Mutually Unbiasedness between Maximally Entangled Bases and Unextendible Maximally Entangled Systems in $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}}$

The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system \(\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)\) are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in \(\mathbb {C}^{2}\otimes \mathbb {C}^{4}\) and \(\mathbb {C}^{2}\otimes \mathbb {C}^{8}\), respectively.     

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.

     

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.     

Quantum limit for the atom-light interferometer

The standard quantum limit is calculated for the atom-light interferometer. It is shown that the smallest detectable phase is $$\delta \phi _{\min } = \frac{1}{2}[N_{atoms} + 4N_{photons} )/N_{atoms} N_{photons} ]^{1/2} .$$ Therefore, in practical experiments, the accuracy is limited by the square root of the number of atoms. We propose a novel correlated atom-photon state interferometer which makes a transition to the Heisenberg limit, δφ_min ∝ 1/N _atoms, as the atoms approach a Bose condensate. Such an interferometer may serve as a sensitive probe of the onset of Bose condensation. Finally, we point out that the correlated atom-photon state preparation scheme we propose may be used in a different way to approach the Heisenberg limit for non-Bose-condensed atoms.     

Magnetic phase transitions and iron valence in the double perovskite Sr 2 FeOsO 6

The insulating and antiferromagnetic double perovskite Sr₂FeOsO₆ has been studied by ⁵⁷Fe Mössbauer spectroscopy between 5 and 295 K. The iron atoms are essentially in the Fe^3?+? high spin $( {t_{2\mathrm{g}}{^3} e_\mathrm{g}{^2} } )$ and thus the osmium atoms in the Os $^{5+} ( {t_{2\mathrm{g}}{^3} } )$ state. Two magnetic phase transitions, which according to neutron diffraction studies occur below T _N?= 140 K and T ₂?= 67 K, give rise to magnetic hyperfine patterns, which differ considerably in the hyperfine fields and thus, in the corresponding ordered magnetic moments. The evolution of hyperfine field distributions, average values of the hyperfine fields, and magnetic moments with temperature suggests that the magnetic state formed below T _N is strongly frustrated. The frustration is released by a magneto-structural transition which below T ₂ leads to a different spin sequence along the c-direction of the tetragonal crystal structure.     

The temperature dependence of the hyperfine interaction of the transition metal osmium in a gadolinium host matrix

Integral perturbed angular correlations of the 931-155keVγγ-cascade of¹⁸⁸Os in Gd have been measured. With this technique the combined magnetic and electric hyperfine interaction of the 155 keV level of¹⁸⁸Os as an impurity in a Gd host has been studied as a function of temperature. The result for the electric field gradient of Os in Gd at 300 K is: $$\left| {V_{zz} \left( {Os:\underline {Gd} } \right)} \right| = \left( {12.8_{ - 1.9}^{ + 3.1} } \right) \cdot 10^{17} {V \mathord{\left/ {\vphantom {V {cm^2 }}} \right. \kern-\nulldelimiterspace} {cm^2 }}.$$ For the magnetic hyperfine field at 4.2 K the value $$H_{hf} \left( {Os:\underline {Gd} } \right) = - 134\left( {26} \right)kG$$ was obtained. Sign and magnitude of the magnetic hyperfine field suggest the existence of a localized moment of about ?0.4 µ _B at the site of Os in Gd. With increasing temperature the magnetic hyperfine field decreases much stronger than the magnetization of the host. Possible explanations for this anomalous temperature dependence are discussed.     

Mutually Unbiased Maximally Entangled Bases for the Bipartite System ℂd⊗ℂdk

The construction of maximally entangled bases for the bipartite system \(\mathbb {C}^{d}\otimes \mathbb {C}^{d}\) is discussed firstly, and some mutually unbiased bases with maximally entangled bases are given, where 2≤d≤5. Moreover, we study a systematic way of constructing mutually unbiased maximally entangled bases for the bipartite system \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{k}}\).     

