Decoherence of Photon-Added Schrödinger Cat States in Thermal Environment

We analyze the statistical properties and decoherence of the field states produced by adding any number of photons to the Schrodinger Cat states (SCSs) in a thermal environment. It is found that the normalization factor of PA-SCS is the Hermite polynomial of the coherent factor α. The statistical properties and decoherence is discussed by deriving analytically the time evolution of the Q-function, Wigner function and the photon-number distribution. It is shown that the single photon-added even SCSs, the Wigner function is always positive in the whole phase space when κt exceeds the threshold value $\frac{1}{2}\ln [ ( 2\bar{n}+2 ) / ( 2\bar{n}+1 ) ] $ . This implies that the single photon-added even SCSs possesses a robustness due to the presence of photon-addition, which can also be seen from the point of view of the volume of negative region of Wigner function.     

Small BGK Waves and Nonlinear Landau Damping

Consider a 1D Vlasov-poisson system with a fixed ion background and periodic condition on the space variable. First, we show that for general homogeneous equilibria, within any small neighborhood in the Sobolev space ${W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)}${W^{s,p}\left( p >1 ,s <1 +\frac{1}{p}\right)} of the steady distribution function, there exist nontrivial travelling wave solutions (BGK waves) with arbitrary minimal period and traveling speed. This implies that nonlinear Landau damping is not true in W^s,p( s < 1 +\frac1p){W^{s,p}\left( s <1 +\frac{1}{p}\right)} space for any homogeneous equilibria and any spatial period. Indeed, in a W^s,p(s < 1 +\frac1p){W^{s,p}\left(s <1 +\frac{1}{p}\right)} neighborhood of any homogeneous state, the long time dynamics is very rich, including travelling BGK waves, unstable homogeneous states and their possible invariant manifolds. Second, it is shown that for homogeneous equilibria satisfying Penrose’s linear stability condition, there exist no nontrivial travelling BGK waves and unstable homogeneous states in some ${W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)}${W^{s,p}\left( p >1 ,s >1 +\frac{1}{p}\right)} neighborhood. Furthermore, when p = 2, we prove that there exist no nontrivial invariant structures in the ${H^{s}\left( s > \frac{3}{2}\right) }${H^{s}\left( s > \frac{3}{2}\right) } neighborhood of stable homogeneous states. These results suggest the long time dynamics in the ${W^{s,p}\left( s >1 +\frac{1}{p}\right) }${W^{s,p}\left( s >1 +\frac{1}{p}\right) } and particularly, in the ${H^{s}\left( s > \frac{3}{2}\right) }${H^{s}\left( s > \frac{3}{2}\right) } neighborhoods of a stable homogeneous state might be relatively simple. We also demonstrate that linear damping holds for initial perturbations in very rough spaces, for a linearly stable homogeneous state. This suggests that the contrasting dynamics in W ^{s, p} spaces with the critical power s=1+\frac1p{s=1+\frac{1}{p}} is a truly nonlinear phenomena which can not be traced back to the linear level.     

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.     

On the Best Constant in the Moser-Onofri-Aubin Inequality

Let S ² be the 2-dimensional unit sphere and let J _α denote the nonlinear functional on the Sobolev space H ¹(S ²) defined by

Exciton emission dynamics in single InAs/GaAs quantum dots due to the existence of plasmon-field-induced metastable states in the wetting layer

A very long lifetime exciton emission with non-single exponential decay characteristics has been reported for single InA-s/GaAs quantum dot(QD) samples,in which there exists a long-lived metastable state in the wetting layer(WL)through radiative field coupling between the exciton emissions in the WL and the dipole field of metal islands.In this article we have proposed a new three-level model to simulate the exciton emission decay curve.In this model,assuming that the excitons in a metastable state will diffuse and be trapped by QDs,and then emit fluorescence in QDs,a stretchedlike exponential decay formula is derived as I(t)=At~(β-1)e~(-(rt)β),which can describe well the long lifetime decay curve with an analytical expression of average lifetime     

