5 research outputs found
Gaussian distribution of short sums of trace functions over finite fields
We show that under certain general conditions, short sums of -adic
trace functions over finite fields follow a normal distribution asymptotically
when the origin varies, generalizing results of Erd\H{o}s-Davenport,
Mak-Zaharescu and Lamzouri. In particular, this applies to exponential sums
arising from Fourier transforms such as Kloosterman sums or Birch sums, as we
can deduce from the works of Katz. By approximating the moments of traces of
random matrices in monodromy groups, a quantitative version can be given as in
Lamzouri's article, exhibiting a different phenomenon than the averaging from
the central limit theorem.Comment: 42 page
Experimental certification of millions of genuinely entangled atoms in a solid
Quantum theory predicts that entanglement can also persist in macroscopic
physical systems, albeit difficulties to demonstrate it experimentally remain.
Recently, significant progress has been achieved and genuine entanglement
between up to 2900 atoms was reported. Here we demonstrate 16 million genuinely
entangled atoms in a solid-state quantum memory prepared by the heralded
absorption of a single photon. We develop an entanglement witness for
quantifying the number of genuinely entangled particles based on the collective
effect of directed emission combined with the nonclassical nature of the
emitted light. The method is applicable to a wide range of physical systems and
is effective even in situations with significant losses. Our results clarify
the role of multipartite entanglement in ensemble-based quantum memories as a
necessary prerequisite to achieve a high single-photon process fidelity crucial
for future quantum networks. On a more fundamental level, our results reveal
the robustness of certain classes of multipartite entangled states, contrary
to, e.g., Schr\"odinger-cat states, and that the depth of entanglement can be
experimentally certified at unprecedented scales.Comment: 11 pages incl. Methods and Suppl. Info., 4 figures, 1 table. v2:
close to published version. See also parallel submission by Zarkeshian et al
(1703.04709
Macroscopic quantum entanglement between an optomechanical cavity and a continuous field in presence of non-Markovian noise
Probing quantum entanglement with macroscopic objects allows to test quantum
mechanics in new regimes. One way to realize such behavior is to couple a
macroscopic mechanical oscillator to a continuous light field via radiation
pressure. In view of this, the system that is discussed comprises an
optomechanical cavity driven by a coherent optical field in the unresolved
sideband regime where we assume Gaussian states and dynamics. We develop a
framework to quantify the amount of entanglement in the system numerically.
Different from previous work, we treat non-Markovian noise and take into
account both the continuous optical field and the cavity mode. We apply our
framework to the case of the Advanced Laser Interferometer Gravitational-Wave
Observatory (Advanced LIGO) and discuss the parameter regimes where
entanglement exists, even in the presence of quantum and classical noises