Efficient numerical methods for computing ground states and dynamics of dipolar Bose–Einstein condensates

New efficient and accurate numerical methods are proposed to compute ground states and dynamics of dipolar Bose–Einstein condensates (BECs) described by a three-dimensional (3D) Gross–Pitaevskii equation (GPE) with a dipolar interaction potential. Due to the high singularity in the dipolar interaction potential, it brings significant difficulties in mathematical analysis and numerical simulations of dipolar BECs. In this paper, by decoupling the two-body dipolar interaction potential into short-range (or local) and long-range interactions (or repulsive and attractive interactions), the GPE for dipolar BECs is reformulated as a Gross–Pitaevskii–Poisson type system. Based on this new mathematical formulation, we prove rigorously existence and uniqueness as well as nonexistence of the ground states, and discuss the existence of global weak solution and finite time blow-up of the dynamics in different parameter regimes of dipolar BECs. In addition, a backward Euler sine pseudospectral method is presented for computing the ground states and a time-splitting sine pseudospectral method is proposed for computing the dynamics of dipolar BECs. Due to the adoption of new mathematical formulation, our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem. Extensive numerical examples in 3D are reported to demonstrate the efficiency and accuracy of our new numerical methods for computing the ground states and dynamics of dipolar BECs.     

Transverse instability of straight vortex lines in dipolar Bose-Einstein condensates

The physics of vortex lines in dipolar condensates is studied. Because of the nonlocality of the dipolar interaction, the 3D character of the vortex plays a more important role in dipolar gases than in typical short-range interacting ones. In particular, the dipolar interaction significantly affects the stability of the transverse modes of the vortex line. Remarkably, in the presence of a periodic potential along the vortexline, the spectrum of transverse modes shows a rotonlike minimum, which eventually destabilizes the straight vortex when the BEC as a whole is still stable, opening the possibility for new scenarios for vortex-line configurations in dipolar gases.     

Dipolar Bose–Einstein condensate soliton on a two-dimensional optical lattice

Using a three-dimensional mean-field model we study one-dimensional dipolar Bose–Einstein condensate (BEC) solitons on a weak two-dimensional (2D) square and triangular optical lattice (OL) potentials placed perpendicular to the polarization direction. The stabilization against collapse and expansion of these solitons for a fixed dipolar interaction and a fixed number of atoms is possible for short-range atomic interaction lying between two critical limits. The solitons collapse below the lower limit and escapes to infinity above the upper limit. One can also stabilize identical tiny BEC solitons arranged on the 2D square OL sites forming a stable 2D array of interacting droplets when the OL sites are filled with a filling factor of 1/2 or less. Such an array is unstable when the filling factor is made more than 1/2 by occupying two adjacent sites of OL. These stable 2D arrays of dipolar superfluid BEC solitons are quite similar to the recently studied dipolar Mott insulator states on 2D lattice in the Bose–Hubbard model by Capogrosso-Sansone et al. [B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, G. Pupillo, Phys. Rev. Lett. 104 (2010) 125301].     

Dipolar and J encoded DQ MAS spectra under rotational resonance

A two-dimensional (2D) double-quantum (DQ) experiment under rotational resonance (R(2)) conditions is introduced for evaluating dipolar couplings in rotating solids. The contributions from the R(2)-recoupled dipolar interaction and the J coupling can be conveniently separated in the resulting 2D R(2)-DQ spectrum, so that the unknown dipolar coupling can readily be extracted, provided that the values of the involved J coupling constants are known. Since the measured parameters are integral intensity ratios between suitably chosen absorption peaks in the 2D spectrum, the proposed method is characterized by a reduced sensitivity to relaxation parameters. The effect of rotor-modulated terms, including chemical shift anisotropy, is efficiently averaged out by synchronizing the excitation/reconversion time with the rotor period. All of these features are demonstrated theoretically by the example of two model systems, namely, isolated spin-pairs and a three-spin system. The results of the theoretical models are applied to both (13)C and (1)H nuclei to extract dipolar couplings in uniformly (13)C labeled L-alanine and a crosslinked natural rubber.     

Landau damping in a dipolar Bose–Fermi mixture in the Bose–Einstein condensation(BEC) limit

By using a mean-field approximation which describes the coupled oscillations of condensate and noncondensate atoms in the collisionless regime, Landau damping in a dilute dipolar Bose–Fermi mixture in the BEC limit where Fermi superfluid is treated as tightly bounded molecules, is investigated. In the case of a uniform quasi-two-dimensional(2D)case, the results for the Landau damping due to the Bose–Fermi interaction are obtained at low and high temperatures. It is shown that at low temperatures, the Landau damping rate is exponentially suppressed. By increasing the strength of dipolar interaction, and the energy of boson quasiparticles, Landau damping is suppressed over a broader temperature range.     

