Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

On the Relationship between Resolution Enhancement and Multiphoton Absorption Rate in Quantum Lithography

The proposal of quantum lithography [Boto et al., Phys. Rev. Lett. 85, 2733 (2000)] is studied via a rigorous formalism. It is shown that, contrary to Boto et al.'s heuristic claim, the multiphoton absorption rate of a ``NOON'' quantum state is actually lower than that of a classical state with otherwise identical parameters. The proof-of-concept experiment of quantum lithography [D'Angelo et al., Phys. Rev. Lett. 87, 013602 (2001)] is also analyzed in terms of the proposed formalism, and the experiment is shown to have a reduced multiphoton absorption rate in order to emulate quantum lithography accurately. Finally, quantum lithography by the use of a jointly Gaussian quantum state of light is investigated, in order to illustrate the trade-off between resolution enhancement and multiphoton absorption rate.Comment: 14 pages, 7 figures, submitted, v2: rewritten in response to referees' comments, v3: rewritten and extended, v4: accepted by Physical Review

Unified Treatment of Heterodyne Detection: the Shapiro-Wagner and Caves Frameworks

A comparative study is performed on two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator called Shapiro-Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator $\hat \psi$ which allows a unified formulation of both cases. For the Shapiro-Wagner scheme, where $[\hat \psi, \hat \psi^{\dag}] =0$, quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by $[\hat \psi, \hat \psi^{\dag}] \neq 0$, within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.Comment: 25 pages. Just very minor editorial cosmetic change

Husimi's Q(α)Q(\alpha) function and quantum interference in phase space

We discuss a phase space description of the photon number distribution of non classical states which is based on Husimi's $Q(\alpha)$ function and does not rely in the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution . This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.Comment: 15 pages, 12 figures, typos correcte

Does nonlinear metrology offer improved resolution? Answers from quantum information theory

A number of authors have suggested that nonlinear interactions can enhance resolution of phase shifts beyond the usual Heisenberg scaling of 1/n, where n is a measure of resources such as the number of subsystems of the probe state or the mean photon number of the probe state. These suggestions are based on calculations of `local precision' for particular nonlinear schemes. However, we show that there is no simple connection between the local precision and the average estimation error for these schemes, leading to a scaling puzzle. This puzzle is partially resolved by a careful analysis of iterative implementations of the suggested nonlinear schemes. However, it is shown that the suggested nonlinear schemes are still limited to an exponential scaling in \sqrt{n}. (This scaling may be compared to the exponential scaling in n which is achievable if multiple passes are allowed, even for linear schemes.) The question of whether nonlinear schemes may have a scaling advantage in the presence of loss is left open. Our results are based on a new bound for average estimation error that depends on (i) an entropic measure of the degree to which the probe state can encode a reference phase value, called the G-asymmetry, and (ii) any prior information about the phase shift. This bound is asymptotically stronger than bounds based on the variance of the phase shift generator. The G-asymmetry is also shown to directly bound the average information gained per estimate. Our results hold for any prior distribution of the shift parameter, and generalise to estimates of any shift generated by an operator with discrete eigenvalues.Comment: 8 page

Minimum-error discrimination between mixed quantum states

We derive a general lower bound on the minimum-error probability for {\it ambiguous discrimination} between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out a upper bound on the minimum-error probability for ambiguous discrimination of mixed quantum states. Therefore, some sufficient conditions are obtained for the minimum-error probability attaining this bound. Finally, under the condition of the minimum-error probability attaining this bound, we compare the minimum-error probability for {\it ambiguously} discriminating arbitrary $m$ mixed quantum states with the optimal failure probability for {\it unambiguously} discriminating the same states.Comment: A further revised version, and some results have been adde

Adaptive phase estimation is more accurate than non-adaptive phase estimation for continuous beams of light

We consider the task of estimating the randomly fluctuating phase of a continuous-wave beam of light. Using the theory of quantum parameter estimation, we show that this can be done more accurately when feedback is used (adaptive phase estimation) than by any scheme not involving feedback (non-adaptive phase estimation) in which the beam is measured as it arrives at the detector. Such schemes not involving feedback include all those based on heterodyne detection or instantaneous canonical phase measurements. We also demonstrate that the superior accuracy adaptive phase estimation is present in a regime conducive to observing it experimentally.Comment: 15 pages, 9 figures, submitted to PR

Propagation of squeezed radiation through amplifying or absorbing random media

We analyse how nonclassical features of squeezed radiation (in particular the sub-Poissonian noise) are degraded when it is transmitted through an amplifying or absorbing medium with randomly located scattering centra. Both the cases of direct photodetection and of homodyne detection are considered. Explicit results are obtained for the dependence of the Fano factor (the ratio of the noise power and the mean current) on the degree of squeezing of the incident state, on the length and the mean free path of the medium, the temperature, and on the absorption or amplification rate.Comment: 8 pages, 4 figure

Instruments and channels in quantum information theory

While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the typical inequalities for the quantum and classical relative entropies, many bounds on the classical information extracted in a quantum measurement, of the type of Holevo's bound, are obtained in a unified manner.Comment: 12 pages, revtex

Light atom quantum oscillations in UC and US

High energy vibrational scattering in the binary systems UC and US is measured using time-of-flight inelastic neutron scattering. A clear set of well-defined peaks equally separated in energy is observed in UC, corresponding to harmonic oscillations of the light C atoms in a cage of heavy U atoms. The scattering is much weaker in US and only a few oscillator peaks are visible. We show how the difference between the materials can be understood by considering the neutron scattering lengths and masses of the lighter atoms. Monte Carlo ray tracing is used to simulate the scattering, with near quantitative agreement with the data in UC, and some differences with US. The possibility of observing anharmonicity and anisotropy in the potentials of the light atoms is investigated in UC. Overall the observed data is well accounted for by considering each light atom as a single atom isotropic quantum harmonic oscillator.Comment: 10 pages, 8 figure