Spinor Bose-Einstein Condensate Research Articles

Correlation-based entanglement criterion in bipartite multiboson systems

We describe a criterion for the detection of entanglement between two multi-boson systems. The criterion is based on calculating correlations of Gell-Mann matrices with a fixed boson number on each subsystem. This applies naturally to systems such as two entangled spinor Bose-Einstein condensates. We apply our criterion to several experimentally motivated examples, such as an $ S^z S^z $ entangled BECs, ac Stark shift induced two-mode squeezed BECs, and photons under parametric down conversion. We find that entanglement can be detected for all parameter regions for the most general criterion. Alternative criteria based on a similar formalism are also discussed together with their merits.

Generation and storage of spin-nematic squeezing in a spinor Bose-Einstein condensate

We propose a scheme to store the spin-nematic squeezing in a spinor Bose-Einstein condensate by applying periodic microwave pulses. For a proper pulse period and phase shift, the squeezing can be enhanced and maintained for a long time, which realizes the storage of the maximal squeezing. We also propose a method to generate the spin-nematic squeezed vacuum, which is associated with negligible occupation of the squeezed modes, through an adiabatic sweep of the magnetic field. It is shown that the method can be readily implemented for both ferromagnetic and antiferromagnetic condensates.

Coherent zero-field magnetization resonance in a dipolar spin-1 Bose-Einstein condensate

With current magnetic field shielding and high precision detection in dipolar spinor Bose-Einstein condensates, it is possible to experimentally detect the low or zero field nonsecular dipolar dynamics. Here we analytically investigate the zero-field nonsecular magnetic dipolar interaction effect, with an emphasis on magnetization dynamics in a spin-1 Bose-Einstein condensate under the single spatial mode approximation within the mean field theory. Due to the biaxial nature of the dipolar interaction, a novel resonance occurs in the condensate magnetization oscillation, contrast to the previous assumption of a conserved magnetization in strong magnetic fields. Furthermore, we propose a dynamical-decoupling detection method for such a resonance, which cancels the stray magnetic fields in experiments but restores the magnetization dynamics. Our results shed new lights on the dipolar systems and may find potential applications beyond cold atoms.

Analytic models for the density of a ground-state spinor condensate

We demonstrate that the ground state of a trapped spin-1 and spin-2 spinor ferromagnetic Bose-Einstein condensate (BEC) can be well approximated by a single decoupled Gross-Pitaevskii (GP) equation. Useful analytic models for the ground-state densities of ferromagnetic BECs are obtained from the Thomas-Fermi approximation (TFA) to this decoupled equation. Similarly, for the ground states of spin-1 anti-ferromagnetic and spin-2 anti-ferromagnetic and cyclic BECs, some of the spin component densities are zero which reduces the coupled GP equation to a simple reduced form. Analytic models for ground state densities are also obtained for anti-ferromagnetic and cyclic BECs from the TFA to the respective reduced GP equations. The analytic densities are illustrated and compared with the full numerical solution of the GP equation with realistic experimental parameters.

Localizing spin dynamics in a spin-1 Bose-Einstein condensate via magnetic pulses

Spin exchange interaction between atoms in a spin-1 Bose-Einstein condensate causes atomic spin evolving periodically under the single spatial mode approximation in the mean field theory. By applying fast magnetic pulses according to a two-step or a four-step control protocol, we find analytically that the spin dynamics is significantly suppressed for an arbitrary initial state. Numerical calculations under single mode approximation are carried out to confirm the validity and robustness of these protocols. This localization method can be readily utilized to improve the sensitivity of a magnetometer based on spin-1 Bose-Einstein condensates.

The management of matter rogue waves in F = 1 spinor Bose–Einstein condensates

We have reported the matter rogue wave solutions for the nonautonomous three coupled Gross–Pitaevskii (GP) equation which governs the pulse propagation in spin-1 Bose–Einstein condensates (BECs) with time modulated nonlinearities. The system under consideration has attractive mean-field interactions and ferromagnetic spin-exchange interactions. The exact solutions have been worked out by using similarity and scaling transformations. We have depicted the controllable center characteristics of rogue waves for the kink-type spin-exchange interactions. Additionally, we have also discussed the management of rogue waves for the hyperbolic form of spin-exchange interactions by invoking isospectral Hamiltonian deformation technique.

