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Phase diagram and dynamics of dark-bright vector solitons in spin-orbit-coupled Bose–Einstein condensate
Chaos, Solitons and Fractals 111 (2018) 62–67
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Phase diagram and dynamics of dark-bright vector solitons in spin-orbit-coupled Bose–Einstein condensate T.F. Xu a,b, W.L. Li a, Zai-Dong Li c,d,∗, C. Zhang e a
Hebei Key Laboratory of Microstructural Material Physics, School of Science, Yanshan University, Qinhuangdao 066004, China Department of Physics, University of California, San Diego, California 92093, USA c Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China d Key Laboratory of Electronic Materials and Devices of Tianjin, School of Electronics and Information Engineering, Hebei University of Technology, Tianjin 300401, China e Henan Key Laboratory of Photovoltaic Materials and School of Physics and Electronics, Henan University, Kaifeng 475004, China b
a r t i c l e
i n f o
Article history: Received 17 January 2018 Revised 9 April 2018 Accepted 9 April 2018
Keywords: Spin-orbit coupling Vector soliton Bose–Einstein condensate Dark-bright soliton
a b s t r a c t We investigate the dynamics of dark-bright vector soliton solutions in a spin-orbit coupled Bose–Einstein condensate with repulsive interaction by the imaginary time evolution. The phase diagram is obtained numerically in spin-orbit coupled Bose–Einstein condensate. We find that the spin-orbit coupling favors miscibility, and the energy detuning between the Raman beam and atom dominates the separation phase of the Bose gases. We also find that the spin-orbit coupled strength affects interaction types (attractive or repulsive) between the two dark-bright solitons.
1. Introduction Spin-orbit coupling (SOC) is an interaction between spin with motion of particle, which plays a special role in many areas. The SOC for electrons in condensed matter system is particularly important for many condensed matter fields, such as spin Hall effect, topological insulator, spintronic devices, etc [1–3]. The experimental realization of synthetic SOC by the two photon Raman process in ultracold atom gases has offered an ideal platform to study novel properties of Bose–Einstein condenses (BECs) [4–6]. Meanwhile, the SOC effect in Bose system is the key factor for half vortices [7], monopoles [8], domain walls [9–11], bright solitons [12,13], dark solitons [14], gap solitons [15], and skyrmions [16] etc. Matter-wave vector solitons as the nonlinear state have attracted much attention in BECs with the tunable intra- and intercomponent repulsive(attractive) atomic interactions, which is expressed by the solution of coupled Schrödinger equation at the mean-field level. According to the type of short-range interaction, these matter wave solitons can be divided into two types, i.e., dark soliton and bright soliton. The former is characterized by a concave density and a non-trial phase jump across their density notch in repulsive interaction system, while a bright soliton with a con-
∗
Corresponding author. E-mail addresses: [email protected] (T.F. Xu), [email protected] (Z.-D. Li).
https://doi.org/10.1016/j.chaos.2018.04.014 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
© 2018 Elsevier Ltd. All rights reserved.
vex density peak against a negligible background in attractive interaction system [17–19]. Compared with scalar BECs, the multicomponent BECs enrich the ample novel nonlinear structures, such as bright-bright solitons [20–22], dark-dark solitons [23], darkbright solitons, which can be thought of as symbiotic solitons [24–27]. Recently, the dynamical properties of dark-bright solitons are carefully investigated by the multiscale expansion method in a reduced SOC-BEC system. The results show that the oscillator frequency of dark-bright solitons in a harmonic trap is close to the regular oscillator frequency without the SOC [28]. Due to the potential application in many fields, dark-bright solitons had attracted intensive attentions [17]. In this paper, we investigate dark-bright solitons in a quasi one dimensional SOC-BECs with repulsive interaction. In particular, we are interested in the effects of the SOC strength and Raman detuning of the system on the densities distribution of different component wave functions and others fundamental characteristic of dark-bright solitons. It will be shown that the SOC strength and the detuning between the Raman beam and energy states of the atoms have the opposite effects on the width of dark-bright solitons and the height of bright soliton amplitude. The paper is organized as follows. In Section 2, we introduce the model equation for dark-bright solitons in a SOC Bose system. Section 3 is devoted to compute the different phase distribution via numerical method for different detunings and SOC strengths. It is aimed to give a di-
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rect insight of the different state densities in an interval of parameters. In Section 4, the dark-bright solitons are calculated by numerical method for various parameter interval. In Section 5 we calculate the dynamic evolution of dark-bright soliton and interaction between two stationary dark-bright solitons for different SOC strengths. Section 6 is a brief summary. 2. Model equation We consider a quasi-one-dimensional SOC-BEC described by the coupled nonlinear Schrödinger equations [12]:
ih ¯
1 ∂ φ= ( px − h¯ ασy )2 + h¯ δσz − g (φ † · φ ) φ , ∂t 2m
(1)
where φ = (φ1 , φ2 )T , φ † · φ = |φ1 |2 + |φ2 |2 , φ 1 and φ 2 are the two component wave functions, m is the atom mass, σ is Pauli matrix, α represents the strength of SOC, is the coupling strength be tween Raman beams with atom levels, g is attractive atomic interactions. To find dark-bright solitons numerically, we start from the dimensionless coupled Schrödinger equations,
1 ∂2 ∂ ∂ 2 2 φD = − + δ + g ( | φ | + | φ | ) φD + γ φB , D B ∂t 2 ∂ x2 ∂x 1 ∂2 ∂ ∂ 2 2 i φB = − − δ + g ( | φ | + | φ | ) φB − γ φD . D B ∂t 2 ∂ x2 ∂x
i
(2)
where φ D and φ B are the dark or bright soliton solution, respectively, δ is the energy detuning between the Raman beam and the atoms, g is the interaction strength between atoms in the unit of Ng m/( h ¯ )/ h ¯ , γ represents the SOC strength in the unit of
