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The control of flying qubits carried by itinerant photons is ubiquitous in quantum communication networks. In addition to their logical states, the shape of flying qubits must also be tailored to match the remote receiver. In this paper, we introduce the quantum optimal control theory to the design of flying-qubit shaping protocols. A gradient-based algorithm is proposed for the generation of arbitrary-shape flying qubits with general non-ideal emitters. Simulations show that, as a joint control with the traditionally used tunable coupler, coherent driving fields can be applied to the shaping when the coupling strength is fixed or limited. The optimized control protocols can effectively suppress unwanted level leakage and multi-photon radiation. The method provides a systematic approach to high-fidelity control of flying qubits using realistic quantum devices., Comment: 9 pages,6 figures

We give another proof of a theorem of Rabinowitz and Stredulinsky obtaining infinite transition solutions for an Allen--Cahn equation. Rabinowitz and Stredulinsky have constructed infinite transition solutions as locally minimal solutions, but it is still an interesting question to establish these solutions by other method. Our result may attract the interest of constructing solutions with the shape of locally minimal solutions of Rabinowitz and Stredulinsky for problems defined on descrete group.

We establish a variational construction of minimal and without self-intersection solutions for an Allen-Cahn equation, especially for those corresponding to irrational rotation vectors. These consequences generalize the results of rational cases of Rabinowitz and Stredulinsky to irrational cases.

We study a generalized Frenkel-Kontorova model and obtain periodic and heteroclinic mountain pass solutions. Heteroclinic mountain pass solution in the second laminations is new to the generalized Frenkel-Kontorova model. Our proof follows that of Bolotin and Rabinowitz for an Allen-Cahn equation, which is different with heat flow method for finding critical point of Frenkel-Kontorova model in the literature. The proofs depend on suitable choices of functionals and working spaces. We also study the multiplicity of these mountain pass solutions.

We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results., Comment: 22 pages

We study heteroclinic solutions of a generalized Frenkel-Kontorova model. Using the methods of Rabinowitz and Stredulinsky, we prove that if the rotation vector of the configuration is rational and if there is an adjacent pair of periodic configurations, then there is a solution that is heteroclinic in one fixed direction and periodic in other directions. Furthermore, if the above heteroclinic solutions have an adjacent pair, then there is a solution that is heteroclinic in two directions and periodic in other directions. The procedure can be repeated to produce more complex solutions. Thus we obtain a variational construction for these minimal and Birkhoff solutions., Comment: 40 pages, 4 figures

We study rays and co-rays in the Wasserstein space $P_p(\mathcal{X})$ ($p > 1$) whose ambient space $\mathcal{X}$ is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient space. We show the existence of co-rays for any prescribed initial probability measure. We introduce Busemann functions on the Wasserstein space and show that co-rays are negative gradient lines in some sense.