Quantum-phase dynamics of two-component Bose–Einstein condensates: Collapse–revival of macroscopic superposition states

We investigate the long-time dynamics of two-component dilute gas Bose–Einstein condensates with relatively different two-body interactions and Josephson couplings between the two components. Although in certain parameter regimes the quantum state of the system is known to evolve into macroscopic superposition, i.e., Schrödinger cat state, of two states with relative atom number differences between the two components, the Schrödinger cat state is also found to repeat the collapse and revival behavior in the long-time region. The dynamical behavior of the Pegg–Barnett phase difference between the two components is shown to be closely connected with the dynamics of the relative atom number difference for different parameters. The variation in the relative magnitude between the Josephson coupling and intra- and inter-component two-body interaction difference turns out to significantly change not only the size of the Schrödinger cat state but also its collapse–revival period, i.e., the lifetime of the Schrödinger cat state.     

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     

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     

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     

Dynamics of entanglement in one-dimensional Ising chains with two- and three-body interactions

The evolution of entanglement in a one-dimensional Ising chain with both two-body and three-body interactions, under two types of initial states, is numerically simulated. We analyse three problems concerning the dynamics of pairwise entanglement: (i) the possibility of generating large entanglement from an initial separable state by the use of a selective irradiation scheme; (ii) the effect of three-body interaction on the generation of entanglement from an initial separable state; (iii) the effect of three-body interaction on the decay of the entanglement from a state with only (m,n)-pair maximal entangled, and the rest in product form. It is shown that a large pairwise concurrence C_mn can be obtained when the resonant, transverse radio-frequency fields are selectively switched on from the mth to nth spins. Three-body interaction will decrease the oscillation amplitude of the nearest neighbour concurrence, while the oscillation amplitude of remote pairwise concurrence will be greatly increased with the consideration of three-body interactions. For an initial (m,n)-pair maximal entangled state, a slow decay of the pairwise concurrence C_mn is found with the introduction of three-body interactions.     

Spin squeezing in a bimodal condensate: spatial dynamics and particle losses

We propose an analytical method to study the entangled spatial and spin dynamics of interacting bimodal Bose-Einstein condensates. We show that at particular times during the evolution spatial and spin dynamics disentangle and the spin squeezing can be predicted by a simple two-mode model. We calculate the maximum spin squeezing achievable in experimentally relevant situations with Sodium or Rubidium bimodal condensates, including the effect of the dynamics and of one, two and three-body losses.     

Quantum entanglement and classical bifurcations in a coupled two-component Bose-Einstein condensate

We investigate the relation between quantum states and classical fixed-point bifurcations in a coupled two-component Bose-Einstein condensate (BEC). It is shown that the classical bifurcations are closely related to a topological change of the corresponding quantum levels, and such a structure change can be manifested in the entanglement properties of the corresponding quantum states. That is, the entanglement of the quantum states displays some peaks near the classical bifurcation points.     

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     

Zero-Threshold Resolvent Asymptotics of Three-Body Schr?dinger Operators

We analyze the spectral properties for three-body Schr?dinger operators at the threshold zero and give some results on the asymptotics of resolvent under the condition that zero is a regular point for all two-body subhamiltonians.     

Conditional Schrödinger cats generation and detection by quantum non-demolition measurements

 We show that a Schr?dinger cat state can be generated using a crossed Kerr interaction between two optical modes and an appropriate conditional measurement on one of the modes. The effect of damping on the generation process is analyzed in detail and two possible detection schemes for the cat state are presented. Received: 15 May 1996     

1D stability analysis of filtering and controlling the solitons in Bose-Einstein condensates

We present one-dimensional (1D) stability analysis of a recently proposed method to filter and control localized states of the Bose–Einstein condensate (BEC), based on novel trapping techniques that allow one to conceive methods to select a particular BEC shape by controlling and manipulating the external potential well in the three-dimensional (3D) Gross–Pitaevskii equation (GPE). Within the framework of this method, under suitable conditions, the GPE can be exactly decomposed into a pair of coupled equations: a transverse two-dimensional (2D) linear Schr?dinger equation and a one-dimensional (1D) longitudinal nonlinear Schr?dinger equation (NLSE) with, in a general case, a time-dependent nonlinear coupling coefficient. We review the general idea how to filter and control localized solutions of the GPE. Then, the 1D longitudinal NLSE is numerically solved with suitable non-ideal controlling potentials that differ from the ideal one so as to introduce relatively small errors in the designed spatial profile. It is shown that a BEC with an asymmetric initial position in the confining potential exhibits breather-like oscillations in the longitudinal direction but, nevertheless, the BEC state remains confined within the potential well for a long time. In particular, while the condensate remains essentially stable, preserving its longitudinal soliton-like shape, only a small part is lost into “radiation”.     

