---
abstract: 'We obtain a representation of pairing energies in phase space, for the Lipkin-Meshkov-Glick and boson Bardeen-Cooper-Schrieffer pairing [mean-field like]{} Hamiltonians. This is done by means of a probability distribution of the quantum state in phase space. In fact, we prove a correspondence between the points at which this probability distribution vanishes and the pairing energies. In principle, the vanishing of this probability distribution is experimentally accessible and additionally gives a method to visualize pairing energies across the model control parameter space. This result opens new ways to experimentally approach quantum pairing systems.'
author:
- 'M. Calixto$^1$, O. Castaños$^{2}$ and E. Romera$^{3}$'
title: Searching for pairing energies in phase space
---

*Introduction*.- The concept of *pairing energies* is a fundamental ingredient in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [@BCS]. The BCS Hamiltonian eigenvalues can be written as a sum of the energies of quasiparticles which are the called pairing energies. The BCS theory has also been widely applied to describe pairing correlations between nucleons in finite nuclei [@Bohr]. Here, we will consider two models describing pairing correlations: the Lipkin-Meshkov-Glick (LMG) and a more general bosonic BCS-like model. The LMG Hamiltonian [@lipkin] is a nuclear mean field plus state-dependent pairing interaction model, used to describe the quantum phase transition from spherical to deformed shapes in nuclei, which can be solved analytically [@Pan99]. The LMG model has been also employed in other fields of physics to describe many body systems such as quantum optics (to generate spin squeezed states and to describe many particle entangled states) [@QO] or in condensed matter physics (to characterize Bose-Einstein condensates and Josephson junctions) [@CMP]. The bosonic BCS-like model is also an exactly solvable pairing model that has been proposed to model quantum phase transitions to a fragmented state for repulsive pairing interactions [@dukelskiprl1; @dukelskiprl2; @dukelskinp05]. Boson models of arbitrary angular momentum involving repulsive pairing interactions have been used to support the validity of the interacting boson model in nuclear structure [@pittel].

On the other hand, we can represent a quantum state $\psi$ in phase-space by means of a quasiprobability distribution used in quantum mechanics, the so-called Husimi distribution $Q_\psi$ of a quantum state $\psi$ (called $Q$-function in quantum optics), which can be defined as the squared overlap between $\psi$ and an arbitrary coherent state, (a more precise definition will be given below) and which plays a fundamental role in many branches of quantum physics, mainly in the study of quantum optics in phase space. Here we shall focus on the zeros of this phase-space representation of $\psi$, their physical meaning and their relation to pairing energies of the LMG pairing model. It is known that the zeros of the phase-space distribution $Q_\psi$ determine the quantum state $\psi$ [@Leb1990]. The zeros of $Q_\psi$ are experimentally measurable for some cases, which provides a way to reconstruct the associated quantum state $\psi$, a basic problem in quantum information theory. For example, for spin systems, the vanishing of the $Q_{\psi}$ at a point in phase space can be tested in principle, by means of a Stern-Gerlach apparatus \[see later for a discussion on experimental setups\]. Also, the time evolution of coherent states of light in a Kerr medium is visualized by measuring $Q_{\psi}$ by cavity state tomography, observing quantum collapses and revivals and confirming the non-classical properties of the transient states [@nature2013]. Moreover, the zeros of this phase-space probability have been shown as an indicator of the regular or chaotic behavior in quantum maps, theoretically, in a a variety of quantum problems such as molecular systems [@molecular] , atomic physics [@atomic], quantum Billiards [@billiars] or in condensed matter physics [@cmp] (see also [@Leb1990; @teo; @bengtsson] and references therein), and experimentally [@nature2009] in the kicked top. They have also been considered as an indicator of metal-insulator phase transitions [@aulbach2004] and quantum phase transitions in the Dicke, vibron and LMG models [@qpt]. Additionally, in Refs. [@vidalprl; @Vidal], the eigenfunctions of the LMG Hamiltonian were analyzed in terms of the zeros of the Majorana polynomial (proportional to the Husimi amplitude), leading to exact expressions for the density of states in the thermodynamic, mean field, limit.

All these examples support the important operational and physical meaning of the $Q_{\psi}$ zeros and its experimental accessibility.

In this work, we establish an exact correspondence between the zeros of the phase-space probability distribution $Q_{\psi}$ and the pairing energies of the LMG model. As a byproduct, a reconstruction of the wave function in terms of the zeros of $Q_{\psi}$ is also provided. Besides, an analysis of the pairing energies’ degeneracy is done in terms of the zeros’ multiplicity. Finally, we show that the link between pairing energies and zeros of the phase-space probability distribution $Q_{\psi}$ can be generalized to other pairing models like, for example, the bosonic BCS [mean-field like]{} model.

