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Bright soliton dynamics in Spin Orbit-Rabi coupled Bose-Einstein condensates

We investigate the dynamics of a spin-orbit (SO) coupled BECs in a time dependent harmonic trap and show the dynamical system to be completely integrable by constructing the Lax pair. We then employ gauge transformation approach to witness the rapid oscillations of the condensates for a relatively smaller value of SO coupling in a time independent harmonic trap compared to their counterparts in a transient trap. Keeping track of the evolution of the condensates in a transient trap during its transition from confining to expulsive trap, we notice that they collapse in the expulsive trap. We further show that one can manipulate the scattering length through Feshbach resonance to stretch the lifetime of the confining trap and revive the condensate. Considering a SO coupled state as the initial state, the numerical simulation indicates that the reinforcement of Rabi coupling on SO coupled BECs generates the striped phase of the bright solitons and does not impact the stability of the condensates despite destroying the integrability of the dynamical system.


Introduction
The advent of Bose-Einstein condensates (BECs) in rubidium [1] and the subsequent experimental identification of bright [2][3][4] and dark solitons [5,6] for attractive and repulsive binary interaction, respectively contributed to a resurgence in the investigation of ultra cold matter. At ultralow temperatures, the macroscopic wave function of BECs can be described by the mean field Gross-Pitaevskii (GP) equation, which is essentially a variant of the celebrated nonlinear Schrödinger (NLS) equation [7]. It is worth pointing out at this juncture that the behavior of single (scalar) component BECs is influenced by the external trapping potential and binary interatomic interaction. Experimental realization of vector (or two component) BECs in which two (or more) internal states or different atoms can be populated has given a fillip to the investigation of multi-component BECs. In contrast to the single component BECs, the multi-component BECs exhibit rich dynamics by virtue of inter-species and intra species binary interaction which can be either attractive or repulsive. This extra freedom associated with multi-component BECs enables them to display novel and rich phenomenon like multidomain walls [8][9][10], spin switching soliton pairs (either brightbright [11][12][13], dark-dark [14] or bright-dark, etc) which can never be witnessed in single component BECs.
Recently, in a landmark experiment, Spielman group at NIST have engineered a synthetic spin-orbit (SO) coupling for a BEC [15]. In the experiment, two Raman laser beams were used to couple a two component BEC consisting of (predominantly) two hyperfine states of 87 Rb. The momentum transfer between laser beams and atoms contributes to the rich possibility of creating synthetic electric and magnetic fields. Recent investigations have explored the possibility of identifying tunable spin orbit coupled BECs [16][17][18][19][20][21] with various trapping potentials and stable regimes of condensates have been observed. The identification of stripe phase in the investigation of spin orbit coupled BECs [22] has only reignited the enthusiasm in this domain of interest as the striped phase is believed to be an indication of the phase transition taking place in a spin orbit coupled BEC. Also, the influence of spin-orbit coupling in a onedimensional BEC, particularly the interplay of the SOC, Raman coupling, and nonlinearity induced precession of the soliton's spin has been studied recently [23]. The existence of "Striped phase", which essentially consists of a linear combination of plane waves and this has contributed to the idea of using ultra cold atoms for the implementation of a quantum simulator [24].
At this juncture, it should be mentioned that eventhough nonlinear excitations like vortices [25], Skymions [26] and bright solitons [27] have been generated in a SO coupled BEC, the dynamics of SO coupled BECs in a time dependent harmonic trap governed by a two coupled GP equation has not been exactly solved analytically. In this paper, we construct the linear eigenvalue problem of SO coupled GP equation in a transient harmonic trap and show that it is completely integrable. We then generate bright solitons solutions and track their evolution in a transient harmonic trap. We observe that the addition of SO coupling contributes to the rapid oscillations of real and imaginary parts of the order parameter and this occurs at a relatively lower value of SO coupling parameter in a time independent harmonic trap compared to their counterparts in a transient trap. Tracing the evolution of the bright solitons (or the condensates) during its transition from confining to expulsive trap, we notice that the condensates collapse suddenly in the expulsive trap. By employing Feshbach resonance management, we show that one can retrieve the condensates by stretching the lifespan of confining trap. Then considering a SO coupled state as the initial state, we numerically study the impact of Rabi coupling on the condensates. The results of our investigation indicate that the reinforcement of Rabi coupling on SO coupled BECs generates striped solitons and leaves no impact on the stability of BECs. We also emphasize that all the above occurs despite the transition of the dynamical system to the nonintegrable regime.