One-range Addition Theorems for Noninteger n Slater Functions using Complete Orthonormal Sets of Exponential Type Orbitals in Standard Convention

By the use of complete orthonormal sets of ${\psi ^{(\alpha^{\ast})}}$ -exponential type orbitals ( ${\psi ^{(\alpha^{\ast})}}$ -ETOs) with integer (for α ^* = α) and noninteger self-frictional quantum number α ^*(for α ^* ≠ α) in standard convention introduced by the author, the one-range addition theorems for ${\chi }$ -noninteger n Slater type orbitals ${(\chi}$ -NISTOs) are established. These orbitals are defined as follows $$\begin{array}{ll}\psi _{nlm}^{(\alpha^*)} (\zeta ,\vec {r}) = \frac{(2\zeta )^{3/2}}{\Gamma (p_l ^* + 1)} \left[{\frac{\Gamma (q_l ^* + )}{(2n)^{\alpha ^*}(n - l - 1)!}} \right]^{1/2}e^{-\frac{x}{2}}x^{l}_1 F_1 ({-[ {n - l - 1}]; p_l ^* + 1; x})S_{lm} (\theta ,\varphi )\\ \chi _{n^*lm} (\zeta ,\vec {r}) = (2\zeta )^{3/2}\left[ {\Gamma(2n^* + 1)}\right]^{{-1}/2}x^{n^*-1}e^{-\frac{x}{2}}S_{lm}(\theta ,\varphi ),\end{array}$$ where ${x=2\zeta r, 0<\zeta <\infty , p_l ^{\ast}=2l+2-\alpha ^{\ast}, q_l ^{\ast}=n+l+1-\alpha ^{\ast}, -\infty <\alpha ^{\ast} <3 , -\infty <\alpha \leq 2,_1 F_1 }$ is the confluent hypergeometric function and ${S_{lm} (\theta ,\varphi )}$ are the complex or real spherical harmonics. The origin of the ${\psi ^{(\alpha ^{\ast})} }$ -ETOs, therefore, of the one-range addition theorems obtained in this work for ${\chi}$ -NISTOs is the self-frictional potential of the field produced by the particle itself. The obtained formulas can be useful especially in the electronic structure calculations of atoms, molecules and solids when Hartree–Fock–Roothan approximation is employed.     

The Essence of Operators’ Weyl Ordering Scheme and New Operator Identities for Converting Q−P Ordering to Weyl Ordering

We find new operator formulas for converting Q?P and P?Q ordering to Weyl ordering, where Q and P are the coordinate and momentum operator. In this way we reveal the essence of operators’ Weyl ordering scheme, e.g., Weyl ordered operator polynomial ${_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}}$ , $$\begin{aligned} {_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}} =&\sum_{l=0}^{\min (m,n)} \biggl( \frac{-i\hbar }{2} \biggr) ^{l}l!\binom{m}{l}\binom{n}{l}Q^{m-l}P^{n-l} \\ =& \biggl( \frac{\hbar }{2} \biggr) ^{ ( m+n ) /2}i^{n}H_{m,n} \biggl( \frac{\sqrt{2}Q}{\sqrt{\hbar }},\frac{-i\sqrt{2}P}{\sqrt{\hbar }} \biggr) \bigg|_{Q_{\mathrm{before}}P} \end{aligned}$$ where ${}_{:}^{:}$ ${}_{:}^{:}$ denotes the Weyl ordering symbol, and H _m,n is the two-variable Hermite polynomial. This helps us to know the Weyl ordering more intuitively.     

Localization for a Random Walk in Slowly Decreasing Random Potential

We consider a continuous time random walk X in a random environment on ?⁺ such that its potential can be approximated by the function V:?⁺→? given by $V(x)=\sigma W(x) -\frac {b}{1-\alpha}x^{1-\alpha}$ where σW a Brownian motion with diffusion coefficient σ>0 and parameters b, α are such that b>0 and 0<α<1/2. We show that P-a.s. (where P is the averaged law) $\lim_{t\to\infty} \frac{X_{t}}{(C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}}=1$ with $C^{*}=\frac{2\alpha b}{\sigma^{2}(1-2\alpha)}$ . In fact, we prove that by showing that there is a trap located around $(C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}$ (with corrections of smaller order) where the particle typically stays up to time t. This is in sharp contrast to what happens in the “pure” Sinai’s regime, where the location of this trap is random on the scale ln² t.     