*Pairing Hamiltonians and the LMG model*.- Many models describing pairing correlations in condensed matter and nuclear physics are defined by a Hamiltonian of the form $$H=\sum_{k,\sigma}\varepsilon_k^\sigma c_{k\sigma}^\dag c_{k\sigma}+
\sum_{kl,\sigma\tau}
\gamma_{kk'll'}^{\sigma\sigma'\tau\tau'}
c_{k\sigma}^\dag c_{k'\sigma'}^\dag c_{l\tau}c_{l'\tau'}\label{ham1}$$ where $c_{k\sigma}^\dag$ ($c_{k\sigma}$) creates (destroys) a fermion in the $k$ state of the $\sigma=\pm$ level with energy $\varepsilon_k^\sigma$. The states $(k' \sigma')$ and $(l' \tau')$ are usually taken to be the conjugate (time-reversed) of $(k \sigma)$ and $(l \tau)$, but other possibilities can also be considered [@dukelskinp05]. [The pairing Hamiltonian can be mapped onto a nonlinear spin system in terms of Anderson pseudospin operators [@Anderson].]{} Indeed, for some particular choices of $\varepsilon_k^l$ and couplings $\gamma$, the Hamiltonian can be written in terms of the $su(2)$ quasispin collective operators $$J_+=\sum_k c_{k+}^\dag c_{k-}=J_-^\dag, \,
 J_z=\frac{1}{2}\sum_{k\sigma}\sigma c_{k\sigma}^\dag c_{k\sigma}.$$ This is the case of the LMG model, which assumes that the nucleus is a system of fermions which can occupy two levels $\sigma=\pm$ with the same degeneracy ($2j$), separated by an energy $\varepsilon$. In the quasispin formalism, the LMG model Hamiltonian is [@lipkin]: $$H = \varepsilon J_z+\frac{\lambda}{2}(J_+^2+J_-^2)+\frac{\gamma}{2}(J_+J_-+J_-J_+).
\label{ham2}$$ The $\lambda$ term annihilates pairs of particles in one level and creates pairs in the other level and the $\gamma$ term scatters one particle up while another is scattered down. The total angular momentum $\vec{J}^2=j(j+1)$ and the total number of particles $N=2j$ are conserved. This symmetry reduces the size of the largest matrix to be diagonalized from $2^N$ to $N+1$. For a Dicke state $|j,m\rangle$, the eigenvalue $m$ of $J_z$ gives the number $n=m+j$ of excited particle-hole pairs. $H$ also commutes with the parity operator $\hat{P}=e^{i\pi(J_z+j)}$, so that temporal evolution does not connect states with different parity. It will be useful for us to make use of the boson (Schwinger) realization of the $su(2)$ operators in terms of two bosons $a$ and $b$: $$J_+=b^\dag a, \, J_-=a^\dag b, \, J_z=\frac{1}{2}(b^\dag b-a^\dag a),$$ and the Dicke states $|j,m\rangle$ in terms of Fock states $|n_a=j-m,n_b=j+m\rangle$, with $n_a$ and $n_b$ the occupancy number of levels $a$ and $b$.

*Exact solvability of the LMG model*.- Making use of the previous Schwinger realization, the LMG model has been proved to be exactly solvable by mapping it to a $SU(1,1)$ Richardson-Gaudin integrable model [@dukelskinp05]. Introducing the new parameters $\gamma_x=\frac{2j-1}{\varepsilon}(\gamma+\lambda)$ and $\gamma_y=\frac{2j-1}{\varepsilon}(\gamma-\lambda)$ and using $t=\sqrt{|\gamma_x/\gamma_y|}$, the unnormalized eigenvectors of the LMG Hamiltonian are found to be $$|\psi_{M,\nu}\rangle=\displaystyle\prod_{ \alpha=1}^M \left(\frac{a^{\dag}a^{\dag}}{e_ {\alpha}+t}+\frac{b^{\dag}b^{\dag}}{e_ {\alpha}-t}\right)|\nu_a,\nu_b\rangle
\label{wf1}$$ where $e_{\alpha}\in\mathbb C$ are the so-called spectral parameters or *pairing energies*, and $\nu_a$ and $\nu_b$ are the seniorities. For integer $j$, the seniorities are equal to $\nu_a=\nu_b\equiv\nu=0$ or $1$, and the total number of pairs is $M=j-\nu$. The $M$ pairing energies $e_{\alpha}$ can be determined by solving a coupled set of Richardson (nonlinear) equations [@dukelskinp05; @lerma2013] and the eigenvalues of the LMG Hamiltonian are given in terms of pairing energies.

*Phase-space probability distribution and correspondence between its zeros and pairing energies*.- Given a general state $|\psi\rangle=\sum_{n_a,n_b} c_{n_a,n_b}|n_a,n_b\rangle$, with $n_a+n_b=2j$ the total occupancy number, the so-called Husimi distribution $Q_\psi$ is defined as the squared modulus of the overlap $${Q_\psi}(\zeta)=|\langle \zeta|\psi\rangle|^2
\label{husiz}$$ between $|\psi\rangle$ and an arbitrary spin-$j$ coherent state $$\begin{aligned}
|\zeta\rangle&=&\frac{1}{\sqrt{(2j)!}}
\frac{(a^{\dag}+ \zeta b^{\dag})^{2j}}{(1+|\zeta|^2)^j}|0,0\rangle,\nonumber\\
&=&(1+|\zeta|^2)^{-j}\sum_{m=-j}^j{\binom{2j}{j+m}}^{1/2}\zeta^{j+m}|j,m\rangle,
\label{cohsu2}\end{aligned}$$ where $\zeta=\tan(\theta/2)e^{-i\phi}$ is given in terms of the polar $\theta$ and azimuthal $\phi$ angles on the Riemann sphere. This phase-space probability distribution function is non-negative and normalized according to $\int_{\mathbb S^2} {Q_\psi}(\zeta) d\Omega(\zeta)=1$ (for normalized $\psi$), with integration measure (the solid angle) $d\Omega(\zeta)=\frac{2j+1}{4\pi}\sin\theta d\theta d\phi$.

