Starting from this expression, we first calculate
the upper bound for r11 as a function of r00 for a
fully separable state with k = 1 (21). The bound,
which is tight, is shown as the solid thick blue
line in Fig. 4. Any state corresponding to a
point outside the blue shaded region necessarily contains at least two-particle entanglement.
The calculation can be extended to larger k, and
the green lines show the bounds for k = 8, 12, and
16. The experimental state with the highest fidelity
of 0.42 lies ~1 SD above the k = 12 curve, indicating that it contains at least 13 entangled atoms.
In our present setup, fidelity is limited by de-
coherence due to differential light shifts in the
dipole trap (21) and a probability of ~0.2 for
spontaneous emission from j1〉 during the QND
detection. For large atom numbers, spontaneous
emission from j0〉 will eventually become domi-
nant. Simulations show that, with state-of-the-art
optical cavities, the entanglement process can be
scaled up to ensembles with N > 104. Because
our method relies only on coherent evolution and
collective QND measurement, it can be adapted
to many systems and, in particular, to other forms
of cavity quantum electrodynamics, including ad-
dressable qubits such as ions in optical cavities
(31) or superconducting qubits in microwave
cavities (32), as long as they are indistinguishable
by cavity measurement. Furthermore, by includ-
ing multiple rotations and QND detection inter-
vals, or by combining it with other entanglement
schemes such as cavity-induced spin squeezing
(6), our scheme can be extended to a large range
of entangled states. In combination with the in-
herent single-particle resolution, this makes it
possible to investigate the fundamental limits of
metrologically relevant forms of entanglement
and could considerably enhance the precision of
interferometric devices based on quantum me-
trology. In addition, our method is well suited to
investigate quantum Zeno dynamics (33), where
permanent QND observation of a degenerate ei-
genvalue limits the coherent evolution of the sys-
tem to a given subspace, enabling preparation
of a large variety of entangled states (34).
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Acknowledgments: We thank G. Semerjian for help with the
calculations on the entanglement criterion and B. Huard for
discussions. This work was supported by the European Union
Information and Communication Technologies project QIBEC
(Quantum Interferometry with Bose-Einstein Condensates)
(GA 284584) and the Integrating Project AQUTE (Atomic
Quantum Technologies) (GA 247687). F.H. acknowledges a
scholarship by Institut Francilien de Recherche sur les Atomes
Froids. Author contributions: F.H. and J.V. performed the
experiment; R.G. made contributions in its early stages; and
F.H., J.V., J.R., and J.E. contributed to data analysis and
interpretation, as well as to the manuscript.
Materials and Methods
Figs. S1 to S6
25 November 2013; accepted 12 March 2014
Published online 27 March 2014;
Unfolding the Laws of Star Formation:
The Density Distribution of
Jouni Kainulainen,1 Christoph Federrath,2 Thomas Henning1
The formation of stars shapes the structure and evolution of entire galaxies. The rate and efficiency of this
process are affected substantially by the density structure of the individual molecular clouds in which stars
form. The most fundamental measure of this structure is the probability density function of volume
densities (r-PDF), which determines the star formation rates predicted with analytical models. This function
has remained unconstrained by observations. We have developed an approach to quantify r-PDFs and
establish their relation to star formation. The r-PDFs instigate a density threshold of star formation and
allow us to quantify the star formation efficiency above it. The r-PDFs provide new constraints for star
formation theories and correctly predict several key properties of the star-forming interstellar medium.
The formation of stars is an indivisible com- ponent of our current picture of galaxy evolution. It also represents the first step
in defining where new planetary systems can
form. The physics of how the interstellar me-
dium (ISM) is converted into stars is strongly
affected by the density structure of individual
molecular clouds (1). This structure directly af-
fects the star-formation rates (SFRs) and efficien-
cies (SFEs) predicted by analytic models (2–5).
Inferring this structure observationally is chal-
lenging because observations only probe pro-
jected column densities. Hence, the key parameters
of star-formation models remain unconstrained.
Here, we present a technique that allows us to
quantify the grounding measure of the molec-
ular cloud density structure: the probability den-
sity function of their volume density (r-PDF).
The SFRs of molecular clouds are esti-
mated in analytic theories from the amount of
gas in the clouds above a critical density, rcrit
SFR ¼ ecore
where s = ln(r/r0) is the logarithmic, mean-normalized density, and scrit = ln(rcrit/r0). We use
the number density, n ¼ r=mmp, where m is the
mean molecular mass and mp is the proton mass,
as the measure of density. The parameter ecore in
Eq. 1 is the core-to-star efficiency, giving the fraction of gas above scrit that collapses into a star. The
tff(r) is the free-fall time of pressureless gas that
approximates the star-formation time scale, and f
is the ratio of the free-fall time to the actual star-formation time scale. The critical density, commonly referred to as the (volume) density threshold of
star formation, indicates that stars form only above
that density. Generally, the critical density depends
on gas properties (2–5), but theoretical considerations suggest that it could be relatively constant
under typical molecular cloud conditions (5).
1Max-Planck-Institute for Astronomy, Königstuhl 17, 69117
Heidelberg, Germany. 2Monash Centre for Astrophysics, School
of Mathematical Sciences, Monash University, Vic 3800, Australia.
*Corresponding author. E-mail: email@example.com