The model and Lax pair
We consider a spin-orbit coupled quasi-one dimensional BEC in a parabolic trap with longitudinal and transverse frequencies ω x ≪ ω ⊥ . Assuming equal contributions of Rashba [28] and Dresselhaus [29] SO coupling (as in the experiment of Ref. [15]), which can be described at sufficiently low temperatures by a set of coupled GP equation of the form [30][31][32]: i In the above equation, the term ±ik L ∂/∂x represents the momentum transfer between the laser beams and atoms due to SO coupling, γ(t) represents binary attractive interaction, and the linear cross coupling parameter Ω denotes Rabi coupling and V(x, t) = λ(t) 2 x 2 /2, where λ(t) = ω x /ω ⊥ , is the time dependent trap frequency Switching off the Rabi coupling (Ω = 0), and employing the following transformation Equation (1) can be written in a simpler form by eliminating the SO coupling term as, i We emphasize that the model governed by equations (1) is exactly integrable if either Rabi coupling (Zeeman splitting) (Ω) or SO coupling (ik L ) is taken into account, but not both of them [33].
The above coupled equations (3a) and (3b) admit the following Lax-pair with In the above, φ(x, t) ≡ c(t)x 2 /2, and [. . .] x denotes differentiation with respect to x. The compatibility condition (1) without Rabi coupling (Ω = 0), while the spectral parameter ζ(t) obeys the following equation: with and It may be noted that a similar Riccati equation (9) has been employed to solve GP-type equations [34][35][36][37][38]. In fact, the identification of the Riccati-type equation (9) gives the first signature of complete integrability of Equation (1) with (Ω = 0). Equation (9), which determines the parabolic potential strength, λ 2 (t), demonstrates that it is related to the interaction strength, γ(t), through the integrability condition, which can be derived by simply substituting Equation (10) in Equation (9): Thus, the system of coupled GP equations (3) is completely integrable for suitable choices of λ(t) and γ(t), which are consistent with equation (11). For condensates in a time independent harmonic trap, λ(t) = λ 0 (a constant), Equation (11) yields γ(t) = γ 0 exp (λ 0 t), where γ 0 is an arbitrary constant. Considering a vacuum seed solution, i.e., q 1 (x, t) = q 2 (x, t) = 0, and employing gauge transformation approach [39][40][41][42], we obtain where δ and χ are arbitrary real parameters, and ε 1 and ε 2 are coupling constants subject to the constraint |ε 1 | 2 + |ε 2 | 2 = 1. A formal solution to the coupled GP equation (1), in the absence of Rabi term, can be straightforwardly written using the transformation (2). The solutions given by Equations (2) illustrate the momentum transfer between laser and atoms when a vector BEC with order parameters q 1 and q 2 driven by bright solitons given by Equations (12) is irradiated with a laser beam.