Taking into account that $\langle 0| a^n a^{\dag m}|0\rangle=n!\delta_{nm}$ (and a similar equation for $b$), after a little bit of algebra we can calculate the Husimi amplitude of the eigenvector in terms of the pairing energies $e_{\alpha}$ as $$\langle\psi_{j-\nu,\nu}|\zeta\rangle=\frac{\sqrt{(2j)!}}{(1+|\zeta|^2)^j}\zeta^{\nu}
\prod_{\alpha=1}^{j-\nu}\left(\frac{1}{\bar{e}_{\alpha}+t}+\frac{\zeta^2}{\bar{e}_{\alpha}-t}\right).\label{Husamp}$$ \[$\bar{e}_\alpha$ denotes complex conjugate\]. Therefore, the zeros of this phase-space representation of the quantum state $\psi$ can be analytically related with the pairing energies by the equation: $$\zeta_{\alpha}^2={\frac{t-\bar{e}_{\alpha}}{\bar{e}_{\alpha}+t}}, \quad
\alpha=1,...,j-\nu
\label{relacion}$$ This is the main result of this letter.

As already said, it is known that one can represent each quantum state $\psi$ by means of the zeros of its phase-space representation $Q_{\psi}$ [@Leb1990]. Here we have also found an explicit expression of the Hamiltonian eigenstates in terms of the zeros of their phase-space representation $Q_{\psi}$ when plugging $e_\alpha=t({1-\bar\zeta_\alpha^2})/({1+\bar\zeta_\alpha^2})$ in eq. .

For a general state $|\psi\rangle=\sum_{m=-j}^jc_m|j,m\rangle$, the $Q_{\psi}$ amplitude has the form $$\langle\psi|\zeta\rangle=\frac{\sum_{m=-j}^{j} \bar c_m \sqrt{\binom{2j}{j+m}} \,\zeta^{j+m}}{(1+|\zeta|^2)^j},\label{Qamp}$$ which is proportional to the so-called Majorana polynomial [@Majorana], $P_j(\zeta)=\sum_{m=-j}^{j} \bar c_m {\binom{2j}{j+m}}^{1/2} \,\zeta^{j+m}=0$, in the variable $\zeta$. This polynomial and its zeros have been used in [@Vidal] to find the energy spectra and density of states of the LMG Hamiltonian in the thermodynamic $j\to\infty$ limit, as well as first order finite size corrections. Here we exploit the exact solvability of the LMG model (by mapping it to a $SU(1,1)$ Richardson-Gaudin integrable model [@dukelskinp05]) and achieve exact solutions for finite $(2j)$-size, by establishing a [connection between the pairing energies $e_\alpha$ and the zeros of the Husimi function of the eigenstates of the LMG model.]{} Finite zeros of $Q_\psi$ are then calculated by solving the $2j$-degree Majorana polynomial equation $P_j(\zeta)=0$. Besides, we have an extra zero of $\langle\psi|\zeta\rangle$ at $\zeta=\infty$ ($\theta=\pi$: South pole) when $c_j=0$ (in particular, when $\psi$ belongs to the odd parity sector with $\nu=1$). In total, the $Q_{\psi}$ amplitude $\langle\psi|\zeta\rangle$ has $2j$ zeros counted with their multiplicity. Since $\pm \zeta_\alpha$ yield the same $e_\alpha$ for $\alpha=1,...,j-\nu$, then one always has $2j-2\nu$ zeros attached to $j-\nu$ pairing energies. For $\nu=1$, the extra $2\nu$ zeros are $\zeta=0$ and $\zeta=\infty$, as can be seen from eq. . All these values determine the eigenfunction of the Hamiltonian. This can be done for any eigenstate of the Hamiltonian, that is finding the corresponding zeros of the Husimi function.

In particular, as an example, we have computed numerically the coefficients $c_m$ of the ground state of the LMG Hamiltonian , and the corresponding zeros of $Q_{\psi}$, for the trajectory $\gamma_x =-\gamma_y+10$ in the control parameter space. Figure (\[figpairons\]) shows pairing energies $e_\alpha$ (first and second panels) and zeros $\zeta_{\alpha}=\tan{(\theta_{\alpha}/2)}e^{-i\phi_{\alpha}}$ of the $Q_{\psi}$ function (third and fourth panels) for the ground state and the trajectory $\gamma_x =-\gamma_y+10$. We have plotted the range $0<\gamma_x<10$. This Figure shows a clear example of the mapping between the pairing energies $e_{\alpha}$ and the zeros $\zeta_{\alpha}$ of $Q_\psi$.

![ \[figpairons\] (Color online) Ground state real and imaginary parts (first and second panels) of pairing energies and polar and azimuthal angles of zeros (third and fourth panels) of the phase-space probability distribution $Q_{\psi}$ representing the ground state, for $j=10$, associated to the trajectory $\gamma_x =-\gamma_y+10 $ for $0<\gamma_x<10$. The values of $\gamma_x$ where pairing energies or zeros degenerate are indicated by grey vertical lines.](fig1 "fig:"){width="8cm"} ![ \[figpairons\] (Color online) Ground state real and imaginary parts (first and second panels) of pairing energies and polar and azimuthal angles of zeros (third and fourth panels) of the phase-space probability distribution $Q_{\psi}$ representing the ground state, for $j=10$, associated to the trajectory $\gamma_x =-\gamma_y+10 $ for $0<\gamma_x<10$. The values of $\gamma_x$ where pairing energies or zeros degenerate are indicated by grey vertical lines.](fig2 "fig:"){width="8cm"} ![ \[figpairons\] (Color online) Ground state real and imaginary parts (first and second panels) of pairing energies and polar and azimuthal angles of zeros (third and fourth panels) of the phase-space probability distribution $Q_{\psi}$ representing the ground state, for $j=10$, associated to the trajectory $\gamma_x =-\gamma_y+10 $ for $0<\gamma_x<10$. The values of $\gamma_x$ where pairing energies or zeros degenerate are indicated by grey vertical lines.](fig3 "fig:"){width="8cm"} ![ \[figpairons\] (Color online) Ground state real and imaginary parts (first and second panels) of pairing energies and polar and azimuthal angles of zeros (third and fourth panels) of the phase-space probability distribution $Q_{\psi}$ representing the ground state, for $j=10$, associated to the trajectory $\gamma_x =-\gamma_y+10 $ for $0<\gamma_x<10$. The values of $\gamma_x$ where pairing energies or zeros degenerate are indicated by grey vertical lines.](fig4 "fig:"){width="8cm"}

