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Register a new userOn the absence of stationary currents
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We review proofs of a theorem of Bloch on the absence of macroscopic stationary currents in quantum systems. The standard proof shows that the current in 1D vanishes in the large volume limit under rather general conditions. In higher dimension, the total current across a cross-section does not need to vanish in gapless systems but it does vanish in gapped systems. We focus on the latter claim and give a self-contained proof motivated by a recently introduced index for many-body charge transport in quantum lattice systems having a conserved $U(1)$-charge.
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We develop an abstract model of atomic clocks that fully describes the dynamics of repeated synchronization between a classical oscillator and a quantum reference. We prove existence of a stationary state of the model and study its dependence on the control scheme, the interrogation time and the stability of the oscillator. For unbiased atomic clocks, we derive a fundamental bound on atomic clocks long time stability for a given local oscillator noise. In particular, we show that for a local oscillator noise with integrated frequency variance scaling as $T^alpha$ for short times $T$, the optimal clock time variance scales as $F^{-(alpha +1)/(alpha +2)}$ with respect to the quantum Fisher information, $F$, associated to the quantum reference.
We set up and study a coupled problem on stationary non-isothermal flow of electrorheological fluids. The problem consist in finding functions of velocity, pressure and temperature which satisfy the motion equations, the condition of incompressibility, the equation of the balance of thermal energy and boundary conditions. We introduce the notions of a $P$-generalized solution and generalized solution of the coupled problem. In case of the $P$-generalized solution the dissipation of energy is defined by the regularized velocity field, which leads to a nonlocal model. Under weak conditions, we prove the existence of the $P$ -generalized solution of the coupled problem. The existence of the generalized solution is proved under the conditions on smoothness of the boundary and on smallness of the data of the problem
Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Slomczynski and Zyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Luders-von Neumann instruments and partitions.
We show that every finite-dimensional quantum system with Markovian time evolution has an autonomous unitary dilation which can be dynamically decoupled. Since there is also always an autonomous unitary dilation which cannot be dynamically decoupled, this highlights the role of dilations in the control of quantum noise. We construct our dilation via a time-dependent version of Stinespring in combination with Howlands clock Hamiltonian and certain point-localised states, which may be regarded as a C*-algebraic analogue of improper bra-ket position eigenstates and which are hence of independent mathematical and physical interest.
A number of papers over the past eight years have claimed to solve the fractional Schr{o}dinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schr{o}dinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schr{o}dinger equation for general fractional parameters $alpha$. On a more positive note, we present a solution to the fractional Schr{o}dinger equation for the one-dimensional harmonic oscillator with $alpha=1$.
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