Gleichzeitige Messung von Hyperfeinstruktur,Stark-Effekt und Zeeman-Effekt des39K19F mit einer Molekülstrahlresonanzapparatur

An electric Molecular-Beam-Resonance-Spectrometer has been used to measure simultanously the Zeeman- and Stark-effect splitting of the hyperfine structure of³⁹K¹⁹ F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. The observed (Δm _J =±1)-transitions were induced electrically. Completely resolved spectra of KF in theJ=1 rotational state have been measured. The obtained quantities are: The electric dipolmomentμ _e l of the molecul forv=0,1 and 2; the rotational magnetic dipolmomentμ _J forv=0,1; the difference of the magnetic shielding (σ_⊥ ? σ_∥) by the electrons of both nuclei as well as the difference of the molecular susceptibility (ξ_⊥ ? ξ_∥). The numerical values are

$$\begin{array}{*{20}c} {\mu _{e1} = 8,585(4)deb,} \\ {\frac{{(\mu _{e1} )_{\upsilon = 1} }}{{(\mu _{e1} )_{\upsilon = 0} }} = 1,0080,} \\ {{{\mu _J } \mathord{\left/ {\vphantom {{\mu _J } J}} \right. \kern-\nulldelimiterspace} J} = ( - )2352(10) \cdot 10^{ - 6} \mu _B ,} \\ {(\sigma _ \bot - \sigma _\parallel )F = ( - )2,19(9) \cdot 10^{ - 4} ,} \\ {(\sigma _ \bot - \sigma _\parallel )K = ( - )12(9) \cdot 10^{ - 4} ,} \\ {(\xi _ \bot - \xi _\parallel ) = 3 (1) \cdot 10^{ - 30} {{erg} \mathord{\left/ {\vphantom {{erg} {Gau\beta ^2 }}} \right. \kern-\nulldelimiterspace} {Gau\beta ^2 }}} \\ \end{array} $$     

Untersuchung von elektrischer Hyperfeinstruktur-Wechselwirkung in Hafniumammoniumhexafluorid-Einkristallen durch γγ-Winkelkorrelationsmessungen

The angular correlation of the 133 keV-482 keV-yy-cascade in the decay of Hf¹⁸¹ is strongly attenuated if solid sources of hafniumammoniumhexafluoride are used. The unperturbed correlation was observed however when a single crystal of hafniumammoniumhexafluoride was used whose main axis pointed into the direction of one of the two detectors. This proves that the perturbation is static and that the maximum component of the electric field gradient at the position of the hafnium nucleus coincides with the direction of the main axis of the crystal. The anisotropy of the angular correlation was measured as a function of the direction of the crystal axis. The results agree with the theoretical predicted functions for a strong electric quadrupole interaction. Then we combined the intrinsic electric field with an external magnetic field. The magnetic field direction was chosen parallel to thez-axis of the electric field gradient and perpendicular to the plane of the detectors. The theory for axially symmetric field gradients predicts a maximum of the anisotropy of the angular correlation for a magnetic field strength at which resonance exists between electric and magnetic precession. For a strong electric interaction the maxium anisotropy has half the value of the unperturbed correlation. In our case the electric quadrupole interaction was so strong that we could not reach the resonance even when we applied external magnetic fields up to 48000 gauss. The observed anisotropies were too large however to be fitted by theoretical curves which were calculated under the assumption that the field gradient has axial symmetry. Therefore we developed the theory for non-axially symmetric electric field gradients. Now a fit was possible and gave unique solutions for the strength of the electric hyperfine interaction as well as for the asymmetry coefficient of the electric field gradient tensor. The accuray of these results was not very high but the strength of the electric hyperfine interaction was found to be small enough to make a direct observation of the electric spin rotation by the differential angular correlation method possible. The observed pattern confirmed the non-axially symmetry of the electric field gradient and we derived the following parameters:

$$\omega _{E_0 } = \left( {570 \pm 30} \right)MHz\left( {\omega _{E_0 } = electric interaction frequency = \frac{{6eQ \cdot \left| {V_{zz} } \right|}}{{4I \cdot \left( {2I - 1} \right) \cdot \rlap{--} h}}} \right)$$     

Phase Transition in Ferromagnetic Ising Models with Non-uniform External Magnetic Fields

In this article we study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field. We show that the Ising model on the hypercubic lattice with a summable magnetic field has a first-order phase transition and, for any positive (resp. negative) and bounded magnetic field, the model does not present the phase transition phenomenon whenever lim?inf?h _i >0, where \(\mathbf{h}=(h_{i})_{i\in \mathbb{Z}^{d}}\) is the external magnetic field.     

Entangled Projector and a New Approach for Obtaining the Entangled Wigner Operator

Like the coordinate projector |q〉〈q|=δ(q?Q), where Q is coordinate operator, we find that $\pi\delta( \eta_{1}-\frac{Q_{1}-Q_{2}}{\sqrt{2}}) \delta( \eta_{2}-\frac{P_{1}+P_{2}}{\sqrt{2}}) $ is an entangled projector |η〉〈η|, where |η〉 is the bipartite entangled state and η=η ₁+iη ₂. We then derive the entangled Wigner operator in terms of the properties of the entangled projector. This seems a new approach for obtaining the entangled Wigner operator.     