*Multiplicity of zeros and degeneracy of pairing energies*.- Additionally one can appreciate the presence of collapses (vertical grey lines in Fig. \[figpairons\]) for certain values of the parameter $\gamma_x$ of the LMG Hamiltonian. These collapses or places where several zeros of the Husimi function or pairing energies join are determined by the expression $$\gamma_x=5\pm\sqrt{5^2 -\left(\frac{2j-1}{2j-1-2k}\right)^2}\,,k=0,...,j-1 \, ,
\label{new1}$$ for the trajectory $\gamma_x =-\gamma_y+10$. This expression is obtained at the intersections between the straigthline $\gamma_x=-\gamma_y+10$ and the family of hyperbolas $\gamma_x\gamma_y=[({2j-1})/({2j-1-2k})]^2$ with $k=0,...,j-1$. Notice that the point $\gamma_x$ given in (\[new1\]) must be real and then it exists if [$k\leq \lfloor 2/5(2 j -1) \rfloor$]{}. The symbol $\lfloor x \rfloor$ means the smallest integer part of $x$. This family of points $(\gamma_x, \gamma_y=5)$ has the property that each element verifies that the ground state amplitude of the Husimi function, $\langle \zeta|\psi\rangle$, has $j-k$ different zeros, one of them of multiplicity $2(k+1)$ and the others $j-k-1$ of multiplicity $2$. Therefore when the path $\gamma_x =-\gamma_y+10$, crosses each point of the above family, the zeros of the ground state $Q_{\psi}$ function join with the aforementioned multiplicity. We want to stress that due to the relation given in Eq. (\[relacion\]) the same behavior is displayed by the pairing energies. For the trajectory $\gamma_x=\gamma_y$ with $\gamma_x>-1$, one verifies that the ground state Husimi function has only one zero (the north pole in the Riemann sphere) of multiplicity $2j$, so that all the zeros of the $Q_{\psi}$ function (and thus all the pairing energies) join at the point $\gamma_x=\gamma_y =5$. The coefficients appearing in the family of hyperbolas, that is $-({2j-1})/({2j-1-2k})$, define the $\gamma_x$ values where there are crossings between the even and odd energy levels, when one takes $\gamma_x=\gamma_y$ (these points are explicitly calculated in [@octavio2005]). [Concerning the physical interpretation of the pairons’ collapses, there are several analysis in the literature, for example in the context of the SU(2) Richardson-Gaudin exact solution of a gas of spinless fermions [@collapses]. In this context, pairons’ collapses are representing bound pairs because the pairons are real and negative. In our case, as it can be seen in Figure \[figpairons\], all the particles form bound pairs at the point $\gamma_x=5$, while at the right- and left-hand sides of $\gamma_x=5$ the number of bound pairs decreases and the number of dispersion states increases. Notice that for $\gamma_x=0$ all the Cooper pairs are deconfined (in the sense of Ref. [@collapses]).]{}

*Experimental setups for state reconstruction and the determination of zeros and pairing energies*.- Concerning the physical meaning and the experimental determination of the zeros of $Q_\psi$ for a spin system like LMG, let us discuss two possible experimental setups. The first one is related to the fact that the coherent state $|\zeta\rangle$ is the rotation of $|j,-j\rangle$ about the axis $\vec{r}=(\sin\phi,-\cos\phi,0)$ in the $x$-$y$ plane by an angle $\theta$, that is $|\zeta\rangle=\exp(-i\theta\vec{r}\cdot\vec{J})|j,-j\rangle$. A possible procedure for quantum-state reconstruction is explained in References [@Amiet; @Mann]. Basically, the phase-space distribution $Q_\psi(\zeta)=|\langle \zeta|\psi\rangle|^2$ is precisely the probability to measure $m=-j$ in $\psi$, which can be determined by means of, for example, a Stern-Gerlach apparatus oriented along $\vec{R}(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ (see e.g. Ref. [@Amiet]). Repeting this procedure (with a large number of identically prepared systems) for a finite number of orientations $(\theta,\phi)$, one can determine the function $Q_\psi$ and, therefore, its zeros. Actually, the zeros $\zeta_0$ of $Q_\psi$ are just those orientations $\vec{R}(\theta_0,\phi_0)\leftrightarrow\zeta(\theta_0,\phi_0)$ of the Stern-Gerlach apparatus for which the probability of the outcome $m=-j$ vanishes. Simulations of the LMG model in a Bose-Einstein condensate in a double-well potential are known (see e.g. [@Chen]). In this context, the polar angle $\theta$ is related to the population imbalance $j\cos\theta$ (the mean value of $J_z$), and the azimuthal angle $\phi$ is the relative phase of the two spatially separated Bose-Einstein condensates. Both quantities can be determined in terms of matter wave interference experiments as is shown in Refs. [@prl92; @prl95; @science307].

Therefore, the reconstruction of any state $|\psi\rangle$ and the possibility to measure pairing energies in terms of the zeros of $Q_\psi$ are then experimentally accessible.

