∼ 2 × 105 Cs atoms in
the |F = 3,mF = 3〉 ground state confined in a crossed-
beam dipole trap generated by a broad-band fiber laser
with a wavelength near 1064nm [19, 20]. The BEC al-
lows us to efficiently produce molecules on a narrow Fesh-
bach resonance at 19.84 G [21] in an optimized scheme
as described in Ref. [22]. With an efficiency of typically
25% we produce a pure molecular ensemble with up to
2.5×104 ultracold molecules all in state |g〉, initially close
to quantum degeneracy [21]. The following experiments
are performed on the molecular ensemble in free fall.
During the initial expansion to a 1/e-diameter of about
28µm along the radial and about 46µm along the axial
direction the peak density is reduced to 1× 1011cm−3 so
that molecule-molecule interactions [3] can be neglected
on the timescale of the experiment.
The molecules can now be transferred to any one of
the molecular states shown in Fig. 1 with near 100%
efficiency by controlled ‘jumping’ or adiabatic following
at the various crossings [23]. When the magnetic field
strength is decreased, the molecules first encounter the
crossing at 13.6G. At all ramp rates used in our present
experiments the passage through this crossing takes place
in a fully adiabatic way. The molecules are thus trans-
ferred from |g〉 to |g′〉 along the upper branch of the cross-
ing. They then encounter the next crossing at a magnetic
field of Bc ≈ 11.4G. We accidentally found this weak
crossing in our previous magnetic moment measurements
[3, 23]. This allowed the identification of the l-wave state
|l〉 [18].
This crossing between |g′〉 and |l〉 plays a central role
in the present experiment. It can be used as a tunable
‘beam splitter’, which allows adiabatic transfer, coherent
splitting, as will be shown below, or diabatic transfer for
the molecular states involved, depending on the chosen
magnetic ramp rate near the crossing. We find that a crit-
ical ramp rate of Rc ≈ 14G/ms leads to a 50/50-splitting
into |g′〉 and |l〉 [23]. Using the well-known Landau-Zener
formula and an estimate for the difference in magnetic
moment for states |g′〉 and |l〉 [18] we determine the cou-
pling strength V between |g′〉 and |l〉 to ∼h×15 kHz.
We state-selectively detect the molecules by ramp-
ing up the magnetic field to bring the molecules above
the threshold for dissociation. There the quasi-bound
molecules decay into the atomic scattering continuum.
For state |g〉 dissociation is observed for magnetic fields
above the 19.84G position of the corresponding Feshbach
resonance. Fig. 1 (b) shows a typical absorption image of
the resulting atom cloud [21]. For state |l〉 dissociation is
observed above 16.5G. The molecular states can thus be
easily discriminated by the different magnetic field values
needed for dissociation. Moreover, the expansion pattern
is qualitatively different from the one connected to state
|g〉. The absorption image in Fig. 1 (c) shows an ex-
panding ‘bubble’ of atoms with a relatively large kinetic
energy of about kB×20µK per atom. Here, kB is Boltz-
mann’s constant. We find that significant dissociation
occurs only when the state |l〉 couples to a quasi-bound
g-wave state about h×0.7 MHz above threshold [24]. The
resulting bubble is not fully spherically symmetric, which
indicates higher partial-wave contributions [25]. The dif-
ferent absorption patterns allow us to clearly distinguish
between the two different dissociation channels in a sin-
gle absorption picture when the magnetic field is ramped
up to ∼ 22 G. These dissociation channels serve as the
interferometer ‘output ports’.
The interferometer is based on two subsequent pas-
sages through the crossing following the scheme illus-
trated in Fig. 2. For an initial magnetic field above the
crossing a downward magnetic-field ramp brings the ini-
tial molecular state into a coherent superposition of |g′〉
and |l〉. After the ramp the field is kept constant at a
FIG. 3: (color online). Series of dissociation patterns showing about one oscillation period with ∆E/h = 155 kHz at a hold field
of 11.19G. From one picture to the next the hold time τ is increased by 1µs. The first and the last of the absorption images
mainly show dissociation of l-wave molecules, whereas the central image shows predominant dissociation of g-wave molecules.
