Generalized parton distributions in impact parameter space

The kinematical factor in the positivity bound (36) is incorrect. The bound correctly reads Our corrected result agrees with inequality (25) in [1], taking into account the different normalization conventions here and there.Published online: 9 October 2003Erratum published online: 10 October 2003     

Unifying the theory of integration within normal-, Weyl- and antinormal-ordering of operators and the s-ordered operator expansion formula of density operators

By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinormally ordered and Weyl ordered product of operators as its special cases).The s-ordered operator expansion(denoted by...) formula of density operators is derived,which is ρ = 2 1 s ∫ d2βπβ|ρ |β exp { 2 s 1(s|β|2 β a + βa a a) }.The s-parameterized quantization scheme is thus completely established.     

The transfer of the quantum correlation from two-mode nonclassical state field to the supercurrents in two distant SQUID rings

We have considered two distant mesoscopic superconducting quantum interference device (SQUID) rings A and B in the presence of two-mode nonclassical state fields and investigated the correlation of the supercurrents in the two rings using the normalized correlation function $C_{\rm AB}$. We show that when the parameter $\alpha$ is very small for the separable state with the density matrix $\hat {\rho } = (\left| {\alpha , - \alpha } \right\rangle \left\langle {\alpha , - \alpha } \right| + \left| { - \alpha ,\alpha } \right\rangle \left\langle { - \alpha ,\alpha } \right|) / 2$ and entangled coherent state (ECS) $\left| u \right\rangle = N_1 (\left| {\alpha , - \alpha } \right\rangle + \left| { - \alpha ,\alpha } \right\rangle )$ fields, the dynamic behaviours of the normalized correlation function $C_{\rm AB}$ are similar, but it is quite different for the entangled coherent state $\left| {u}' \right\rangle = N_2 (\left| {\alpha , - \alpha } \right\rangle - \left| { - \alpha ,\alpha } \right\rangle )$ field. When the parameter $\alpha $ is very large, the dynamic behaviours of $C_{\rm AB}$ are almost the same for the separable state, entangled coherent state $\left| u \right\rangle $ and $\left| {u}' \right\rangle $ fields. For the two-mode squeezed vacuum state field the maximum of $C_{\rm AB}$ increases monotonically with the squeezing parameter $r$, and as $r \to \infty $, $C_{\rm AB} \to 1$. This means that the supercurrents in the two rings A and B are quantum mechanically correlated perfectly. It is concluded that not all the quantum correlations in the two-mode nonclassical state field can be transferred to the supercurrents; and the transfer depends on the state of the two-mode nonclassical state field prepared.     

Parity-violating neutron spin rotation in hydrogen and deuterium

We calculate the (parity-violating) spin-rotation angle of a polarized neutron beam through hydrogen and deuterium targets, using pionless effective field theory up to next-to-leading order. Our result is part of a program to obtain the five leading independent low-energy parameters that characterize hadronic parity violation from few-body observables in one systematic and consistent framework. The two spin-rotation angles provide independent constraints on these parameters. Our result for np spin rotation is $\frac{1} {\rho }\frac{{d\varphi _{PV}^{np} }} {{dl}} = \left[ {4.5 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {2g^{\left( {^3 S_1 - ^3 P_1 } \right)} + g^{\left( {^3 S_1 - ^3 P_1 } \right)} } \right) - \left[ {18.5 \pm 1.9} \right] rad MeV^{ - \frac{1} {2}} \left( {g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 2} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right)$\frac{1} {\rho }\frac{{d\varphi _{PV}^{np} }} {{dl}} = \left[ {4.5 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {2g^{\left( {^3 S_1 - ^3 P_1 } \right)} + g^{\left( {^3 S_1 - ^3 P_1 } \right)} } \right) - \left[ {18.5 \pm 1.9} \right] rad MeV^{ - \frac{1} {2}} \left( {g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 2} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right), while for nd spin rotation we obtain $\frac{1} {\rho }\frac{{d\varphi _{PV}^{nd} }} {{dl}} = \left[ {8.0 \pm 0.8} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^1 P_1 } \right)} + \left[ {17.0 \pm 1.7} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^3 P_1 } \right)} + \left[ {2.3 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {3g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 1} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right)$\frac{1} {\rho }\frac{{d\varphi _{PV}^{nd} }} {{dl}} = \left[ {8.0 \pm 0.8} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^1 P_1 } \right)} + \left[ {17.0 \pm 1.7} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^3 P_1 } \right)} + \left[ {2.3 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {3g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 1} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right), where the g ^(X-Y), in units of $MeV^{ - \frac{3} {2}}$MeV^{ - \frac{3} {2}}, are the presently unknown parameters in the leading-order parity-violating Lagrangian. Using naıve dimensional analysis to estimate the typical size of the couplings, we expect the signal for standard target densities to be $\left| {\frac{{d\varphi _{PV} }} {{dl}}} \right| \approx \left[ {10^{ - 7} \ldots 10^{ - 6} } \right]\frac{{rad}} {m}$\left| {\frac{{d\varphi _{PV} }} {{dl}}} \right| \approx \left[ {10^{ - 7} \ldots 10^{ - 6} } \right]\frac{{rad}} {m} for both hydrogen and deuterium targets. We find no indication that the nd observable is enhanced compared to the np one. All results are properly renormalized. An estimate of the numerical and systematic uncertainties of our calculations indicates excellent convergence. An appendix contains the relevant partial-wave projectors of the three-nucleon system.     