*Extension to other bosonic BCS-like models*.- The correspondence between pairing energies and zeros of the probability distribution representing a state in phase space is also present in other higher-dimensional BCS models, although the relation is a little bit more subtle. Let us consider for example the bosonic counterpart of , where $c_{k\sigma}$ are now boson $b_{k\sigma}$ annihilation operators. We shall consider $L+1$ scalar bosons $b_0, b_1, \dots, b_{L}$. In the case of uniform couplings $\gamma_{kkll'}=\gamma/4$, the complete set of eigenstates of this model is given by $$|\psi_{M,\nu}\rangle=\displaystyle\prod_{ \alpha=1}^M \left(\sum_{\ell=0}^{L}\frac{b_\ell^{\dag}b_\ell^{\dag}}{2\varepsilon_\ell-e_ {\alpha}}
\right)|\nu\rangle,
\label{wf2}$$ where $|\nu\rangle=|\nu_0,\dots,\nu_L\rangle$ is a state of $\nu=\sum_{\ell=0}^L\nu_\ell$ unpaired bosons and $\nu_\ell=0,1$ are the seniorities. The total number of particles is $N=2M+\nu$, with $M$ the number of paired bosons. As for the LMG model, each eigenstate $|\psi_{M,\nu}\rangle$ is completely determined by a set of $M$ pairing energies $e_\alpha$ (which are solutions of a set of coupled non-linear Richardson’s equations [@dukelskinp05]) and their energies are given by $E_{M,\nu}(e)=\sum_{\ell=0}^L\varepsilon_\ell
\nu_\ell +\sum_{\alpha=1}^M e_\alpha$. The $Q_{\psi}$ probability distribution of any quantum state $\psi$ for this model is the squared modulus of the overlap between $|\psi\rangle$ and a general $SU(L+1)$ coherent state $$|\zeta\rangle=\frac{1}{\sqrt{N!}}
\frac{(b_0^\dag+\sum_{\ell=1}^L\zeta_\ell b^{\dag}_\ell)^{N}}{(1+\sum_{\ell=1}^L|\zeta_\ell|^2)^{N/2}}|0\rangle,
\label{cohsuL}$$ which is a generalization of for $\zeta=(\zeta_1,\dots,\zeta_L)\in \mathbb C^L$. The $Q_{\psi}$ amplitude of an eigenstate turns out to be $$\begin{aligned}
\langle \psi_{M,\nu}|\zeta\rangle&=&
\frac{\sqrt{N!}\prod_{\ell=1}^L\zeta_\ell^{\nu_\ell}}{(1+\sum_{\ell=1}^L|\zeta_\ell|^2)^{N/2}}\label{final3}\\
&&\times\displaystyle\prod_{\alpha=1}^M\left(
\frac{1}{2\varepsilon_0-\bar e_{\alpha}}+\sum_{\ell=1}^L
\frac{\zeta_\ell^2}{2\varepsilon_\ell-\bar e_{\alpha}}\right).\nonumber\end{aligned}$$ Therefore, the zeros $\zeta_{\ell,\alpha}, \alpha=1,\dots,M$ of $\langle \psi_{M,\nu}|\zeta\rangle$ lie now on $L$-dimensional complex ellipsoids $$\frac{\zeta_{1,\alpha}^2}{\xi_{1,\alpha}^2}+\dots+\frac{\zeta_{L,\alpha}^2}{\xi_{L,\alpha}^2}=1, \;\;
\xi_{\ell,\alpha}^2=\frac{2\varepsilon_\ell-\bar e_\alpha}{\bar e_\alpha-2\varepsilon_0},$$ with semi-principal axes of complex “length” $\xi_{\ell,\alpha}$. Still, there is a correspondence between pairing energies and complex ellipsoids in phase space, where the $Q_{\psi}$ distribution vanishes. Moreover, the occurrence (or absence) of $\zeta_\ell=0$ as a zero of means seniority $\nu_\ell=1$ (or $\nu_\ell=0$).

*Conclusions*.- Exploiting the exact solvability of the LMG model (as a particular case of $SU(1,1)$ Richardson-Gaudin integrable model), we have revealed an interesting relation between pairing energies and zeros of a phase space probability distribution representing the quantum state in finite-size pairing [mean-field like]{} models. Zeros are experimentally accessible and this gives a method to find pairing energies’ multiplicities across the control parameter space. As a byproduct, knowing the zeros, one can reconstruct the corresponding state. These results are proven to be valid for a large class of paring systems and, in principle, they could be extended to other systems where particles pairs emerge.

Acknowledgments {#acknowledgments .unnumbered}
===============

This work was supported by CONACyT-México (under project 101541), the Spanish MICINN (under projects FIS2011-24149 and FIS2011-29813-C02-01), Junta de Andalucía project FQM1861, and the University of Granada project CEI-BioTIC-PV8.

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Houses in Ancient Carthage

By Natasha Sheldon

It was believed that the Romans erased all traces of Punic Carthage. But preserved under the Roman forum are the unique remains of the homes of ordinary Carthaginians.

Ancient Carthage was convincingly destroyed by the victorious armies of Scipio Aemilianus in 146BC. When Julius Caesar decided to reinstate the city as a colony ninety eight years later, it had to be completely rebuilt.

However, the Romans did not quite manage to destroy everything. Archaeologists excavating the Byrsa Hill have discovered the remains of houses, streets and shops under the Roman foundations of the forum that reveal a great deal about Punic civilization.

Carthage at the Time of the Punic Wars.

By the time of the final Punic Wars in 150BC, Carthage had expanded to within 400m of the Tophet. The Byrsa hill, the site of the original settlement, still represented the centre of urban life. On its peak were the city’s most important buildings and principle temples. From here, streets radiated down the hill, following the natural contours.

The buildings discovered under what became the Roman forum date from this time. Overlooking the harbour areas, they did not belong to the rich but rather merchants and ordinary public officials.

Preservation Under the Roman Forum.

