# Showing papers in "Physical Review Letters in 1996"

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TL;DR: A simple derivation of a simple GGA is presented, in which all parameters (other than those in LSD) are fundamental constants, and only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked.

Abstract: Generalized gradient approximations (GGA’s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. [S0031-9007(96)01479-2] PACS numbers: 71.15.Mb, 71.45.Gm Kohn-Sham density functional theory [1,2] is widely used for self-consistent-field electronic structure calculations of the ground-state properties of atoms, molecules, and solids. In this theory, only the exchange-correlation energy EXC › EX 1 EC as a functional of the electron spin densities n"srd and n#srd must be approximated. The most popular functionals have a form appropriate for slowly varying densities: the local spin density (LSD) approximation Z d 3 rn e unif

117,932 citations

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TL;DR: It is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of {rho}, has only non-negative eigenvalues.

Abstract: A quantum system consisting of two subsystems is separable if its density matrix can be written as $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{A}^{}{w}_{A}{\ensuremath{\rho}}_{A}^{\ensuremath{'}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}},$ where ${\ensuremath{\rho}}_{A}^{\ensuremath{'}}$ and ${\ensuremath{\rho}}_{A}^{\ensuremath{'}\ensuremath{'}}$ are density matrices for the two subsystems, and the positive weights ${w}_{A}$ satisfy $\ensuremath{\Sigma}{w}_{A}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of \ensuremath{\rho}, has only non-negative eigenvalues Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability

3,936 citations

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TL;DR: A mechanism for depression of the plasma frequency into the far infrared or even GHz band is proposed: Periodic structures built of very thin wires dilute the average concentration of electrons and considerably enhance the effective electron mass through self-inductance.

Abstract: The plasmon is a well established collective excitation of metals in the visible and near UV, but at much lower frequencies dissipation destroys all trace of the plasmon and typical Drude behavior sets in. We propose a mechanism for depression of the plasma frequency into the far infrared or even GHz band: Periodic structures built of very thin wires dilute the average concentration of electrons and considerably enhance the effective electron mass through self-inductance. Computations replicate the key features and confirm our analytic theory. The new structure has novel properties not observed before in the GHz band, including some possible impact on superconducting properties.

3,690 citations

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TL;DR: With properly chosen parameters, the model provides a remarkably accurate ``roadmap'' of nanotube behavior beyond Hooke's law.

Abstract: Carbon nanotubes subject to large deformations reversibly switch into different morphological patterns. Each shape change corresponds to an abrupt release of energy and a singularity in the stress-strain curve. These transformations, simulated using a realistic many-body potential, are explained by a continuum shell model. With properly chosen parameters, the model provides a remarkably accurate ``roadmap'' of nanotube behavior beyond Hooke's law.

2,403 citations

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TL;DR: The new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators is presented, and a relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.

Abstract: We present the new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators. To characterize this phenomenon, we use the analytic signal approach based on the Hilbert transform and partial Poincar\'e maps. For coupled R\"ossler attractors, in the synchronous regime the phases are locked, while the amplitudes vary chaotically and are practically uncorrelated. Coupling a chaotic oscillator with a hyperchaotic one, we observe another new type of synchronization, where the frequencies are entrained, while the phase difference is unbounded. A relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.

2,298 citations

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IBM

^{1}, Université de Montréal^{2}, Tel Aviv University^{3}, Kenyon College^{4}, University of California, Los Angeles^{5}, Williams College^{6}TL;DR: Upper and lower bounds on the yield of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$ are given, showing $D(M)>0$ if $\bra{Psi-}M\ket-}>\half$.

Abstract: Two separated observers, by applying local operations to a supply of not-too-impure entangled states (e.g., singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity (e.g., near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D\left(M\right)$ of pure singlets $(|{\ensuremath{\Psi}}^{\ensuremath{-}}〉)$ distillable from mixed states $M$, showing $D\left(M\right)g0$ if $〈{\ensuremath{\Psi}}^{\ensuremath{-}}|M|{\ensuremath{\Psi}}^{\ensuremath{-}}〉g\frac{1}{2}$.

