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arXiv:2507.13683 · Physical review. E

Nonlinear management of the miscibility-immiscibility transition in binary Bose-Einstein condensates

Bakhtiyor B. Baizakov, Boris A. Malomed, Mario Salerno

Published Jul 16, 2025• 3 citations
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Abstract

We investigate application of nonlinearity management (NM, i.e., periodic variation of the strength of the intercomponent repulsion) to the miscibility-immiscibility (MIM) transition across the critical point of a two-component Bose-Einstein condensate, both with and without linear mixing [Rabi coupling (RC)] between the components. To this end, we first identify, by means of a variational approximation and numerical solution, diverse stationary domain-wall (DW) structures supported by the system in the absence of management. The approximate analytical solutions for the DWs are found to be in excellent agreement with their numerical counterparts. An analytical estimate is also produced for the upshift of the MIM transition caused by the pressure of the trapping potential in the case of a confined system. An exact DW solution is produced for the system including the Pöschl-Teller potential, which is stable (unstable) if the potential is repulsive (attractive). Further, we find the spectrum of linear excitations in the spatially uniform mixed state, and thus establish parameter regions where the system is stable or unstable against demixing. In particular, RC upshifts the critical strength of the intercomponent repulsion for the onset of the MIM transition. Eigenfrequencies of excitations on top of DW states are identified from numerical simulations through monitoring the evolution of perturbed states. Weak NM applied at the DW eigenfrequency reveals features of the nonlinear resonance. Stronger NM, under which the system periodically crosses the MIM-transition point, restricts the miscibility.

DOI
10.1103/c655-fb5s

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We investigate application of nonlinearity management (NM, i.e., periodic variation of the strength of the intercomponent repulsion) to the miscibility-immiscibility (MIM) transition across the critical point of a two-component Bose-Einstein condensate, both with and without linear mixing [Rabi coupling (RC)] between the components. To this end, we first identify, by means of a variational approximation and numerical solution, diverse stationary domain-wall (DW) structures supported by the system in the absence of management. The approximate analytical solutions for the DWs are found to be in excellent agreement with their numerical counterparts. An analytical estimate is also produced for the upshift of the MIM transition caused by the pressure of the trapping potential in the case of a confined system. An exact DW solution is produced for the system including the Pöschl-Teller potential, which is stable (unstable) if the potential is repulsive (attractive). Further, we find the spectrum of linear excitations in the spatially uniform mixed state, and thus establish parameter regions where the system is stable or unstable against demixing. In particular, RC upshifts the critical strength of the intercomponent repulsion for the onset of the MIM transition. Eigenfrequencies of excitations on top of DW states are identified from numerical simulations through monitoring the evolution of perturbed states. Weak NM applied at the DW eigenfrequency reveals features of the nonlinear resonance. Stronger NM, under which the system periodically crosses the MIM-transition point, restricts the miscibility.
A one-dimensional mean-field model of a two-component condensate in the parabolic trap is considered, with the components corresponding to different spin states of the same atom. We demonstrate that the linear coupling (interconversion) between them, induced by a resonant electromagnetic wave, can drive the immiscible binary condensate into a miscible state. This transition is predicted in an analytical form by means of a variational approximation (for an infinitely long system), and is confirmed by direct numerical solutions of the symmetric and asymmetric models (the asymmetry accounts for a possible difference in the chemical potential between the components). We define an order parameter of the system as an off-centre shift of the centre of mass of each component. A numerically found dependence of the order parameter on the linear-coupling strength reveals a second-kind phase transition (in an effectively finite system). The phase transition looks very similar in two different regimes, namely, with a fixed number of atoms, or a fixed chemical potential. An additional transition is possible between double- and single-humped density distributions in the weak component of a strongly asymmetric system. We also briefly consider dynamical states, with the two components oscillating relative to each other. In this case, the components periodically separate even if they should be mixed in the static configuration.
We investigate the effects of dipole-dipole (DD) interactions on immiscibility-miscibility transitions (IMT's) in two-component Bose-Einstein condensates (BEC's) trapped in the harmonic-oscillator (HO) potential, with the components linearly coupled by a resonant electromagnetic field (accordingly, the components represent two different spin states of the same atom). The problem is studied by means of direct numerical simulations. Different mutual orientations of the dipolar moments in the two components are considered. It is shown that, in the binary BEC formed by dipoles with the same orientation and equal magnitudes, the IMT cannot be induced by the DD interaction alone, being possible only in the presence of the linear coupling between the components, while the miscibility threshold is affected by the DD interactions. However, in the binary condensate with the two dipolar components polarized in opposite directions, the IMT can be induced without any linear coupling. Further, we demonstrate that those miscible and immiscible localized states, formed in the presence of the DD interactions, which are unstable evolve into robust breathers, which tend to keep the original miscibility or immiscibility, respectively. An exception is the case of a very strong DD attraction, when narrow stationary modes are destroyed by the instability. The binary BEC composed of copolarized dipoles with different magnitudes are briefly considered as well.
The dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analyzing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady-state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.
We consider a model of a two-component condensate in one dimension, where the components represent two different spin states of the same atom. We demonstrate that linear interconversion between them, induced by a resonant electromagnetic wave, can make the immiscible binary condensate miscible. This transition is predicted by means of a variational approximation, and is then confirmed by direct numerical solutions of the symmetric and asymmetric models (the asymmetry is accounted for by a chemical-potentil difference between the components). The dependence of the corresponding order parameters on the linear-coupling strengths reveals a typical phase transition of the second kind. However, in dynamic states the species may remain separated, even if they would get mixed in a static configuration.
We investigate the effect of higher-order interactions induced by shape-dependent confinement in the modulational instability (MI) of a binary mixture of Bose-Einstein condensates. For this, we present and compute both analytically and numerically a system of coupled Gross-Pitaevskii equations with residual nonlinearity that rule the dynamics of the mixture. Using the linear stability approach, we obtain the instability criteria of the mixture and find that the MI can be excited in miscible condensates and altered in immiscible condensates due to the effect of residual nonlinearity. Direct numerical calculations are performed to support the analytical predictions, and a good agreement is found. The space-time evolution of the condensate density is displayed in both cases when the mixture is miscible and immiscible, showing the generation of bright solitons for modes predicted to be unstable.
We investigate the non-equilibrium dynamics across the miscible–immiscible phase separation in a binary mixture of Bose–Einstein condensates. The excitation spectra reveal that the Landau critical velocity vanishes at the critical point, where the superfluidity spontaneously breaks down. We analytically extract the dynamical critical exponent z = 2 and static correlation length critical exponent v = 1/2 from the Landau critical velocity. Moreover, by simulating the real-time dynamics across the critical point, we find the average domain number and the average bifurcation delay show universal scaling laws with respect to the quench time. We then numerically extract the static correlation length critical exponent v = 1/2 and the dynamical critical exponent z = 2 according to Kibble–Zurek mechanism. The scaling exponents ( v = 1/2, z = 2) in the phase separation driven by quenching the atom–atom interaction are different from the ones ( v = 1/2, z = 1) in the phase separation driven by quenching the Rabi coupling strength (2009 Phys. Rev. Lett. 102 070401; 2011 Phys. Rev. Lett. 107 230402). Our study explores the connections between the spontaneous superfluidity breakdown and the spontaneous defect formation in the phase separation dynamics.