The houses survived by chance. Left in ruins after the Roman’s razed the city, they lay abandoned until the rebuilding commenced.

The new Roman forum was symbolically placed over the old Carthaginian civic centre. However, unlike their predecessors, the Roman’s did not choose to work in harmony with the hill. Instead, they flattened the summit of the Byrsa and supported the new forum was supported by arched foundations erected over the Punic remains.

Surplus earth from the excavation of Bysra hill was pushed downwards over the Carthaginian’s ruined homes, covering them completely. They lay preserved until the 20th century when archaeologists rediscovered the remains of streets and blocks of houses with walls surviving up to 3m in places.

The Streets.

Each block of houses was separated by a 5-7metre wide street. Unpaved and of beaten earth, they were extremely steep, with a 1:7 gradient. As the construction of this area was in sympathy with the topography, steps were built periodically built to allow pedestrians to move between different levels. It would have been impossible to have used wheeled transport.

The Houses.

The houses were identified as Punic by finds and the techniques used to build them. Bronze Punic coins were found in the ruins as well as black glazed Campanian pottery that was brought back from Italy with Hannibal. The common floor style was pavimenta punica, a style dated to the last 50 years of Carthage’s history. Consisting of a grey base over which shards of green and yellow pot were laid, it was uniquely Punic.

Each house was an elongated rectangle, with a back and front entrance opening out onto the streets. Commonly, a side corridor ran uninterrupted through the building, connecting both entrance ways. Depressions have been found in these corridors indicating the location for eliminating household waste into stone lined soak ways built in the streets.

Some buildings were purely residential, with large, often ornate rooms flanking the front and back entrances. Others were dual purpose, with shops on one of the street fronts.

All were built around a central courtyard which was the main source of light for the house and also the site of its water cistern. The courtyard opened directly onto the house’s main corridor or else was partitioned off from the rest of the house. One house with a partitioned courtyard had a number of small rooms built directly onto it. Some seemed to be for storage purposes but one was a bathroom, complete with water and waste pipes and a terracotta mosaic floor.

Appian described Punic houses as being six stories high. Although it is impossible to verify whether this was actually the case, there is evidence of internal stairways so certainly the houses were more than one story high.

Sources.

The author's 2006 visit to Carthage.

Lancel, Serge (1992). Carthage: A History. Blackwell: Oxford and Cambridge

CopyrightNatashaSheldon2010. All photographs (unless otherwise stated) are the property of either myself or Neil Bate and can be used on request. All articles are the intellectual property of Natasha Sheldon and may not be published or reproduced in any print or electronic format without permission of the author.


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Football in Mauritius

Football is the most popular sport in Mauritius. The national governing body is the Mauritius Football Association. Internationally, Mauritius is represented by Club M in senior competitions, and by the U-17 and U-20 teams in youth competitions. The top domestic football league in Mauritius is the Mauritian League, and the top knockout tournament is the Mauritian Cup.

History
Football in Mauritius has had a tumultuous history. The Mauritian League was founded in 1935, and the Mauritius Football Association (MFA) was founded in 1952, affiliated to FIFA in 1962 and to the Confederation of African Football (CAF) in 1963. The founding members of the association were FC Dodo, Faucon Flacq SC (now renamed Flacq SC), CSC, Hounds, Royal College of Curepipe, and Saint Joseph College, all of which were based out of Curepipe and, with the exception of Flacq SC, have since ceased to exist. Troubles have arisen for the MFA, ranging from perpetual financial shortages to constant changes of presidents, and it has been under fire from various quarters for failing to stem the decline of football in Mauritius. Although football continues to be the most popular sport in the country, the majority of domestic league matches attract very few spectators (sometimes less than a handful), earning the clubs and the MFA very little income and leaving the players without the motivation to showcase their skills.

The focus of the sport's popularity has almost completely shifted to the English Premier League, which is considered far more exciting due to the high quality of football on display and the presence of numerous world-famous stars. National and International television networks beam Premier League matches into the living rooms multiple times every week, adding to the erosion of interest in local football.

The 1999 major restructuring of the MPL was regarded as the beginning of the decline in Mauritian football. Due to a riot on May 23, 1999, between Fire Brigade Sports Club (now renamed as Pamplemousses SC) and Scouts Club (renamed as Port Louis Sporting Club), which lasted for three days and killed seven people, the government imposed an 18-month ban on all footballing activities in the country, with only the national team permitted to play during this period. The restructure, which was an effort to de-ethnicize local clubs, required the "regionalization" of clubs, with teams now being formed on the basis of region instead of ethnicity or religion. Although the move achieved the desired results in terms of reduction in violent conflicts, it also removed the traditional rivalry among fans, resulting in loss of interest and support. In fact, many locals believe that the resurrection of Mauritian football lies in permitting the formation of clubs on the basis of ethnicity or religion.

On the international stage, the national team, known as Club M, has not had much success, only winning the Indian Ocean Games twice and qualifying for the Africa Cup of Nations once in 1974. It has plummeted down the FIFA World Rankings in the past few years to the lowest it has ever been. The future looks bright though, as in 2006 four Regional Technical Centers (CTRs) were created to improve youth football in Mauritius, with hopes of revitalizing the national team. There is hope that Mauritian football can grow to new heights if the MPL continues to grow and new talent is found in the CTRs.

Mauritius national football team

The Mauritius national football team is the national football team of Mauritius. Club M won the Indian Ocean Island Games in 2003.

Mauritian League

The Mauritian League, otherwise known as the Barclays League, is the top league of Mauritian football. The MFA Second Division is the second highest level of Mauritian football.

Mauritian Cup

The Mauritian Cup is the top knockout tournament of Mauritian football.

