Dipolar condensed atomic mixtures and miscibility under rotation

By considering symmetric and asymmetric dipolar coupled mixtures (with dysprosium and erbium isotopes), we report a study on relevant anisotropic effects, related to spatial separation and miscibility, due to dipole-dipole interactions (DDIs) in rotating binary dipolar Bose-Einstein condensates. The binary mixtures are kept in strong pancake-like traps, with repulsive two-body interactions modeled by an effective two-dimensional (2D) coupled Gross-Pitaevskii equation. The DDI are tuned from repulsive to attractive by varying the dipole polarization angle. A clear spatial separation is verified in the densities for attractive DDIs, being angular for symmetric mixtures and radial for asymmetric ones. Also relevant is the mass-imbalance sensibility observed by the vortex-patterns in symmetric and asymmetric-dipolar mixtures. In an extension of this study, here we show how the rotational properties and spatial separation of these dipolar mixture are affected by a quartic term added to the harmonic trap of one of the components.

sition. The recent investigations in ultracold laboratories with two-component dipolar Bose-Einstein Condensates (BEC), on stability and miscibility properties, became quite interesting due to the number of control parameters that can be explored in new experimental setups.
The parameters which can be controlled are given by the strengths of dipoles, the number of atoms in each component, the inter-and intra-species scattering lengths, as well as confining trap geometries. The stability and pattern formation have been studied in Ref. [4], by considering dipolar-dipolar interactions (DDIs) in two-component dipolar BEC systems. Rotational properties of two-component dipolar BEC in concentrically coupled annular traps were also studied in Ref. [5], by assuming a mixture with only one dipolar component.
Following previous studies with rotating binary dipolar mixtures and their miscibility properties [6][7][8], the miscible-immiscible transition (MIT) of the dipolar mixtures with 162,164 Dy and 168 Er were also recently studied by us in Ref. [9]. For these coupled dipolar systems, the miscible-immiscible stable conditions were analyzed within a full 3D formalism, by considering repulsive contact interactions, within pancake-and cigar-type trap configurations.
The rotational properties and vortex-lattice pattern structures of these dipolar mixtures were further investigated by us in Refs. [10][11][12], by changing the inter-to intra-species scattering lengths, as well as the polarization angles of the dipoles. Among the observed characteristics of these strong dipolar binary systems, relevant for further investigations are the possibilities to alter the effective time-averaged DDI from repulsive to attractive, by tuning the polarization angle ϕ of both interacting dipoles from zero to 90 • , respectively. In Ref. [11], vortex pattern structures were studied by considering rotating binary mixtures confined by squared optical lattices, whereas in Ref. [12], by tuning ϕ, our investigation was mainly concerned with rotational properties together with spatial separations of the binary mixtures. Motivated by the above mentioned studies, considering that the interplay between DDIs and contact interactions can bring us different interesting effects in the MIT, showing richer vortex-lattice structures in rotating binary dipolar systems, in the present contribution we are also reporting some new results obtained for the properties of dipolar mixtures confined by a strongly pancake-like two-dimensional (2D) rotating harmonic trap. The effect of a weak quartic perturbation in the x-y plane, applied to the first dipolar component, is studied by tuning the polarization angles of the dipoles together with the contact inter-species interactions.
Next section, the model formalism is presented with our parametrization and numerical procedure. In section 3, after an analysis of the main results for symmetric and asymmetric bi-nary dipolar mixtures confined by strong pancake-like harmonic traps, we present new results obtained when considering the effect of a weak quartic perturbation added to the harmonic trap of one of the components. Finally, a summary with our conclusions is given in section 4.