130 140 150 160 170 180
=11.24 G
94(1) kHz
=11.20 G
132(1) kHz
=11.17 G
169(1) kHz
 hold time (µs)
FIG. 4: Interferometer fringes for magnetic hold fields B0 be-
low the crossing of |g′〉 and |l〉. The g-wave molecular fraction
is plotted as a function of the hold time τ . Sinusoidal fits give
the oscillation frequency as indicated.
hold field B0 below the crossing for a variable hold time
τ . A differential phase φ is then accumulated between the
two components, which linearly increases with the prod-
uct of the binding energy difference ∆E and the hold
time τ . The magnetic field is then ramped back up, and
the second passage creates a new superposition state de-
pending on φ. For a 50/50-splitting ratio, this can lead to
complete destructive or constructive interference in the
two output ports and thus to high-contrast fringes as a
function of τ or ∆E. These fringes, resulting from two
passages through the same crossing, are analogous to the
well-known Stückelberg oscillations in collision physics
[14, 26] or in the physics with Rydberg atoms [27, 28].
Note that our realization of a ‘Stückelberg interferome-
ter’ gives full control over the interferometer dynamics
by appropriate choice of ramp rates and magnetic offset
fields.
A typical ramp sequence, as shown in Fig. 2 (b), starts
with a sample of |g′〉 molecules at a magnetic field of
11.6G about 250mG above the crossing. At the critical
ramp rate Rc we ramp the magnetic field within about
50µs to a hold field B0 below the crossing. After the vari-
able hold time τ we reverse the ramp and transverse the
crossing a second time at the critical ramp rate. The out-
put of the interferometer is detected by rapidly ramping
the magnetic field up to 22G and by imaging the pattern
of dissociating |l〉 and |g〉 molecules.
For one period of oscillation the dependence of the dis-
sociation pattern on the hold time τ is demonstrated by
the series of absorption images shown in Fig. 3. The hold
time is increased in steps of 1µs while the entire prepa-
ration, ramping, and detection procedure is repeated for
each experimental cycle, lasting about 20 s. The molecu-
lar population oscillates from being predominantly l-wave
to predominatly g-wave and back. For a quantitative
analysis of the molecular population in each output port
we fit the images with a simple model function [29] and
extract the fraction of molecules in each of the two out-
put ports. Fig. 4 shows the g-wave molecular population
as a function of hold time τ for various hold fields B0
corresponding to different ∆E. The existence of these
Stückelberg oscillations confirms that coherence is pre-
served by the molecular beam splitter. Their high con-
trast ratio shows that near 50/50-splitting is achieved.
Sinusoidal fits to the data allow for an accurate determi-
nation of the oscillation frequency and hence of ∆E.
Fig. 5 shows ∆E as a function of magnetic field
strength near the avoided crossing. For magnetic fields
below the crossing we obtain ∆E as described before. For
magnetic fields above the crossing, we invert the interfer-
ometric scheme. Molecules are first transferred from |g′〉
into |l〉 using a slow adiabatic ramp. The field is then
ramped up above the crossing with a rate near Rc, kept
constant for the variable time τ at the hold field B0 and
then ramped down to close the interferometer. An adia-
batic ramp through the crossing maps population in |g′〉
onto |l〉 and vice versa. Detection then proceeds as be-
fore.
For both realizations of the interferometer we obtain
high-contrast fringes even when it is not operated in the
Landau-Zener regime and the fast ramps are stopped
right at the crossing (see inset to Fig. 5). This allows
us to measure ∆E in the crossing region. A fit to the
data with a hyperbolic fit function according to the stan-
dard Landau-Zener model yields Bc = 11.339(1) G for
the position of the crossing, ∆µ = 0.730(6)µB for the
difference in magnetic moment of the two states involved
(µB is Bohr’s magneton), and V = h× 14(1) kHz for
the coupling strength. While the measured ∆µ agrees
reasonably well with the result from an advanced theo-
retical model of the Cs2 dimer [18], Bc and V cannot be
obtained from these calculations with sufficient accuracy.