α h¯ /m. The sign of γ , δ and g are positive for the whole paper. The scaling energy unit is , length is h ¯ /(m) and time is 1/. For the sake of simplicity the intra and inter interaction are set to be equal to g [14]. The normalization wave function is deter∞ mined via −∞ (|φD |2 + |φB |2 )dx = N. N is in direct proportion to the total atom numbers. The dark-bright solitons in the SOC-BECs can be experimentally achieved by the counterflow of two hyperfine states of 87 Rb. The physical parameters can be similar as those in the experiment. The SOC γ depends on the laser wave length and the relative angle between the two Raman beams; while the energy difference δ can be easily tuned by changing the relative frequency of the two lasers. The units of the length and time are 1.7 μm and 4 ms, respectively, the total atom number is N = 5 ∗ 104 level [4,27]. Eq. (2) are the starting point of the numerical calculations throughout this paper. By solving Eq. (2) numerically, one is able to obtain the darkbright solitons solution. To solve Eq. (2), we can first differentiate it by using split-time-step Crank–Nicolson method along with the homogeneous Neumann boundary condition, and then evaluate several thousands of steps in imaginary time until the invariable energy is reached. Every soliton(dark or bright) solution of Eq. (2) lives on a background which may be rest or highly oscillatory. Different from bright solitons cases, asymptotic behavior of dark soliton on the two sides of boundary is two different cases, that is, φ D → A ± eikx , where A+ and A− are not the same complex constants and k is a real constant. According to the value of k, the background of soliton is rest (k = 0) or oscillatory (k = 0). It is not an easy task to find a stable dark-bright soliton in the case of highly oscillatory background, so we set the rest background case for simplicity [29]. 3. Phase diagram in SOC Bose gas mixtures In this section, we study the dark-bright soliton solutions in the SOC Bose gas mixtures. The dark-bright solitons have been created experimentally by filling the dark state with different hyperfine
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states of 87 Rb atoms. The phase of dark soliton can be imprinted and transferred bright soliton state atoms into dark one by the two-photon resonance process simultaneously [26]. When the SOC effect is presented in the Bose system, more ample phenomena occur. Here we carefully checked the different solutions of Eq. (2) in diverse parameter regions. The numerical result shows there exist three kinds of soliton solutions, i.e., dark soliton states, darkbright solitons and stripe solitons, as shown in Fig. 1. We choose the wavefuction as the order of these phase transitions and use the properties of bright soliton solutions of Eq. (2) to distinguish the three different states of the system. When the amplitude of bright soliton of the dark-bright soliton solutions is less than certain value (e.g. 1% of the amplitude of dark soliton background), which is almost invisible, the state is defined as a single dark soliton. When the amplitude of bright soliton increases with the increasing SOC constant γ and detuning δ until the second peaks of the bright soliton of vector dark-bright solitons emerge, we call it as dark-bright soliton state. Otherwise, we define the states as stripe solitons, which have been intensively investigated in many literatures. Here we just focus the properties of vector dark-bright solitons in the SOC-BECs. In fact, the dark and dark-bright soliton states form the separation phase and stripe soliton is a mixture phase of SOC Bose gases. From Fig. 1 we observe that |φ D, B |2 evolves from one darkbright soliton distribution to many nodes one when the SOC constant γ is increased, which are typical strip solitons in 1D system [30–32] and are mixture states of different hyperfine states. Particularly the mixture states are different from the same component strip solitons, which are total bright or dark soliton [13,14,33]. However, we can find that the emergence condition of the stripe solitons in Fig. 1 is the same to the case of region II of Fig. 1 in Ref.[13] when regarding to the linear dispersion relation of Eq. (2). At this time, the lowest energy of the stripe solitons may have either a positive or negative momentum in the linear dispersion relation or they can be a linear superposition of both modes, thus forming the stripe phase. With the increasing of the detuning δ , the components |φ D, B |2 take a transformation from stripe soliton to one dark soliton state or dark-bright soliton, as shown in Fig. 1. This result shows that the SOC favors miscibility and the detuning between the Raman beam and atom energy levels dominates separation phase of the Bose gases. It should be pointed out that the presented phase diagram is different from the early result which mainly concentrate the different inter and intra interaction effect on the half-quantum vortex condensate or spin-spiral condensate in two dimension [34]. And it is not the same to another important result which focus on spin polarization and momentum as a function of Raman coupling constant and critical density [35]. The stripe phase locates at the bottom of phase diagram and the separation phase lies on the top of Fig. 1, which is consisted with the recently experimental result [36]. As the SOC γ and detuning δ are set to zero, Eq. (2) reduces to the integrable Manakov limit when the intra and inter interactions are equal to the same constant [37]. Kivshar et al. has studied the dark-dark soliton solutions of coupled nonlinear equation in the normal group-velocity dispersion region, which is equal to repulsive interaction between atoms in BEC system[38]. A general dark soliton solution may have different intensities for two polarization modes. At this time the solution can be written in the form
φD1 = U0 (cos θ1 tanh Z + i sin θ1 ) exp(i 1 ), φD2 = V0 (cos θ2 tanh Z + i sin θ2 ) exp(i 2 ).
(3)
where Z = U0 cos θ1 (x + t/W − τ0 ), 1 = k1 x + (U02 + V02 + k21 )t, 2 = k2 x + (U02 + V02 + k22 )t and the parameters U0 , V0 , W, k1 and k2 are coupled by the following relations: U0 cos θ1 = V0 cos θ2 , W −1 = U0 sin(θ1 + θ2 )/ cos θ2 + (k1 + k2 ), k2 − k1 = sin(θ1 − θ2 )/ cos θ2 . Here, φD1 and φD2 are the two component of dark solitons solu-
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Fig. 1. The SOC γ and detuning δ dependent phase diagram in the SOC Bose gas mixtures. The mark “o” corresponding to Manakov case, is used in Ref.[37] to discuss dark-dark solitons. The interaction strength is g = 1.0. The normalization constant is N = 5 for all the following case, in addition to Fig. 2.