Non-quantum liquefaction of coherent gases

We show that a gas-to-liquid phase transition at zero temperature may occur in a coherent gas of bosons in the presence of competing nonlinear effects. This situation can take place in atomic systems like Bose-Einstein condensates in alkali gases with two-body and three-body interactions of opposite signs, as well as in laser beams which propagate through optical media with Kerr (focusing) and higher order (defocusing) nonlinear responses. The liquefaction process takes place in the absence of any quantum effect and can be formulated in the framework of a mean field theory, in terms of the minimization of a thermodynamic potential. We study from a thermodynamic point of view all the stationary solutions of the cubic-quintic nonlinear Schrödinger equation which describes our system. We show that solitonic localized solutions connect the gaseous and liquid phases. Furthermore, we also perform a numerical simulation in the presence of linear gain and three-body recombination where a rich dynamics, including the emergence of self-organization behavior, is found.     

Universal properties of a trapped two-component fermi gas at unitarity

We treat the trapped two-component Fermi system, in which unlike fermions interact through a two-body short-range potential having no bound state but an infinite scattering length. By accurately solving the Schr?dinger equation for up to N=6 fermions, we show that no many-body bound states exist other than those bound by the trapping potential, and we demonstrate unique universal properties of the system: Certain excitation frequencies are separated by 2variant Planck's over 2piomega, the wave functions agree with analytical predictions and a virial theorem is fulfilled. Further calculations up to N=30 determine the excitation gap, an experimentally accessible universal quantity, and it agrees with recent predictions based on a density functional approach.     

Finite-well potential in the 3D nonlinear Schrödinger equation: application to Bose-Einstein condensation

Using variational and numerical solutions we show that stationary negative-energy localized (normalizable) bound states can appear in the three-dimensional nonlinear Schr?dinger equation with a finite square-well potential for a range of nonlinearity parameters. Below a critical attractive nonlinearity, the system becomes unstable and experiences collapse. Above a limiting repulsive nonlinearity, the system becomes highly repulsive and cannot be bound. The system also allows nonnormalizable states of infinite norm at positive energies in the continuum. The normalizable negative-energy bound states could be created in BECs and studied in the laboratory with present knowhow.     

Dynamics of texture for spinor condensates

We study a dynamics of texture for a two-component spinor bose condensate. This is carried out by adopting a time dependent Landau-Ginzburg Lagrangian for a spinor order parameter. By using a polar form of the spinor order parameter, we obtain a field equation for the texture. In particular we consider a one dimensional model in which we can obtain analytic forms for the textures in terms of elliptic functions of several kinds. We find that these solutions are characterized by a modulus parameter, and changes in this parameter cause structural changes of texture.     

原子薛定谔猫态中角动量的压缩及其时间演化

在低Q值腔内，原子相干态在一些特定时刻可以演化为原子薛定谔猫态.讨论了在这种原子薛定谔猫态中原子角动量的涨落和高阶涨落.根据不确定性原理，进一步研究了原子角动量的压缩和高阶压缩性质及其演化.研究表明，原子薛定谔猫态可以被压缩到二阶和六阶，但不能被压缩到四阶.当原子薛定谔猫态中被叠加的原子相干态数为无限多项时，其压缩特性与原子相干态相同. 关键词： 原子相干态 薛定谔猫态 角动量压缩 Bloch态     

PREPARATION OF NONCLASSICAL STATES OF A CAVITY FIELD BASED ON A NONLINEAR JAYNES-CUMMINGS MODEL

It is proven in this paper that some kinds of nonclassical quantum states of the cavity field, such as Schr?dinger cat state (amplitude cat or phase cat), sub-Poissonian photon distribution and Fock state, etc., can be generated by conditional meas urements on atoms in the two-photon Jaynes-Cummings model with a Kerr-like medium. The nonlinear constant plays the key modulation role in the preparation of the quantum states, which gives a new controllable parameter for further experi mental researches in this field.     

Implementing the Deutsch-Jozsa algorithm by using Schrodinger cat states in cavity QED

We propose a scheme to implement the Deutsch-Jozsa algorithm by using Schroedinger cat states in cavity quantum electron-dynamics （QED）. The scheme is based on the Raman interaction of a degenerate three-level A-type atom with a coherent state in a cavity. By using Schroedinger cat states, the atomic spontaneous emission can be minimized and the Hadamard transformation in our scheme is not needed.     

A solution of the Coulomb three-body problem

In this work, a final state wave function is constructed which represents a solution of the three-body Schr?dinger equation. The formulated wave function is superimposed of one basic analytical function with various parameters. The coefficients of these basic functions involved in final state wave function can be easily calculated from a set of linear equations. The coefficients depend only on incident energy of the system. The process can also be prolonged for application to the problems more than three bodies.