National Center of Competence in Research Quantum Science and
Technology, the Army Research Office (ARO) Multidisciplinary
University Research Initiative (MURI) grant no. W911NF11-1-0268,
ARO grant no. W911NF-12-1-0523, the Lockheed Martin
Corporation, and Microsoft Research. Simulations were performed
on clusters at Microsoft Research and ETH Zurich an on
supercomputers of the Swiss Center for Scientific Computing.
We acknowledge hospitality of the Aspen Center for Physics,
supported by NSF grant PHY-1066293.
SUPPLEMENTARY MATERIALS
www.sciencemag.org/content/345/6195/420/DC1
Supplementary Text
Figs. S1 to S10
Tables S1 and S2
References (31–37)
17 February 2014; accepted 10 June 2014
Published online 19 June 2014;
10.1126/science.1252319
QUANTUM METROLOGY
Fisher information and entanglement
of non-Gaussian spin states
Helmut Strobel,1 Wolfgang Muessel,1 Daniel Linnemann,1 Tilman Zibold,1
David B. Hume,1 Luca Pezzè,2 Augusto Smerzi,2 Markus K. Oberthaler1
Entanglement is the key quantum resource for improving measurement sensitivity beyond
classical limits. However, the production of entanglement in mesoscopic atomic systems
has been limited to squeezed states, described by Gaussian statistics. Here, we report on
the creation and characterization of non-Gaussian many-body entangled states. We
develop a general method to extract the Fisher information, which reveals that the
quantum dynamics of a classically unstable system creates quantum states that are not
spin squeezed but nevertheless entangled. The extracted Fisher information quantifies
metrologically useful entanglement, which we confirm by Bayesian phase estimation with
sub–shot-noise sensitivity. These methods are scalable to large particle numbers and
applicable directly to other quantum systems.
Multiparticle entangled states are the key ingredients for advanced quantum tech- nologies (1). Various types have been achieved in experimental settings rang- ing from ion traps (2), photonic systems
(3), and solid state circuits (4) to Bose-Einstein
condensates. For the latter, squeezed states (5, 6)
have been generated (7–12), and a rich class of
entangled non-Gaussian states is predicted to be
obtainable (13), including maximally entangled
Schrödinger cat states (14, 15). The production of
these fragile states in large systems remains a
challenge, and efficient methods for characterization are necessary because full state reconstruction becomes intractable. Here, we generate a
class of non-Gaussian many-particle entangled
states and reveal their quantum properties by
studying the distinguishability of experimental
probability distributions.
A measure of the distinguishability with respect to small phase changes of the state is provided by the Fisher information F (16). It is related
to the highest attainable interferometric phase
sensitivity by the Cramer-Rao bound DqCR ¼ 1=
ffiffiffi
F
p
(17). This limit follows from general statistical
arguments for a measurement device with fluc-
tuating output (18). The Fisher information is lim-
ited by quantum fluctuations of the input state as
well as the performance of the device. Even in the
absence of technical noise, the Fisher informa-
tion of a classical input state is F≤ N because
of the intrinsic granularity of N independent
particles, which translates into the shot-noise
limit Dq ≥ 1=
ffiffiffiffi
N
p
for phase estimation. This clas-
sical bound can be surpassed with a reduction of
the input fluctuations by introducing entangle-
ment between the N particles (5). These states,
known as squeezed states, are fully characterized
by mean and variance of the observable and al-
ready used in precision measurements (19–21). In
contrast, non-Gaussian quantum states can have
increased fluctuations of the observable but
nevertheless allow surpassing shot-noise limited per-
formance. A textbook example is the Schrödinger
cat state characterized by macroscopic fluctua-
tions but achieving the best interferometric per-
formance allowed by quantum mechanics, that
is, at the fundamental Heisenberg limit F ¼ N2
(22). In general, the class of states that are entangled and useful for sub–shot-noise phase estimation is identified by the Fisher information
criterion F > N (13). Exploiting these resources
requires probabilistic methods for phase estimation, such as maximum likelihood or Bayesian
analysis (23), which go beyond standard evaluation of averages.
424 25 JULY 2014 • VOL 345 ISSUE 6195
sciencemag.org SCIENCE
1Kirchhoff-Institut für Physik, Universität Heidelberg, Im
Neuenheimer Feld 227, 69120 Heidelberg, Germany. 2QSTAR
(Quantum Science and Technology in Arcetri), INO-CNR
(Istituto Nazionale di Ottica–Consiglio Nazionale delle
Ricerche), and LENS (European Laboratory for Nonlinear
Spectroscopy), Largo Enrico Fermi 2, 50125 Firenze, Italy.
*Corresponding author. E-mail: fisherinformation@matterwave.de
10 µm
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ac
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.
0 0.5
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10
6
2
−0.5
0° 60°
x10-2
−0.5 0 0.5
D
Fig. 1. Preparation and detection of non-Gaussian entangled states. (A) Array of Bose-Einstein
condensates in an optical lattice potential addressed by microwave and radio frequency fields. (B) The
interplay of nonlinear interaction (blue) and weak Rabi coupling (red) between the internal states ja⟩ and
jb⟩ results in an unstable fixed point in the classical phase space. The state of the system is visualized on
a generalized Bloch sphere with radius J ¼N=2. Gray lines indicate trajectories of the mean-field
equations of motion (18). The initial coherent spin state (green) ideally evolves into a squeezed state
(orange) followed by non-Gaussian states at later evolution times (violet). Edges of shaded areas are
contours of the Husimi distribution for N ¼
380 at 1=e2 of its maximum. (C) Experimental absorption
picture showing the site- and state-resolved optical lattice after a Stern-Gerlach separation. Shaded
boxes indicate the sites with a total atom number in the range of 380 T 15, which are selected for further
analysis. (D) Example histograms of the imbalance z ¼ 2Jz=N after nonlinear evolution of 25 ms and
final rotation (angles indicated in the graphs) compared with the ideal coherent spin state of identical N
(green Gaussian).