A. Model formalism
The coupled dipolar system with condensed two atomic species i = 1, 2, with the respective masses m i (with m 1 ≥ m 2 ) are assumed to be confined in strongly pancake-shaped harmonic traps, with fixed aspect ratios, such that λ = ω i,z /ω i,⊥ = 20 for both species i = 1, 2, where ω i,z and ω i,⊥ are, respectively, the longitudinal and transverse trap frequencies. The coupled Gross-Pitaevskii (GP) equation is cast in a dimensionless format, with energy and length units given, respectively, by ħ hω 1,⊥ and l ⊥ ≡ ħ h/(m 1 ω 1,⊥ ). Correspondingly, the space and time variables are given in units of l ⊥ and 1/ω 1 , respectively, such that r → l ⊥ r and t → τ/ω 1 .
Within these units, and by adjusting both trap frequencies such that m 2 ω 2 2,⊥ = m 1 ω 2 1,⊥ , the dimensionless external 3D trap potential for each one of the species i can be written as On the miscibility conditions for binary trapped dipolar systems, more details and discussion can be found in Refs. [10][11][12]. Large values for λ allow us to reduce to 2D the original 3D formalism by considering the usual factorization of the 3D wave function as ψ i (x, y, τ)χ i (z), where χ i (z) ≡ (λ/π) 1/4 e −λz 2 /2 . The two-body contact interactions related to the scattering lengths a i j , and DDI parameters are defined as [12] where i, j = 1, 2, with N j being the number of atoms and m i j = m i m j /(m i + m j ) the reduced mass of the species i and j. In our numerical analysis, the length unit will be assumed being l ⊥ = 1µm ≈ 1.89 × 10 4 a 0 , with a 0 being the Bohr radius. The corresponding 2D coupled GP equation for the two components ψ i ≡ ψ i (x, y, τ) of the total wave function can be written as y) being the reduced 2D expression for the DDI. The 2D confining potential V i (x, y) is assumed to be harmonic for both components, as in Ref. [12]. However, in the present contribution we are providing an extension to our study reported in Ref. [12], by examining the effect, on the pattern distribution and spatial separation of the dipolar mixture, of a quartic term applied to one of the components, which we define as the more-massive one. So, the trap is given by with κ 1 ≡ κ being a dimensionless positive parameter (in principle, assumed to be small), which increases the trap confinement of the more massive component. Experimentally, the quartic potential together with harmonic trap can be created by using far-detuned laser beam propagating along the axis of the trap, perpendicular to the (x, y) plane. So, the width and strength of the quartic trap can be controlled, respectively, by the width and amplitude of the blue-detuned Gaussian laser beam. More details can be found in the reference [13], where experiments with quartic trap in BEC are discussed. Each component of the wave function is assumed normalized to one, with Ω being the corresponding rotation parameter (in units of ω 1 ), which is assumed to be common for both components.
The 2D DDI presented in the integrand of the second term shown in Eq. (2) can be expressed in the 2D momentum space as the combination of two terms, by considering the orientations of the dipoles ϕ and the projection of the corresponding Fourier transformed V (d) (x, y). One term is perpendicular, with the other parallel to the direction of the dipole inclinations, as described in Refs. [7,8]. By generalizing the description to a polarization field rotating in the (x, y) plane, the two terms can be combined according to the dipole orientations ϕ, with the total 2D momentum-space DDI given by [12] where k 2 ρ ≡ k 2 x + k 2 y , with erfc(x) being the complementary error function of x. The 2D configuration-space effective DDI is obtained by applying the convolution theorem in Eq. (2), performing the inverse 2D Fourier-transform for the product of the DDI and density, such that (4), one should notice that such momentum-space Fourier transform of the dipole-dipole potential changes the sign at some particular large momentum k ρ . However, after applying the convolution theorem with the inverse Fourier transform (by integrating the momentum variables), the corresponding coordinate-space interaction has a definite value, as in the 3D case, which is positive for ϕ ≤ ϕ M , and negative for 90 • ≥ ϕ > ϕ M , where ϕ M ≈ 54.7 • is the so-called "magic angle", in which the DDI is canceled out.

B. Parametrization and numerical approach
The two binary mixtures ( 164 Dy-162 Dy and 168 Er-164 Dy) that we are investigating are called, respectively, "symmetric" and "asymmetric" ones; where these terms are related to the dipolar symmetry of the condensed atoms. The corresponding magnetic dipole moments of the three species are the following: µ = 10µ B for 162,164 Dy, and µ = 7µ B for 168 Er. So, by considering the definitions given in (1), the strengths of the DDI are a We fix both intra-species contact interactions at a 11 = a 22 = 50a 0 , remaining the inter-species one to be explored by varying the ratio parameter δ ≡ a 12 /a 11 . Once selected the polarization angle and δ as the appropriate parameters to alter the miscibility properties of a mixture, we fix other parameters guided by possible realistic settings and stability requirements. For the present approach, we choose Ω = 0.75 for the rotation frequency parameter, larger than the one used in Ref. [12] (Ω = 0.6), in order to improve the observation of vortex-pattern structures and spatial separation.
For the numerical approach to solve the GP formalism (2), the split-step Crank-Nicolson method [14,15] is applied, combined with a standard method for evaluating DDI integrals in momentum space, as described in Ref. [12]. In the search for stable solutions, the numerical simulations were carried out in imaginary time on a grid with a maximum of 464 points in both x − y directions, with spatial and time steps ∆x = ∆ y = 0.05 and ∆t = 0.0005, respectively.
In this approach, both wave-function components are renormalized to one at each time step.

