Emergence of unstable avoided crossing in the collective excitations of spin-1 spin-orbit-coupled Bose-Einstein condensates

We present analytical and numerical results on the collective excitation spectrum of quasi-one-dimensional spin-orbit (SO)-coupled spin-1 spinor ferromagnetic Bose-Einstein condensates. The collective excitation spectrum, using Bogoliubov–de Gennes theory, reveals the existence of a diverse range of phases in the SO-coupling and Rabi coupling $\left(k_L\text{−}\mathrm{\Omega }\right)$ planes. Based on the nature of the eigenvalue of the excitation spectrum, we categorize the $k_L\text{−}\mathrm{\Omega }$ plane into three distinct regions, namely, I, II (IIa and IIb), and III. In region I, a stable mode with phononlike excitations is observed. In region IIa, single- and multiple-band instabilities are noted with a gapped mode, while multiband instability accompanied by a mode corresponding to no gap between low-lying and first-excited states is realized in region IIb, which also provides evidence of unstable avoided crossing between low-lying and first-excited modes, responsible for the $I_o$ type of oscillatory nonequilibrium dynamical pattern formation. The gap between low-lying and first-excited states increases upon increasing the Rabi coupling and decreases upon increasing the SO coupling. Using eigenvector analysis, we confirm the presence of the spin-dipole mode in the spinlike modes in region II. We corroborate the nature of the collective excitation through real-time dynamical evolution of the ground state perturbed with the quench of the trap using the mean-field Gross-Pitaevskii model. This analysis suggests the presence of dynamical instability leading to the disappearance of the zeroth component of the condensate. In region III, mainly encompassing $\mathrm{\Omega }\sim 0$ and finite $k_L$, we observe phononlike excitations in both the first-excited and the low-lying state. The eigenvectors in this region reveal alternative in- and out-of-phase behaviors of the spin components. Numerical analysis reveals the presence of a superstripe phase for small Rabi coupling in this region, wherein the eigenvector indicates the presence of more complicated spinlike-density mixed modes.

Figure 1

Single-particle energy spectrum in momentum space for different sets of SO- and Rabi-coupling strengths $(k_L,\mathrm{\Omega })$: (a) $(0,0)$, (b) $(0.7,0)$, (c) $(0,0.5)$, (d) $(0.7,0.5)$, (e) $(0.7,0.7)$, and (f) $(1.0,0.5)$. The thick red solid, green dash-dotted, and thin blue solid lines represent the energy eigenspectrum for the $\{-1,0,+1\}$ components of the spin, respectively. Varying the coupling parameters leads to the changes in the eigenspectrum of different components, that is, ${\omega }_{-}$ has only the lowest minimum for $\mathrm{\Omega }>k_L^2$, whereas for $\mathrm{\Omega }<k_L^2$ the ${\omega }_{-}$ exhibits double minima of the plane-wave phase, indicating the presence of the stripe phase.

Figure 2

Stability phase diagram in the $k_L\text{−}\mathrm{\Omega }$ plane for the interaction parameters $c_0=0.5$ and $c_2=-0.1$, which are considered for all the simulation runs. Based on the different characteristics of the eigenspectrum and ground state, the phase diagram is divided into regions I, IIa, IIb, and III. While region I is stable, regions IIa, IIb, and III are unstable. The white dotted line with green circles represents the $\mathrm{\Omega }=k_L^2$ curve that separates region I from IIa. The white solid line with blue circles represents $\mathrm{\Omega }\approx 0.1365k_L^2-0.0686$, which separates regions IIa and IIb. The horizontal line with rectangles for $\mathrm{\Omega }\sim 0$ denotes the region III after the cutoff, $k_L^c=0.68$, which is a tricritical point for regions IIa, IIb, and III, indicated by a red triangle.

Figure 3

(a) Excitation spectrum and (b) and (c) eigenvectors for the coupling parameters $k_L=0.5$ and $\mathrm{\Omega }=2.0$. (a) The magenta solid line represents $\mathrm{Re}\left({\omega }_{-}\right)$ and the green dash-dotted line represents $|\mathrm{Im}\left({\omega }_{-}\right)|$. Here solid and dash-dotted lines are the analytical results of the BdG equation (12) and open circles are numerical results obtained by solving Eq. (9). (b) and (c) Eigenvectors corresponding to the eigenspectrum, obtained by solving Eq. (9) numerically: $|u_{+1}{|}^2$ (open red circles), $|u_{-1}{|}^2$ (closed black diamonds), $|u_0{|}^2$ (open magenta hexagons), $|v_0{|}^2$ (open cyan stars), $|v_{+1}{|}^2$ (open green squares), and $|v_{-1}{|}^2$ (closed magenta triangles). The presence of the phonon mode in the eigenspectrum is shown in (a). It has no negative or complex eigenfrequency and thus it is energetically and dynamically stable.

Figure 4

Time evolution of ground-state density profiles illustrating dynamical stability of the (a) $+1$, (b) 0, and (c) $-1$ components of the condensate, with the parameters similar to those in Fig. 3. All three components $|{\psi }_{+1}{|}^2$, $|{\psi }_0{|}^2$, and $|{\psi }_{-1}{|}^2$ depict stable breatherlike dynamics, which validates the dynamical stability of the condensate.

Figure 5

Energy variation in the time evolution for the stable region I. The parameters are similar to those in Fig. 3. During time evolution, the energy of the condensate decreases in the beginning and further shows stable behavior, which signifies that the condensate is energetically stable due to the lack of negative eigenfrequency.

