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.     

Dynamical instability induced by the zero mode under symmetry breaking external perturbation

A complex eigenvalue in the Bogoliubov–de Gennes equations for a stationary Bose–Einstein condensate in the ultracold atomic system indicates the dynamical instability of the system. We also have the modes with zero eigenvalues for the condensate, called the zero modes, which originate from the spontaneous breakdown of symmetries. Although the zero modes are suppressed in many theoretical analyses, we take account of them in this paper and argue that a zero mode can change into one with a pure imaginary eigenvalue by applying a symmetry breaking external perturbation potential. This emergence of a pure imaginary mode adds a new type of scenario of dynamical instability to that characterized by the complex eigenvalue of the usual excitation modes. For illustration, we deal with two one-dimensional homogeneous Bose–Einstein condensate systems with a single dark soliton under a respective perturbation potential, breaking the invariance under translation, to derive pure imaginary modes.     

Vacuum quark condensate,chiral Lagrangian,and Bose–Einstein statistics

In a series of articles it was recently claimed that the quantum chromodynamic (QCD) condensates are not the properties of the vacuum but of the hadrons and are confined inside them. We point out that this claim is incompatible with the chiral Lagrangian and Bose–Einstein statistics of the Goldstone bosons (pions) in chiral limit and conclude that the quark condensate must be the property of the QCD vacuum.     

Bose–Einstein condensates in magnetic waveguides

In this article, we describe an experimental system for generating Bose–Einstein condensates and controlling the shape and motion of a condensate by using miniaturised magnetic potentials. In particular, we describe the magnetic trap setup, the vacuum system, the use of dispenser sources for loading a high number of atoms into the magneto-optical trap, the magnetic transfer of atoms into the microtrap, and the experimental cycle for generating Bose–Einstein condensates. We present first results on outcoupling of condensates into a magnetic waveguide and discuss influences of the trap surface on the ultra-cold ensembles. Received: 21 August 2002 / Revised version: 10 December 2002 / Published online: 26 February 2003 RID="*" ID="*"Corresponding author. Fax: +49-7071/295-829, E-mail: fortagh@pit.uni-tuebingen.de     

An efficient numerical method for computing dynamics of spin F = 2 Bose–Einstein condensates

In this paper, we extend the efficient time-splitting Fourier pseudospectral method to solve the generalized Gross–Pitaevskii (GP) equations, which model the dynamics of spin F = 2 Bose–Einstein condensates at extremely low temperature. Using the time-splitting technique, we split the generalized GP equations into one linear part and two nonlinear parts: the linear part is solved with the Fourier pseudospectral method; one of nonlinear parts is solved analytically while the other one is reformulated into a matrix formulation and solved by diagonalization. We show that the method keeps well the conservation laws related to generalized GP equations in 1D and 2D. We also show that the method is of second-order in time and spectrally accurate in space through a one-dimensional numerical test. We apply the method to investigate the dynamics of spin F = 2 Bose–Einstein condensates confined in a uniform/nonuniform magnetic field.     

Finite-temperature phase-locking transition in three-dimensional arrays of Josephson coupled Bose–Einstein condensates

We consider theoretically a phase-locking transition in Bose–Einstein condensate in an optical lattice in the regime where system can realized as a three-dimensional Josephson junction array. The coherence between adjacent Bose condensates (trapped in the valleys of the periodic potential) caused by the Josephson tunneling can lead to a phase transition with a global phase coherence at certain critical temperature. Using a model Hamiltonian of Josephson weakly coupled Bose condensates we calculate the critical temperature for the three-dimensional system placed in a simple cubic lattice and discuss the result in the context of system parameters and possible experiments.     

Bright matter wave solitons and their collision in Bose–Einstein condensates

We obtain the bright matter wave solitons in Bose–Einstein condensates from a trivial input solution by solving the time dependent Gross–Pitaevskii (GP) equation with quadratic potential and exponentially varying scattering length. We observe that the matter wave density of bright soliton increases with time by virtue of the exponentially increasing scattering length. We also understand that the matter wave densities of bright soliton trains remain finite despite the exchange of atoms during interaction and they travel along different trajectories (diverge) after interaction. We also observe that their amplitudes continue to fluctuate with time. For exponentially decaying scattering lengths, instability sets in the condensates. However, the scattering length can be suitably manipulated without causing the explosion or the collapse of the condensates.     