Influence of dephasing on the entanglement teleportation via a two-qubit Heisenberg XYZ system

We study the entanglement dynamics of an anisotropic two-qubit Heisenberg XYZ system in the presence of intrinsic decoherence. The usefulness of such a system for performance of the quantum teleportation protocol T₀\mathcal{T}_0 and entanglement teleportation protocol T₁\mathcal{T}_1 is also investigated. The results depend on the initial conditions and the parameters of the system. The roles of system parameters such as the inhomogeneity of the magnetic field b and the spin-orbit interaction parameter D, in entanglement dynamics and fidelity of teleportation, are studied for both product and maximally entangled initial states of the resource. We show that for the product and maximally entangled initial states, increasing D amplifies the effects of dephasing and hence decreases the asymptotic entanglement and fidelity of the teleportation. For a product initial state and specific interval of the magnetic field B, the asymptotic entanglement and hence the fidelity of teleportation can be improved by increasing B. The XY and XYZ Heisenberg systems provide a minimal resource entanglement, required for realizing efficient teleportation. Also, in the absence of the magnetic field, the degree of entanglement is preserved for the maximally entangled initial states $\left| {\psi \left. {\left( 0 \right)} \right\rangle = \frac{1} {{\sqrt 2 }}\left( {\left| {\left. {00} \right\rangle \pm } \right|\left. {11} \right\rangle } \right)} \right.$\left| {\psi \left. {\left( 0 \right)} \right\rangle = \frac{1} {{\sqrt 2 }}\left( {\left| {\left. {00} \right\rangle \pm } \right|\left. {11} \right\rangle } \right)} \right.. The same is true for the maximally entangled initial states $\left| {\psi \left. {\left( 0 \right)} \right\rangle = \frac{1} {{\sqrt 2 }}\left( {\left| {\left. {01} \right\rangle \pm } \right|\left. {10} \right\rangle } \right)} \right.$\left| {\psi \left. {\left( 0 \right)} \right\rangle = \frac{1} {{\sqrt 2 }}\left( {\left| {\left. {01} \right\rangle \pm } \right|\left. {10} \right\rangle } \right)} \right., in the absence of spin-orbit interaction D and the inhomogeneity parameter b. Therefore, it is possible to perform quantum teleportation protocol T₀\mathcal{T}_0 and entanglement teleportation T₁\mathcal{T}_1, with perfect quality, by choosing a proper set of parameters and employing one of these maximally entangled robust states as the initial state of the resource.     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

Magnetization and magnetic phase diagrams of a spin-1/2 ferrimagnetic diamond chain at low temperature

We used the Jordan–Wigner transform and the invariant eigenoperator method to study the magnetic phase diagram and the magnetization curve of the spin-1/2 alternating ferrimagnetic diamond chain in an external magnetic field at finite temperature. The magnetization versus external magnetic field curve exhibits a 1/3 magnetization plateau at absolute zero and finite temperatures, and the width of the 1/3 magnetization plateau was modulated by tuning the temperature and the exchange interactions. Three critical magnetic field intensities H_(CB), H_(CE) and H_(CS) were obtained, in which the H_(CB) and H_(CE) correspond to the appearance and disappearance of the 1/3 magnetization plateau, respectively, and the higher H_(CS) correspond to the appearance of fully polarized magnetization plateau of the system. The energies of elementary excitation ωσ,k(σ = 1, 2, 3) present the extrema of zero at the three critical magnetic fields at 0 K, i.e., [hω_(3,k)(HCB)]_(min)= 0, [hω_(2,k)(H_(CE))]_(max)= 0 and [hω _(2,k)(H_(CS))]_(min)= 0, and the magnetic phase diagram of magnetic field versus different exchange interactions at 0 K was established by the above relationships. According to the relationships between the system's magnetization curve at finite temperatures and the critical magnetic field intensities, the magnetic field-temperature phase diagram was drawn. It was observed that if the magnetic phase diagram shows a three-phase critical point, which is intersected by the ferrimagnetic phase, the ferrimagnetic plateau phase, and the Luttinger liquid phase, the disappearance of the1/3 magnetization plateau would inevitably occur. However, the 1/3 magnetization plateau would not disappear without the three-phase critical point. The appearance of the 1/3 magnetization plateau in the low temperature region is the macroscopic manifestations of quantum effect.     