Hybrid Matter-Wave-Microwave Solitons Produced by the Local-Field Effect.

It was recently found that the electric local-field effect (LFE) can lead to a strong coupling of atomic Bose-Einstein condensates (BECs) to off-resonant optical fields. We demonstrate that the magnetic LFE gives rise to a previously unexplored mechanism for coupling a (pseudo-) spinor BEC or fermion gas to microwaves (MWs). We present a theory for the magnetic LFE and find that it gives rise to a short-range attractive interaction between two components of the (pseudo) spinor, and a long-range interaction between them. The latter interaction, resulting from deformation of the magnetic field, is locally repulsive but globally attractive, in sharp contrast with its counterpart for the optical LFE, produced by phase modulation of the electric field. Our analytical results, confirmed by the numerical computations, show that the long-range interaction gives rise to modulational instability of the spatially uniform state, and it creates stable ground states in the form of hybrid matter-wave-microwave solitons (which seem like one-dimensional magnetic monopoles), with a size much smaller than the MW wavelength, even in the presence of arbitrarily strong contact intercomponent repulsion. The setting is somewhat similar to exciton-polaritonic condensates in semiconductor microcavities. The release of matter waves from the soliton may be used for the realization of an atom laser. The analysis also applies to molecular BECs with rotational states coupled by the electric MW field.

Half-Quantum Vortices in an Antiferromagnetic Spinor Bose-Einstein Condensate.

We report on the observation of half-quantum vortices (HQVs) in the easy-plane polar phase of an antiferromagnetic spinor Bose-Einstein condensate. Using insitu magnetization-sensitive imaging, we observe that pairs of HQVs with opposite core magnetization are generated when singly charged quantum vortices are injected into the condensate. The dynamics of HQV pair formation is characterized by measuring the temporal evolutions of the pair separation distance and the core magnetization, which reveals the short-range nature of the repulsive interactions between the HQVs. We find that spin fluctuations arising from thermal population of transverse magnon excitations do not significantly affect the HQV pair formation dynamics. Our results demonstrate the instability of a singly charged vortex in the antiferromagnetic spinor condensate.

Quantum tunneling and entanglement of dipolar spin-1 bosons in double well potentials

We study spinor Bose-Einstein condensates with dipolar interaction in double well potentials. We consider the two-site Bose Hubbard model and find single atom tunneling occurs for the small tunneling strength. We also investigate the entanglement between the two wells and show that the entanglement is due to both number and spin degrees of freedom. The number entanglement is insensitive to the dipolar interaction, while the spin entanglement is suppressed by dipolar interaction. In addition, we show that the single atom tunneling behavior can be manipulated by adiabatically varying the energy offset between the two wells.

Route to non-Abelian quantum turbulence in spinor Bose-Einstein condensates

We have studied computationally the collision dynamics of spin-2 Bose--Einstein condensates initially confined in a triple-well trap. Depending on the intra- and inter-component relative phases of the initial state spinor wave function, the collision of the three condensate segments produces one of many possible vortex-antivortex lattices after which the system transitions to quantum turbulence. We find that the emerging vortex lattice structures can be described in terms of multi-wave interference. We show that the three-segment collisions can be used to systematically produce staggered vortex-antivortex honeycomb lattices of fractional-charge vortices, whose collision dynamics are known to be non-Abelian. Such condensate collider experiments could potentially be used as a controllable pathway to generating non-Abelian superfluid turbulence with networks of vortex rungs.

Quantum gases. Observation of isolated monopoles in a quantum field.