League system

Notable Mauritian footballers

Born in Mauritius

Of Mauritian heritage

References


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Uncharted 2: Among Thieves Review

This review is going to be short simply because it’s hard for me to find anything negative about such an awesome game. Naughty Dog’s Uncharted 2: Among Thieves is my game of the year and certainly the best game to grace the PS3 this year.

Before I really get started with this review pick up a copy of Uncharted: Drake’s Fortune the prequel to Among Thieves. You don’t need to play that first and you can simply enjoy Uncharted 2 but if you have a PS3 you deserve to play that game. It’s also a greatest hit title so you can get it for cheap, no excuses.

In Uncharted 2 you once again play as treasure hunter Nathan Drake and his quest to find the missing fleet of Marco Polo. New and old characters turn up but to avoid spoilers I won’t talk about the story at all but just to let you know that it is very good and all the actors involved deliver outstanding performance, most notable Nolan North who plays the title character Nathan Drake.

Uncharted 2 single player story is one of the best 10-12 hours I’ve ever played. From exploring the streets of Nepal to running and gunning on moving train which is actually moving so you have to adjust your aim accordingly, there is nothing better this year. There is a perfect mix of combat and puzzles to be done and the puzzles are bit harder then the first Uncharted. Throw in a 100 treasures to find you have plenty of content to be satisfied with.

There are multiple difficulties ranging from Very Easy to Crushing so you can beat the game no matter how good you are. Before writing this review I made sure to beat the game on Crushing and damn this game is hard. After beating it I’ll never play it on Crushing again.

To go along with the superb single player is an amazing addictive multiplayer. Naughty Dog created a good deal of variety ranging from deathmatch to the Uncharted style of capture the flag called Plunder, which is great by the way.

To go with good variety of competitive gameplay is the 3-player co-op modes. You have co-op objectives levels from the single player for all three of you to complete. It also has different difficultly settings so three of your friends can try Crushing and be warned, you will die. There is also a survival mode and my favorite Gold Rush where you have to capture the gold and each wave gets harder and harder.

I could go on and on about how good Uncharted 2 is and I would if I wasn’t to busy playing this. You won’t find another game like this and if you own a PS3 and do not own Uncharted 2 something is seriously wrong. Uncharted 2: Among Thieves is GOTY, for me anyways and I don’t think too many people will disagree with me.


New Hampshire: Clean Energy is One of Our Primary Concerns

Excerpt from “New Hampshire: Clean Energy is One of Our Primary Concerns” by Bruce Clendenning, published in The Nature Conservancy.

Aside from politics of the New Hampshire presidential primary, Granite Staters are also talking about clean energy. This worker is installing solar panels at Wire Belt Company of America in Londonderry, New Hampshire. Photo courtesy ReVision Energy.

While attention is fixed on the New Hampshire primary,
Granite Staters of all political stripes support a clean energy future.

This year, like every four years, the nation watches New Hampshire as Granite Staters pepper the presidential candidates about every possible issue. Right now, across our state, they’re asking questions you’d expect about national security, the economy and immigration.

But they’re also asking about energy and climate.

When the votes are all cast and the national press moves on to the next big state, we’ll still be talking about clean energy. Republicans and Democrats alike.

Here are four ways that New Hampshire demonstrates that we don’t just choose presidents. We can lead the way on the critical energy policies facing the nation, if we work together the New Hampshire way.

Bi-Partisan Support. In New Hampshire, support for clean energy and action on climate change is strong and bipartisan. Polling on energy issues and climate routinely shows support for the kinds of policies that will directly reduce climate emissions. Recent examples include an August 2015 survey that found that 68 percent of New Hampshire residents believe that climate change is a serious problem and 66 percent believe that the policies included in the Clean Power Plan represent a good approach to the problem.In December, 72 percent of New Hampshire Republican primary voters stated that it’s important for presidential candidates to have a clean energy plan and 74 percent want increased usage of renewable energy and decreased use of coal.While public support for these energy goals is strong, as in any other state, sometimes our elected officials lag in reading the public sentiment. We expect that as public support remains consistently strong, the “leaders” will follow their constituents toward clean energy solutions.