Dynamical stability of dipolar condensate in a parametrically modulated one-dimensional optical lattice

We study the stabilization properties of dipolar Bose–Einstein condensate in a deep one-dimensional optical lattice with an additional external parametrically modulated harmonic trap potential. Through both analytical and numerical methods, we solve a dimensionless nonlocal nonlinear discrete Gross–Pitaevskii equation with both the short-range contact interaction and the long-range dipole–dipole interaction. It is shown that, the stability of dipolar condensate in modulated deep optical lattice can be controled by coupled effects of the contact interaction, the dipolar interaction and the external modulation. The system can be stabilized when the dipolar interaction, the contact interaction, the average strength of potential and the ratio of amplitude to frequency of the modulation satisfy a critical condition. In addition, the breather state, the diffused state and the attractive-interaction-induced-trapped state are predicted. The dipolar interaction and the external modulation of the lattice play important roles in stabilizing the condensate.     

Broadband Polarization Transfer under Magic-Angle Spinning: Application to Total Through-Space-Correlation NMR Spectroscopy

A pulse sequence is described that leads to a broadband recoupling of the dipolar interaction in magic-angle-spinning solid-state NMR experiments of¹³C spins. The sequence is based on a combination of rotating frame and laboratory frame transfer periods. The recovered dipolar interaction is only weakly dependent on spectral parameters but is a faithful measure for the internuclear distances. Furthermore, a pure zero-quantum term is recovered (of the type found in static “spin-diffusion” experiments). This makes the pulse sequence particularly suited for incorporation into two-dimensional total through-space correlation experiments that deliver simultaneous information about all dipolar couplings in a single 2D experiment. It is found that the necessary decoupling from abundant protons is best performed in two steps: first, the strong homonuclear couplings between the high-γ spins are averaged by Lee–Goldburg irradiation and, second, the heteronuclear dipolar interaction is averaged by the combined application of an RF field to the low-γ spins and magic-angle sample spinning. Phase-inversion and amplitude attenuation in the rotating frame and refocusing pulses in the laboratory frame part of the pulse sequence are introduced to achieve an optimum chemical-shift offset-independence and for the suppression of unwanted double-quantum transitions. The design principles are explained in detail. Finally, the pulse scheme is applied to total-correlation spectroscopy of a uniformly labeled amino acid. The experimental cross-peak intensities are in qualitative agreement with the known crystal structure of the model compound.     

Selective excitation in pulsed EPR of a spin-correlated triplet-radical pair

Experiments are described in which a low-amplitude microwave pulse excites only one out of three allowed transitions of the quinone radical (Q(A)(-)) in a spin-correlated triplet-radical pair 3PQ(A)(-) of the bacterial photosynthetic reaction center. A second high-amplitude pulse produces a FID whose temporal shape is strongly modulated with frequencies determined by electron-electron dipolar interaction in the pair. The FID is detected in both the in-phase and the out-of-phase channels. The out-of-phase FID is a result of switching off the magnetic dipolar interaction between 3P and Q(A)(-) due to decay of 3P during the time interval between the two pulses. Refocusing of FID by an additional non-selective pulse allows a dead-time free measurement of this modulation. The influence of the dead-time problem on the distance determination is discussed.     

Adjusting Conditions for High-resolution Pulsed NMR in Solids

Adjusting conditions of pulse sequences for high-resolution nmr in solids are demonstrated by the example of the WHH4 cycle. For this pulse sequence the well-known adjusting condition [1] occurs as limiting case at sufficiently weak dipolar interaction. The relations between the pulse rotation angle and interval time proved to be dependent on the dipolar interaction. Therefore it is impossible to suppress the dipolar interaction in ordinary solids completely. If the strength of the dipolar interaction varies in a range depending on the allowable inhomogeneity of the rf field a spectrometer for high-resolution nmr in solids can be adjusted in an almost optimal way.     