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, \emph{ J. Phys.} A {\bf 25} (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator $\varDelta \left( q',p'\right) $ ($\mathrm{q}$-number transform) in phase space quantum mechanics, $\iint_{-\infty}^{\infty}\frac{{\rm d}p'{\rm d}q'}{\pi }\varDelta \left( q',p'\right) \e^{-2\i\left( p-p'\right) \left( q-q'\right) }=\delta \left( p-P\right) \delta \left( q-Q\right),$ and its inverse% $ \iint_{-\infty}^{\infty}{\rm d}q{\rm d}p\delta \left( p-P\right) \delta \left( q-Q\right) \e^{2\i\left( p-p'\right) \left( q-q'\right) }=\varDelta \left( q',p'\right),$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.     

Structures of distorted phases and critical and noncritical atomic displacements of elpasolite Rb<Subscript>2</Subscript>KInF<Subscript>6</Subscript> during phase transitions

The structures of all three phases of the Rb₂KInF₆ crystal have been determined from the experimental X-ray diffraction data for the powder sample. The refinement of the profile and structural parameters has been carried out by the technique implemented in the DDM program, which minimizes the differences between the derivatives of the calculated and measured X-ray intensities over the entire profile of the X-ray diffraction pattern. The results obtained have been discussed using the group-theoretical analysis of the complete order-parameter condensate, which takes into account the critical and noncritical atomic displacements and permits the interpretation of the experimental data obtained previously. It has been reliably established that the sequence of changes in the symmetry during phase transitions in Rb₂KInF₆ can be represented as $ Fm\bar 3m\xrightarrow[{0,0,\phi }]{{11 - 9\left( {\Gamma _4^ + } \right)}}{{I114} \mathord{\left/ {\vphantom {{I114} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}}} \right. \kern-\nulldelimiterspace} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}} $ Fm\bar 3m\xrightarrow[{0,0,\phi }]{{11 - 9\left( {\Gamma _4^ + } \right)}}{{I114} \mathord{\left/ {\vphantom {{I114} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}}} \right. \kern-\nulldelimiterspace} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}} .     

Electron tunnelling phase time and dwell time through an associated delta potential barrier

The electron tunnelling phase time τP and dwell time τD through an associated delta potential barrier U(x) = ξδ(x) are calculated and both are in the order of 10^-17～10^-16s. The results show that the dependence of the phase time on the delta barrier parameter ξ can be described by the characteristic length lc = h^2/meξ and the characteristic energy Ec=meξ^2/h^2 of the delta barrier, where me is the electron mass, lc and Ec are assumed to be the effective width and height of the delta barrier with lcEc=ξ, respectively. It is found that TD reaches its maximum and τD = τp as the energy of the tunnelling electron is equal to Ec/2, i.e. as lc =λDB, λDB is de Broglie wave length of the electron.     

Common Eigenvectors of Four Particles’ Compatible Observables

We construct eight operators for a four-particle system, namely one center-of-mass coordinate operator, three relative coordinate operators, one total momentum operator and three mass-weighted relative momentum operators, and give common eigenvectors of four compatible observables , which are composed of four particles’ coordinate and momentum . By compatible we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product (IWOP) of operators, we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation.     

On the Product of Real Spectral Triples

The product of two real spectral triples and , the first of which is necessarily even, was defined by A.Connes as given by and, in the even-even case, by . Generically it is assumed that the real structure obeys the relations , , , where the -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this -sign table. In this Letter, we propose an alternative definition of the product real structure such that the -sign table is also satisfied by the product.     

Some implications of inclusive electro- and neutrino production data

We systematically exploit the reported data on \(F_2^{\gamma p} ,F_2^{\gamma n} ,\sigma ^{vN} ,\sigma ^{\bar vN} ,\left\langle {xy} \right\rangle _{vN} ,\left\langle {xy} \right\rangle _{\bar vN} ,\left\langle {1 - y} \right\rangle _{vN} \) and \(\left\langle {1 - y} \right\rangle _{\bar vN} \) in order to test various versions of the quark parton model and to obtain further predictions.     