2,101 citations

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TL;DR: It is shown that a pair of states which are, in a certain sense, “macroscopically different,” can form a superposition in which the interference phase between the two parts is measurable, providing a highly stabilized “Schrodinger cat” state.

Abstract: A new type of uncertainty relation is presented, concerning the information-bearing properties of a discrete quantum system. A natural link is then revealed between basic quantum theory and the linear error correcting codes of classical information theory. A subset of the known codes is described, having properties which are important for error correction in quantum communication. It is shown that a pair of states which are, in a certain sense, “macroscopically different,” can form a superposition in which the interference phase between the two parts is measurable. This provides a highly stabilized “Schrodinger cat” state. [S0031-9007(96)00779-X]

1,984 citations

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TL;DR: The relaxation processes of electrons and spins systems following the absorption of femtosecondoptical pulses in ferromagnetic nickel have been studied using optical and magneto-optical pump-probetechniques and the experimental results are adequately described by a model including three interacting reservoirs.

Abstract: The relaxation processes of electrons and spins systems following the absorption of femtosecond optical pulses in ferromagnetic nickel have been studied using optical and magneto-optical pump-probe techniques. The magnetization of the film drops rapidly during the first picosecond, but different electron and spin dynamics are observed for delays in the range 0--5 ps. The experimental results are adequately described by a model including three interacting reservoirs (electron, spin, and lattice).

1,626 citations

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TL;DR: A simple one-dimensional scattering theory model with a dynamic frequency-dependent well depth to describe the transmission properties is proposed and highly efficient transmission of light around sharp corners in photonic band-gap waveguides is demonstrated.

Abstract: We demonstrate highly efficient transmission of light around sharp corners in photonic band-gap waveguides. Numerical simulations reveal complete transmission at certain frequencies, and very high transmission $(g95%)$ over wide frequency ranges. High transmission is observed even for 90\ifmmode^\circ\else\textdegree\fi{} bends with zero radius of curvature, with a maximum transmission of 98% as opposed to 30% for analogous conventional dielectric waveguides. We propose a simple one-dimensional scattering theory model with a dynamic frequency-dependent well depth to describe the transmission properties.

1,625 citations

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Abstract: The magnetoresistance (MR) and the field dependent magnetization have been systematically examined in the low temperature ferromagnetic metallic state of single crystal and polycrystalline ${\mathrm{La}}_{2/3}{\mathrm{Sr}}_{1/3}{\mathrm{MnO}}_{3}$. We find that the intrinsic negative MR in single crystal is due to the suppression of spin fluctuations, and magnetic domain boundaries do not dominate the scattering process. In contrast, we demonstrate that the MR in the polycrystalline samples exhibits two distinct regions: large MR at low fields dominated by spin-polarized tunneling between grains and high field MR which is remarkably temperature independent from 5 to 280 K.

1,559 citations

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TL;DR: It is shown that a single-parameter potential is sufficient to reproduce the slow dynamics of proteins obtained with vastly more complex empirical potentials, which inevitably leads to unstable modes which must be eliminated through elaborate methods, and which cast doubts on the validity of the analysis.