Analytical multisoliton solutions for multicomponent Bose–Einstein condensate systems with time- and space-controlled potentials

We construct explicit multisoliton complex solutions for multicomponent Bose–Einstein condensate systems with time- and spatial-coordinate-dependent atomic potentials and interactions. The exact solutions are used to analyze the important solitary matter wave properties such as the profiles of temporal and spatial multimode beams as well as focusing effects. Results demonstrate that soliton complexes can be controlled nonlinearly during the interaction by modulating the external potentials and nonlinearities.     

Beliaev theory of spinor Bose–Einstein condensates

By generalizing the Green’s function approach developed by Beliaev [S.T. Beliaev, Sov. Phys. JETP 7 (1958) 299; S.T. Beliaev, Sov. Phys. JETP 7 (1958) 289], we study effects of quantum fluctuations on the energy spectra of spin-1 spinor Bose–Einstein condensates, in particular, of a ⁸⁷Rb condensate in the presence of an external magnetic field. We find that due to quantum fluctuations, the effective mass of magnons, which characterizes the quadratic dispersion relation of spin-wave excitations, increases compared with its mean-field value. The enhancement factor turns out to be the same for two distinct quantum phases: the ferromagnetic and polar phases, and it is a function of only the gas parameter. The lifetime of magnons in a spin-1 ⁸⁷Rb spinor condensate is shown to be much longer than that of phonons due to the difference in their dispersion relations. We propose a scheme to measure the effective mass of magnons in a spinor Bose gas by utilizing the effect of magnons’ nonlinear dispersion relation on the time evolution of the distribution of transverse magnetization. This type of measurement can be applied, for example, to precision magnetometry.     

Non-autonomous bright matter wave solitons in spinor Bose–Einstein condensates

We investigate the dynamics of bright matter wave solitons in spin-1 Bose–Einstein condensates with time modulated nonlinearities. We obtain soliton solutions of an integrable autonomous three-coupled Gross–Pitaevskii (3-GP) equations using Hirota?s method involving a non-standard bilinearization. The similarity transformations are developed to construct the soliton solutions of non-autonomous 3-GP system. The non-autonomous solitons admit different density profiles. An interesting phenomenon of soliton compression is identified for kink-like nonlinearity coefficient with Hermite–Gaussian-like potential strength. Our study shows that these non-autonomous solitons undergo non-trivial collisions involving condensate switching.     

A variational approach to the modulational-oscillatory instability of Bose–Einstein condensates in an optical potential

We use the time-dependent variational approach to demonstrate how the modulational and oscillatory instabilities can be generated in Bose–Einstein condensates (BECs) trapped in a periodic optical lattice with weak driving harmonic potential. We derive and analyze the ordinary differential equations for the time evolution of the amplitude and phase of the modulational perturbation, and obtain the instability condition of the condensates through the effective potential. The effect of the optical potential on the dynamics of the BECs is shown. We perform direct numerical simulations to support our theoretical findings, and good agreement is found.     

Manipulating nonclassical quantum statistical properties of light field by an f-deformed Bose–Einstein condensate

We consider the interaction between an f-deformed Bose–Einstein condensate and a single-mode quantized light field. By using the Gardiner’s phonon operators, we find that there exists a natural deformation in the model which modifies the Bogoliubov approximation under the condition of large but finite number of particles in condensate. This approach introduces an intrinsically deformed Bose–Einstein condensate, where the deformation parameter, well-defined by the particle number N in condensate, controls the strength of the associated nonlinearity. By introducing the deformed Gardiner’s phonon operators we modify the very dilute-gas approximation through including atomic collisions in condensate. The rate of atomic collisions κ, as a new deformation parameter in the deformed Bose–Einstein condensate, controls the nonlinearity related to the atomic collisions. We show that by controlling the nonlinearities in the f-deformed atomic condensate through the two atomic parameters N and κ, it is possible to generate and manipulate the nonclassical quantum statistical properties of radiation field, such as, the sub-Poissonian photon statistics and quadrature squeezing. Also, it is possible to control the collapses and revivals phenomena in the average number of photons by atomic parameters N and κ.     