Babies fed on mother's milk are known to have fewer medical problems than babies fed mother's milk substitutes. In particular, breast-fed babies suffer fewer infections than babies that are not breast-fed.
In many mammals, mortality is high immediately after parturition and among the newborn. For example, 20 percent of pigs do not survive the first week of life. Approximately one-half of that number die due to gastrointestinal disorders caused by pathogenic strains of one or more bacteria. Further, gastrointestinal disorders contribute to poor growth in many pigs that survive the first week.
The frequency of diarrheal deaths in British infants in 1960 was six times greater for those infants fed cow's milk than those who were breast-fed. [Goldman et al., J. Pediatr., 82:1082 (1979)]In 1979, the incidence of enteric infections was found to be lowest in breast-fed infants in countries where sanitary conditions are poor. [Kabara, Nutrition Reviews, 38:65 (1980)]Whether the above statistics are due to a protective factor in human milk or whether human milk conveys fewer pathogenic organisms has not been determined. [Kabara above; Welsh, J. Pediatr., 94:1 (1979)]
A recent report by Koopman et al., Am. J. Public Health, May 1985, compared data on 143 babies with gastrointestinal illness and 143 babies free of such illness. Their case-control study indicated that babies fed on formula were at 6 times the risk of babies on breast milk and 2.5 times the risk of babies on cows'milk.
Among those substances in human milk reported to provide anti-infective properties are a growth enhancer for lactobacillus, an antistaphlococcal agent, immunoglobulins, complement components, lysozyme, lactoperoxidose, lactoferrin, macrophages, lymphocytes and lipids (Kabara, above). However, whether these factors protect the mammary gland from infection or confer infection resistance on the child is unknown (Kabara, above).
The gut of the breast-fed human infant is enriched with lactobacilli in comparison to the gut of infants who are not breast-fed. This difference has been attributed to a lactobacillus growth factor and the high concentration of lactose in human milk, along with its low bulk, low protein content and low buffering capacity. It has been postulated that the high lactose concentration allows sufficient lactose to pass to the intestine to support the growth of lactose fermenters that produce acid conditions and encourage lactobacilli. The large population of lactobacilli alters growth conditions in the gut creating an environment in which diarrhea-producing organisms are less likely to grow (Welsh, above).
Other possible sources of resistance to enteric infections have not been definitively demonstrated. IgA has been thought to be protective (Welsh, above). Complement is thought to be insignificant because human milk heated to 56.degree. C. for thirty minutes retains its antibacterial activity, and that temperature is normally sufficient to inactivate complement.
Studies indicate that milk, particularly human milk, contains lipids thought to account for antimicrobial activity. Data are reported to indicate a strong correlation between microbial resistance and the level of lipase and monoglycerides (Kabara, above). Further, very short chain fatty acids, that is C.sub.6 or less, affect gram-negative organisms, the organisms that frequently are responsible for enteric infections.
Kabara et al., Antimicrob. Aq. Chemother. 2:23. (1972) advocate adding monoglycerides to improve the lipid formulation of synthetic or animal milk and artificial feeds for human infants and other young mammals to provide the food with more antimicrobial activity as is usually found in human milk.
Kabara et al., above, report several generalities regarding the bacteriostatic activity of fatty acids. Of tested straight chain saturated fatty acids, lauric acid (C.sub.12) was reported to have the most bacteriostatic activity on gram-positive organisms. Bacteriostatic activity reportedly increased with the addition of one and two double bonds to the fatty acid chain, but a third unsaturation was not as effective. A free carboxyl group was reported to be necessary for activity. Esters reportedly had decreased activity.
However, reduction of the carboxyl group to an aldehyde or alcohol, and changing that group into an amide or amine reportedly increased bacteriostatic effects. Further, in contrast to the other derivatives, amines were active against both gram-positive and gram-negative organisms. Specifically, the addition of monolaurin to milk substitutes and animal feed was advocated to provide the best overall antimicrobial activity.
Investigators have also studied the effect of lipid, and in particular, gangliosides, on the body's response to bacteria or bacterial toxins.
Ganglioside fraction GM1 (monosialoganglioside) is considered to be the cell membrane receptor for heat-labile enterotoxins of Escherichia coli (E. coli) and Vibrio cholerae [Olsvik et al., NIPH Ann., 6:5, (1983)]. Immunogenicity studies in rabbits indicated that gangliosides inhibited the primary and secondary immune responses to a common bacterial antigen as measured by hemagglutinin levels. However, gangliosides did not inhibit attachment of the antigen to red blood cells or alter the antigenic determinants. Agarwal et al. J. Immun., 107:1448, (1970) concluded gangliosides were an immunosuppressant.
Otaness et al. Infect. Immun., 40:563. (1983) studied the inhibition of E. coli and Vibrio cholerae enterotoxins by the ganglioside fraction of human milk. They reported that the enterotoxin inhibitor in human milk appeared to be gangliosides, and in particular ganglioside fraction GM1. They concluded that human milk gangliosides may be important in protecting infants against enterotoxin-induced diarrhea at the cell membrane receptor-toxin interaction level, rather than by an effect upon the bacteria themselves.
It would be advantageous to find a substance in human milk that protects the infant from the microorganisms that cause enteric infections, and to use that substance as an additive to milk substitutes for young mammals. It would be particularly desirable to find a relatively more available source of such a substance than human milk that functions protectively in a manner similar to human milk, and that can be added to mother's milk substitutes to enhance the resistance of a young mammal to enteric infections.


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OHSU Named "Leader in LGBT Healthcare"

For the fourth year in a row, Oregon Health & Science University has been recognized as a "Leader in LGBT Healthcare Equality" by the Human Rights Campaign (HRC) Foundation, the educational arm of the country's largest lesbian, gay, bisexual and transgender (LGBT) civil rights organization.

The findings were part of HRC Foundation's Healthcare Equality Index 2014, a unique annual survey that encourages equal care for LGBT Americans, and recognizes healthcare institutions doing the best work.

OHSU earned top marks in meeting non-discrimination and training criteria that demonstrate its commitment to equitable, inclusive care for LGBT patients, and their families, who can face significant challenges in securing the quality health care and respect they deserve.

OHSU was one of a select group of 426 healthcare facilities nationwide – and one of only four in Oregon – to be named Leaders in LGBT Healthcare Equality. Facilities awarded this title meet key criteria, including patient and employee non-discrimination policies that specifically mention sexual orientation and gender identity, a guarantee of equal visitation for same-sex partners and parents, and LGBT health education for key staff members.

OHSU is the largest employer in the Portland area that offers health coverage for employees and their dependents for care related to gender transition. The Center for Diversity and Inclusion coordinated trainings and cultural competency lectures on patient-centered care. OHSU Pride, the employee resource group for LGBTQ staff and their allies, promotes an inclusive workplace that provides opportunities for growth for all people, regardless of sexual orientation or gender identity.

The Healthcare Equality Index offers healthcare facilities unique and powerful resources designed to help provide equal care to a long-overlooked group of patients, as well as assistance in complying with new regulatory requirements and access to high-quality staff training.

For more information about the Healthcare Equality Index 2014, or to download a free copy of the report, visit www.hrc.org/hei.