Abstract: Normal mode analysis (NMA) is a leading method for studying long-time dynamics and elasticity of biomolecules. The method proceeds from complex semiempirical potentials characterizing the covalent and noncovalent interactions between atoms. It is widely accepted that such detailed potentials are essential to the success of NMA’s. We show that a single-parameter potential is sufficient to reproduce the slow dynamics in good detail. Costly and inaccurate energy minimizations are eliminated, permitting direct analysis of crystal coordinates. The technique can be used for new applications, such as mapping of one crystal form to another by means of slow modes, and studying anomalous dynamics of large proteins and complexes. [S0031-9007(96)01063-0] PACS numbers: 87.15.By, 87.15.He Thermal equilibrium fluctuations of the x-ray crystal coordinates of proteins provide a basis for understanding the complex dynamics and elasticity of biological macromolecules [1]. Analysis of the normal modes of globular proteins shows an interesting anomaly. The density of the slow vibrational modes is proportional to their frequency, gsv d, v, rather than gsv d, v 2 as predicted by Debye’s theory [2]. Yet, the atoms in globular proteins are packed as tightly as in solids. We show that a single-parameter potential reproduces the slow elastic modes of proteins obtained with vastly more complex empirical potentials. The simplicity of the potential permits greater insight and understanding of the mechanisms that underlie the slow, anomalous motions in biological macromolecules such as proteins. To date, normal modes of globular proteins have been used to reproduce crystallographic temperature factors [3] and diffuse scatter [4]. Normal mode analyses (NMA’s) shed light on shear and hinge motions necessary for catalytic reactions, and have been used with some success to map one crystal form of a protein into another [5]. Finally, NMA’s yield macroscopic elastic moduli of large protein assemblies, based on their microscopic structure [6]. NMA studies of macromolecules are handicapped, however, by the complex phenomenological potentials used to model the covalent and nonbonded interactions between atom pairs. The necessary initial energy minimization requires a great deal of computer time and memory, and is virtually impossible for even moderately large proteins (with typically thousands of degrees of freedom) with a reasonable degree of accuracy. This inevitably leads to unstable modes which must be eliminated through elaborate methods, and which cast doubts on the validity of the analysis. Moreover, partly because the minimization is carried out in vacuo, the final configuration disagrees with the known crystallographic structure, complicating the interpretation of the results of NMA. A typical example of a semiempirical potential used in molecular dynamics studies and NMA’s has the form [7] Ep › 1 X bonds Kbsb 2 b0d 2 1 1 X angles Kus u2u 0 d 2

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Abstract: Nickel ferrite nanoparticles exhibit anomalous magnetic properties at low temperatures: low magnetization with a large differential susceptibility at high fields, hysteresis loops which are open up to 160 kOe, time-dependent magnetization in 70 kOe applied field, and shifted hysteresis loops after field cooling. We propose a model of the magnetization within these particles consisting of ferrimagnetically aligned core spins and a spin-glass-like surface layer. We find that qualitative features of this model are reproduced by a numerical calculation of the spin distribution. Implications of this model for possible macroscopic quantum tunneling in these materials are discussed.

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TL;DR: It is proposed that these effects are manifestations of thermally assisted, field-tuned resonant tunneling between quantum spin states, and attribute the observation of quantum-mechanical phenomena on a macroscopic scale to tunneling in a large (Avogadro's) number of magnetically identical molecules.

Abstract: We report the observation of steps at regular intervals of magnetic field in the hysteresis loop of a macroscopic sample of oriented M${\mathrm{n}}_{12}$${\mathrm{O}}_{12}$(C${\mathrm{H}}_{3}$COO${)}_{16}$(${\mathrm{H}}_{2}$O${)}_{4}$ crystals. The magnetic relaxation rate increases substantially when the field is tuned to a step. We propose that these effects are manifestations of thermally assisted, field-tuned resonant tunneling between quantum spin states, and attribute the observation of quantum-mechanical phenomena on a macroscopic scale to tunneling in a large (Avogadro's) number of magnetically identical molecules.

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TL;DR: The decoherence phenomenon transforming this superposition into a statistical mixture was observed while it unfolded, providing a direct insight into a process at the heart of quantum measurement.

Abstract: A mesoscopic superposition of quantum states involving radiation fields with classically distinct phases was created and its progressive decoherence observed. The experiment involved Rydberg atoms interacting one at a time with a few photon coherent field trapped in a high $Q$ microwave cavity. The mesoscopic superposition was the equivalent of an `` $\mathrm{atom}+\mathrm{measuring}\mathrm{apparatus}$'' system in which the ``meter'' was pointing simultaneously towards two different directions---a ``Schr\"odinger cat.'' The decoherence phenomenon transforming this superposition into a statistical mixture was observed while it unfolded, providing a direct insight into a process at the heart of quantum measurement.

