Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo’s linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.     

Quantum field theoretical analysis on unstable behavior of Bose-Einstein condensates in optical lattices

We study the dynamics of Bose-Einstein condensates flowing in optical lattices on the basis of quantum field theory. For such a system, a Bose-Einstein condensate shows an unstable behavior which is called the dynamical instability. The unstable system is characterized by the appearance of modes with complex eigenvalues. Expanding the field operator in terms of excitation modes including complex ones, we attempt to diagonalize the unperturbative Hamiltonian and to find its eigenstates. It turns out that although the unperturbed Hamiltonian is not diagonalizable in the conventional bosonic representation the appropriate choice of physical states leads to a consistent formulation. Then we analyze the dynamics of the system in the regime of the linear response theory. Its numerical results are consistent with those given by the discrete nonlinear Schrödinger equation.     

Tunneling of trapped-atom Bose condensates

We obtain the dynamics in number and phase difference, for Bose condensates that tunnel between two wells of a double-well atomic trap, using the (nonlinear) Gross-Pitaevskii equation. The dynamical equations are of the canonical form for the two conjugate variables, and the Hamiltonian corresponds to that of a momentum-shortened pendulum, supporting a richer set of tunneling oscillation modes than for a superconductor Josephson junction, that has a fixed-length pendulum as a mechanical model. Novel modes include ‘inverted pendulum’ oscillations with an average angle of π; and oscillations about a self-maintained population imbalance that we term ‘macroscopic quantum self-trapping’. Other systems with this phase-number nonlinear dynamics include two-component (interconverting) condensates in a single harmonic trap, and He³B superfluids in two containers connected by micropores.     

Decoherence effects in Bose–Einstein condensate interferometry I. General theory

The present paper outlines a basic theoretical treatment of decoherence and dephasing effects in interferometry based on single component Bose–Einstein condensates in double potential wells, where two condensate modes may be involved. Results for both two mode condensates and the simpler single mode condensate case are presented. The approach involves a hybrid phase space distribution functional method where the condensate modes are described via a truncated Wigner representation, whilst the basically unoccupied non-condensate modes are described via a positive P representation. The Hamiltonian for the system is described in terms of quantum field operators for the condensate and non-condensate modes. The functional Fokker–Planck equation for the double phase space distribution functional is derived. Equivalent Ito stochastic equations for the condensate and non-condensate fields that replace the field operators are obtained, and stochastic averages of products of these fields give the quantum correlation functions that can be used to interpret interferometry experiments. The stochastic field equations are the sum of a deterministic term obtained from the drift vector in the functional Fokker–Planck equation, and a noise field whose stochastic properties are determined from the diffusion matrix in the functional Fokker–Planck equation. The stochastic properties of the noise field terms are similar to those for Gaussian–Markov processes in that the stochastic averages of odd numbers of noise fields are zero and those for even numbers of noise field terms are the sums of products of stochastic averages associated with pairs of noise fields. However each pair is represented by an element of the diffusion matrix rather than products of the noise fields themselves, as in the case of Gaussian–Markov processes. The treatment starts from a generalised mean field theory for two condensate modes, where generalised coupled Gross–Pitaevskii equations are obtained for the modes and matrix mechanics equations are derived for the amplitudes describing possible fragmentations of the condensate between the two modes. These self-consistent sets of equations are derived via the Dirac–Frenkel variational principle. Numerical studies for interferometry experiments would involve using the solutions from the generalised mean field theory in calculations for the stochastic fields from the Ito stochastic field equations.     

Decoherence,correlation, and unstable quantum states in semiclassical cosmology

As almost any S-matrix of quantum theory possesses a set of complex poles (or branch cuts), it is shown using one example that this is the case in quantum field theory in curved space-time. These poles can be transformed into complex eigenvalues, the corresponding eigenvectors being Gamow vectors. This formalism, which is heuristic in ordinary Hilbert space, becomes a rigorous one within the framework of a properly chosen rigged Hilbert space. Then complex eigenvalues produce damping or growing factors and a typical two semigroups structure. It is known that the growth of entropy, decoherence, and the appearance of correlations, occur in the universe evolution, but this fact is demonstrated only under a restricted set of initial conditions. It is proved that the damping factors are mathematical tools that allow one to enlarge the set.     