The present interferometer allows us to observe up to
100 oscillations at 200 kHz. Shot-to-shot fluctuations in-
11.1 11.2 11.3 11.4 11.5 11.6
50 75 100 125 150 175
magnetic field (G)
 hold time  (µs)
FIG. 5: Interferometrically measured binding energy differ-
ence ∆E in the region of the crossing between states |g′〉 and
|l〉 as a function of magnetic field. Solid circles: Standard
ramp sequence of the interferometer. Open circles: Inverted
scheme for field values above the crossing. The one-sigma
statistical error from the sinusoidal fit is less than the size of
the symbols. The solid curve is a hyperbolic fit to the experi-
mental data. Inset: Oscillation at 26.6(3) kHz for a hold field
B0 = 11.34 G right on the crossing.
creasingly scramble the phase of the oscillations for longer
hold times until the phase appears to be fully randomized
while large amplitude variations for the molecular pop-
ulations persist. The peak-to-peak amplitude of these
fluctuations decays slowly and is still 50% of the initial
contrast after 1 ms. We attribute this phase scram-
bling to magnetic field noise that causes shot-to-shot
variations of ∆E, the same, however, for each molecule.
The large amplitude of these fluctuations indicates that
phase coherence is preserved within the molecular sam-
ple. We attribute the gradual loss of peak-to-peak am-
plitude to spatial magnetic field inhomogeneities. We
expect that straightforward technical improvements re-
garding the magnetic field stability and homogeneity and
applying the interferometer to trapped molecular samples
will allow us extend the hold times far into the millisec-
ond range. It will then be possible to measure ultraweak
crossings with coupling strengths well below h×1 kHz.
We have demonstrated a molecular Stückelberg inter-
ferometer with full control over the interferometer dy-
namics. The interferometer can be used as a spectros-
copic tool as it allows precise measurements of binding
energy differences of molecular states. In particular, the
technique can be employed to measure feeble interactions
between molecular states, such as parity non-conserving
interactions [30]. The sensitivity to detect ultraweak level
crossings, combined with long storage times in optical
molecule traps [3] or lattices [5, 6, 7], may allow to de-
tect interaction phenomena on the h× 1 Hz scale. In
view of the rapid progress in various preparation methods
for cold molecular samples, new tools for precision mea-
surements on molecular samples, such as our Stückelberg
interferometer, will open up exciting avenues for future
research.
We thank E. Tiesinga for discussions and for the-
oretical support. We acknowledge financial support
by the Austrian Science Fund (FWF) within SFB 15
(project part 16) and by the EU within the Cold
Molecules TMR Network under contract No. HPRN-CT-
2002-00290. M.M. and C.C. acknowledge support by
DOC [PhD-Program of the Austrian Academy of Science]
and the FWF Lise-Meitner program, respectively.
[1] For a review, see T. Köhler, K. Góral, and P.S. Julienne,
Rev. Mod. Phys. 78, 1311 (2006).
[2] See e.g. Ultracold Fermi Gases, Proceedings of the In-
ternational School of Physics “Enrico Fermi”, Course
CLXIV, Varenna, 20 - 30 June 2006, edited by M. In-
guscio, W. Ketterle, and C. Salomon, in press.
[3] C. Chin et al., Phys. Rev. Lett. 94, 123201 (2005).
[4] J.R. Abo-Shaeer et al. Phys. Rev. Lett. 94, 040405
(2005).
[5] K. Winkler et al., Nature 441, 853 (2006).
[6] G. Thalhammer et al., Phys. Rev. Lett. 96, 050402
(2006).