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Abstract: A new density-functional approach to calculate the excitation spectrum of many-electron systems is proposed. It is shown that the full linear density response of the interacting system, which has poles at the exact excitation energies, can rigorously be expressed in terms of the response function of the noninteracting (Kohn-Sham) system and a frequency-dependent exchange-correlation kernel. Using this expression, the poles of the full response function are obtained by systematic improvement upon the poles of the Kohn-Sham response function. Numerical results are presented for atoms.

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TL;DR: It is shown that nonlinear rescalings of a Gaussian linear stochastic process cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity.

Abstract: Current tests for nonlinearity compare a time series to the null hypothesis of a Gaussian linear stochastic process. For this restricted null assumption, random surrogates can be constructed which are constrained by the linear properties of the data. We propose a more general null hypothesis allowing for nonlinear rescalings of a Gaussian linear process. We show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity. An iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution.

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TL;DR: An extensive ab initio database of CO chemisorption energies calculated within density functional theory (DFT) using the generalized gradient approximation (GGA) is presented and it is demonstrated that the trends can be understood using a simple two-level model describing the coupling of the CO 5s and 2p p states to the metal d va.

Abstract: A database of ab initio calculations of the chemisorption energy of CO over Ni(111), Cu(111), Ru(0001), Pd(111), Ag(111), Pt(111), Au(111), $\mathrm{Cu}{}_{3}$Pt(111), and some metallic overlayer structures is presented. The trends can be reproduced with a simple model describing the interaction between the metal $d$ states and the CO $2{\ensuremath{\pi}}^{*}$ and 5 $\ensuremath{\sigma}$ states, renormalized by the metal $\mathrm{sp}$ continuum. Our model rationalizes the results by Rodriguez and Goodman [Science 257, 897 (1992)] showing a strong correlation between the CO chemisorption energy and the surface core level shift.

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TL;DR: Coulomb oscillations in vertical quantum dots containing a tunable number of electrons starting from zero are measured, as predicted by Hund’s rule, to favor the filling of parallel spins.

Abstract: We study atomiclike properties of artificial atoms by measuring Coulomb oscillations in vertical quantum dots containing a tunable number of electrons starting from zero. At zero magnetic field the energy needed to add electrons to a dot reveals a shell structure for a two-dimensional harmonic potential. As a function of magnetic field the current peaks shift in pairs, due to the filling of electrons into spin-degenerate single-particle states. When the magnetic field is sufficiently small, however, the pairing is modified, as predicted by Hund’s rule, to favor the filling of parallel spins. [S00319007(96)01418-4]

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Abstract: The vacuum stress between closely spaced conducting surfaces, due to the modification of the zero-point fluctuations of the electromagnetic field, has been conclusively demonstrated. The measurement employed an electromechanical system based on a torsion pendulum. Agreement with theory at the level of 5% is obtained.

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TL;DR: The growth of multilayer arrays of coherently strained islands self-organizes into a more regular three-dimensional arrangement, providing a possible route to obtain the size uniformity needed for electronic applications of quantum dot arrays.

Abstract: We investigate the growth of multilayer arrays of coherently strained islands, which may serve as ``quantum dots'' in electronic devices. A simple model reproduces the observed vertical correlation between islands in successive layers. However, the arrangement of islands is not simply repeated from layer to layer. Instead, the island size and spacing grow progressively more uniform. In effect, the structure ``self-organizes'' into a more regular three-dimensional arrangement, providing a possible route to obtain the size uniformity needed for electronic applications of quantum dot arrays.

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TL;DR: The concept of cooperative rearrangement was introduced in an attempt to unify these two views of the glass transition by demonstrating that such cooperativity, coupled with a thermodynamic glass transition, resulted naturally insystem dynamics such as those described by the WLFequation for temperatures near freezing.