Quantum superchemistry: dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates

We show that in certain parameter regimes there is a macroscopic dynamical breakdown of the Gross-Pitaevskii equation. Stochastic field equations for coupled atomic and molecular condensates are derived using the functional positive- P representation. These equations describe the full quantum state of the coupled condensates and include the commonly used Gross-Pitaevskii equation as the noiseless limit. The full quantum theory includes the spontaneous processes which will become significant when the atomic population is low. The experimental signature of the quantum effects will be the time scale of the revival of the atomic population after a near total conversion to the molecular condensate.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

Nontrivial Eigenvalues of the Liouvillian of an Open Quantum System

We present methods of finding complex eigenvalues of the Liouvillian of an open quantum system. The goal is to find eigenvalues that cannot be predicted from the eigenvalues of the corresponding Hamiltonian. Our model is a T-type quantum dot with an infinitely long lead. We suggest the existence of the non-trivial eigenvalues of the Liouvillian in two ways: one way is to show that the original problem reduces to the problem of a two-particle Hamiltonian with a two-body interaction and the other way is to show that diagram expansion of the Green’s function has correlation between the bra state and the ket state. We also introduce the integral equations equivalent to the original eigenvalue problem.     

一维双组分玻色-爱因斯坦凝聚体系的量子隧穿特性

利用双模近似方法研究了一维双组分玻色-爱因斯坦凝聚体(Bose-Einstein condensates,BECs)的量子隧穿特性.从描述三维双组分BECs系统的Gross-Pitaevskii方程(GPE)出发,得到了描述一维体系的GP方程.把体系波函数写成原子数和相位指数的乘积,得到描述体系隧穿特性的费曼方程.数值求解费曼方程,研究了原子之间相互作用(双组分BECs体系原子之间的相互作用包括组分内部原子之间的相互作用和不同组分原子之间的相互作用)对隧穿特性的影响.结果显示,当原子之间的相互作用较弱时,体系发生量子隧穿现象,表现为原子数在平衡位置附近作周期振荡;随着原子之间相互作用增强,体系经历一个临界状态,进入自俘获状态,即由于原子之间相互作用的存在,在对称双势阱中演化的BECs可以呈现出原子数高度的不对称分布,好像绝大数原子被其中一个势阱俘获.从隧穿到自俘获原子之间的相互作用存在一个临界值,从而体系的能量也对应一个临界值,根据体系的哈密顿函数,就能求出相互作用临界值的表达式.     

Fully quantized theory of four-wave mixing with bosonic matter waves

We solve exactly a set of fully quantized coupled equations that describe the quantum dynamics, including the evolution of quantum fluctuations, entanglements, and correlations, of four-wave mixing (FWM) with matter waves. The analytical solution reveals Rabi oscillations, collapses and revivals, and agrees with FWM experiments. It also applies to optical FWM and the Rayleigh superradiance in Bose-Einstein condensates.     

The dynamics of triple-well trapped Bose-Einstein condensates with atoms feeding and loss effects

In this paper, we consider the macroscopic quantum tunnelling and self-trapping phenomena of Bose-Einstein condensates （BECs） with three-body recombination losses and atoms feeding from thermal cloud in triple-well potential. Using the three-mode approximation, three coupled Gross-Pitaevskii equations （GPEs）, which describe the dynamics of the system, are obtained. The corresponding numerical results reveal some interesting characteristics of BECs for different scattering lengths. The self-trapping and quantum tunnelling both are found in zero-phase and ：r-phase modes. Furthermore, we observe the quantum beating phenomenon and the resonance character during the self-trapping and quantum tunnelling. It is also shown that the initial phase has a significant effect on the dynamics of the system.     