[7] T. Volz et al., Nature Physics 2, 692 (2006).
[8] K. Winkler et al., Phys. Rev. Lett. 98, 043201 (2007).
[9] J. J. Hudson, B. E. Sauer, M. R. Tarbutt, and E. A.
Hinds, Phys. Rev. Lett. 89, 023003 (2002).
[10] E. R. Hudson, H. J. Lewandowski, B. C. Sawyer, and J.
Ye, Phys. Rev. Lett. 96, 143004 (2006).
[11] C. Chin and V. V. Flambaum, Phys. Rev. Lett. 96,
230801 (2006).
[12] S. Bize et al., J. Phys. B 38, S449 (2005).
[13] M. M Boyd et al., Phys. Rev. Lett. 98, 083002 (2007).
[14] E.C.G. Stückelberg, Helv. Phys. Acta 5, 369 (1932).
[15] N.F. Ramsey, Molecular Beams (Oxford University Press,
London, 1956).
[16] C. Chin et al., Phys. Rev. A 70, 032701 (2004).
[17] Using the notation |f , mf ; l, ml〉 of Ref. [16], the
three states |g〉, |g′〉, and |l〉 correspond to |4, 4; 4, 2〉,
|6, 6; 4, 0〉, and |6, 3; 8, 3〉, respectively.
[18] E. Tiesinga (private communication).
[19] T. Kraemer et al., Appl. Phys. B 79, 1013 (2004).
[20] T. Weber et al., Science 299, 232 (2003).
[21] J. Herbig et al., Science 301, 1510 (2003).
[22] M. Mark et al., Europhys. Lett. 69, 706 (2005).
[23] M. Mark et al., manuscript in preparation.
[24] S. Knoop et al., manuscript in preparation.
[25] S. Dürr, T. Volz, and G. Rempe, Phys. Rev. A 70,
031601(R) (2004).
[26] E. E. Nikitin and S. Ya. Umanskii, Theory of Slow Atomic
Collisions (Springer, Berlin, 1984).
[27] M. C. Baruch and T. F. Gallagher, Phys. Rev. Lett. 68,
3515 (1992).
[28] S. Yoakum, L. Sirko, and P. M. Koch, Phys. Rev. Lett.
68, 1919 (1992).
[29] We model the dissociation pattern with appropriately
chosen spherical harmonic functions to account for the
angular distribution [25].
[30] E. D. Commins, Adv. At. Mol. Opt. Phys. 40, 1 (1999).

  We report on the realization of a time-domain `St\"uckelberg interferometer',
which is based on the internal state structure of ultracold Feshbach molecules.
Two subsequent passages through a weak avoided crossing between two different
orbital angular momentum states in combination with a variable hold time lead
to high-contrast population oscillations. This allows for a precise
determination of the energy difference between the two molecular states. We
demonstrate a high degree of control over the interferometer dynamics. The
interferometric scheme provides new possibilities for precision measurements
with ultracold molecules.

cond-mat/somewhere
‘Stückelberg interferometry’ with ultracold molecules
M. Mark,1 T. Kraemer,1 P. Waldburger,1 J. Herbig,1 C. Chin,1,3 H.-C. Nägerl,1 R. Grimm1,2
Institut für Experimentalphysik und Forschungszentrum für Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria
Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
Physics Department and James Franck Institute, University of Chicago, Chicago, IL 60637, USA
(Dated: August 18, 2021)
We report on the realization of a time-domain ‘Stückelberg interferometer’, which is based on
the internal state structure of ultracold Feshbach molecules. Two subsequent passages through a
weak avoided crossing between two different orbital angular momentum states in combination with
a variable hold time lead to high-contrast population oscillations. This allows for a precise determi-
nation of the energy difference between the two molecular states. We demonstrate a high degree of
control over the interferometer dynamics. The interferometric scheme provides new possibilities for
precision measurements with ultracold molecules.