Two Types of Maximally Entangled Bases and Their Mutually Unbiased Property in ℂd⊗ℂd′

We first construct a new maximally entangled basis in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\ (k\in Z^{+})\) which is diffrent from the one in Tao et al. (Quantum Inf. Process. 14, 2291 (2015)), then we generalize such maximally entangled basis into arbitrary bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{d^{\prime }}\). We also study the mutual unbiased property of the two types of maximally entangled bases in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\). In particular, explicit examples in \(\mathbb {C}^{2} \otimes \mathbb {C}^{4}\), \(\mathbb {C}^{2} \otimes \mathbb {C}^{8}\) and \(\mathbb {C}^{3} \otimes \mathbb {C}^{3}\) are presented.     

Experimental and theoretical determination of transition dipole moments of ethyl 5-(4-aminophenyl)- and 5-(4-dimethylaminophenyl)-3-amino-2,4-dicyanobenzoate

The absorption and fluorescence transition dipole moments ( $\hat M_{ge}$ and $\hat M_{eg}$ ) for ethyl 5-(4-aminophenyl)-3-amino-2, 4-dicyanobenzoate (EAADCy) and ethyl 5-(4-dimethylaminophenyl)-3-amino-2, 4-dicyanobenzoate (EDMAADCy) have been determined on the basis of the steady-state and time-resolved spectroscopic measurements and semiempirical quantum-chemical calculations. The values of the transition dipole moments of perpendicular and flattened forms of the investigated molecules were estimated as a function of the solvent polarity. Noted differences between the absorption and emission transition dipole moments (i.e., ${{\hat M_{ge} } \mathord{\left/ {\vphantom {{\hat M_{ge} } {\hat M_{eg} }}} \right. \kern-0em} {\hat M_{eg} }} \ne 1$ ) confirm that the change of the electronic and molecular structure take place in the excited state.     

Gleichzeitige Messung von Hyperfeinstruktur,Starkeffekt und Zeemaneffekt des133Cs19F mit einer Molekülstrahl-Resonanzapparatur

An electric molecular beam resonance spectrometer has been used to measure simultaneously the Zeeman- and Stark-effect splitting of the hyperfine structure of¹³³Cs¹⁹F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. Electrically induced (Δ m _J =±1)-transitions have been measured in theJ=1 rotational state, υ=0, 1 vibrational state. The obtained quantities are: The electric dipolmomentμ _el of the molecule for υ=0, 1; the rotational magnetic dipolmomentμ _J for υ=0, 1; the anisotropy of the magnetic shielding (σ _⊥-σ‖) by the electrons of both nuclei as well as the anisotropy of the molecular susceptibility (ξ _⊥-ξ‖), the spin rotational interaction constantsc _Cs andc _F, the scalar and the tensor part of the nuclear dipol-dipol interaction, the quadrupol interactioneqQ for υ=0, 1. The numerical values are:

$$\begin{gathered} \mu _{el} \left( {\upsilon = 0} \right) = 73878\left( 3 \right)deb \hfill \\ \mu _{el} \left( {\upsilon = 1} \right) - \mu _{el} \left( {\upsilon = 0} \right) = 0.07229\left( {12} \right)deb \hfill \\ \mu _J /J\left( {\upsilon = 0} \right) = - 34.966\left( {13} \right) \cdot 10^{ - 6} \mu _B \hfill \\ \mu _J /J\left( {\upsilon = 1} \right) = - 34.823\left( {26} \right) \cdot 10^{ - 6} \mu _B \hfill \\ \left( {\sigma _ \bot - \sigma _\parallel } \right)_{Cs} = - 1.71\left( {21} \right) \cdot 10^{ - 4} \hfill \\ \left( {\sigma _ \bot - \sigma _\parallel } \right)_F = - 5.016\left( {15} \right) \cdot 10^{ - 4} \hfill \\ \left( {\xi _ \bot - \xi _\parallel } \right) = 14.7\left( {60} \right) \cdot 10^{ - 30} erg/Gau\beta ^2 \hfill \\ c_{cs} /h = 0.638\left( {20} \right)kHz \hfill \\ c_F /h = 14.94\left( 6 \right)kHz \hfill \\ d_T /h = 0.94\left( 4 \right)kHz \hfill \\ \left| {d_s /h} \right|< 5kHz \hfill \\ eqQ/h\left( {\upsilon = 0} \right) = 1238.3\left( 6 \right) kHz \hfill \\ eqQ/h\left( {\upsilon = 1} \right) = 1224\left( 5 \right) kHz \hfill \\ \end{gathered} $$     