光晶格中双组分偶极玻色-爱因斯坦凝聚体的调制不稳定性

利用线性稳定性分析的方法,对光晶格中双组分偶极玻色-爱因斯坦凝聚体(Bose-Einstein condensates,简称BECs)的调制不稳定性进行了研究.得到了光晶格中双组分偶极BECs原子系统调制不稳定性区域的分布与在位相互作用和由偶极-偶极相互作用所导致的格点间BECs相互作用之间的关系.结果显示,格点间BECs的相互作用对光晶格中双组分偶极BECs的调制不稳定性有较大的影响,这可为实际应用中如何操控双组分偶极BECs提供有用的信息. 关键词： 光晶格 双组分玻色-爱因斯坦凝聚体 调制不稳定性 偶极-偶极相互作用     

Merging and splitting dynamics between two bright solitons in dipolar Bose-Einstein condensates

We numerically study the interaction dynamics of two bright solitons with zero initial velocities in the one-dimensional dipolar Bose-Einstein condensates. Under different dipolar strengths, the two bright solitons can merge into a breathing wave, and then split or propagate constantly after several oscillating periods. We quantitatively study the breathing frequency of wave after merging and the asymmetry property of solitons after splitting, and analyze their formation mechanism by the system's energy evolution. Also, the change of initial phase difference brings distinct effects on the soliton interaction. Our results provide insight into the new dynamical phenomena in dipolar systems and enrich the understanding for interaction between dipolar solitons.     

Tunneling magnetoresistance in small dot arrays with perpendicular anisotropy

The tunneling magnetoresistance (TMR) of a small magnetic dot array with perpendicular anisotropy, is studied by using a resistor network model. Because of the competition between dipolar interaction and perpendicular anisotropy, the TMR ratio can be up to a maximum value (~26%) as predicted by a theoretical model. At moderate dipolar interaction strength, the perpendicular TMR ratio exhibits abrupt jumps due to the switching of magnetic moments in the array when the applied field (normal to the array plane) decreases from a saturation field. This novel character does not occur if the dipolar interaction between particles is quite strong. Furthermore, the effect of the array size N on TMR is also studied and the result shows that TMR ratio fluctuates when N increases for a moderate dipolar interaction strength. When the applied field h_e is parallel to the array plane, the in-plane TMR curve seems insensitive to the dipolar interaction strength, but the maximum TMR ratio (~26%) can also be obtained at h_e=0.     

Band structures of a slowly rotating dipolar Bose-Einstein condensate with a quantized vortex along a one-dimensional optical lattice

We derive the effective Gross-Pitaevskii equation for a slowly rotating dipolar Bose-Einstein condensate (BEC) with a quantized vortex along a one-dimensional optical lattice and calculate its band structures. The band structure of a slowly rotating BEC in a lattice becomes interesting when dipole-dipole interaction (DDI) is involved. Under rotation, a dipolar rotating term emerges from the DDI potential. The dipolar rotating term makes a BEC with an attractive DDI more stable than one with a repulsive DDI. The dipolar rotating term changes and generalizes the definition for the type of BEC, which cannot be simply determined by an s-wave scattering length or an effective contact interaction term. The dipolar rotating term also makes the band structure fascinating and tunable. A so-called swallowtail band structure, i.e., a multi-valued solution due to nonlinear interaction, can either elongate or shrink as the band index increases, in contrast to a non-rotating dipolar BEC system with a monotonic dependence. With the dipolar rotating term, various band structures as well as an attractive BEC without collapse can be easily achieved. We demonstrate that a rotating dipolar BEC system subject to an optical lattice combines features of a crystal and a superfluid and promises wide applications.     

The role of dipolar interactions for the magnetic properties of ferromagnetic nanoring

We investigate numerically the effects of the dipolar interactions on magnetic properties in small ferromagnetic nanorings using a Monte Carlo technique. Our simulated results show that the strength of dipolar interaction in the magnetic nanoring has an important influence on the magnetization reversal processes and further the coercivity and the remanence. As the dipolar interaction increases, the transition of magnetization reversal processes from the onion-rotation state to the vortex state can occur, which results in an increase in coercivity and a decrease in remanence. On the other hand, it is found that the coercivity and the remanence depend more strongly on the strength of dipolar coupling for the relatively small size nanoring than for the large size nanoring in width. This can be attributed to the stable vortex state without core in smaller width nanoring in contrast to the metastable vortex state with core in larger width nanoring, induced by strong dipolar interactions. Additionally, the temperature dependence of coercivity and remanence in magnetic nanoring is also studied at a fixed dipolar interaction.     

Li NMR in lithium borate glasses

In anticipation of using fluctuations in the nuclear dipolar and quadrupolar interaction as a probe of lithium ion motion in lithium borate glasses, the static values of these interactions were measured using a variety of echo techniques. The static quadrupolar echo spectrum of ⁷Li and a calculation of the dipolar interaction in crystalline Li₂B₄O₇ (same chemical composition as the glass under study) were used to estimate the strength of the two interactions. These indicate that the dipolar and quadrupolar interactions for ⁶Li will be of similar size and the dipolar interaction will be dominated by the unlike spin interaction between the ⁶Li and the ¹⁰B, ¹¹B spins. An appropriate theoretical model is proposed and explicit expressions for the echo amplitude are calculated in terms of the dipolar and quadrupolar second moments. This single spin model takes into account the quadrupolar interaction but treats the dipolar interaction as an effective magnetic field. Experimental results are presented which show the essential validity of the model and measurements lead to reasonable values for the dipolar and quadrupolar second moments. The relative merits of the various echo techniques are discussed.     

