The repulsive chemotaxis-consumption system
u
t
=
∇
·
(
D
(
u
)
∇
u
)
+
∇
·
(
u
∇
v
)
,
0
=
Δ
v
-
u
v
,
is considered along with the boundary conditions
(
D
(
u
)
∇
u
+
u
∇
v
)
·
ν
|
∂
Ω
=
0
and
v
|
...∂
Ω
=
M
in a ball
Ω
⊂
R
2
. Under the assumption that
D
suitably generalizes the function
0
≤
ξ
↦
(
ξ
+
1
)
-
α
for some
α
>
0
, it is firstly shown that for each nontrivial radially symmetric
u
0
∈
W
1
,
∞
(
Ω
)
, one can find
M
⋆
(
u
0
)
>
0
with the property that whenever
M
>
M
⋆
(
u
0
)
, a corresponding initial-boundary value problem admits a classical solution blowing up in finite time. This is complemented by a second statement which asserts that when
inf
D
and
M
are positive, for any such initial data a global bounded classical solution exists.
Dynamics of large complex systems, such as relaxation towards equilibrium in classical statistical mechanics, often obeys a master equation that captures essential information from the complexities. ...Here, we find that thermalization of an isolated many-body quantum state can be described by a master equation. We observe sudden quench dynamics of quantum Ising-like models implemented in our quantum simulator, defect-free single-atom tweezers in conjunction with Rydberg-atom interaction. Saturation of their local observables, a thermalization signature, obeys a master equation experimentally constructed by monitoring the occupation probabilities of prequench states and imposing the principle of the detailed balance. Our experiment agrees with theories and demonstrates the detailed balance in a thermalization dynamics that does not require coupling to baths or postulated randomness.
We propose and demonstrate three-dimensional rearrangements of single atoms. In experiments performed with single
Rb atoms in optical microtraps actively controlled by a spatial light modulator, we ...demonstrate various dynamic rearrangements of up to N = 9 atoms including rotation, 2D vacancy filling, guiding, compactification, and 3D shuffling. With the capability of a phase-only Fourier mask to generate arbitrary shapes of the holographic microtraps, it was possible to place single atoms at arbitrary geometries of a few μm size and even continuously reconfigure them by conveying each atom. For this purpose, we loaded a series of computer-generated phase masks in the full frame rate of 60 Hz of the spatial light modulator, so the animation of phase mask transformed the holographic microtraps in real time, driving each atom along the assigned trajectory. Possible applications of this method of transformation of single atoms include preparation of scalable quantum platforms for quantum computation, quantum simulation, and quantum many-body physics.
Establishing a reliable method to form scalable neutral-atom platforms is an essential cornerstone for quantum computation, quantum simulation and quantum many-body physics. Here we demonstrate a ...real-time transport of single atoms using holographic microtraps controlled by a liquid-crystal spatial light modulator. For this, an analytical design approach to flicker-free microtrap movement is devised and cold rubidium atoms are simultaneously rearranged with 2N motional degrees of freedom, representing unprecedented space controllability. We also accomplish an in situ feedback control for single-atom rearrangements with the high success rate of 99% for up to 10 μm translation. We hope this proof-of-principle demonstration of high-fidelity atom-array preparations will be useful for deterministic loading of N single atoms, especially on arbitrary lattice locations, and also for real-time qubit shuttling in high-dimensional quantum computing architectures.
The incompressible fractional Navier–Stokes equations in the rotational framework is considered. We establish the global existence result and temporal decay estimate for a unique smooth solution when ...the speed of rotation is sufficiently rapid. It is found that the strong rotational effect enhances the temporal decay rate of a certain norm of the velocity.
A chemotaxis system with Newtonian attraction and fractional dissipation of order
α
∈
(
0
,
2
)
is considered in
R
N
. For initial data belonging to
L
1
∩
H
4
but small in
L
N
α
,
N
=
2
,
3
, the ...temporal decay and the asymptotic behavior of a global classical solution are established. In particular, we derive a precise decay estimate for higher Sobolev norms.
Strong mutual interaction which correlates elementary excitations of quantum matter plays a key role in a range of emergent phenomena, from binding and condensation to quantum thermalization and ...many-body localization. Here, we employ a Rydberg quantum simulator to experimentally demonstrate strongly correlated spin transport in anisotropic Heisenberg magnets, where the magnon-magnon interaction can be tuned 2 orders of magnitude larger than the magnon hopping strength. In our approach, the motion of magnons is controlled by an induced spin-exchange interaction through far off-resonant driving, which enables coherent transport of a single Rydberg excitation across a chain of ground-state atoms. As the most prominent signature of a giant anisotropy, we show that nearby Rydberg excitations form distinct types of magnon-bound states, where a tightly bound pair exhibits frozen dynamics in a fragmented Hilbert space, while a loosely bound pair propagates and establishes correlations beyond a single lattice site. Our scheme complements studies using resonant dipole-dipole interactions between Rydberg states and opens the door to exploring quantum thermodynamics with ultrastrong interactions and kinetic constraints.
Cell–cell adhesion is an inherently nonlocal phenomenon. Numerous partial differential equation models with nonlocal term have been recently presented to describe this phenomenon, yet the ...mathematical properties of nonlocal adhesion model are not well understood. Here we consider a model with two kinds of nonlocal cell–cell adhesion, satisfying no-flux conditions in a multidimensional bounded domain. We show global-in-time well-posedness of the solution to this model and obtain the uniform boundedness of solution.
This study investigates the Cauchy problem of an incompressible magnetohydrodynamic system in the rotational framework. Under the assumption that the rotation speed is sufficiently large, the Cauchy ...problem is shown to be globally well-posed in
H
s
(
R
3
)
×
(
L
2
∩
L
q
)
(
R
3
)
for
1
2
<
s
<
3
4
and
3
<
q
<
min
{
6
3
-
2
s
,
27
6
+
4
s
}
.
We present Rydberg-atom-array experiments performed to find the maximum independent sets of Platonic graphs. Three Platonic graphs—the tetrahedron, cube, and octahedron of Platonic solids—are ...constructed with atoms and Rydberg interatomic interactions, representing, respectively, the vertices and edges of the graphs. In particular, the three-dimensional Platonic graph structures are transformed onto the two-dimensional plane by using Rydberg quantum wires that couple otherwise uncoupled long-distance atoms. The maximum independent sets of the graphs correspond to the antiferrolike many-body ground-state spin configurations of the as-constructed Rydberg-atom arrays, which are successfully probed by quasiadiabatic control of the Rydberg-atom arrays from the paramagnetic phase to their antiferrolike phases. Our small-scale quantum simulations using fewer than 18 atoms are limited by experimental imperfections, which can be easily improved upon by the state-of-the-art Rydberg-atom technologies for scales of more than 1000 atoms. Our quantum-wire approach is expected to pave a new route toward large-scale quantum simulations.
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