PACS numbers: 34.50.-s, 05.30.Jp, 32.80.Pj, 67.40.Hf
The creation of molecules on Feshbach resonances in
atomic quantum gases has opened up a new chapter in
the field of ultracold matter [1]. Molecular quantum gases
are now readily available in the lab for various applica-
tions. Prominent examples are given by the creation of
strongly interacting many-body systems based on molec-
ular Bose-Einstein condensates [2], experiments on few-
body collision physics [3], the realization of molecular
matter-wave optics [4], and by the demonstration of ex-
otic pairs in optical lattices [5]. Recent experimental
progress has shown that full control over all degress of
freedom can be expected for such molecules [6, 7, 8]. Ul-
tracold molecular samples with very low thermal spread
and long interaction times could greatly increase the sen-
sitivity in measurements of fundamental physical prop-
erties such as the existence of an electron dipole moment
[9] and a possible time-variation of the fine-structure con-
stant [10, 11].
Most of today’s most accurate and precise measure-
ments rely on interferometric techniques applied to ultra-
cold atomic systems. For example, long coherence times
in atomic fountains or in optical lattices allow ultrapre-
cise frequency metrology [12, 13]. Molecules, given their
rich internal structure, greatly extend the scope of possi-
ble precision measurements. Molecular clocks, for exam-
ple, may provide novel access to fundamental constants
and interaction effects, different from atomic clocks. The
fast progress in preparing cold molecular samples thus
opens up fascinating perspectives for precision interfer-
ometry. Recently, the technique of Stark deceleration has
allowed a demonstration of Ramsey interferometry with a
cold and slowed molecular beam [10]. Ultracold trapped
molecular ensembles are expected to further enhance the
range of possible measurements.
In this Letter, we report on the realization of an inter-
nal-state interferometer with ultracold Cs2 molecules. A
weak avoided crossing is used as a ‘beam splitter’ for
molecular states as a result of partial Landau-Zener tun-
10 12 14 16 18 20
magnetic field (G)
FIG. 1: (color online). (a) Molecular energy structure below
the dissociation threshold showing all molecular states up ℓ=
8. The relevant states for the present experiment (solid lines)
are labeled |g〉, |g′〉, and |l〉. Molecules in state |g〉 or |l〉
are detected upon dissociation as shown in (b) and (c). The
crossing used for the interferometer is the one between |g′〉 and
|l〉 near 11.4G. Initially, ultracold molecules are generated in
state |g〉 on the Feshbach resonance at 19.8G.
neling when it is traversed by means of an appropriately
chosen magnetic field ramp. Using the avoided crossing
twice, first for splitting, and then for recombination of
molecular states, leads to the well-known ‘Stückelberg
oscillations’ [14]. We thus call our scheme a ‘Stückelberg
interferometer’. Our realization of this interferometer al-
lows full control over the interferometer dynamics. In
particular, the hold time between splitting and recombi-
nation can be freely chosen. In analogy to the well-known
Ramsey interferometer [15] the acquired interferometer
phase is mapped onto the relative populations of the two
output states that can be well discriminated upon molec-
ular dissociation. To demonstrate the performance of the
Stückelberg interferometer we use it for precision molec-
ular spectroscopy to determine the position and coupling
strength of the avoided crossing.
http://arxiv.org/abs/0704.0653v1
hold time 
FIG. 2: (a) Scheme of the ‘Stückelberg interferometer’. By
ramping the magnetic field over the avoided crossing at Bc
at a rate near the critical ramp rate Rc the population in the
initial molecular state is coherently split. ∆E is the bind-
ing energy difference at the given hold field B0. After the
hold time τ a reverse ramp coherently recombines the two
populations. The populations in the two ‘output ports’ are
then determined as a function of acquired phase difference
φ ∝ ∆E × τ . (b) Corresponding magnetic field ramp.
