Efficient Scheme for Macroscopic Superpositions and Entanglement of High-Order Squeezed Vacuum States in Cavity QED

We propose an etticient scheme for generating the macroscopic superpositions and the entanglement between the high-order squeezed vacuum states by considering the multi-photon interaction of N two-level atoms in a cavity with high quality factor, assisted by a strong driving field. Through specific choices of the cavity detuning, a number of multi-party entangled states between the atoms and the high-order squeezed vacuum states and among the high-order squeezed vacuum states of the cavities can be prepared, including also the macroscopic ＂Schrodinger cats＂ of the high- order squeezed vacuum states, the entangled states of the macroscopic ＂Schrodinger cats＂, and so on. Possible extension and application of our scheme are discussed. Our scheme is reachable within the current techniques in the cavity QED.     

Macroscopic Superpositions and Entanglement of Mesoscopic Squeezed Vacuum States in Dissipative Cavity QED System

A scheme has been proposed for generating the macroscopic superpositions and the entanglement between the mesoscopic squeezed vacuum states by considering the two-photon interaction of N two-level atoms in a dissipative cavity with high quality factor assisted by a strong driving field. A number of multiparty entangled states between the atoms and the squeezed vacuum states, among the atoms, and among the squeezed vacuum states, can be prepared by virtue of a specific choice of the cavity detuning and the detuning of applied coherent field under the dissipation condition. The corresponding analytical expressions of the influence can be given. Moreover, we can also give a series of macroscopic entangled states between the usual coherent states and the squeezed vacuum states using the combination of the dissipative one-photon interaction Hamiltonian with the dissipative two-photon interaction Hamiltonian. We also discuss the experimental feasibility. Our scheme can be realized in the current techniques on the cavity QED.     

Entangling Mesoscopic Squeezed Vacuum States and Fock States in Dissipative Cavity QED System

A scheme is proposed for generating the mesoscopic entanglement between the mesoscopic squeezed vacuum states and Fock states {|0〉,|2〉} by considering both the two-photon interaction of N two-level atoms in cavities with high quality factor assisted by a strong driving field. Moreover, we derive the dissipative interaction models for two-photon interaction. The corresponding analytical expressions of the fidelities can be given. Our scheme can be realized in the current techniques on the cavity QED.     

Entangling Mesoscopic Squeezed Vacuum States and Mesoscopic Coherent States in Dissipative Cavity QED System

A scheme has been proposed for generating the macroscopic entanglement between the mesoscopic squeezed vacuum states and mesoscopic coherent states by considering both the two-photon interaction and the single photon interaction of N two-level atoms in cavities with high quality factor assisted by a strong driving field. Moreover, we derive the dissipative interaction models for single photon interaction and two-photon interaction, respectively. The corresponding analytical expressions of the fidelities can be given. Our scheme can be realized in the current techniques on the cavity QED.     

利用介观LC电路制备薛定谔猫态

提出了一种制备相干态的叠加态，即薛定谔猫态的方案. 该方案是基于将外加冲激信号作用于介观LC电路系统而设计的. 在该薛定谔猫态下，介观电路系统有非经典的量子压缩效应. 关键词： 介观LC电路 冲激信号 薛定谔猫态     

Coherent control of the self-trapping transition

We discuss the dynamics of two weakly coupled Bose-Einstein condensates in a double-well potential, contrasting the mean-field picture to the exact N-particle evolution. On the mean-field level, a self-trapping transition occurs when the scaled interaction strength exceeds a critical value; this transition essentially persists in small condensates comprising about 1000 atoms. When the double-well is modulated periodically in time, Floquet-type solutions to the nonlinear Schr?dinger equation take over the role of the stationary mean-field states. These nonlinear Floquet states can be classified as “unbalanced” or “balanced”, depending on whether or not they entail long-time confinement of most particles to one well. Since the emergence of unbalanced Floquet states depends on the amplitude and frequency of the modulating force, we predict that the onset of self-trapping can efficiently be controlled by varying these parameters. This prediction is verified numerically by both mean-field and N-particle calculations. Received 5 November 2000 and Received in final form 16 February 2001     