Quantum Electrodynamics of Atomic Resonances

A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass m, finitely many excited states and an electric dipole moment, \({\vec{d}_0 = -\lambda_{0} \vec{d}}\), where \({\| d^{i}\| = 1, i = 1, 2, 3,}\) and \({\lambda_0}\) is proportional to the elementary electric charge. The interaction of the atom with the radiation field is described with the help of the Ritz Hamiltonian, \({-\vec{d}_0 \cdot \vec{E}}\), where \({\vec{E}}\) is the electric field, cut off at large frequencies. A mathematical study of the Lamb shift, the decay channels and the life times of the excited states of the atom is presented. It is rigorously proven that these quantities are analytic functions of the momentum \({\vec{p}}\) of the atom and of the coupling constant \({\lambda_0}\), provided \({\vert\vec{p} \vert < mc}\) and \({\vert \Im \vec{p} \vert}\) and \({\vert \lambda_{0} \vert}\) are sufficiently small. The proof relies on a somewhat novel inductive construction involving a sequence of ‘smooth Feshbach–Schur maps’ applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast.     

g-factors of rotational states in176Hf,177Hf,and180Hf

g-factors of rotational states in¹⁷⁶Hf and¹⁸⁰Hf were measured with the twelve detector IPAC-apparatus of our laboratory [1]. The natural radioactivity 3.78·10¹⁰y¹⁷⁶Lu and the 5.5 h isomer^180mHf were used which populate the ground-state rotational bands of¹⁷⁶Hf and¹⁸⁰Hf. The integral rotations ofγ-γ directional correlations in strong external magnetic fields and in static hyperfine fields of (Lu→Hf)Fe₂ and HfFe₂ were observed. The following results were obtained: $$\begin{array}{l} ^{176} Hf: g\left( {4_1^ + } \right) = + 0.334\left( {38} \right) \\ ^{180} Hf: g\left( {2_1^ + } \right) = + 0.305\left( {14} \right) \\ g\left( {4_1^ + } \right) = + 0.358\left( {43} \right) \\ {{ g\left( {6_1^ + } \right)} \mathord{\left/ {\vphantom {{ g\left( {6_1^ + } \right)} {g\left( {4_1^ + } \right)}}} \right. \kern-\nulldelimiterspace} {g\left( {4_1^ + } \right)}} = + 0.95\left( {12} \right) \\ \end{array}$$ . The hyperfine field in (Lu→Hf)Fe₂ was calibrated by observing the integral rotation of the 9/2^? first excited state of¹⁷⁷Hf populated in the decay of 6.7d¹⁷⁷Lu. Theg-factor of this state was redetermined in an external magnetic field as $$^{177} Hf: g\left( {{9 \mathord{\left/ {\vphantom {9 {2^ - }}} \right. \kern-\nulldelimiterspace} {2^ - }}} \right) = + 0.228\left( 7 \right)$$ . Finally theg-factor of the 2 ₁ ⁺ state of¹⁷⁶Hf was derived from the measuredg(2 ₁ ⁺ ) of¹⁸⁰Hf by use of the precisely known ratiog(2 ₁ ⁺ ,¹⁷⁶Hf)/g(2 ₁ ⁺ ,¹⁸⁰Hf) [2] as $$^{176} Hf: g\left( {2_1^ + } \right) = + 0.315\left( {30} \right)$$ .     

Effect of low-level Ge additions on the superconducting transition in PbTe:Tl

A study is reported of the effect of low-level germanium additions (∼0.01–0.1 at. %) on the parameters of the superconducting transition, viz. the critical temperature T _c, the second critical magnetic field H _c2, and in PbTe doped with 2 at. % Tl, which are derived from the dependence of the electrical resistivity of a sample on temperature (0.4–4.2 K) and magnetic field (0–1.3 T). The discontinuity revealed by experimental data is related to the onset of a Ge-induced structural phase transition. Fiz. Tverd. Tela (St. Petersburg) 40, 1204–1205 (July 1998)     

Investigation of radiation damage by larmor precession of112In in Ag

The temperature and magnetic field dependence of the spin rotation patterns for two isomeric states of¹¹²In in Ag have been investigated. The results obtained are consistent with a weak static electric field gradient from the radiation damage defects: . The quadrupole moment of the 6⁺ isomeric state in¹¹²In was measured: |Q(6⁺)|=0.75(15) b.     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.