On Exotic Lagrangian Tori in CP^2

We construct an exotic monotone Lagrangian torus in CP^2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian isotopic to the known Clifford and Chekanov tori.Comment: 62 page

Infinitely many monotone Lagrangian tori in del Pezzo surfaces

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $\mathbb{C}P^2 \# k \overline{\mathbb{C}P^2}$, for k=0,3,4,5,6,7,8. We name these tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ also have infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ . Finally, the Lagrangian tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$ inside a del Pezzo surface $X$ can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus $\Sigma$. We argue that $\Theta^{n_1,n_2,n_3}_{p,q,r}$ give rise to infinitely many exact Lagrangian tori in $X \setminus \Sigma$, even after attaching the positive end of a symplectization to the boundary of $X \setminus \Sigma$.Comment: 28 pages, 20 figure

Continuum families of non-displaceable Lagrangian tori in (CP1)2m(\mathbb{C}P^1)^{2m}

We construct a family of Lagrangian tori $\Theta^n_s$ $\subset$ $(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $\Theta^n_{1/2} = \Theta^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n = 2m$ and $s \ge 1/2$ these tori are non-displaceable. Then by considering $\Theta^{k_1}_{s_1}$ $\times$ $\cdots$ $\times$ $\Theta^{k_l}_{s_l}$ $\times$ $(S^2_{\mathrm{eq}})^{n - \sum_i k_i}$ $\subset$ $(\mathbb{C}P^1)^n$, with $s_i \in [1/2,1)$ and $k_i \in 2\mathbb{Z}_{>0}$, $\sum_i k_i \le n$ we get several $l$-dimensional families of non-displaceable Lagrangian tori. We also show that there exists partial symplectic quasi-states $\zeta^{\mathfrak{b}_s}_{\textbf{e}_s}$ and linearly independent homogeneous Calabi quasimorphims $\mu^{\mathfrak{b}_s}_{\textbf{e}_s}$ or which $\Theta^{2m}_s$ are $\zeta^{\mathfrak{b}_s}_{\textbf{e}_s}$-superheavy and $\mu^{\mathfrak{b}_s}_{\textbf{e}_s}$-superheavy. We also prove a similar result for $(\mathbb{C}P^2 3\bar{\mathbb{C}P^2}, \omega_\epsilon)$, where $\{\omega_\epsilon; 0 < \epsilon < 1\}$ is a family of symplectic forms in $\mathbb{C}P^2 3\bar{\mathbb{C}P^2}$, for which $\omega_{1/2}$ is monotone.Comment: 17 pages, 1 Figur

Low-area Floer theory and non-displaceability

We introduce a new version of Floer theory of a non-monotone Lagrangian submanifold which only uses least area holomorphic disks with boundary on it. We use this theory to prove non-displaceability theorems about continuous families of Lagrangian tori in the complex projective plane and other del Pezzo surfaces.Comment: 32 pages, 9 figures; v2: major improvements, added a new result concerning del Pezzos, corrected some mistakes; v3: explained transversality for annuli, commented on higher dimensions; accepted versio

Computable Measures for the Entanglement of Indistinguishable Particles

We discuss particle entanglement in systems of indistinguishable bosons and fermions, in finite Hilbert spaces, with focus on operational measures of quantum correlations. We show how to use von Neumann entropy, Negativity and entanglement witnesses in these cases, proving interesting relations. We obtain analytic expressions to quantify quantum correlations in homogeneous D-dimensional Hamiltonian models with certain symmetries.Comment: 9 page

Optimal estimation of quantum processes using incomplete information: variational quantum process tomography

We develop a quantum process tomography method, which variationally reconstruct the map of a process, using noisy and incomplete information about the dynamics. The new method encompasses the most common quantum process tomography schemes. It is based on the variational quantum tomography method (VQT) proposed by Maciel et al. in arXiv:1001.1793[quant-ph].Comment: 3 pages, one figure. Revised version, including numerical example

Resonant interaction between an ultrashort pulse train and a two-level system: frequency domain analysis

We investigate the problem of two-level atoms driven by an ultrashort pulse train in the frequency domain. At low intensity regime, we obtain a perturbative analytical solution that allows us to discuss the role of the mode number of the frequency comb near or at resonance on the temporal evolution of the atomic coherence. At high intensities, the effect of the number of modes is analyzed in the steady-state regime through numerical calculations.Comment: 8 pages, 8 figure

Quantumness of correlations in indistinguishable particles

We discuss a general notion of quantum correlations in fermionic or bosonic indistinguishable particles. Our approach is mainly based on the identification of the algebra of single-particle observables, which allows us to devise an activation protocol in which the \textit{quantumness of correlations} in the system leads to a unavoidable creation of entanglement with the measurement apparatus. Using the distillable entanglement, or the relative entropy of entanglement, as entanglement measure, we show that our approach is equivalent to the notion of minimal disturbance in a single-particle von Neumann measurement, also leading to a geometrical approach for its quantification.Comment: 7 pages, 2 figure

Asymptotic behavior of Vianna's exotic Lagrangian tori Ta,b,cT_{a,b,c} in CP2\mathbb{CP}^2 as a+b+c→∞a+b+c \to \infty

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ associated to Markov triples $(a,b,c)$ described in \cite{Vi14}. We first prove that the Gromov capacity of the complement $\mathbb{CP}^2 \setminus T_{a,b,c}$ is greater than or equal to $\frac13$ of the area of the complex line for all Markov triple $(a,b,c)$. We then prove that there is a representative of the family $\{T_{a,b,c}\}$ whose loci completely miss a metric ball of nonzero size and in particular the loci of the union of the family is not dense in $\mathbb{CP}^2$.Comment: 24 pages, 7 figures;v2) typos corrected, English improved, More references added;v3) 32 pages, 16 figures, the third author newly added, previous main theorem improved, density question resolved and many more results on relative ball packings adde

Entanglement of indistinguishable particles as a probe for quantum phase transitions in the extended Hubbard model

We investigate the quantum phase transitions of the extended Hubbard model at half-filling with periodic boundary conditions employing the entanglement of particles, as opposed to the more traditional entanglement of modes. Our results show that the entanglement has either discontinuities or local minima at the critical points. We associate the discontinuities to first order transitions, and the minima to second order ones. Thus we show that the entanglement of particles can be used to derive the phase diagram, except for the subtle transitions between the phases SDW-BOW, and the superconductor phases TS-SS.Comment: Improved text and dicussions. Added finite size scaling analysis. 9 pages, 8 figures. Accepted for publication in Phys. Rev.