Quantum hydrodynamics

Quantum hydrodynamics in superfluid helium and atomic Bose–Einstein condensates (BECs) has been recently one of the most important topics in low temperature physics. In these systems, a macroscopic wave function (order parameter) appears because of Bose–Einstein condensation, which creates quantized vortices. Turbulence consisting of quantized vortices is called quantum turbulence (QT). The study of quantized vortices and QT has increased in intensity for two reasons. The first is that recent studies of QT are considerably advanced over older studies, which were chiefly limited to thermal counterflow in ⁴${}^4$He, which has no analog with classical traditional turbulence, whereas new studies on QT are focused on a comparison between QT and classical turbulence. The second reason is the realization of atomic BECs in 1995, for which modern optical techniques enable the direct control and visualization of the condensate and can even change the interaction; such direct control is impossible in other quantum condensates like superfluid helium and superconductors. Our group has made many important theoretical and numerical contributions to the field of quantum hydrodynamics of both superfluid helium and atomic BECs. In this article, we review some of the important topics in detail. The topics of quantum hydrodynamics are diverse, so we have not attempted to cover all these topics in this article. We also ensure that the scope of this article does not overlap with our recent review article (arXiv:1004.5458), “Quantized vortices in superfluid helium and atomic Bose–Einstein condensates”, and other review articles.     

Approximate solutions for half-dark solitons in spinor non-equilibrium Polariton condensates

In this work I generalize and apply an analytical approximation to analyze 1D states of non-equilibrium spinor polariton Bose–Einstein condensates (BEC). Solutions for the condensate wave functions carrying black solitons and half-dark solitons are presented. The derivation is based on the non-conservative Lagrangian formalism for complex Ginzburg–Landau type equations (cGLE), which provides ordinary differential equations for the parameters of the dark soliton solutions in their dynamic environment. Explicit expressions for the stationary dark soliton solution are stated. Subsequently the method is extended to spin sensitive polariton condensates, which yields ordinary differential equations for the parameters of half-dark solitons. Finally a stationary case with explicit expressions for half-dark solitons is presented.     

Stability of Attractive Bose–Einstein Condensates

We propose the critical nonlinear Schrödinger equation with a harmonic potential as a model of attractive Bose–Einstein condensates. By an elaborate mathematical analysis we show that a sharp stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental data. Moreover with this threshold we prove that a ground state of the condensate exists and is orbital stable. We also evaluate the minimum of the condensate energy.     

Fictitious magnetic resonance by quasielectrostatic field

We propose a new kind of spin manipulation method using a fictitious magnetic field generated by a quasielectrostatic field. The method can be applicable to every atom with electron spins and has distinct advantages of small photon scattering rate and local addressability. By using a CO₂ laser as a quasielectrostatic field, we have experimentally demonstrated the proposed method by observing the Rabi oscillation of the ground state hyperfine spin F=1 of the cold ⁸⁷Rb atoms and the Bose–Einstein condensate.     

Magnetic resonance imaging of Bose–Einstein condensates

As a new method for measuring the spatial distribution of Bose–Einstein condensates, the magnetic resonance imaging (MRI) method is proposed and studied in detail. The basic concepts, the resolution limit and the formalism of the MRI method are presented. It is expected that a resolution higher than that in optical imaging methods can be obtained by using the MRI method. Results of simulation of expected MRI signals for Bose–Einstein condensates containing dark solitons are also presented. Received: 27 September 2001 / Revised version: 24 October 2001 / Published online: 17 January 2002     

Production of a chromium Bose–Einstein condensate

The recent achievement of Bose–Einstein condensation of chromium atoms [1] has opened longed-for experimental access to a degenerate quantum gas with long-range and anisotropic interaction. Due to the large magnetic moment of chromium atoms of 6 μ_B, in contrast to other Bose–Einstein condensates (BECs), magnetic dipole-dipole interaction plays an important role in a chromium BEC. Many new physical properties of degenerate gases arising from these magnetic forces have been predicted in the past and can now be studied experimentally. Besides these phenomena, the large dipole moment leads to a breakdown of standard methods for the creation of a chromium BEC. Cooling and trapping methods had to be adapted to the special electronic structure of chromium to reach the regime of quantum degeneracy. Some of them apply generally to gases with large dipolar forces. We present here a detailed discussion of the experimental techniques which are used to create a chromium BEC and allow us to produce pure condensates with up to 10⁵ atoms in an optical dipole trap. We also describe the methods used to determine the trapping parameters.     

Modulated amplitude waves with non-trivial phase in quasi-1D inhomogeneous Bose–Einstein condensates

We consider a 1D nonlinear Schrödinger equation (NLSE) which describes the mean field dynamics of an elongated Bose–Einstein condensate and prove the existence of modulated amplitude waves with non-trivial phase and minimal spatial period tending to infinite. The proof combines the theory of local continuation of non-degenerate periodic solutions with a property of the Ermakov–Pinney equation.