tion without SOC effect version of Eq. (2) [38], k1 and k2 are the wave number of the background, θ 1 and θ 2 are the so-called dark solitons phase angle, −W −1 is the velocity. In order to compare the results obtained by the imaginary-time method with Kivshar’s analytic solutions, we calculate the darkdark soliton solutions numerically. For simplicity we set that the background of dark-dark solitons are rest (k1 = k2 = 0), the height of dark solitons are U0 = V0 = 1, the dark solitons phase angle are θ1 = θ2 = 0, the results are shown in Fig. 2. It should be mentioned that we only give the first component φD1 of the coupled equation solutions and the other soliton solution shares the same density mode. The two solutions are degenerated in this case. So, we only plot the first soliton component. It turns out that the analytic curve of |φD1 |2 agrees well with the soliton obtained by the imaginary-time method without the SOC effect [see Fig. 2]. Based on the excellent agreement between two methods, the imaginarytime method is mainly used for the following calculations. At the same time, we numerically solve the coupled Eq. (2) without the SOC effect items using split-time-step Crank–Nicolson method in real time. The results are shown in Fig. 2(b), the dynamic of the soliton is highly robust and it perfectly conserves its amplitude and phase. The amplitude of dark solitons propagates with changing slightly their shapes and emerges a little ripples when the SOC is considered, which we will discuss below. 4. Dark-bright solitons for SOC-BECs It is well known that the general solutions are dark solitons for the repulsive interaction in a single component Bose system when potential trap is absent. However, owing to the intra- and intercomponent repulsive interactions, BEC with multiple components admits dark-bright soliton solutions [39,40]. It is found that the inter-component repulsive interaction plays the important roles in the formation of excitation states for multiple component BEC. Dark-bright soliton only exists when an effective potential well is created by the dark soliton through the inter-component interaction; However, the density of bright solitons are not only decided by the effective potential well, but also influenced by these parameters: the detuning δ and the strength of SOC. In the following we choose a proper initial state, plug it into the coupled equation and finally get the three soliton solutions. As is shown in Fig. 3 (a), the ratio of bright soliton in total particle density is very small when the detuning is bigger (δ = 25), which can be regarded as total separation phase in SOC-BEC. The ampli-
Fig. 2. (a) The first component dark soliton solution without SOC. The solid lines are numerical method results, and the dotted lines are the analytic dark soliton density in Ref.[38]. (b) The real-time evolution result of the first dark solitons by the coupled nonlinear Schrödinger equation Eq. (2) without SOC effect.
Fig. 3. The three kinds states of the SOC-BEC in Fig. 1. Panel (a), (b) and (c) show the amplitudes of one dark solution, dark-bright soliton and stripe soliton respectively. Top(bottom) panel represents the first(second) component. The parameters is δ = 25, 5, 1 respectively, other parameters are γ = 2.5 and g = 1.0.
tudes of bright soliton become big with the decrease of the detuning (δ = 5), see Fig. 3 (b). At the same time, the density notch edge in dark soliton emerges some ripples. We also check the phase of dark soliton, which has no changes and still slip across their density notch. When the detuning further reduced (δ = 1), the effective potential well can not hold the bright solitons, the system in Eqs. (2) admits the stripe soliton solution and the particle density tends to the same value for the two components (Fig. 3 (c)). Different from gap stripe soliton in nonlinear periodic system [41], these stripe solitons are observed in the absence of external potential trap. It should be pointed out that the soliton amplitudes in Fig. 3 (a1)–(c1) are multiplied by a negative unit. It can not change the previous conclusion because the phase of bright soliton at its convex density peak in Fig. 3 (a1)–(c1) stay the same. We also find that the number of stripe soliton nodes grows with the increase of SOC γ for the same detuning δ . It is consistent with the result of bright-bright soliton solution in BEC with the SOC attractive interaction [12]. This also suggests that no-node theorem is excluded for dark-bright vector solitons in the spin-orbit coupled Bose-Einstein condensates with repulsive interaction [34,42,43].
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Fig. 4. The amplitude and width of these dark-bright solitons versus main parameters of the SOC-BEC system. Panel (a), (b) and (c) show the relation between the amplitudes, widths of the dark-bright solitons and the coupling strength g, the detuning δ and the SOC γ respectively. Top(bottom) panel represents the amplitude(width) of the dark-bright (bright) soliton component. The other parameters of panel (a) are γ = 1.0, δ = 1.0, panel (b) are γ = 1.0, g = 1.0 and panel (c) are δ = 1.0 and g = 1.0.
Our numerical results confirm the experimental observations that the energy detuning between the Raman beam and atoms really impacts SOC-BEC density distribution [4]. The detuning has an noneligible heating effect at zero detuning and finite detuning has contributed to suppress the effect [41]. To gain an intrinsic understanding of the effect of SOC-BEC on dark-bright vector solitons, we further investigate the relation of the amplitude and width of darkbright solitons with main parameters of the SOC-BEC system. We first study the effect of the repulsive interaction among atoms on the amplitude and width of dark-bright solitons. It is well known that the amplitude of bright soliton decreases and the width increases with the repulsive increasing interaction g in case of single component repulsive interaction in harmonic potential well. It can be understood in the following manner. The increasing interaction g means that repulsive interaction among atoms is enhanced and the final state need more space to build naturally. As can be seen in Fig. 4(a), the amplitude of bright soliton grows with the increasing of the interaction g, and the width decreases along with the increasing of the interaction g. This result shows that the bright soliton component in dark-bright solitons takes the unique properties with atoms interaction in SOC-BEC, which is totally different from single component repulsive interaction BEC in a harmonic trap. It causes the compressed concave density space of dark soliton, which leads to the decreasing width for dark-bright soliton and increasing height for the bright soliton component. As shown in Fig. 4, the amplitudes of dark solitons are almost invariant in these parameters region, except the amplitude increases slowly in Fig. 4 (b1) and (c1). However, the bright soliton amplitude decreases with the increasing of detuning and almost reach to zero(see Fig. 4 (b1). This means that the large detuning favors only one dark soliton state for the SOC-BEC system(see Fig. 3 (a)). The bright soliton amplitude becomes stronger when the SOC grows, as shown in Fig. 4 (c1). Our numerical results show that the width of dark soliton is almost equal to the one of bright soliton, and we can use the width of bright soliton to denote the width of the whole dark-bright soliton. The width is defined by full width at half maximum of corresponding bright solitons. The dark-bright width changes rapidly with the increasing of detuning and the width is decreased with the increasing of the SOC in Fig. 4 (b2) and (c2). Compared Fig. 4 (b) with (c), we can easily see that the influences of the detuning δ and SOC γ on the height
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Fig. 5. Typical density distribution of dark-bright solitons for different parameters region. These parameters in panel (a) are δ = 1.0 and γ = 1.0, panel (b) are δ = 1.0 and g = 1.0, and panel (c) are γ = 1.0 and g = 1.0.