The energy structure of weakly bound Cs2 dimers in
the relevant range of low magnetic field strength is shown
in Fig. 1 [16]. Zero binding energy corresponds to the
threshold of dissociation into two free Cs atoms in the
lowest hyperfine sublevel |F = 3,mF = 3〉 and thus to
the zero-energy collision limit of two atoms. The states
relevant for this work are labelled by |g〉, |g′〉, and |l〉 [17].
While |g〉 and |g′〉 are g-wave states with orbital angular
momentum ℓ=4, the state |l〉 is an l-wave state with a
high orbital angular momentum of ℓ= 8 [18]. Coupling
with ∆ℓ=4 between s-wave atoms and g-wave molecules
and between g- and l-wave states is a result of the strong
indirect spin-spin interaction between two Cs atoms [16].
The starting point for our experiments is a Bose-
Einstein condensate (BEC) with ∼ 2 × 105 Cs atoms in
the |F = 3,mF = 3〉 ground state confined in a crossed-
beam dipole trap generated by a broad-band fiber laser
with a wavelength near 1064nm [19, 20]. The BEC al-
lows us to efficiently produce molecules on a narrow Fesh-
bach resonance at 19.84 G [21] in an optimized scheme
as described in Ref. [22]. With an efficiency of typically
25% we produce a pure molecular ensemble with up to
2.5×104 ultracold molecules all in state |g〉, initially close
to quantum degeneracy [21]. The following experiments
are performed on the molecular ensemble in free fall.
During the initial expansion to a 1/e-diameter of about
28µm along the radial and about 46µm along the axial
direction the peak density is reduced to 1× 1011cm−3 so
that molecule-molecule interactions [3] can be neglected
on the timescale of the experiment.
The molecules can now be transferred to any one of
the molecular states shown in Fig. 1 with near 100%
efficiency by controlled ‘jumping’ or adiabatic following
at the various crossings [23]. When the magnetic field
strength is decreased, the molecules first encounter the
crossing at 13.6G. At all ramp rates used in our present
experiments the passage through this crossing takes place
in a fully adiabatic way. The molecules are thus trans-
ferred from |g〉 to |g′〉 along the upper branch of the cross-
ing. They then encounter the next crossing at a magnetic
field of Bc ≈ 11.4G. We accidentally found this weak
crossing in our previous magnetic moment measurements
[3, 23]. This allowed the identification of the l-wave state
|l〉 [18].
This crossing between |g′〉 and |l〉 plays a central role
in the present experiment. It can be used as a tunable
‘beam splitter’, which allows adiabatic transfer, coherent
splitting, as will be shown below, or diabatic transfer for
the molecular states involved, depending on the chosen
magnetic ramp rate near the crossing. We find that a crit-
ical ramp rate of Rc ≈ 14G/ms leads to a 50/50-splitting
into |g′〉 and |l〉 [23]. Using the well-known Landau-Zener
formula and an estimate for the difference in magnetic
moment for states |g′〉 and |l〉 [18] we determine the cou-
pling strength V between |g′〉 and |l〉 to ∼h×15 kHz.
We state-selectively detect the molecules by ramp-
ing up the magnetic field to bring the molecules above
the threshold for dissociation. There the quasi-bound
molecules decay into the atomic scattering continuum.
For state |g〉 dissociation is observed for magnetic fields
above the 19.84G position of the corresponding Feshbach
resonance. Fig. 1 (b) shows a typical absorption image of
the resulting atom cloud [21]. For state |l〉 dissociation is
observed above 16.5G. The molecular states can thus be
easily discriminated by the different magnetic field values
needed for dissociation. Moreover, the expansion pattern
is qualitatively different from the one connected to state
|g〉. The absorption image in Fig. 1 (c) shows an ex-
panding ‘bubble’ of atoms with a relatively large kinetic
energy of about kB×20µK per atom. Here, kB is Boltz-
mann’s constant. We find that significant dissociation
occurs only when the state |l〉 couples to a quasi-bound
g-wave state about h×0.7 MHz above threshold [24]. The
resulting bubble is not fully spherically symmetric, which
indicates higher partial-wave contributions [25]. The dif-
ferent absorption patterns allow us to clearly distinguish
between the two different dissociation channels in a sin-
gle absorption picture when the magnetic field is ramped
up to ∼ 22 G. These dissociation channels serve as the
interferometer ‘output ports’.