Abstract: We report the first measurements of the glass transition temperature ${T}_{g}$ for thin freely standing polystyrene (PS) films. We have used Brillouin light scattering to measure ${T}_{g}$ for freely standing films of different thicknesses. We find that ${T}_{g}$ decreases linearly with film thickness $h$ for $h\ensuremath{\le}700\AA{}$, with a reduction of 70 K for a film with $h\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}290\AA{}$. These measurements characterize unambiguously the effects of the free surface on ${T}_{g}$ of thin polymer films. Results are compared to similar results for supported PS films [Keddie et al., Europhys. Lett. 27, 59 (1994)], and we find that their measured values are influenced strongly by the substrate.

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TL;DR: It is demonstrated experimentally that one can transmit one of three messages, i.e., 1 {open_ quote}{open_quote}trit{close_quote}{close_ quote} {approx_equal}1.58 bit, by manipulating only one of two entangled particles.

Abstract: Classically, sending more than one bit of information requires manipulation of more than one two-state particle. We demonstrate experimentally that one can transmit one of three messages, i.e., 1 ``trit'' $\ensuremath{\approx}1.58\mathrm{bit}$, by manipulating only one of two entangled particles. The increased channel capacity is proven by transmitting ASCII characters in five trits instead of the usual 8 bits.

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Bell Labs

^{1}TL;DR: A model which incorporates the physics of dynamic Jahn-Teller and double-exchange effects is presented and solved via a dynamical mean field approximation to reproduce the behavior of the resistivity and magnetic transition temperature observed in Sr_x MnO_3.

Abstract: A model for the doped rare-earth manganites such as ${\mathrm{La}}_{1\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{MnO}}_{3}$ incorporating the physics of dynamic Jahn-Teller and double-exchange effects is presented and solved via a dynamical mean field approximation. The interplay of these two effects as the electron phonon coupling is varied reproduces the observed behavior of the resistivity and magnetic transition temperature.

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TL;DR: The concept of quantum privacy amplification and a cryptographic scheme incorporating it which is provably secure over a noisy channel is introduced and implemented using technology that is currently being developed.

Abstract: Existing quantum cryptographic schemes are not, as they stand, operable in the presence of noise on the quantum communication channel. Although they become operable if they are supplemented by classical privacy-amplification techniques, the resulting schemes are difficult to analyze and have not been proved secure. We introduce the concept of quantum privacy amplification and a cryptographic scheme incorporating it which is provably secure over a noisy channel. The scheme uses an “entanglement purification” procedure which, because it requires only a few quantum controllednot and single-qubit operations, could be implemented using technology that is currently being developed. [S0031-9007(96)01288-4] Quantum cryptography [1 ‐ 3] allows two parties (traditionally known as Alice and Bob) to establish a secure random cryptographic key if, first, they have access to a quantum communication channel, and second, they can exchange classical public messages which can be monitored but not altered by an eavesdropper (Eve). Using such a key, a secure message of equal length can be transmitted over the classical channel. However, the security of quantum cryptography has so far been proved only for the idealized case where the quantum channel, in the absence of eavesdropping, is noiseless. That is because, under existing protocols, Alice and Bob detect eavesdropping by performing certain quantum measurements on transmitted batches of qubits and then using statistical tests to determine, with any desired degree of confidence, that the transmitted qubits are not entangled with any third system such as Eve. The problem is that there is in principle no way of distinguishing entanglement with an eavesdropper (caused by her measurements) from entanglement with the environment caused by innocent noise, some of which is presumably always present. This implies that all existing protocols are, strictly speaking, inoperable in the presence of noise, since they require the transmission of messages to be suspended whenever an eavesdropper (or, therefore, noise) is detected. Conversely, if we want a protocol that is secure in the presence of noise, we must find one that allows secure transmission to continue even in the presence of eavesdroppers. To this end, one might consider modifying the existing pro

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TL;DR: It is demonstrated that the long time dynamics of a single hydrogen bond in ambient liquid water is indeed characterized by significant nonexponential relaxation, and this complex relaxation is essentially uncorrelated to the specific bonding patterns near the tagged hydrogen bond.