and width of dark-bright solitons are completely opposite. It can be seen that typical density distribution of dark-bright solitons are shown in Fig. 5 with different parameters. In view of Fig. 4 (a) and (c), we conclude that the interaction parameter g and SOC γ in the SOC-BEC system have a similar effect on particle distribution between the dark and bright solitons. On the contrary, the detuning between the Raman beam and energy levels of the atoms has an opposite effect on the densities of the dark and bright solitons. These stable dark-bright soliton distributions are a balance results between the SOC effects, nonlinear repulsive interaction and the detuning, kinetic energy of atoms. For the literature in Ref.[28], they mainly focus on the dynamical properties of the dark-bright soliton in the reduced SOC-BECs. Especially, the dark soliton component power ratio almost stays the same value with the increasing SOC strength, while the bright soliton power ratio decreases with the increasing SOC after the long time evolution, see Fig. 3 in Ref.[28]. In our work, the final stationary state is derived by the imaginary time evolution method, see Fig. 4. We keep the particle number unchanged during the process. Moreover, the SOC item of Eq. (2) is nondiagonal term of Hamilton, which has an important effect on the final results. 5. The dynamics of two dark-bright solitons for different spin-orbit coupling parameters In this section, we study the dynamics of the two dark-bright solitons in the SOC-BEC system. The soliton dynamic properties are significantly different from those without spin orbit coupling system, where the fundamental properties can be hold in different parameters region. Due to the lack of the Galilean invariance in the SOC-BEC system, the density of dark-bright solitons takes the oscillation behaviors [12]. We numerically solve Eq. (2) using split-time-step Crank-Nicolson method in real time. We can first get a single dark-bright wave functions solution by imaginary time evolution method. Then we construct two dark-bright solitons by separation a distance of the two same single dark-bright solitons central as the initial state of Eq. (2) for imaginary time evolution. Another method of constructing two dark-bright solution is gotten by multiplying two dark solitons and adding two bright solitons centered at varying distances, then minimize the system energy
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ticle properties of solitons, a stable space-time soliton evolution is the consequence of balance between repulsive and attractive interaction. Therefore, the reducing SOC strength is equal to low the repulsive interaction between atoms, which can lead to the transition from repulsive interaction to attractive interaction. We also check the case of three dark-bright solitons interaction, which come to the similar conclusion. 6. Summary
Fig. 6. Dynamical evolution of two stationary dark-bright solitons for different parameter in SOC-BEC system. Panel (a) and (b) represent the density and phase of the dark-bright solitons, respectively. The SOC is γ = 0.85 in (a1) and (a2) and γ = 1.0 in (a3) and (a4). Other parameters are δ = 1.0 and g = 1.0.
numerically [40]. Using the two dark-bright solitons as initial state, we obtain the evolution results in real time domain. It is well known that the phase of convex density peak in bright soliton does not change, while the phase of the concave density in their notch has a non-trial jump in one dark-bright soliton for real time evolution when the SOC is not present. A dark-bright soliton maintains its shape and phase in the parameter region, no mixing dynamics occurred and the robust dark-bright soliton is highly stable. In contract, one can find that the density of dark-bright solitons will sharply oscillate if the background is not rest. The darkbright solitons interactions have been studied extensively in the literatures [44,45]. The interaction depends on the relative phase between the two bright solitons of the vector dark-bright solitons [44]. With the presentation of the SOC, we find that the spin-orbit coupled strength affects interaction types (attractive or repulsive) between the two dark-bright solitons in the case of in-phase. We turn to explore the interaction dynamics between the two dark-bright solitons. The interaction between the two dark-bright solitons can be attractive (In Fig. 6 (a1) and (a2)) or repulsive (see Fig. 6 (a3) and (a4)), which is totally decided by the strength of SOC in this parameter interval. When γ = 0.85 the solitons move inward in the beginning, collide each other and then attracted again. However, when the SOC increases slightly(γ = 1.0), the dark-bright solitons show the repulsive behavior. Repulsive dynamics is manifested between the two dark-bright solitons, which can be interpreted by one kind of constructive interference behavior, and this result is consisted with the literature of [40]. The case of attractive interaction between the two solitons in fact is the natural result of the reduced effective repulsive interaction. As mentioned above, the interaction g and SOC have similar effects on the width and amplitude of bright solitons, see Fig. 4 (a2) and (c2). It should also be pointed out that the top left corner in Fig. 6 (a3) is a result of dark-bright evolution close to the Neumann boundary. The dark soliton amplitudes in Fig. 6 (a) show in colorbar at top right panel, the bright amplitude is about one-tenth of the dark one. It should be pointed out that the single component excited states by the imaginary evolution method generally hold their shape when they are checked for the real time evolution. But the interaction between the different components exists attractive or repulsive behaviors for multi component solitons when these solutions evolve in real time domain [40,46]. Usually, the type of interaction is decided by the distance and phase between the multi component solitons [47]. Here we find the kind of interaction which is produced by the SOC strength. Due to the par-
We have investigated the dark-bright soliton properties of repulsive interacting bosons in one-dimensional SOC-BEC system. We mainly focus on the influence of the detuning and the SOC of two component solitons. The phase diagram shows there exist miscible or separated phases in the different SOC and energy detuning parameter regimes. Our results verify that the SOC favors miscibility and the energy detuning between the Raman beam and atom dominates separation phase of the Bose gases. Three kinds of solitons states (one dark soliton, dark-bright soliton, stripe soliton) are found in the proper detuning and SOC region. The dynamics of solitons show that the shape of individual components can be kept in long real time evolution. Especially, we find that the interaction between the two dark-bright solitons can be attractive or repulsive which is decided by the strength of SOC. Acknowledgements We thank Biao Wu for useful discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11304270, 11475144, 61504039 and 61774001), and the Key Project of Scientific and Technological Research in Hebei Province, China (Grant No. ZD2015133). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Phase diagram and dynamics of dark-bright vector solitons in spin-orbit-coupled Bose–Einstein condensate T.F. Xu a,b, W.L. Li a, Zai-Dong Li c,d,∗, C. Zhang e a
Hebei Key Laboratory of Microstructural Material Physics, School of Science, Yanshan University, Qinhuangdao 066004, China Department of Physics, University of California, San Diego, California 92093, USA c Department of Applied Physics, Hebei University of Technology, Tianjin 300401, China d Key Laboratory of Electronic Materials and Devices of Tianjin, School of Electronics and Information Engineering, Hebei University of Technology, Tianjin 300401, China e Henan Key Laboratory of Photovoltaic Materials and School of Physics and Electronics, Henan University, Kaifeng 475004, China b
a r t i c l e
i n f o
Article history: Received 17 January 2018 Revised 9 April 2018 Accepted 9 April 2018
Keywords: Spin-orbit coupling Vector soliton Bose–Einstein condensate Dark-bright soliton
a b s t r a c t We investigate the dynamics of dark-bright vector soliton solutions in a spin-orbit coupled Bose–Einstein condensate with repulsive interaction by the imaginary time evolution. The phase diagram is obtained numerically in spin-orbit coupled Bose–Einstein condensate. We find that the spin-orbit coupling favors miscibility, and the energy detuning between the Raman beam and atom dominates the separation phase of the Bose gases. We also find that the spin-orbit coupled strength affects interaction types (attractive or repulsive) between the two dark-bright solitons.