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

Fluctuations and a rigorous uncertainty relation of trigonometric operators of the phase and the number of photons of an electromagnetic field for general quantum superpositions of coherent states

The quantum-statistical properties of states of an electromagnetic field of general superpositions of coherent states of the form of N _α,β(α?+e ^iξ β? are investigated. Formulas for the fluctuations (variances) of Hermitian trigonometric phase field operators ? ≡ côs φ, ? ≡ sîn φ (the so-called “Susskind–Glogower operators”) are found. Expressions for the rigorous uncertainty relations (Cauchy inequalities) for operators of the number of photons and trigonometric phase operators, as well as for operators ? and ?, are found and analyzed. The states of amplitude \({N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i\varphi }}\rangle + {e^{i\xi }}\left| {{{\sqrt {{n_\beta }e} }^{i\varphi }}\rangle } \right.} \right.} \right)\), φ = φ_α = φ_β, and phase \({N_{\alpha ,\beta }}\left( {\left| {{{\sqrt {ne} }^{i{\varphi _\alpha }}}\rangle + {e^{i\xi }}\left| {{{\sqrt {ne} }^{i{\varphi _\beta }}}\rangle } \right.} \right.} \right)\), n = n _α = n _β, superpositions of coherent states are considered separately. The types of quantum superpositions of meso- and macroscales (n _α, n _β » 1) are found for which the sines and/or cosines of the phase of the field can be measured accurately, since, under certain conditions, the quantum fluctuations of these quantities are close to zero. A simultaneous accurate measurement of cosφ and sinφ is possible for amplitude superpositions, while an accurate measurement of one of these trigonometric phase functions is possible in the case of certain phase superpositions. Amplitude superpositions of coherent states with a vacuum state are quantum states of the field with a “maximum” level of the quantum uncertainty both in the case of a mesoscopic scale and in the case of a macroscopic scale of the field with an average number of photons n _α/β ≈ 0, n _β/α » 1.     

Generalized measure permutation formulas in Feynman’s operational calculi

Measure permutation formulas in Feynman’s operational calculi for noncommuting operators give relationships between the two operators \(\mathcal{T}_{\mu 1,\mu 2} f\left( {\tilde A,\tilde B} \right)\) and \(\mathcal{T}_{\mu 2,\mu 1} f\left( {\tilde A,\tilde B} \right)\) . We develop generalized and iterated measure permutation formulas in the Jefferies-Johnson theory of Feynman’s operational calculi. In particular, we apply our formulas to derive an identity for a function of the Pauli matrices.     

Cohomology of $${\mathfrak{osp}}(1|2)$$ Acting on Linear Differential Operators on the Supercircle S1|1

We compute the first cohomology spaces of the Lie superalgebra with coefficients in the superspace of linear differential operators acting on weighted densities on the supercircle S ^1|1. The structure of these spaces was conjectured in (Gargoubi et al. in Lett Math Phys 79:5165, 2007). In fact, we prove here that the situation is a little bit more complicated.      

Lifetime vibrational interference during the NO 1s-1π* resonant excitation studied by the NO+(A 1Π → X 1Σ+) fluorescence

Dispersed fluorescence from fragments formed after the de-excitation of the 1s^-1π^* resonances of N^*O and NO^* has been measured in the spectral range of 118–142 nm. This range is dominated by lines of atomic nitrogen and oxygen fragments and by the bands in the NO⁺ ion which result from the participator Auger decay of the 1s^-1π^* resonances. Ab-initio calculations of the transition probabilities between vibrational levels during the reaction NO N^*O ⇒ NO were used to explain the observed intensity dependence for the fluorescence bands on the exciting-photon energy across the resonances and on both v^′ and v^′′ vibrational quantum numbers. The multiplet structure of the 1s^-1π^* resonance and lifetime vibrational interference explain the observed exciting-photon energy dependence of the fluorescence intensity. A strong spin-orbit coupling between singlet and triplet states of NO⁺ is proposed to reduce additional cascade population of the state via radiative transitions from the and states and to explain remaining differences between measured and calculated integral fluorescence intensities.     

Options for angular momentum projection II: A finite sum representation of the Hill-Wheeler projector

An exact and finite sum representation of the Hill-Wheeler projection operator is obtained under the provision that the state on which the operator acts can be expanded as $$\left| {\psi _\alpha } \right\rangle = \sum\limits_{J = J_{\min } }^{J_{\max } } {c_J^\alpha \left| {J,\alpha } \right\rangle .} $$ The result provides a definite advantage over numerical integration methods, especially if high spin states are considered.     

Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

We consider the block band matrices, i.e. the Hermitian matrices $H_N$ , $N=|\Lambda |W$ with elements $H_{jk,\alpha \beta }$ , where $j,k \in \Lambda =[1,m]^d\cap \mathbb {Z}^d$ (they parameterize the lattice sites) and $\alpha , \beta = 1,\ldots , W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha \beta }$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha _1\beta _1}H_{j_2k_2,\alpha _2\beta _2}\rangle =\delta _{j_1k_2}\delta _{j_2k_1} \delta _{\alpha _1\beta _2}\delta _{\beta _1\alpha _2} J_{j_1k_1},$ where $J=1/W+\alpha \Delta /W$ , $\alpha < 1/4d$ . This matrices are the special case of Wegner’s $W$ -orbital models. Assuming that the number of sites $|\Lambda |$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda _0|< \sqrt{2}$ .