Topological defects play important roles throughout nature, appearing in contexts as diverse as cosmology, particle physics, superfluidity, liquid crystals, and metallurgy. Point defects can arise naturally as magnetic monopoles resulting from symmetry breaking in grand unified theories. We devised an experiment to create and detect quantum mechanical analogs of such monopoles in a spin-1 Bose-Einstein condensate. The defects, which were stable on the time scale of our experiments, were identified from spin-resolved images of the condensate density profile that exhibit a characteristic dependence on the choice of quantization axis. Our observations lay the foundation for experimental studies of the dynamics and stability of topological point defects in quantum systems.

Coherent spin-mixing dynamics in thermalRb87spin-1 and spin-2 gases

We study the non-equilibrium coherent spin mixing dynamics in ferromagnetic spin-1 and antiferromagnetic spin-2 thermal gases of ultracold $^{87}$Rb atoms. Long lasting spin population oscillations with magnetic field dependent resonances are observed in both cases. Our observations are well reproduced by Boltzmann equations of the Wigner distribution function. Compared to the equation of motion of spinor Bose-Einstein condensates, the only difference here is a factor of two increase in the spin-dependent interaction, which is confirmed directly in the spin-2 case by measuring the relation between the oscillation amplitude and the sample's density.

An Inequality for Spinor Bose–Einstein Condensates

An inequality for spin-$F$ Bose-Einstein condensates (BECs) $ F^2(\rho^2-|\Theta|^2)-\boldsymbol{M}^2\ge0 $ is reported, where $\rho$, $\Theta$, and $\boldsymbol{M}$ represent the density, singlet pair amplitude, and magnetization vector, respectively. The distribution of high-symmetry spinors in the allowed region by the inequality is elucidated with using the Majorana representation. The result is illustrated by the example of spin-2 BECs.

Topological defects and inhomogeneous spin patterns induced by the quadratic Zeeman effect in spin-1 Bose-Einstein condensates

We obtain analytically two kinds of inhomogeneous spin domain configurations endowed with time-dependent periodic domain walls in spin-1 Bose-Einstein condensate, which result from the positive and negative quadratic Zeeman effects, respectively. It is shown analytically that the topological defects are given by the unmagnetized atoms, at which the order parameter is of polar phase, but besides them, the order parameter is of axisymmetrybroken phase. The quadratic Zeeman effect can be tuned to induce the dynamical phase transition of the spin domain, and its sign can affect the topological structure of the spin pattern. These features arise from the axisymmetry breaking due to the pointwise-different population exchange between the sublevels determined uniquely by the quadratic Zeeman effects. The related experimental observations for the spin-1 87 Rb and 23 Na condensates are also discussed.

Sound waves and modulational instabilities on continuous-wave solutions in spinor Bose-Einstein condensates

We analyze sound waves (phonons, i.e. Bogoliubov excitations) propagating on continuous-wave (cw) solutions of repulsive $F=1$ spinor Bose-Einstein condensates (BECs) such as $^{23}\mathrm{Na}$ (which is antiferromagnetic or polar) and $^{87}\mathrm{Rb}$ (which is ferromagnetic). Zeeman splitting by a uniform magnetic field is included. All cw solutions to ferromagnetic BECs with vanishing ${M}_{F}=0$ particle density and nonzero components in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ fields are subject to modulational instability (MI). Modulational instability increases with increasing particle density. Modulational instability also increases with differences in the components' wave numbers; this effect is larger at lower densities but becomes insignificant at higher particle densities. Continuous-wave solutions to antiferromagnetic (polar) BECs with vanishing ${M}_{F}=0$ particle density and nonzero components in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ fields do not suffer MI if the wave numbers of the components are the same. If there is a wave-number difference, MI initially increases with increasing particle density and then peaks before dropping to zero beyond a given particle density. The cw solutions with particles in both ${M}_{F}=\ifmmode\pm\else\textpm\fi{}1$ components and nonvanishing ${M}_{F}=0$ components do not have MI if the wave numbers of the components are the same, but do exhibit MI when the wave numbers are different. Direct numerical simulations of a continuous wave with weak white noise confirm that weak noise grows fastest at wave numbers with the largest MI and show some of the results beyond small-amplitude perturbations. Phonon dispersion curves are computed numerically; we find analytic solutions for the phonon dispersion in a variety of limiting cases.