Manipulating a Bose-Einstein condensate with a single photon

We propose a method to create macroscopic superpositions, so-called Schr?dinger cat states, of different motional states of an ideal Bose-Einstein condensate. The scheme is based on the scattering of a freely expanding condensate by the light field of a high-finesse optical cavity in a quantum superposition state of different photon numbers. The atom-photon interaction creates an entangled state of the motional state of the condensate and the photon number, which can be converted into a pure atomic Schr?dinger cat state by operations only acting on the cavity field. We discuss in detail the fully quantised theory and propose an experimental procedure to implement the scheme using short coherent light pulses. Received 26 June 2000 and Received in final form 2nd October 2000     

Convergence in Higher Mean of a Random Schrödinger to a Linear Boltzmann Evolution

We study the macroscopic scaling and weak coupling limit for a random Schrödinger equation on $\mathbb{Z}^3We study the macroscopic scaling and weak coupling limit for a random Schr?dinger equation on . We prove that the Wigner transforms of a large class of “macroscopic” solutions converge in r ^th mean to solutions of a linear Boltzmann equation, for any 1 ≤ r < ∞. This extends previous results where convergence in expectation was established.     

Preparation and decoherence of superpositions of electromagnetic field states

In a recent experiment the progressive decoherence of a mesoscopic superposition of two coherent field states in a high-Q cavity, known as Schr?dinger cat state, has been measured for the first time [Brune et al., Phys. Rev. Lett. 77, 4887 (1996)]. Here, the full master equation governing the coupled dissipative dynamics of the atom-field system studied in the experiment is formulated and solved numerically for the experimental parameters. The model simulated avoids the approximations underlying an analytically solvable model which is based on a harmonic expansion of the energies of the dressed atomic states and on a treatment of their dynamics within the adiabatic approximation. In particular, the numerical simulations reveal that the coupling of the cavity field mode to its environment causes important decoherence effects already during the initial preparation phase of the Schr?dinger cat state. This phenomenon is investigated in detail with the help of a measure for the purity of states. Moreover, the Hilbert-Schmidt distance of the intended target state, the Schr?dinger cat, to the state that is actually prepared in the experiment is determined. Received 13 September 2000 and Received in final form 22 December 2000     

The Fundamental Solution and Strichartz Estimates for the Schrödinger Equation on Flat Euclidean Cones

We study the Schrödinger equation on a flat euclidean cone ${\mathbb{R}_+ \times \mathbb{S}^1_\rho}We study the Schr?dinger equation on a flat euclidean cone \mathbbR₊ ×\mathbbS¹_r{\mathbb{R}_+ \times \mathbb{S}^1_\rho} of cross-sectional radius ρ > 0, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the “geometric front,” the part of the solution coming from formal application of the method of images, and the “diffractive front” emerging from the cone tip. As an application, we prove Strichartz estimates for the Schr?dinger propagator on this class of cones.     

Monte Carlo wavefunction approach to the dissipative quantum-phase dynamics of two-component Bose-Einstein condensates

We investigate the relaxation effects on the dynamics of two-component dilute gas Bose-Einstein condensates (BEC) with relatively different two-body interactions and Josephson couplings between the two components. Three types of relaxation effects, i.e., one- and three-body losses and a pure phase relaxation caused by elastic two-body collision between condensed and noncondensed atoms, are examined on the dynamical behavior of a macroscopic superposition, i.e., Schr?dinger cat state, of two states with atom-number differences between the two components, which is known to be created by the time evolution in certain parameter regimes. Although three-body losses show a relatively large suppression of the revival behavior of Schr?dinger cat state and the Pegg-Barnett phase-difference distribution between the two components for a small-size Schr?dinger cat state, one- and three-body loss effects are not shown to directly depend on the size of Schr?dinger cat state. In contrast, the pure-phase relaxation effects, causing a reduction of phase-difference distribution and then decaying the Schr?dinger cat state, significantly increase with the increase of the size of Schr?dinger cat state. These features suggest that a detection of damped collapse-revival behavior is highly possible for medium-size Schr?dinger cat states in small-size two-component BECs.     