6Li NMR in lithium borate glasses

In anticipation of using fluctuations in the nuclear dipolar and quadrupolar interaction as a probe of lithium ion motion in lithium borate glasses, the static values of these interactions were measured using a variety of echo techniques. The static quadrupolar echo spectrum of ⁷Li and a calculation of the dipolar interaction in crystalline Li₂B₄O₇ (same chemical composition as the glass under study) were used to estimate the strength of the two interactions. These indicate that the dipolar and quadrupolar interactions for ⁶Li will be of similar size and the dipolar interaction will be dominated by the unlike spin interaction between the ⁶Li and the ¹⁰B, ¹¹B spins. An appropriate theoretical model is proposed and explicit expressions for the echo amplitude are calculated in terms of the dipolar and quadrupolar second moments. This single spin model takes into account the quadrupolar interaction but treats the dipolar interaction as an effective magnetic field. Experimental results are presented which show the essential validity of the model and measurements lead to reasonable values for the dipolar and quadrupolar second moments. The relative merits of the various echo techniques are discussed.     

Ground states for lattices of ferromagnetic granules with dipolar magnetic interaction

The magnetic phase diagrams of 2D and 3D regular lattices formed by nonspherical single-domain ferromagnetic granules featuring a dipolar magnetic interaction are studied. The energy of a magnetic state of such systems is calculated using an approximate expression for the pair interaction of nonspherical granules. The character of the magnetic ground state of the system is determined by three geometric parameters: (i) the eccentricity of granules; (ii) the ratio of periods of the rectangular (2D) or tetragonal (3D) lattice; and (iii) the ratio of a lattice period to a granule size. In contrast to the case of lattices formed by point (or spherical) magnetic moments, in which the ground state is always antiferromagnetic or frustrated (for triangular lattices), the ground state of a 2D lattice composed of nonspherical granules can be ferromagnetic. The magnetic phase diagrams of the systems studied are constructed in the space of the above geometric parameters.     

Bound states and Cooper pairs of molecules in 2D optical lattices bilayer

We investigate the formation of Cooper pairs, bound dimers and the dimer‐dimer elastic scattering of ultracold dipolar Fermi molecules confined in a 2D optical lattice bilayer configuration. While the energy and their associated bound states are determined in a variational way, the correlated two‐molecule pair is addressed as in the original Cooper formulation. We demonstrate that the 2D lattice confinement favors the formation of zero center mass momentum bound states. Regarding the Cooper pairs binding energy, this depends on the molecule populations in each layer. Maximum binding energies occur for non‐zero (zero) pair momentum when the Fermi system is polarized (unpolarized). We find an analytic expression for the dimer‐dimer effective interaction in the deep BEC regime. The present analysis represents a route for addressing the BCS‐BEC crossover in dipolar Fermi gases confined in 2D optical lattices within the current experimental panorama.     

Monte Carlo calculation of the magnetization behavior and the magnetocaloric effect of interacting particles

The magnetization behavior and the magnetic entropy change of a system made up of ferromagnetically interacting particles are calculated by using Monte Carlo simulation. The effect of the magnetic anisotropy of particles and the dipolar–dipolar interaction between particles on the magnetization and the magnetic entropy change of the system are discussed. It is found that there is no spontaneous magnetization, both the magnetic anisotropy of particles and the dipolar–dipolar interaction between particles restrains the system's magnetizing in the external magnetic field. The magnetic entropy change decreases with the increase in temperature in the system without the dipolar–dipolar interaction; however, the dipolar–dipolar interaction between particles makes the magnetic entropy change of the system have maximum value at low temperatures.     

2D interactions and binary crystals of dipolar and quadrupolar nematic colloids

In this Letter, we demonstrate that the symmetry of the elastic interaction between the dipolar and quadrupolar colloidal particles in the nematic liquid crystal leads to a novel variety of 2D nematic "binary" colloidal crystals, which have not been observed in any colloidal system. The dipolar-quadrupolar interaction is highly anisotropic and shows a power-law dependence when the particles approach each other along the director field with a pair-binding energy of the order of several thousands of k(B)T for 4 microm diameter colloids.