1. Introduction Spin-orbit coupling (SOC) is an interaction between spin with motion of particle, which plays a special role in many areas. The SOC for electrons in condensed matter system is particularly important for many condensed matter fields, such as spin Hall effect, topological insulator, spintronic devices, etc [1–3]. The experimental realization of synthetic SOC by the two photon Raman process in ultracold atom gases has offered an ideal platform to study novel properties of Bose–Einstein condenses (BECs) [4–6]. Meanwhile, the SOC effect in Bose system is the key factor for half vortices [7], monopoles [8], domain walls [9–11], bright solitons [12,13], dark solitons [14], gap solitons [15], and skyrmions [16] etc. Matter-wave vector solitons as the nonlinear state have attracted much attention in BECs with the tunable intra- and intercomponent repulsive(attractive) atomic interactions, which is expressed by the solution of coupled Schrödinger equation at the mean-field level. According to the type of short-range interaction, these matter wave solitons can be divided into two types, i.e., dark soliton and bright soliton. The former is characterized by a concave density and a non-trial phase jump across their density notch in repulsive interaction system, while a bright soliton with a con-
∗
Corresponding author. E-mail addresses: [email protected] (T.F. Xu), [email protected] (Z.-D. Li).
https://doi.org/10.1016/j.chaos.2018.04.014 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
© 2018 Elsevier Ltd. All rights reserved.
vex density peak against a negligible background in attractive interaction system [17–19]. Compared with scalar BECs, the multicomponent BECs enrich the ample novel nonlinear structures, such as bright-bright solitons [20–22], dark-dark solitons [23], darkbright solitons, which can be thought of as symbiotic solitons [24–27]. Recently, the dynamical properties of dark-bright solitons are carefully investigated by the multiscale expansion method in a reduced SOC-BEC system. The results show that the oscillator frequency of dark-bright solitons in a harmonic trap is close to the regular oscillator frequency without the SOC [28]. Due to the potential application in many fields, dark-bright solitons had attracted intensive attentions [17]. In this paper, we investigate dark-bright solitons in a quasi one dimensional SOC-BECs with repulsive interaction. In particular, we are interested in the effects of the SOC strength and Raman detuning of the system on the densities distribution of different component wave functions and others fundamental characteristic of dark-bright solitons. It will be shown that the SOC strength and the detuning between the Raman beam and energy states of the atoms have the opposite effects on the width of dark-bright solitons and the height of bright soliton amplitude. The paper is organized as follows. In Section 2, we introduce the model equation for dark-bright solitons in a SOC Bose system. Section 3 is devoted to compute the different phase distribution via numerical method for different detunings and SOC strengths. It is aimed to give a di-
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rect insight of the different state densities in an interval of parameters. In Section 4, the dark-bright solitons are calculated by numerical method for various parameter interval. In Section 5 we calculate the dynamic evolution of dark-bright soliton and interaction between two stationary dark-bright solitons for different SOC strengths. Section 6 is a brief summary. 2. Model equation We consider a quasi-one-dimensional SOC-BEC described by the coupled nonlinear Schrödinger equations [12]:
ih ¯
1 ∂ φ= ( px − h¯ ασy )2 + h¯ δσz − g (φ † · φ ) φ , ∂t 2m
(1)
where φ = (φ1 , φ2 )T , φ † · φ = |φ1 |2 + |φ2 |2 , φ 1 and φ 2 are the two component wave functions, m is the atom mass, σ is Pauli matrix, α represents the strength of SOC, is the coupling strength be tween Raman beams with atom levels, g is attractive atomic interactions. To find dark-bright solitons numerically, we start from the dimensionless coupled Schrödinger equations,
1 ∂2 ∂ ∂ 2 2 φD = − + δ + g ( | φ | + | φ | ) φD + γ φB , D B ∂t 2 ∂ x2 ∂x 1 ∂2 ∂ ∂ 2 2 i φB = − − δ + g ( | φ | + | φ | ) φB − γ φD . D B ∂t 2 ∂ x2 ∂x
i
(2)
where φ D and φ B are the dark or bright soliton solution, respectively, δ is the energy detuning between the Raman beam and the atoms, g is the interaction strength between atoms in the unit of Ng m/( h ¯ )/ h ¯ , γ represents the SOC strength in the unit of
α h¯ /m. The sign of γ , δ and g are positive for the whole paper. The scaling energy unit is , length is h ¯ /(m) and time is 1/. For the sake of simplicity the intra and inter interaction are set to be equal to g [14]. The normalization wave function is deter∞ mined via −∞ (|φD |2 + |φB |2 )dx = N. N is in direct proportion to the total atom numbers. The dark-bright solitons in the SOC-BECs can be experimentally achieved by the counterflow of two hyperfine states of 87 Rb. The physical parameters can be similar as those in the experiment. The SOC γ depends on the laser wave length and the relative angle between the two Raman beams; while the energy difference δ can be easily tuned by changing the relative frequency of the two lasers. The units of the length and time are 1.7 μm and 4 ms, respectively, the total atom number is N = 5 ∗ 104 level [4,27]. Eq. (2) are the starting point of the numerical calculations throughout this paper. By solving Eq. (2) numerically, one is able to obtain the darkbright solitons solution. To solve Eq. (2), we can first differentiate it by using split-time-step Crank–Nicolson method along with the homogeneous Neumann boundary condition, and then evaluate several thousands of steps in imaginary time until the invariable energy is reached. Every soliton(dark or bright) solution of Eq. (2) lives on a background which may be rest or highly oscillatory. Different from bright solitons cases, asymptotic behavior of dark soliton on the two sides of boundary is two different cases, that is, φ D → A ± eikx , where A+ and A− are not the same complex constants and k is a real constant. According to the value of k, the background of soliton is rest (k = 0) or oscillatory (k = 0). It is not an easy task to find a stable dark-bright soliton in the case of highly oscillatory background, so we set the rest background case for simplicity [29]. 