Abstract: In a series of molecular dynamics calculations we simulate the dynamics of forming and breaking a hydrogen bond in liquid water at room temperature. We show that this dynamics is clearly nonexponential, yet virtually uncorrelated with the fluctuations of neighboring bonds.

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TL;DR: Necessary and sufficient conditions for the occurrence of generalized synchronization of unidirectionally coupled dynamical systems are given in terms of asymptotic stability and the existence of generalized synchronized systems in the case of parameter mismatch between coupled systems leads to a new interpretation of recent experimental results.

Abstract: Necessary and sufficient conditions for the occurrence of generalized synchronization of unidirectionally coupled dynamical systems are given in terms of asymptotic stability The relation between generalized synchronization, predictability, and equivalence of dynamical systems is discussed All theoretical results are illustrated by analytical and numerical examples In particular, the existence of generalized synchronization in the case of parameter mismatch between coupled systems leads to a new interpretation of recent experimental results Furthermore, the possible application of generalized synchronization for attractor reconstruction in nonlinear time series analysis is discussed

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TL;DR: Electronic energy transfer between close packed quantum dots using cw and time resolved photoluminescence is demonstrated and is consistent with long-range resonance transfer of electronic excitations from the more electronically confined states of the small dots to the higher excitedStates of the large dots.

Abstract: We demonstrate electronic energy transfer between close packed quantum dots using cw and time resolved photoluminescence. Optically clear and thin, close packed quantum dot solids were prepared from mixtures of small and large CdSe quantum dots (38.5 and 62 \AA{}, $\ensuremath{\sigma}l4.5%$). Quenching of the luminescence (lifetime) of the small dots accompanied by enhancement of the luminescence (lifetime) of the large dots is consistent with long-range resonance transfer of electronic excitations from the more electronically confined states of the small dots to the higher excited states of the large dots.

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TL;DR: This investigation of the excited levels of the atom-cavity system reveals nonlinear quantum features at extremely low field strengths.

Abstract: We have observed the Rabi oscillation of circular Rydberg atoms in the vacuum and in small coherent fields stored in a high Q cavity. The signal exhibits discrete Fourier components at frequencies proportional to the square root of successive integers. This provides direct evidence of field quantization in the cavity. The weights of the Fourier components yield the photon number distribution in the field. This investigation of the excited levels of the atom-cavity system reveals nonlinear quantum features at extremely low field strengths.

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TL;DR: For the first time it is possible to handle surfaces consisting of complex particles close enough to interact strongly, and a fully retarded implementation of Maxwell's equations on adaptive meshes allows treatment of large particles as well as small.

Abstract: We present an implementation of Maxwell's equations on adaptive meshes in order to study interaction of light with metal surfaces. For the first time it is possible to handle surfaces consisting of complex particles close enough to interact strongly. A fully retarded implementation allows treatment of large particles as well as small. By way of example we model a rough silver surface as an array of half-cylinders embedded in a silver surface. Very localized plasmon modes, created by strong electromagnetic coupling between touching metallic objects, dominate the surface enhanced Raman scattering response.

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TL;DR: The latent heat of fusion for Sn particles formed by evaporation on inert substrate with radii ranging from 5 to 50 nm has been measured directly using a novel scanning nanocalorimeter and a particle-size-dependent reduction of $\ensuremath{\Delta}{H}_{m}$ has been observed.

Abstract: For the first time, the latent heat of fusion $\ensuremath{\Delta}{H}_{m}$ for Sn particles formed by evaporation on inert substrate with radii ranging from 5 to 50 nm has been measured directly using a novel scanning nanocalorimeter. A particle-size-dependent reduction of $\ensuremath{\Delta}{H}_{m}$ has been observed. An ``excluded volume'' is introduced to describe the latent heat of fusion from the enhanced surface melting of small particles. Melting point depression has also been found by our nanocalorimetric technique.