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Construction of Capacity-Achieving Lattice Codes: Polar Lattices
In this paper, we propose a new class of lattices constructed from polar
codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR)
of the additive white Gaussian-noise (AWGN) channel. Our construction follows
the multilevel approach of Forney \textit{et al.}, where we construct a
capacity-achieving polar code on each level. The component polar codes are
shown to be naturally nested, thereby fulfilling the requirement of the
multilevel lattice construction. We prove that polar lattices are
\emph{AWGN-good}. Furthermore, using the technique of source polarization, we
propose discrete Gaussian shaping over the polar lattice to satisfy the power
constraint. Both the construction and shaping are explicit, and the overall
complexity of encoding and decoding is for any fixed target error
probability.Comment: full version of the paper to appear in IEEE Trans. Communication
Bath-induced decay of Stark many-body localization
We investigate the relaxation dynamics of an interacting Stark-localized
system coupled to a dephasing bath, and compare its behavior to the
conventional disorder-induced many body localized system. Specifically, we
study the dynamics of population imbalance between even and odd sites, and the
growth of the von Neumann entropy. For a large potential gradient, the
imbalance is found to decay on a time scale that grows quadratically with the
Wannier-Stark tilt. For the non-interacting system, it shows an exponential
decay, which becomes a stretched exponential decay in the presence of finite
interactions. This is different from a system with disorder-induced
localization, where the imbalance exhibits a stretched exponential decay also
for vanishing interactions. As another clear qualitative difference, we do not
find a logarithmically slow growth of the von-Neumann entropy as it is found
for the disordered system. Our findings can immediately be tested
experimentally with ultracold atoms in optical lattices
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