3. Phase diagram in SOC Bose gas mixtures In this section, we study the dark-bright soliton solutions in the SOC Bose gas mixtures. The dark-bright solitons have been created experimentally by filling the dark state with different hyperfine
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states of 87 Rb atoms. The phase of dark soliton can be imprinted and transferred bright soliton state atoms into dark one by the two-photon resonance process simultaneously [26]. When the SOC effect is presented in the Bose system, more ample phenomena occur. Here we carefully checked the different solutions of Eq. (2) in diverse parameter regions. The numerical result shows there exist three kinds of soliton solutions, i.e., dark soliton states, darkbright solitons and stripe solitons, as shown in Fig. 1. We choose the wavefuction as the order of these phase transitions and use the properties of bright soliton solutions of Eq. (2) to distinguish the three different states of the system. When the amplitude of bright soliton of the dark-bright soliton solutions is less than certain value (e.g. 1% of the amplitude of dark soliton background), which is almost invisible, the state is defined as a single dark soliton. When the amplitude of bright soliton increases with the increasing SOC constant γ and detuning δ until the second peaks of the bright soliton of vector dark-bright solitons emerge, we call it as dark-bright soliton state. Otherwise, we define the states as stripe solitons, which have been intensively investigated in many literatures. Here we just focus the properties of vector dark-bright solitons in the SOC-BECs. In fact, the dark and dark-bright soliton states form the separation phase and stripe soliton is a mixture phase of SOC Bose gases. From Fig. 1 we observe that |φ D, B |2 evolves from one darkbright soliton distribution to many nodes one when the SOC constant γ is increased, which are typical strip solitons in 1D system [30–32] and are mixture states of different hyperfine states. Particularly the mixture states are different from the same component strip solitons, which are total bright or dark soliton [13,14,33]. However, we can find that the emergence condition of the stripe solitons in Fig. 1 is the same to the case of region II of Fig. 1 in Ref.[13] when regarding to the linear dispersion relation of Eq. (2). At this time, the lowest energy of the stripe solitons may have either a positive or negative momentum in the linear dispersion relation or they can be a linear superposition of both modes, thus forming the stripe phase. With the increasing of the detuning δ , the components |φ D, B |2 take a transformation from stripe soliton to one dark soliton state or dark-bright soliton, as shown in Fig. 1. This result shows that the SOC favors miscibility and the detuning between the Raman beam and atom energy levels dominates separation phase of the Bose gases. It should be pointed out that the presented phase diagram is different from the early result which mainly concentrate the different inter and intra interaction effect on the half-quantum vortex condensate or spin-spiral condensate in two dimension [34]. And it is not the same to another important result which focus on spin polarization and momentum as a function of Raman coupling constant and critical density [35]. The stripe phase locates at the bottom of phase diagram and the separation phase lies on the top of Fig. 1, which is consisted with the recently experimental result [36]. As the SOC γ and detuning δ are set to zero, Eq. (2) reduces to the integrable Manakov limit when the intra and inter interactions are equal to the same constant [37]. Kivshar et al. has studied the dark-dark soliton solutions of coupled nonlinear equation in the normal group-velocity dispersion region, which is equal to repulsive interaction between atoms in BEC system[38]. A general dark soliton solution may have different intensities for two polarization modes. At this time the solution can be written in the form
φD1 = U0 (cos θ1 tanh Z + i sin θ1 ) exp(i 1 ), φD2 = V0 (cos θ2 tanh Z + i sin θ2 ) exp(i 2 ).
(3)
where Z = U0 cos θ1 (x + t/W − τ0 ), 1 = k1 x + (U02 + V02 + k21 )t, 2 = k2 x + (U02 + V02 + k22 )t and the parameters U0 , V0 , W, k1 and k2 are coupled by the following relations: U0 cos θ1 = V0 cos θ2 , W −1 = U0 sin(θ1 + θ2 )/ cos θ2 + (k1 + k2 ), k2 − k1 = sin(θ1 − θ2 )/ cos θ2 . Here, φD1 and φD2 are the two component of dark solitons solu-
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Fig. 1. The SOC γ and detuning δ dependent phase diagram in the SOC Bose gas mixtures. The mark “o” corresponding to Manakov case, is used in Ref.[37] to discuss dark-dark solitons. The interaction strength is g = 1.0. The normalization constant is N = 5 for all the following case, in addition to Fig. 2.