Soliton dynamics of spin-orbit-Rabi coupled BEC in a transient and time independent trap
Choosing a transient trap shown in figure 1, we show the real, imaginary and absolute value of the macroscopic order parameters ψ 1 and ψ 2 without and with spin orbit coupling in figures 2(a) and 2(b). Comparison of figures 2(a) and 2(b) indicates that the addition of SO coupling contributes to the rapid oscillations of the real and imaginary parts of order parameters ψ 1 and ψ 2 . The fact that the phase of ψ 1 and ψ 2 oscillates with space and time as it is evident from (2) contributes to oscillating nature of spin orbit coupled real and imaginary parts of order parameter. Keeping track of the evolution of the condensates in the transient trap, we observe that the condensate collapses during time evaluation as shown in figure 3(b) for t = 10. In other words, the oscillating nature of real and imaginary parts of order parameter is sustained as long as the trap remains confining in nature (λ(t) 2 > 0) and the condensates collapse as soon as the trap becomes expulsive (λ(t) 2 < 0). In addition to the expulsive trap, the fact that the scattering length γ(t) is driven by exp(0.02t 2 ) [by virtue of (11)] contributes to the collapse of BECs in quasi one dimension. However, by employing Feshbach resonance and manipulating the trap frequency appropriately, one can increase the longevity of the confining trap and recover the condensates as shown in figure 4. For a quasi one dimensional attractive BEC confined in a harmonic trap with frequencies ω x = 2π × 20Hz and ω ⊥ = 2π × 1000Hz, the trap frequency becomes (ω x /ω ⊥ ) 2 = 0.0004. The fact that the condensates are recovered for c(t) = 0.0004t (see figure 4) is consistent with the results observed recently in [27]. The stabilization of the condensates by manipulating the trapping frequency through Feshbach resonance is also numerically verified and the corresponding density profiles are shown in figure 5. It should also be emphasized that the present integrable model offers the luxury of retrieving the condensates by tuning the time dependent trap frequency through Feshbach resonance. Switching off the time dependence of the trap, one observes the same oscillatory behavior for a relatively small value of SO coupling and larger trap frequency as shown in figure 6. However, the subsequent time evolution leads to collapse of the condensates which can once again be Re ψ 1 , Im stabilized employing Feshbach resonance. We then study the impact of Rabi coupling on bright solitons numerically by considering the spin orbit coupled profile in (2) at t = 0 as the initial condition. The numerical simulations are employed using a split-step Crank-Nicolson method [43][44][45][46]. The initial profile is refined to be a stationary state using imaginary time propagation, which is then evolved with realtime propagation by adding the correct phase ξ(x, 0) and θ(x, 0) from (13). Figure 7 depicts the initial profile (Ω = 0) from (2) at t = 0 and final profile of the stationary state as obtained from imaginary time propagation with Ω = 10. Figures 8 and 9 show the numerically simulated density profile of the SO coupled condensates in the presence of Rabi coupling in a transient and time independent harmonic trap respectively. The presence of stripes in figures 8 and 9 is a clear signature of SO coupling consistent with the results of [27]. The observation of striped phase which is an indication of the phase transition taking place in spin orbit coupled BECs is also consistent with the results of spin orbit coupled spin-1 BECs [22]. Eventhough we have shown the system governed by (1) to be completely integrable only in the absence of Rabi coupling (Ω = 0), we observe that the addition of Rabi coupling does not impact the stability of the condensates as it is evident from figures 8 and 9. In other words, eventhough the addition of Rabi coupling to SO coupled condensates takes the dynamical system to the nonintegrable regime, one does not obviously observe its signature on the condensates and they dwell in the stable regime. The addition of different trap strengths (or different interaction strengths) merely changes the trajectories of bright solitons without impacting the stability of the condensates either as shown in figures 9(a)-(c). The tunability of transient harmonic trap to produce stable condensates is reminiscent of the tunability of stable SOC-BECs in a double well potential with the adjustable Raman frequency [21].

Conclusion
In this paper, we have unearthed the Lax-pair of the SO coupled BECs in a transient harmonic trap and shown them to be completely integrable. We observe that the addition of SO coupling contributes to the rapid oscillations in the real and imaginary parts of order parameter. Tracing their evolution in a transient harmonic trap, we observe that the condensates collapse in the expulsive trap. However, by employing Feshbach resonance management and manipulating trap frequency appropriately, we are able to retrieve the condensates. The oscillatory behavior in the condensates occurs at a relatively small SO coupling parameter in a time independent harmonic trap compared to its counterpart in a transient trap. Then, considering a SO coupled state as the ground state, we study numerically the impact of Rabi coupling on BECs. We observe that the reinforcement of Rabi coupling in a SO coupled BEC gives rise to striped solitons and one does not see any signature of instability in the condensates. It would be interesting to study the impact of SO coupling by considering a Rabi coupled BEC as the initial state and the results will be published later.