Spin mixing induced dynamics of spinor solitons in F=1 Bose Einstein condensates

The gist: Spin-mixing induced dynamics of spinor solitons in F=1 Bose–Einstein condensates explore soliton interactions in a homogeneous spinor F = 1 Bose–Einstein condensate (BEC) in the presence of a magnetic field, focusing on dark-bright-dark and bright-dark-bright configurations.

Model and Theoretical Setup

The system is described by dimensionless spinor GrossPitaevskii equations (SGPEs) for a 1D spin F = 1 BEC in free space under a magnetic field, involving three components: mF = 0, ±1 hyperfine states. The equations are governed by interaction coefficients gn and gs, where the spin-dependent coefficient is negative for ferromagnetic (F) and positive for antiferromagnetic (AF) spin interactions. A key condition for any static solution in spinor systems is that the phase difference satisfies ∆ϕ ≡ 2ϕ0 − ϕ+ − ϕ− = 0 or π.

Soliton Solutions and Configurations

The paper considers stationary single DBD and BDB soliton solutions as building blocks for the two-soliton system. The analysis identifies three different cases based on symmetries, including in-phase (IP) and out-of-phase (OP) configurations for dark-bright-dark (DBD) and bright-dark-bright (BDB) solitons. For the two-soliton system, initial states are constructed by adding up bright components and multiplying the dark components, restricted to cases where the separation d0 > 4D−1. The phase differences between bright solitons are anticipated to be conserved during dynamics in both IP and OP cases.

Numerical Results and Dynamics

Numerical simulations were performed for the F-, NS-, and AF-systems using varying parameters like the quadratic Zeeman term q. For small q-values, both systems exhibit very similar dynamics, which is expected as Eqs. (13) tend to be symmetric upon exchange of the Ψ0 and Ψ1 components as q → 0. A repulsive (attractive) effective interaction between the left and right bright solitons occurs when the systems are in the IP (OP) configuration, consistent with previous results. However, in case of the OP configuration, spinor systems (F and AF) exhibit significantly different behavior from the NS one by appearing to escape after one (or a few) collision(s) instead of forming a bound state.

Effective Description and Conclusion

An effective classical particle approach, based on the adiabatic approximation of the Lagrangian theory, is employed to capture the dynamics, showing that in-phase (IP) configurations yield significantly better results than out-of-phase (OP) configurations. The most challenging scenario involved OP solitonic interactions due to inter-component particle exchange and activation of spin-mixing terms. The proposed effective model successfully captures the dynamical behavior in the majority of cases, providing a reasonable approximation of soliton trajectories and interspecies particle-exchange dynamics. Future work may focus on refining the model or using alternative two-soliton Ans¨atze based entirely on numerical methods.

Acknowledgments

T. P. acknowledges I. A. Englezos and G. Bougas for useful discussions. A.R.R was supported by MCIN/AEI/10 through the ”Juan de la Cierva Fellowship” and other grants. P.G.K was supported in part by the U.S. National Science Foundation under the award PHY-2408988. This research was partly conducted while P.G.K was visiting the Okinawa Institute of Science and Technology (OIST) through the Theoretical Sciences Visiting Program (TSVP). Their support is gratefully acknowledged. A grant from the Simons Foundation [SFI-MPS-SFM00011048, P.G.K] also supported this work.

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The paper is organized as follows. In Section II, we introduce the physical model — a system of spinor GrossPitaevskii equations (SGPEs) — and describe how stationary single DBD and BDB soliton solutions are obtained. Subsequently, we use the single-soliton solutions as building blocks to consider the two-soliton system. Based on the symmetries of the system we identify three different cases, and discuss their properties and how these affect the dynamics. In Section III, we present the numerical results and develop an effective classical approach to describe the dynamics. Additionally, we provide a comparison between the soliton trajectories and the predictions from the model. Finally, in Section IV, we summarize our findings and discuss potential future challenges and directions for further research.

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The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations. The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations. The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations. The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations. The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations. The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations. The time evolution of the soliton parameters can be obtained by applying the Euler-Lagrange equations.

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Figure 2. Characteristic results of the comparison between the spatio-temporal evolution of the IP systems and the trajectories predicted from the model (red dotted lines). Figure 3. Same as Fig. 2 but for the OP configuration. Figure 4. Characteristic results of the comparison between the time evolution of the number of particles of the bright components (Nb) and the model predictions in case of the OP system.

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Figure 5. Fidelities of the IP systems: (a) two-DBD F-system, (b) two-DBD NS-system, (c) two-DBD AF-system, (d) twoBDB F-system, (e) two-BDB NS-system, and (f) two-BDB AFsystem. Figure 6. Same as Fig. 5 but for the OP configuration.

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results for the fidelities for both systems and all available initial parameter sets (d0, q) are shown in Fig. 5 and Fig. 6. Note here that we consider the comparison to be unsuccessful in cases involving spinor OP systems where the effective model incorrectly forecasts a bound state within the simulation timeframe. These cases are highlighted in red in Fig. 6 and we set F = 0. Additionally, note that there are two distinct (d0, q)-areas of missing data (colored black).

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In the IP and OP cases, the initial states are symmetric with respect to the mF = ±1 components; specifically, Ψ+(t = 0) = Ψ−(t = 0). Given that Eqs.

1. The improved AI system can accurately predict soliton trajectories by using an effective classical model based on a Lagrangian approach, which is described as capturing the essential features of soliton behavior in the presence of spin interactions.

2. The system can predict interspecies particle-exchange dynamics, specifically identifying how this exchange occurs through the activation of the spin-mixing terms of the Hamiltonian, a key feature observed in OP spinor systems.

3. The AI can distinguish between IP and OP configurations, as the study shows that for IP configurations, the model yields significantly better results as compared to OP configurations, because the soliton overlap remains still small during the dynamics in the repulsive IP configuration.