Power Law Subordinacy and Singular Spectra.¶II. Line Operators

We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schr?dinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19]. Received: 21 October 1999 / Accepted: 21 December 1999     

Generation of superpositions of coherent states on a circle

We propose a simple method to obtain a superposition of coherent states on a circle, including Schr?dinger cat states as a special case, via conditional measurement of the state of three level atoms interacting with a one mode cavity field. In the low amplitude limit, very good approximation of Fock states can also be generated in this way. Received: 8 October 1998 / Accepted: 30 October 1998     

Generation of Quantum States for Atomic Ensemble via Cavity QED

A scheme is proposed for generating quantum states of atomic ensemble. In this scheme, a beam of three-level atoms in the Λ configuration is trapped in a cavity, then squeezed vacuum state and squeezed coherent state of the atomic ensemble are prepared by choosing different initial states of the system. The scheme is based on the off-resonant interaction between the atom and cavities, so the high-level of the atom is eliminated adiabatically.     

Deterministic generation of squeezing for a cavity field with a collection of driven atoms

We propose an efficient scheme for realizing squeezing for a cavity mode. In the scheme, a collection of ladder-type three-level atoms are trapped in a cavity and driven by two classical fields. Under certain conditions, the cavity field deterministically evolves to a squeezed state. The scheme can also be used for conditional generation of superpositions of different squeezed vacuum states.     

Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on {\mathbb R^2}

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from ${\mathbb {R}^2}We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schr?dinger flow as special cases) for degree m equivariant maps from \mathbb R²{\mathbb {R}^2} to \mathbb S²{\mathbb {S}^2} . If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work (Gustafson et al., Duke Math J 145(3), 537–583, 2008) down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m = 3, involving (among other tools) a “normal form” for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schr?dinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m = 2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even “eternal oscillation”.     

On the Integrated Density of States for Schrödinger Operators on ℤ2 with Quasi Periodic Potential

In this paper we consider discrete Schr?dinger operators on the lattice ℤ² with quasi periodic potential. We establish new regularity results for the integrated density of states, as well as a quantitative version of a “Thouless formula”, as previously considered by Craig and Simon, for real energies and with rates of convergence. The main ingredient is a large deviation theorem for the Green's function that was recently established by Bourgain, Goldstein, and the author. For the integrated density of states an argument of Bourgain is used. Finally, we establish certain fine properties of separately subharmonic functions of two variables that might be of independent interest. Received: 13 February 2001 / Accepted: 21 May 2001     

Continuation of the Schrödinger Equation for Higher Angular-Momentum States to D Dimensions and Interdimensional Degeneracies

The application of the techniques of dimensional scaling, and in particular the 1/D expansion, to higher angular-momentum states of multi-electron atoms requires the generalized Euler angles, which multiply with increasing D to be “factored out” of the wave function. The factorization must be performed in a way that produces from the Schr?dinger equation a tractable set of differential equations which admit continuation in the dimension D. In two recent works the authors have achieved the necessary factorization of the wave function by generalizing the Schwartz expansion to N electrons in D dimensions. The present paper applies the N-electron D-dimensional Schwartz expansion to the two-electron problem in D dimensions. The resulting set of coupled differential equations in the internal variables admit continuation in D, enabling the methods of dimensional scaling to be applied to higher-angular-momentum states. In addition, the coupled differential equations clearly show the complete spectrum of exact interdimensional degeneracies of the two-electron system. Received November 6, 1995; accepted in final form February 10, 1996