energetically forbidden. The two atoms preferentially propagate through the lattice together,
as reflected in increasing weights on the diagonal of the correlation matrix. For the strongest
interactions, the particles form a repulsively bound
pair with effective single-particle behavior (38).
The two-particle dynamics may be described as a
quantum walk of the bound pair (19, 20) at a decreased tunneling rate Jpair, which reduces to the
second-order tunneling (39) Jpair = (2J2/U) << J
for large values of u.
The formation of repulsively bound pairs and
their coherent dynamics can be observed in
two-particle Bloch oscillations. We focused on
the dynamics of two particles initially prepared
on the same site with a gradient D ≈ 0.5 (Fig. 4).
In the weakly interacting regime (u = 0.3), both
particles undergo symmetric Bloch oscillations
as in the single-particle case, and we observed a
high-quality revival after one Bloch period. For
intermediate interactions (u = 2.4), the density
evolution is very complex: In this regime where
J, U, and E are similar in magnitude, states both
with and without double occupancy are energetically allowed and contribute to the dynamics. The skew to the right against the applied
force is caused by resonant long-range tunneling of single particles over several sites (40, 41)
and agrees with numerical simulation. When the
interactions are sufficiently strong (u = 3.5), the
pairs of atoms are tightly bound by the repulsive
interaction and behave like a single composite
particle. However, the effective gradient has doubled with respect to the single-particle case, and
the pairs perform Bloch oscillations at twice the
fundamental frequency and reduced spatial amplitude. The frequency doubling of Bloch oscillations was predicted for electron systems (42)
and cold atoms (20, 40) and has recently been
simulated with photons in a waveguide array
(24). Throughout the breathing motion, the
repulsively bound pairs themselves undergo coherent dynamics and delocalize without unbinding. The clean revival after half a Bloch period
directly demonstrates the entanglement of atom
pairs during the oscillation.
Quantum walks of ultracold atoms in optical
lattices offer an ideal starting point for the “
bottom-up” study of many-body quantum dynamics. The
present two-particle implementation provides intuitive access to essential features of many-body
systems, such as localization caused by interactions or fermionization of bosons. Such microscopic features, when scaled to larger system sizes,
manifest in emergent phenomena—for example,
quantum phase transitions, quasi-particles, or
superfluidity—as observed in other cold atom experiments. The particle-by-particle assembly of
interacting systems may give access to the crossover from few- to many-body physics and may
reveal the microscopic details of disordered
quantum systems (21) and many-body quench
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Ultracold Atoms, the Army Research Office with funding from the
DARPA OLE program and a MURI program, an Air Force Office of
Scientific Research MURI program, the Gordon and Betty Moore
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through the NDSEG program (M.E. T.), a NSF Graduate Research
Fellowship (M.R.), and the Pappalardo Fellowship in Physics (Y.L.).
Materials and Methods
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25 August 2014; accepted 4 February 2015
Coulomb crystallization of highly
L. Schmöger,1,2 O. O. Versolato,1,2 M. Schwarz,1,2 M. Kohnen,2 A. Windberger,1
B. Piest,1 S. Feuchtenbeiner,1 J. Pedregosa-Gutierrez,3 T. Leopold,2
P. Micke,1,2 A. K. Hansen,4† T. M. Baumann,5 M. Drewsen,4 J. Ullrich,2
P. O. Schmidt,2,6 J. R. Crespo López-Urrutia1‡
Control over the motional degrees of freedom of atoms, ions, and molecules in a field-free
environment enables unrivalled measurement accuracies but has yet to be applied to
highly charged ions (HCIs), which are of particular interest to future atomic clock designs
and searches for physics beyond the Standard Model. Here, we report on the Coulomb
crystallization of HCIs (specifically 40Ar13+) produced in an electron beam ion trap and
retrapped in a cryogenic linear radiofrequency trap by means of sympathetic motional cooling
through Coulomb interaction with a directly laser-cooled ensemble of Be+ ions. We also
demonstrate cooling of a single Ar13+ ion by a single Be+ ion—the prerequisite for quantum logic
spectroscopy with a potential 10−19 accuracy level. Achieving a seven-orders-of-magnitude
decrease in HCI temperature starting at megakelvin down to the millikelvin range removes the
major obstacle for HCI investigation with high-precision laser spectroscopy.
Methods to simultaneously control both internal electronic and motional degrees of freedom of individual atoms, mole- cules, and low-charge-state ions in traps (1) have enabled unparalleled measure-
ment accuracies. Prime examples include optical
atomic clocks operating at an accuracy level of a
few parts per 10−18 (2, 3), which is sufficient to
measure subtle effects of relativity (4), achieve
sensitivity to geodesic gravitational potential dif-
ferences of Earth, and set upper limits on possible
temporal or spatial variations of fundamental
constants (5, 6). Most laser spectroscopy work on
trapped samples has explored a small class of
atoms and atomic ions—namely, the hydrogen
atom, the alkali/alkaline-earth atoms/ions, and a