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     

Bound on the Speed of Computation from Generalized Salecker-Wigner Inequalities

Basing on generalized Salecker-Wigner inequalities, we show that the accuracy of a simple computer sets a limit on the speed of computation ν<10⁴⁴ sec^?1. The product of the amount of information I and the speed ν of the computer is limited as $I\nu^{2}<\frac{1}{4} [1-4t^{2}_{\mathrm{ p}}/\tau^{2}]t^{-2}_{\mathrm{ p}}<\frac{1}{4} t^{-2}_{\mathrm{ p}}\sim\frac{1}{4}\times10^{88}~\mathrm{sec}^{-2} $ . For application or comparing, the case of black hole is discussed.     

Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations

We consider the quantum mechanics on the noncommutative plane with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar }{2}, {\Delta } x_{1} {\Delta } p_{2} \ge \frac {\eta }{2}\). We show that the model has two essentially different phases which is determined by \(\kappa = 1 + \frac {1}{\hbar ^{2} } (\eta ^{2} - \theta \bar {\theta })\). We construct a operator \(\hat {\pi }_{i}\) commuting with \(\hat {x}_{j} \) and discuss the harmonic oscillator model in two dimensional non-commutative space for three case κ > 0, κ = 0, κ < 0. Finally, we discuss the thermodynamics of a particle whose hamiltonian is related to the harmonic oscillator model in two dimensional non-commutative space.     

Global Regularity for the Navier-Stokes-Maxwell System with Fractional Diffusion

In this paper, we study the global regularity for the Navier-Stokes-Maxwell system with fractional diffusion. Existence and uniqueness of global strong solution are proved for \(\alpha \geqslant \frac {3}{2}\). When 0 < α < 1, global existence is obtained provided that the initial data \(\|u_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|E_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|B_{0}\|_{H^{\frac {5}{2}-2\alpha }}\) is sufficiently small. Moreover, when \(1<\alpha <\frac {3}{2}\), global existence is obtained if for any ε >?0, the initial data \(\|u_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|E_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|B_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}\) is small enough.     

Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator

Let be the spectrum of in L ²(ℝ), where q is an even almost-periodic complex-valued function with bounded primitive and derivative. It is known that , where is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to the first asymptotic correction is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of Δμ _n .      

Entanglement Involved in Time Evolution of Two-Mode Squeezed State in Single-Mode Diffusion Channel

We derive the evolution law of an initial two-mode squeezed vacuum state \( \text {sech}^{2}\lambda e^{a^{\dag }b^{\dagger }\tanh \lambda }\left \vert 00\right \rangle \left \langle 00\right \vert e^{ab\tanh \lambda }\) (a pure state) passing through an a-mode diffusion channel described by the master equation

$$\frac{d\rho \left( t\right) }{dt}=-\kappa \left[ a^{\dagger}a\rho \left( t\right) -a^{\dagger}\rho \left( t\right) a-a\rho \left( t\right) a^{\dagger}+\rho \left( t\right) aa^{\dagger}\right] , $$

since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from this channel is a two-mode mixed state. Performing partial trace over the b-mode of ρ(t) yields a new chaotic field, \(\rho _{a}\left (t\right ) =\frac {\text {sech}^{2}\lambda }{1+\kappa t \text {sech}^{2}\lambda }:\exp \left [ \frac {- \text {sech}^{2}\lambda }{1+\kappa t\text {sech}^{2}\lambda }a^{\dagger }a \right ] :,\) which exhibits higher temperature and more photon numbers, showing the diffusion effect. Besides, measuring a-mode of ρ(t) to find n photons will result in the collapse of the two-mode system to a new Laguerre polynomial-weighted chaotic state in b-mode, which also exhibits entanglement.