A. Dipolar mixtures confined by identical harmonic pancake-like traps
We focus our study in the two coupled mixtures given by 168 Er-164 Dy and 164 Dy-162 Dy, motivated by recent experimental studies with dipolar BEC systems. In our investigation, we have considered harmonic strongly pancake-like trap, as detailed in Ref. [12]. First, a detailed analysis of ground state and stability properties was performed in the absence of rotation. In this respect, we understand that our theoretical predictions can be helpful in verifying misci- As discussed above, being dipolar symmetric, with a 11 = a 22 = 131a 0 , this 164 Dy-162 Dy BEC mixture exposes more miscible properties. As verified in Ref. [12], this mixture in the rotating harmonic trap [with κ i = 0 in Eq. (3)] shows triangular, squared, rectangular-shaped, double core, striped, and with domain wall vortex lattices regarding the ratio between inter-and intra-species contact interaction. Also, this mixture shows complete spatial separation at large polarization angles, where the DDI is purely attractive. By modifying the external confinement of one of the components, we can introduce some external asymmetry to the mixture. So, in this contribution, for this binary system we start by adding a very weak quartic term in the first component of the mixture, in order to analyze the miscibility and complete spatial separation of the coupled system. We consider two different miscible cases, with δ = 1 and 1.45. For these particular cases, striped and domain wall vortex structures are observed [12], respectively, when both species are under identical rotating harmonic pancake-like traps, with λ = 20 and Ω = 0.6. By adding a quartic term to the trap, as explained in section II, we have also reduced the number of atoms to N i = 5000 and increased the frequency to Ω = 0.75 in order to improve our observation on the corresponding rotational structure and spatial separation.
In this case, ring lattice structures can be verified, also verified even for single component radially phase-separated, changing the previous patterns observed in Ref. [12] for κ = 0. Such similar behavior for non-dipolar mixtures was also analyzed theoretically recently in Ref. [16].
To improve our understanding of the phase separation and the effect of the added quartic trap, we studied the dipolar binary system by increasing the strength κ. We observed that, for κ ≥ 0.1, with large repulsive inter-species interaction δ = 1.45, the spatial phase separations of the densities change completely from the previous angular to radial ones. This behavior is indicated in Fig. 2, where the phase-separated case, displayed for ϕ = 90 • with κ = 0.05, is being compared with the κ = 0.08 case. So, when κ ≥ 0.1, only radial spatial separation can be observed in the binary mixture.

C. Dipolar asymmetric 168 Er-164 Dy mixture, with a quartic trap applied to 168 Er
In this subsection we consider the dipolar asymmetric 168 Er-164 Dy BEC system, to study the effect of a quartic dipolar trap applied to the first component ( 168 Er) of the mixture. As reported in Ref. [12] for this dipolar asymmetric case, when κ i = 0 in the rotating confining harmonic trap given by Eq. rations obtained without the quartic term interaction. Even a weak quartic trap is enough to modify the angular spatial separation to radial ones in the dipolar 164 Dy-162 Dy mixture, for attractive dipolar interactions. In the asymmetric 168 Er-164 Dy dipolar BEC mixture, where we have already radial spatial separation for attractive dipolar interactions even without the quartic term, with the 168 Er element surrounding the other element, we have observed that, for the addition of enough large quartic term to the 168 Er element, there is an exchange of the two coupled densities, with this element moving to the center. So, for asymmetric mixture with repulsive inter-species interaction and attractive DDI, strong quartic trap (κ ≥ 1) will prevent exchanges between both densities, which will remain completely radial-separated spatially.