The interferometer is based on two subsequent pas-
sages through the crossing following the scheme illus-
trated in Fig. 2. For an initial magnetic field above the
crossing a downward magnetic-field ramp brings the ini-
tial molecular state into a coherent superposition of |g′〉
and |l〉. After the ramp the field is kept constant at a
FIG. 3: (color online). Series of dissociation patterns showing about one oscillation period with ∆E/h = 155 kHz at a hold field
of 11.19G. From one picture to the next the hold time τ is increased by 1µs. The first and the last of the absorption images
mainly show dissociation of l-wave molecules, whereas the central image shows predominant dissociation of g-wave molecules.
130 140 150 160 170 180
=11.24 G
94(1) kHz
=11.20 G
132(1) kHz
=11.17 G
169(1) kHz
 hold time (µs)
FIG. 4: Interferometer fringes for magnetic hold fields B0 be-
low the crossing of |g′〉 and |l〉. The g-wave molecular fraction
is plotted as a function of the hold time τ . Sinusoidal fits give
the oscillation frequency as indicated.
hold field B0 below the crossing for a variable hold time
τ . A differential phase φ is then accumulated between the
two components, which linearly increases with the prod-
uct of the binding energy difference ∆E and the hold
time τ . The magnetic field is then ramped back up, and
the second passage creates a new superposition state de-
pending on φ. For a 50/50-splitting ratio, this can lead to
complete destructive or constructive interference in the
two output ports and thus to high-contrast fringes as a
function of τ or ∆E. These fringes, resulting from two
passages through the same crossing, are analogous to the
well-known Stückelberg oscillations in collision physics
[14, 26] or in the physics with Rydberg atoms [27, 28].
Note that our realization of a ‘Stückelberg interferome-
ter’ gives full control over the interferometer dynamics
by appropriate choice of ramp rates and magnetic offset
fields.
A typical ramp sequence, as shown in Fig. 2 (b), starts
with a sample of |g′〉 molecules at a magnetic field of
11.6G about 250mG above the crossing. At the critical
ramp rate Rc we ramp the magnetic field within about
50µs to a hold field B0 below the crossing. After the vari-
able hold time τ we reverse the ramp and transverse the
crossing a second time at the critical ramp rate. The out-
put of the interferometer is detected by rapidly ramping
the magnetic field up to 22G and by imaging the pattern
of dissociating |l〉 and |g〉 molecules.
For one period of oscillation the dependence of the dis-
sociation pattern on the hold time τ is demonstrated by
the series of absorption images shown in Fig. 3. The hold
time is increased in steps of 1µs while the entire prepa-
ration, ramping, and detection procedure is repeated for
each experimental cycle, lasting about 20 s. The molecu-
lar population oscillates from being predominantly l-wave
to predominatly g-wave and back. For a quantitative
analysis of the molecular population in each output port
we fit the images with a simple model function [29] and
extract the fraction of molecules in each of the two out-
put ports. Fig. 4 shows the g-wave molecular population
as a function of hold time τ for various hold fields B0
corresponding to different ∆E. The existence of these
Stückelberg oscillations confirms that coherence is pre-
served by the molecular beam splitter. Their high con-
trast ratio shows that near 50/50-splitting is achieved.
Sinusoidal fits to the data allow for an accurate determi-
nation of the oscillation frequency and hence of ∆E.