tion without SOC effect version of Eq. (2) [38], k1 and k2 are the wave number of the background, θ 1 and θ 2 are the so-called dark solitons phase angle, −W −1 is the velocity. In order to compare the results obtained by the imaginary-time method with Kivshar’s analytic solutions, we calculate the darkdark soliton solutions numerically. For simplicity we set that the background of dark-dark solitons are rest (k1 = k2 = 0), the height of dark solitons are U0 = V0 = 1, the dark solitons phase angle are θ1 = θ2 = 0, the results are shown in Fig. 2. It should be mentioned that we only give the first component φD1 of the coupled equation solutions and the other soliton solution shares the same density mode. The two solutions are degenerated in this case. So, we only plot the first soliton component. It turns out that the analytic curve of |φD1 |2 agrees well with the soliton obtained by the imaginary-time method without the SOC effect [see Fig. 2]. Based on the excellent agreement between two methods, the imaginarytime method is mainly used for the following calculations. At the same time, we numerically solve the coupled Eq. (2) without the SOC effect items using split-time-step Crank–Nicolson method in real time. The results are shown in Fig. 2(b), the dynamic of the soliton is highly robust and it perfectly conserves its amplitude and phase. The amplitude of dark solitons propagates with changing slightly their shapes and emerges a little ripples when the SOC is considered, which we will discuss below. 4. Dark-bright solitons for SOC-BECs It is well known that the general solutions are dark solitons for the repulsive interaction in a single component Bose system when potential trap is absent. However, owing to the intra- and intercomponent repulsive interactions, BEC with multiple components admits dark-bright soliton solutions [39,40]. It is found that the inter-component repulsive interaction plays the important roles in the formation of excitation states for multiple component BEC. Dark-bright soliton only exists when an effective potential well is created by the dark soliton through the inter-component interaction; However, the density of bright solitons are not only decided by the effective potential well, but also influenced by these parameters: the detuning δ and the strength of SOC. In the following we choose a proper initial state, plug it into the coupled equation and finally get the three soliton solutions. As is shown in Fig. 3 (a), the ratio of bright soliton in total particle density is very small when the detuning is bigger (δ = 25), which can be regarded as total separation phase in SOC-BEC. The ampli-
Fig. 2. (a) The first component dark soliton solution without SOC. The solid lines are numerical method results, and the dotted lines are the analytic dark soliton density in Ref.[38]. (b) The real-time evolution result of the first dark solitons by the coupled nonlinear Schrödinger equation Eq. (2) without SOC effect.
Fig. 3. The three kinds states of the SOC-BEC in Fig. 1. Panel (a), (b) and (c) show the amplitudes of one dark solution, dark-bright soliton and stripe soliton respectively. Top(bottom) panel represents the first(second) component. The parameters is δ = 25, 5, 1 respectively, other parameters are γ = 2.5 and g = 1.0.
tudes of bright soliton become big with the decrease of the detuning (δ = 5), see Fig. 3 (b). At the same time, the density notch edge in dark soliton emerges some ripples. We also check the phase of dark soliton, which has no changes and still slip across their density notch. When the detuning further reduced (δ = 1), the effective potential well can not hold the bright solitons, the system in Eqs. (2) admits the stripe soliton solution and the particle density tends to the same value for the two components (Fig. 3 (c)). Different from gap stripe soliton in nonlinear periodic system [41], these stripe solitons are observed in the absence of external potential trap. It should be pointed out that the soliton amplitudes in Fig. 3 (a1)–(c1) are multiplied by a negative unit. It can not change the previous conclusion because the phase of bright soliton at its convex density peak in Fig. 3 (a1)–(c1) stay the same. We also find that the number of stripe soliton nodes grows with the increase of SOC γ for the same detuning δ . It is consistent with the result of bright-bright soliton solution in BEC with the SOC attractive interaction [12]. This also suggests that no-node theorem is excluded for dark-bright vector solitons in the spin-orbit coupled Bose-Einstein condensates with repulsive interaction [34,42,43].
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Fig. 4. The amplitude and width of these dark-bright solitons versus main parameters of the SOC-BEC system. Panel (a), (b) and (c) show the relation between the amplitudes, widths of the dark-bright solitons and the coupling strength g, the detuning δ and the SOC γ respectively. Top(bottom) panel represents the amplitude(width) of the dark-bright (bright) soliton component. The other parameters of panel (a) are γ = 1.0, δ = 1.0, panel (b) are γ = 1.0, g = 1.0 and panel (c) are δ = 1.0 and g = 1.0.
Our numerical results confirm the experimental observations that the energy detuning between the Raman beam and atoms really impacts SOC-BEC density distribution [4]. The detuning has an noneligible heating effect at zero detuning and finite detuning has contributed to suppress the effect [41]. To gain an intrinsic understanding of the effect of SOC-BEC on dark-bright vector solitons, we further investigate the relation of the amplitude and width of darkbright solitons with main parameters of the SOC-BEC system. We first study the effect of the repulsive interaction among atoms on the amplitude and width of dark-bright solitons. It is well known that the amplitude of bright soliton decreases and the width increases with the repulsive increasing interaction g in case of single component repulsive interaction in harmonic potential well. It can be understood in the following manner. The increasing interaction g means that repulsive interaction among atoms is enhanced and the final state need more space to build naturally. As can be seen in Fig. 4(a), the amplitude of bright soliton grows with the increasing of the interaction g, and the width decreases along with the increasing of the interaction g. This result shows that the bright soliton component in dark-bright solitons takes the unique properties with atoms interaction in SOC-BEC, which is totally different from single component repulsive interaction BEC in a harmonic trap. It causes the compressed concave density space of dark soliton, which leads to the decreasing width for dark-bright soliton and increasing height for the bright soliton component. As shown in Fig. 4, the amplitudes of dark solitons are almost invariant in these parameters region, except the amplitude increases slowly in Fig. 4 (b1) and (c1). However, the bright soliton amplitude decreases with the increasing of detuning and almost reach to zero(see Fig. 4 (b1). This means that the large detuning favors only one dark soliton state for the SOC-BEC system(see Fig. 3 (a)). The bright soliton amplitude becomes stronger when the SOC grows, as shown in Fig. 4 (c1). Our numerical results show that the width of dark soliton is almost equal to the one of bright soliton, and we can use the width of bright soliton to denote the width of the whole dark-bright soliton. The width is defined by full width at half maximum of corresponding bright solitons. The dark-bright width changes rapidly with the increasing of detuning and the width is decreased with the increasing of the SOC in Fig. 4 (b2) and (c2). Compared Fig. 4 (b) with (c), we can easily see that the influences of the detuning δ and SOC γ on the height
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Fig. 5. Typical density distribution of dark-bright solitons for different parameters region. These parameters in panel (a) are δ = 1.0 and γ = 1.0, panel (b) are δ = 1.0 and g = 1.0, and panel (c) are γ = 1.0 and g = 1.0.