     

A Note on the Integrability of Hamiltonian Systems

We consider a family of Hamiltonian systems

Thermodynamic implications of the gravitationally induced particle creation scenario

A rigorous thermodynamic analysis has been done as regards the apparent horizon of a spatially flat Friedmann–Lemaitre–Robertson–Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate \(\Gamma \). Assuming a perfect fluid equation of state \(p=(\gamma -1)\rho \) with \(\frac{2}{3} \le \gamma \le 2\), the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied, and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain \(\Gamma \) explicitly. Moreover, a lower bound for the fluid temperature \(T_f\) has also been found which is given by \(T_f \ge 8\left( \frac{\frac{3\gamma }{2}-1}{\frac{2}{\gamma }-1}\right) H^2\). It has been shown that the GSL is satisfied for \(\frac{\Gamma }{3H} \le 1\). Further, when \(\Gamma \) is constant, thermodynamic equilibrium is always possible for \(\frac{1}{2}<\frac{\Gamma }{3H} < 1\), while for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma -2}{3\gamma -2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\), equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, \(\Gamma \le H\). When \(\Gamma \) is not a constant, thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1-\frac{\Gamma }{3H}\right) ^2\), however, such a condition is by no means necessary for the attainment of equilibrium.     

Empirical formulas for the fermion spectra and Yukawa matrices

We present empirical relations that connect the dimensionless ratios of low energy fermion masses for the charged lepton, up-type quark and down-type quark sectors and the CKM elements: and . Explaining these relations from first principles imposes strong constraints on the search for the theory of flavor. We present a simple set of normalized Yukawa matrices, with only two real parameters and one complex phase, which accounts with precision for these mass relations and for the CKM matrix elements and also suggests a simpler parametrization of the CKM matrix. The proposed Yukawa matrices accommodate the measured CP-violation, giving a particular relation between standard model CP-violating phases, . According to this relation the measured value of is close to the maximum value that can be reached, for . Finally, the particular mass relations between the quark and charged lepton sectors find their simplest explanation in the context of grand unified models through the use of the Georgi-Jarlskog factor.Received: 31 July 2004, Revised: 22 September 2004, Published online: 9 November 2004     

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\(\frac{n}{2} < \delta < \alpha\).

     

More corrections to the sixth-order anomalous magnetic moment of the muon

The contribution to the sixth-order muon anomaly from second-order electron vacuum polarization is determined analytically to orderm _e/m _μ. The result, including the contributions from graphs containing proper and improper fourth-order electron vacuum polarization subgraphs, is $$\begin{gathered} \left( {\frac{\alpha }{\pi }} \right)^3 \left\{ {\frac{2}{9}\log ^2 } \right.\frac{{m_\mu }}{{m_e }} + \left[ {\frac{{31}}{{27}}} \right. + \frac{{\pi ^2 }}{9} - \frac{2}{3}\pi ^2 \log 2 \hfill \\ \left. { + \zeta \left( 3 \right)} \right]\log \frac{{m_\mu }}{{m_e }} + \left[ {\frac{{1075}}{{216}}} \right. - \frac{{25}}{{18}}\pi ^2 + \frac{{5\pi ^2 }}{3}\log 2 \hfill \\ \left. { - 3\zeta \left( 3 \right) + \frac{{11}}{{216}}\pi ^4 - \frac{2}{9}\pi ^2 \log ^2 2 - \frac{1}{9}log^4 2 - \frac{8}{3}a_4 } \right] \hfill \\ + \left[ {\frac{{3199}}{{1080}}\pi ^2 - \frac{{16}}{9}\pi ^2 \log 2 - \frac{{13}}{8}\pi ^3 } \right]\left. {\frac{{m_e }}{{m_\mu }}} \right\} \hfill \\ \end{gathered} $$ . To obtain the total sixth-order contribution toa _μ?a _e, one must add the light-by-light contribution to the above expression.