Fig. 5 shows ∆E as a function of magnetic field
strength near the avoided crossing. For magnetic fields
below the crossing we obtain ∆E as described before. For
magnetic fields above the crossing, we invert the interfer-
ometric scheme. Molecules are first transferred from |g′〉
into |l〉 using a slow adiabatic ramp. The field is then
ramped up above the crossing with a rate near Rc, kept
constant for the variable time τ at the hold field B0 and
then ramped down to close the interferometer. An adia-
batic ramp through the crossing maps population in |g′〉
onto |l〉 and vice versa. Detection then proceeds as be-
fore.
For both realizations of the interferometer we obtain
high-contrast fringes even when it is not operated in the
Landau-Zener regime and the fast ramps are stopped
right at the crossing (see inset to Fig. 5). This allows
us to measure ∆E in the crossing region. A fit to the
data with a hyperbolic fit function according to the stan-
dard Landau-Zener model yields Bc = 11.339(1) G for
the position of the crossing, ∆µ = 0.730(6)µB for the
difference in magnetic moment of the two states involved
(µB is Bohr’s magneton), and V = h× 14(1) kHz for
the coupling strength. While the measured ∆µ agrees
reasonably well with the result from an advanced theo-
retical model of the Cs2 dimer [18], Bc and V cannot be
obtained from these calculations with sufficient accuracy.
The present interferometer allows us to observe up to
100 oscillations at 200 kHz. Shot-to-shot fluctuations in-
11.1 11.2 11.3 11.4 11.5 11.6
50 75 100 125 150 175
magnetic field (G)
 hold time  (µs)
FIG. 5: Interferometrically measured binding energy differ-
ence ∆E in the region of the crossing between states |g′〉 and
|l〉 as a function of magnetic field. Solid circles: Standard
ramp sequence of the interferometer. Open circles: Inverted
scheme for field values above the crossing. The one-sigma
statistical error from the sinusoidal fit is less than the size of
the symbols. The solid curve is a hyperbolic fit to the experi-
mental data. Inset: Oscillation at 26.6(3) kHz for a hold field
B0 = 11.34 G right on the crossing.
creasingly scramble the phase of the oscillations for longer
hold times until the phase appears to be fully randomized
while large amplitude variations for the molecular pop-
ulations persist. The peak-to-peak amplitude of these
fluctuations decays slowly and is still 50% of the initial
contrast after 1 ms. We attribute this phase scram-
bling to magnetic field noise that causes shot-to-shot
variations of ∆E, the same, however, for each molecule.
The large amplitude of these fluctuations indicates that
phase coherence is preserved within the molecular sam-
ple. We attribute the gradual loss of peak-to-peak am-
plitude to spatial magnetic field inhomogeneities. We
expect that straightforward technical improvements re-
garding the magnetic field stability and homogeneity and
applying the interferometer to trapped molecular samples
will allow us extend the hold times far into the millisec-
ond range. It will then be possible to measure ultraweak
crossings with coupling strengths well below h×1 kHz.
We have demonstrated a molecular Stückelberg inter-
ferometer with full control over the interferometer dy-
namics. The interferometer can be used as a spectros-
copic tool as it allows precise measurements of binding
energy differences of molecular states. In particular, the
technique can be employed to measure feeble interactions
between molecular states, such as parity non-conserving
interactions [30]. The sensitivity to detect ultraweak level
crossings, combined with long storage times in optical
molecule traps [3] or lattices [5, 6, 7], may allow to de-
tect interaction phenomena on the h× 1 Hz scale. In
view of the rapid progress in various preparation methods
for cold molecular samples, new tools for precision mea-
surements on molecular samples, such as our Stückelberg
interferometer, will open up exciting avenues for future
research.
We thank E. Tiesinga for discussions and for the-
oretical support. We acknowledge financial support
by the Austrian Science Fund (FWF) within SFB 15
(project part 16) and by the EU within the Cold
Molecules TMR Network under contract No. HPRN-CT-
2002-00290. M.M. and C.C. acknowledge support by
DOC [PhD-Program of the Austrian Academy of Science]
and the FWF Lise-Meitner program, respectively.
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