and width of dark-bright solitons are completely opposite. It can be seen that typical density distribution of dark-bright solitons are shown in Fig. 5 with different parameters. In view of Fig. 4 (a) and (c), we conclude that the interaction parameter g and SOC γ in the SOC-BEC system have a similar effect on particle distribution between the dark and bright solitons. On the contrary, the detuning between the Raman beam and energy levels of the atoms has an opposite effect on the densities of the dark and bright solitons. These stable dark-bright soliton distributions are a balance results between the SOC effects, nonlinear repulsive interaction and the detuning, kinetic energy of atoms. For the literature in Ref.[28], they mainly focus on the dynamical properties of the dark-bright soliton in the reduced SOC-BECs. Especially, the dark soliton component power ratio almost stays the same value with the increasing SOC strength, while the bright soliton power ratio decreases with the increasing SOC after the long time evolution, see Fig. 3 in Ref.[28]. In our work, the final stationary state is derived by the imaginary time evolution method, see Fig. 4. We keep the particle number unchanged during the process. Moreover, the SOC item of Eq. (2) is nondiagonal term of Hamilton, which has an important effect on the final results. 5. The dynamics of two dark-bright solitons for different spin-orbit coupling parameters In this section, we study the dynamics of the two dark-bright solitons in the SOC-BEC system. The soliton dynamic properties are significantly different from those without spin orbit coupling system, where the fundamental properties can be hold in different parameters region. Due to the lack of the Galilean invariance in the SOC-BEC system, the density of dark-bright solitons takes the oscillation behaviors [12]. We numerically solve Eq. (2) using split-time-step Crank-Nicolson method in real time. We can first get a single dark-bright wave functions solution by imaginary time evolution method. Then we construct two dark-bright solitons by separation a distance of the two same single dark-bright solitons central as the initial state of Eq. (2) for imaginary time evolution. Another method of constructing two dark-bright solution is gotten by multiplying two dark solitons and adding two bright solitons centered at varying distances, then minimize the system energy
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ticle properties of solitons, a stable space-time soliton evolution is the consequence of balance between repulsive and attractive interaction. Therefore, the reducing SOC strength is equal to low the repulsive interaction between atoms, which can lead to the transition from repulsive interaction to attractive interaction. We also check the case of three dark-bright solitons interaction, which come to the similar conclusion. 6. Summary
Fig. 6. Dynamical evolution of two stationary dark-bright solitons for different parameter in SOC-BEC system. Panel (a) and (b) represent the density and phase of the dark-bright solitons, respectively. The SOC is γ = 0.85 in (a1) and (a2) and γ = 1.0 in (a3) and (a4). Other parameters are δ = 1.0 and g = 1.0.
numerically [40]. Using the two dark-bright solitons as initial state, we obtain the evolution results in real time domain. It is well known that the phase of convex density peak in bright soliton does not change, while the phase of the concave density in their notch has a non-trial jump in one dark-bright soliton for real time evolution when the SOC is not present. A dark-bright soliton maintains its shape and phase in the parameter region, no mixing dynamics occurred and the robust dark-bright soliton is highly stable. In contract, one can find that the density of dark-bright solitons will sharply oscillate if the background is not rest. The darkbright solitons interactions have been studied extensively in the literatures [44,45]. The interaction depends on the relative phase between the two bright solitons of the vector dark-bright solitons [44]. With the presentation of the SOC, we find that the spin-orbit coupled strength affects interaction types (attractive or repulsive) between the two dark-bright solitons in the case of in-phase. We turn to explore the interaction dynamics between the two dark-bright solitons. The interaction between the two dark-bright solitons can be attractive (In Fig. 6 (a1) and (a2)) or repulsive (see Fig. 6 (a3) and (a4)), which is totally decided by the strength of SOC in this parameter interval. When γ = 0.85 the solitons move inward in the beginning, collide each other and then attracted again. However, when the SOC increases slightly(γ = 1.0), the dark-bright solitons show the repulsive behavior. Repulsive dynamics is manifested between the two dark-bright solitons, which can be interpreted by one kind of constructive interference behavior, and this result is consisted with the literature of [40]. The case of attractive interaction between the two solitons in fact is the natural result of the reduced effective repulsive interaction. As mentioned above, the interaction g and SOC have similar effects on the width and amplitude of bright solitons, see Fig. 4 (a2) and (c2). It should also be pointed out that the top left corner in Fig. 6 (a3) is a result of dark-bright evolution close to the Neumann boundary. The dark soliton amplitudes in Fig. 6 (a) show in colorbar at top right panel, the bright amplitude is about one-tenth of the dark one. It should be pointed out that the single component excited states by the imaginary evolution method generally hold their shape when they are checked for the real time evolution. But the interaction between the different components exists attractive or repulsive behaviors for multi component solitons when these solutions evolve in real time domain [40,46]. Usually, the type of interaction is decided by the distance and phase between the multi component solitons [47]. Here we find the kind of interaction which is produced by the SOC strength. Due to the par-
We have investigated the dark-bright soliton properties of repulsive interacting bosons in one-dimensional SOC-BEC system. We mainly focus on the influence of the detuning and the SOC of two component solitons. The phase diagram shows there exist miscible or separated phases in the different SOC and energy detuning parameter regimes. Our results verify that the SOC favors miscibility and the energy detuning between the Raman beam and atom dominates separation phase of the Bose gases. Three kinds of solitons states (one dark soliton, dark-bright soliton, stripe soliton) are found in the proper detuning and SOC region. The dynamics of solitons show that the shape of individual components can be kept in long real time evolution. Especially, we find that the interaction between the two dark-bright solitons can be attractive or repulsive which is decided by the strength of SOC. Acknowledgements We thank Biao Wu for useful discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11304270, 11475144, 61504039 and 61774001), and the Key Project of Scientific and Technological Research in Hebei Province, China (Grant No. ZD2015133). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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