Quantum Fidelity Decay of Quasi-Integrable Systems

We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.Comment: 11 pages, 11 figures, to be published Phys. Rev.

Semi-optimal Practicable Algorithmic Cooling

Algorithmic Cooling (AC) of spins applies entropy manipulation algorithms in open spin-systems in order to cool spins far beyond Shannon's entropy bound. AC of nuclear spins was demonstrated experimentally, and may contribute to nuclear magnetic resonance (NMR) spectroscopy. Several cooling algorithms were suggested in recent years, including practicable algorithmic cooling (PAC) and exhaustive AC. Practicable algorithms have simple implementations, yet their level of cooling is far from optimal; Exhaustive algorithms, on the other hand, cool much better, and some even reach (asymptotically) an optimal level of cooling, but they are not practicable. We introduce here semi-optimal practicable AC (SOPAC), wherein few cycles (typically 2-6) are performed at each recursive level. Two classes of SOPAC algorithms are proposed and analyzed. Both attain cooling levels significantly better than PAC, and are much more efficient than the exhaustive algorithms. The new algorithms are shown to bridge the gap between PAC and exhaustive AC. In addition, we calculated the number of spins required by SOPAC in order to purify qubits for quantum computation. As few as 12 and 7 spins are required (in an ideal scenario) to yield a mildly pure spin (60% polarized) from initial polarizations of 1% and 10%, respectively. In the latter case, about five more spins are sufficient to produce a highly pure spin (99.99% polarized), which could be relevant for fault-tolerant quantum computing.Comment: 13 pages, 5 figure

Quantum Sensor Miniaturization

The classical bound on image resolution defined by the Rayleigh limit can be beaten by exploiting the properties of quantum mechanical entanglement. If entangled photons are used as signal states, the best possible resolution is instead given by the Heisenberg limit, an improvement proportional to the number of entangled photons in the signal. In this paper we present a novel application of entanglement by showing that the resolution obtained by an imaging system utilizing separable photons can be achieved by an imaging system making use of entangled photons, but with the advantage of a smaller aperture, thus resulting in a smaller and lighter system. This can be especially valuable in satellite imaging where weight and size play a vital role.Comment: 3 pages, 1 figure. Accepted for publication in Photonics Technology Letter

Stable self-similar blow-up dynamics for slightly L2L^2-supercritical generalized KdV equations

In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up dynamic with self-similar blow-up rate in the energy space $H^1$ and give a specific description of the formation of the singularity near the blow-up time.Comment: 38 page

Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene

Consider electromagnetic waves in two-dimensional {\it honeycomb structured media}. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator \LA=-\nabla_\bx\cdot A(\bx) \nabla_\bx, where A(\bx) is $\Lambda_h-$ periodic ($\Lambda_h$ denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(\bx) is $\mathcal{P}\mathcal{C}-$ invariant (A(\bx)=\overline{A(-\bx)}) and $120^\circ$ rotationally invariant (A(R^*\bx)=R^*A(\bx)R, where $R$ is a $120^\circ$ rotation in the plane). We first obtain results on the existence, stability and instability of Dirac points, conical intersections between two adjacent Floquet-Bloch dispersion surfaces. We then show that the introduction through small and slow variations of a {\it domain wall} across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) {\it edge states}. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. These results imply the existence of {\it uni-directional} propagating edge states for two classes of time-reversal invariant media in which $\mathcal{C}$ symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schr\"odinger operators.Comment: 65 pages, 8 figures, To appear in Archive for Rational Mechanics and Analysi

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to an amplitude instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows to predict the stability and instability strength

Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations

We consider a family of dispersion generalized Benjamin-Ono equations (dgBO) which are critical with respect to the L2 norm and interpolate between the critical modified (BO) equation and the critical generalized Korteweg-de Vries equation (gKdV). First, we prove local well-posedness in the energy space for these equations, extending results by Kenig, Ponce and Vega concerning the (gKdV) equations. Second, we address the blow up problem in the spirit of works of Martel and Merle on the critical (gKdV) equation, by studying rigidity properties of the (dgBO) flow in a neighborhood of solitons. We prove that when the model is close to critical (gKdV), solutions of negative energy close to solitons blow up in finite or infinite time in the energy space. The blow up proof requires in particular extensions to (dgBO) of monotonicity results for localized versions of L2 norms by pseudo-differential operator tools.Comment: Submitte

Equilibrium Configuration of Black Holes and the Inverse Scattering Method

The inverse scattering method is applied to the investigation of the equilibrium configuration of black holes. A study of the boundary problem corresponding to this configuration shows that any axially symmetric, stationary solution of the Einstein equations with disconnected event horizon must belong to the class of Belinskii-Zakharov solutions. Relationships between the angular momenta and angular velocities of black holes are derived.Comment: LaTeX, 14 pages, no figure

I=3/2 KπK \pi Scattering in the Nonrelativisitic Quark Potential Model

We study $I=3/2$ elastic $K\pi$ scattering to Born order using nonrelativistic quark wavefunctions in a constituent-exchange model. This channel is ideal for the study of nonresonant meson-meson scattering amplitudes since s-channel resonances do not contribute significantly. Standard quark model parameters yield good agreement with the measured S- and P-wave phase shifts and with PCAC calculations of the scattering length. The P-wave phase shift is especially interesting because it is nonzero solely due to $SU(3)_f$ symmetry breaking effects, and is found to be in good agreement with experiment given conventional values for the strange and nonstrange constituent quark masses.Comment: 12 pages + 2 postscript figures, Revtex, MIT-CTP-210