the quantum resource for improved phase estimation. The presented method does not depend
on the special shape of the probability distributions and is not limited to small particle
numbers. It is therefore broadly applicable to the
efficient characterization of highly entangled
states, relevant for further improvement of atom
interferometers (8, 11, 12, 19, 20, 28–30) toward
the ultimate Heisenberg limit (22). More generally, it can be applied to any phenomenon characterizable by the distinguishability of quantum
states, as in quantum phase transitions (31), quantum Zeno dynamics (32), and quantum information protocols (1).
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We thank J. Tomkovič, E. Nicklas, and I. Stroescu for technical help
and discussions. This work was supported by the Forschergruppe
FOR760, the Deutsche Forschungsgemeinschaft, the Heidelberg
Center for Quantum Dynamics, and the European Commission
small or medium-scale focused research project QIBEC (Quantum
Interferometry with Bose-Einstein condensates, contract no.
284584). W.M. acknowledges support by the Studienstiftung des
deutschen Volkes. D.B.H. acknowledges support from the
Alexander von Humboldt Foundation. L.P. acknowledges financial
support by Ministero dell’Istruzione, dell’Università e della Ricerca
through Fondo per gli Investimenti della Ricerca di Base project
no. RBFR08H058. QSTAR is the Max Planck Institute of Quantum
Optics, LENS, Istituto Italiano di Tecnologia, Università degli
Studi di Firenze Joint Center for Quantum Science and
Technology in Arcetri.
Materials and Methods
Figs. S1 to S5
24 December 2013; accepted 3 June 2014
Invisibility cloaking in a diffusive
light scattering medium
Robert Schittny,1,2 Muamer Kadic,1,3 Tiemo Bückmann,1,2 Martin Wegener1,2,3*
In vacuum, air, and other surroundings that support ballistic light propagation according
to Maxwell’s equations, invisibility cloaks that are macroscopic, three-dimensional, broadband,
passive, and that work for all directions and polarizations of light are not consistent with
the laws of physics. We show that the situation is different for surroundings leading to
multiple light scattering, according to Fick’s diffusion equation. We have fabricated cylindrical
and spherical invisibility cloaks made of thin shells of polydimethylsiloxane doped with
melamine-resin microparticles. The shells surround a diffusively reflecting hollow core, in
which arbitrary objects can be hidden. We find good cloaking performance in a water-based
diffusive surrounding throughout the entire visible spectrum and for all illumination
conditions and incident polarizations of light.
With an invisibility cloak (1–5), light is guided in a detour around an object to be hidden so that it emerges behind as though no object was there. Ideally, this concerns the direction, amplitude, and
timing of light. In vacuum (or air) and for macroscopic objects much larger than the wavelength of light, this geometrical detour means
that the local velocity of light must exceed the
vacuum speed of light somewhere. For broadband operation—in the absence of wavelength
dependence—the phase velocity of light equals
its energy velocity. Furthermore, according to
the theory of relativity, energy and mass are
equivalent, and transport of mass faster than
the vacuum speed of light is not possible. Thus,
ideal passive broadband invisibility cloaking
of macroscopic objects in vacuum (or air) is not
consistent (4, 5) with relativity and Maxwell’s
equations of electromagnetism.
Nevertheless, inspired by transformation optics
(1–3) based on Maxwell’s equations, many interesting cloaking experiments have been performed
(6–12). All of these cloaks, however, are either
narrow in bandwidth or not macroscopic, do not
properly recover the time-of-flight (or phase), work
only for restricted polarizations or directions of
light, or exhibit combinations of these limitations.
In many practical instances, light does not
propagate ballistically—it cannot be described
adequately by the macroscopic Maxwell equations
for continua. For example, in clouds, fog, milk,
frosted glass, or in other systems containing many
randomly distributed scattering centers, every
photon of visible light performs a random walk.
Effectively, this random walk slows down light
propagation with respect to vacuum and scrambles
any incident polarization. Multiple light scattering
can also lead to coherent speckles (13) and to
Anderson localization (14, 15). For a broad range
of settings, however, multiple light scattering is
well described by the diffusion of photons (14).
We demonstrate that passive broadband in-
visibility cloaking of macroscopic objects for all
incident directions and polarizations of light is
possible in the regime of light diffusion.
In 1855, Fick postulated his diffusion equation
(16), which was then followed by a microscopic
derivation on the basis of statistical mechanics
(17). Our diffusion experiments have been inspired
(18, 19) by recent work on thermal cloaking:
Whereas early cloaking structures designed with
coordinate transformations contained many alternating layers of effectively low and high heat
conductivity (20), more recent thermal cloaks
(21, 22) work for just one pair of layers with low
and high heat conductivity. Such core-shell structures have also successfully been used for static
magnetic cloaking (23). Intuitively, the combination of core and shell can be thought of as a single
period of a highly anisotropic laminate metamaterial (24). It is known theoretically (24, 25) that
core-shell geometries can be perfectly undetectable
in the stationary case, assuming a spatially constant
gradient of the temperature (or magnetic field or
photon density, for example) across the cloak. Recently, it has become clear that these cloaks also
work for nonconstant gradients (21, 22). Core-shell
geometries allow for cloaks that are thin as compared with the size of the object they hide (21–23).
sciencemag.org 25 JULY 2014 • VOL 345 ISSUE 6195 427
1Institute of Applied Physics, Karlsruhe Institute of
Technology (KIT), D-76128 Karlsruhe, Germany. 2Deutsche
Forschungsgemeinschaft (DFG)–Center for Functional
Nanostructures (CFN), KIT, D-76128 Karlsruhe, Germany.
3Institute of Nanotechnology, KIT, D-76021 Karlsruhe, Germany.
*Corresponding author. E-mail: firstname.lastname@example.org
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