Coherent spin dynamics in spin-imbalanced ferromagnetic spinor condensates**Project supported by the National Natural Science Foundation of China (Grant No. 11104217), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2014JQ1022), and the Youth Foundation of XUPT, China (Grant No. ZL2013-36).

We study the coherent spin dynamics of a ferromagnetic spinor Bose–Einstein condensate (BEC) in its domain formation process with an arbitrary spin configuration. Through a simplified schematic view of the domain structure, a semiclassical theory that captures the essential dynamics of the system is presented, and the coherent spin mixing dynamics can be understood in terms of oscillation in the phase space diagram. Using the phase diagram analysis method, we identify new phases, including the π phase oscillation and the running phase for the spin-imbalanced ferromagnetic spinor BEC.

Counting rule of Nambu–Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

When continuous symmetry is spontaneously broken, there appear Nambu–Goldstone modes (NGMs) with linear or quadratic dispersion relation, which is called type-I or type-II, respectively. We propose a framework to count these modes including the coefficients of the dispersion relations by applying the standard Gross–Pitaevskii–Bogoliubov theory. Our method is mainly based on (i) zero-mode solutions of the Bogoliubov equation originated from spontaneous symmetry breaking and (ii) their generalized orthogonal relations, which naturally arise from well-known Bogoliubov transformations and are referred to as “σ-orthogonality” in this paper. Unlike previous works, our framework is applicable without any modification to the cases where there are additional zero modes, which do not have a symmetry origin, such as quasi-NGMs, and/or where spacetime symmetry is spontaneously broken in the presence of a topological soliton or a vortex. As a by-product of the formulation, we also give a compact summary for mathematics of bosonic Bogoliubov equations and Bogoliubov transformations, which becomes a foundation for any problem of Bogoliubov quasiparticles. The general results are illustrated by various examples in spinor Bose–Einstein condensates (BECs). In particular, the result on the spin-3 BECs includes new findings such as a type-I–type-II transition and an increase of the type-II dispersion coefficient caused by the presence of a linearly-independent pair of zero modes.

Metastable spin textures and Nambu-Goldstone modes of a ferromagnetic spin-1 Bose-Einstein condensate confined in a ring trap

We investigate the metastability of a ferromagnetic spin-1 Bose-Einstein condensate confined in a quasi-one-dimensional rotating ring trap by solving the spin-1 Gross-Pitaevskii equation. We find analytical solutions that exhibit spin textures. By performing linear stability analysis, it is shown that the solutions can become metastable states. We also find that the number of Nambu-Goldstone modes changes at a certain rotation velocity without changing the continuous symmetry of the order parameter.

Quantum mass acquisition in spinor Bose-Einstein condensates.

Quantum mass acquisition, in which a massless (quasi)particle becomes massive due to quantum corrections, is predicted to occur in several subfields of physics. However, its experimental observation remains elusive since the emergent energy gap is too small. We show that a spinor Bose-Einstein condensate is an excellent candidate for the observation of such a peculiar phenomenon as the energy gap turns out to be 2 orders of magnitude larger than the zero-point energy. This extraordinarily large energy gap is a consequence of the dynamical instability. The propagation velocity of the resultant massive excitation mode is found to be decreased by the quantum corrections as opposed to phonons.

Gauge field for neutral atoms near a current loop

We investigate the spatial distribution of the gauge field for neutral, spin-carrying atoms generated from the spatially inhomogeneous magnetic field near a current loop. When a zero-field ring is formed in the magnetic field, the gauge field around the ring exhibits a spatial structure resembling that of an effective dipole ring. We show that the emergence of the dipole-like spatial structure of the gauge field can be viewed as being the result of a merging of two effective monopoles of opposite charges that correspond to zero-field points in the magnetic field. Finally, we propose an experimental scheme using three concentric current loops to explore the gauge field effects on spinor Bose-Einstein condensates in a versatile manner.