New drift modes in a nonuniform quantum magnetoplasma

We report the existence of two new drift modes in a nonuniform quantum magnetoplasma. By using the electron and ion fluid velocities deduced from the quantum momentum equations, together with the continuity and Poisson equations, we derive the governing equations for the low-frequency drift modes. The equations are then Fourier transformed to obtain linear dispersion relations, which admit new drift modes. The results are relevant for identifying electrostatic fluctuations that can cause cross-field charged particle transport in an inhomogeneous, ultracold magnetized quantum plasma.     

Hyperchaos of two coupled Bose--Einstein condensates with a three-body interaction

We investigate the dynamics of two tunnel-coupled Bose--Einstein condensates (BECs) in a double-well potential. The effects of the three-body recombination loss and the feeding of the condensates from the thermal cloud are studied in the case of attractive interatomic interaction. An imaginary three-body interaction term is considered and a two-mode approximation is used to derive three coupled equations which describe the total atomic numbers of the two condensates, the relative population and relative phase respectively. Theoretical analyses and numerical calculations demonstrate the existence of chaotic and hyperchaotic behaviour by using a periodically time-varying scattering length.     

Effect of quantum fluctuations on bose-einstein condensation of bilayer atomic gases in rapid rotation

We investigate vortex states and quantum fluctuations of bilayer atomic gases in rapid rotation. Among mean-field solutions of the Gross-Pitaevskii equation, we consider two types of vortex configurations: the vortex core positions of the layers are coincident or staggered. It is found that the coincident type is energetically preferred in the practical parameter regime. We also calculate the dispersion relations of collective modes and the filling factor for bosons outside the condensates. In the double layer system, quantum depletion is found to be suppressed due to interlayer tunneling. This means that Bose-Einstein condensation is stabilized compared to the single layer case.     

Reaction diffusion equation in the ultra cold coexisting atomic and molecular condensates and the importance of Feshbach interaction

We consider a gas of bosonic atoms near a Feshbach resonance. The dynamics of the atomic and molecular condensates can be described in the limit of small fluctuations by a set of coupled nonlinear Gross Piteaviskii equations. In the case of a strong atom-molecule conversion, the system has an integral motion for the spatially uniform solutions which exhibits temporal oscillation. The possible consequence of this oscillation may be the existence of Josephson like current in the condensates which has been investigated.     

Diffraction theory of an open resonator embedded in a uniaxial medium

A two-dimensional diffraction theory of an open strip resonator embedded in a umaxially anisotropic medium is presented. Applying the modified Kirchhoff approximation and a Green's function technique, the problem is reduced to the solution of a system of two coupled, homogencous integral equations for the unknown field distributions on the mirrors. The eigenvalues of this system of integral equations are related to the characteristic (complex) frequencies of the modes of the open resonator by an implicit relationship. The eigenvalues as well as the corresponding characteristic frequencies are calculated numerically. From the characteristic frequency of a mode, its resonance frequency, its diffraction loss and its quality factor follow immediatedly. For various resonator configurations the influence of the orientation of the optical axis of the uniaxial medium on the properties of the modes is studied in detail.     

Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity

Microdisk lasers are investigated for their thresholds characteristics. We present a novel approach for studying the threshold gains of the whispering-gallery (WG) and other modes based on solving the boundary value problem for the Maxwell's equations. The novelty is that we consider the real-value pairs of frequencies and material gains as eigenvalues. In the two-dimensional (2D) approximation this problem is reduced to the set of independent transcendental equations. A Newton's method is further used to calculate the thresholds and natural frequencies numerically.     

Bloch Oscillations of Two-Component Bose-Einstein Condensates in Optical Lattices

We study the Bloch oscillations of two-component Bose-Einstein condensates trapped in spin-dependent optical lattices. The influence of the intercomponent atom interaction on the system is discussed in detail Accelerated breakdown of the Bloch oscillations and revival phenomena are found respectively for the repulsive and attractive case. For both the cases, the system will finally be set in a quantum self-trapping state due to dynamical instability.     

Exact analytical solution for quantum spins mixing in spin-1 Bose--Einstein condensates

This paper solves exactly a set of fully quantized coupled equations describing the quantum dynamics of quantum spins mixing in spin-1 Bose-Einstein condensates by deriving the exact explicit analytical expressions for the evolution of creation and annihilation operators.     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996