Figure 6

Stability excitation spectrum eigenvalues and eigenvectors. The SO- and Rabi-coupling strengths are (a i), (b i), and (c i) $(k_L,\mathrm{\Omega })=(2.0,2.0)$ and (a ii), (b ii), and (c ii) $(2.35,4.0)$. (a) Eigenenergy and (b) and (c) corresponding eigenvectors. The lines and symbols have the same denotations as in Fig. 3. The eigenenergy spectrum shows the presence of complex eigenfrequencies in terms of (a i) two bands and (a ii) one band, which indicates the system is dynamically unstable. The eigenvectors display the spin dipole along with the spinlike mode in the $q_x$ momentum direction and are symmetric along the axis.

Figure 7

Temporal evolution of the ground-state density profile of (a) and (d) $|{\psi }_{+1}{|}^2$, (b) and (e) $|{\psi }_0{|}^2$, and (c) and (f) $|{\psi }_{-1}{|}^2$ depicted for different $(k_L,\mathrm{\Omega })$: (a)–(c) $(2.0,2.0)$ and (d)–(f) $(2.35,4.0)$. The density profile shows stripe wave behavior in both cases for (a)–(c) $t<150$ and (d)–(f) $t<180$. Further, it gets fragmented into several small domains. This behavior signifies the change in shape and amplitude of the density profile, where in (d)–(f) the zeroth component loses density while the $\pm 1$ components gain density.

Figure 8

(a) and (b) Total energy of the condensate for the density evolution in the upper and lower panel of Fig. 7 respectively. In both cases the energy of the condensate increases at the beginning of the dynamical evolution, but finally shows well-settled stable behavior; such a sudden change happens due to the external perturbation. This behavior validates the energetic stability of the condensate; however, the system is dynamically unstable.

Figure 9

Excitation spectrum for the coupling strengths (a) $(k_L,\mathrm{\Omega })=(3.1,1.17)$ and (b) $(k_L,\mathrm{\Omega })=(5.0,2.5)$ depicting the low-lying and first-excited branches of the spectrum, respectively. The magenta solid line represents $\mathrm{Re}\left({\omega }_{-}\right)$ and the green dashed line represents $\left|\mathrm{Im}\right({\omega }_{-}\left)\right|$, which is the low-lying branch of the spectrum, while the blue dash-dotted line represents $\mathrm{Re}\left({\omega }_{+}\right)$ and the black dotted line represents $\left|\mathrm{Im}\right({\omega }_{+}\left)\right|$, corresponding to the first-excited branch of the spectrum, which is obtained from Eq. (12). The eigenspectrum in (a) and (b) shows multiple instability bands as we obtained in Fig. 6(a i); however, (a) and (b) show one and two modes, corresponding to no gap existing between the lower branch and the first-excited branch. Insets show close-ups of the no-gap mode between the lower and first-excited state branches. The $y$-axis scale on the right represents the magnitudes of the imaginary eigenfrequencies.

Figure 10

Eigenvectors in (a) and (b) corresponding to Figs. 9 and 9, respectively: $|u_{+1}{|}_{ll}^2$ (closed red circles), $|u_{-1}{|}_{ll}^2$ (open black hexagons), $|u_0{|}_{ll}^2$ (open magenta hexagons), $|v_0{|}_{ll}^2$ (open cyan stars), $|v_{+1}{|}_{ll}^2$ (open green squares), $|v_{-1}{|}_{ll}^2$ (closed magenta triangles), $|u_{+1}{|}_{fe}^2$ (open blue pentagons), $|u_{-1}{|}_{fe}^2$ (closed orange stars), $|u_0{|}_{fe}^2$ (open olive pluses), $|v_0{|}_{fe}^2$ (open red crosses on dotted lines), $|v_{+1}{|}_{fe}^2$ (open cyan diamonds), and $|v_{-1}{|}_{fe}^2$ (closed maroon circles). For $q_x$ the eigenfrequencies and eigenvectors are symmetric about the axis. Here we find that the eigenvectors show the spinlike mode when the complex frequency exhibits the densitylike mode only when it is real.

Figure 11

Dynamics of the ground-state density profile of (a) and (d) $|{\psi }_{+1}{|}^2$, (b) and (e) $|{\psi }_0{|}^2$, and (c) and (f) $|{\psi }_{-1}{|}^2$ components for different $(k_L,\mathrm{\Omega })$: (a)–(c) $(3.1,1.17)$ and (d)–(f) $(5.0,2.5)$. The density profile has stripe wave behavior in both cases for (a)–(c) $t<100$ units and (d)–(f) $t<50$ units. For $t>100$, (a)–(c) show nonperiodic oscillation and nonuniform density in space-time, whereas the zeroth component disappears. For $t>50$, (d)–(f) show that the density profile further fragments into several small domains. This behavior signifies the change in shape and amplitude of the density profile, along with the disappearance of the zeroth component, which confirms the dynamical instability.

Figure 12

Total energies of the condensate for the dynamical evolution from Fig. 11. From the energy saturation, we understand that the system is energetically stable but dynamically unstable.

Figure 13

(a) Low-lying and (b) first-excited branches of the excitation spectrum for the coupling strengths $k_L=5.0$ and $\mathrm{\Omega }=0.0$. The lines and symbols have the same denotations as in Fig. 9. Insets show close-ups of the complex frequency modes. The mode corresponding to no gap between the low-lying- and first-excited-state branches appears along with the symmetric roton mode.

Figure 14

Eigenvectors in (a) and (b) corresponding to Figs. 13 and 13, respectively. The lines and symbols have the same denotations as in Fig. 10. In this region, both the low-lying and first-excited states assume the value $\omega =0$ for $q_x=0$ with eigenvectors that are in phase with the low-lying and first-excited states, which indicates the existence of the phonon mode for these branches.

Figure 15

Time evolution of superstripe ground-state density profiles of the (a) $|{\psi }_{+1}{|}^2$, (b) $|{\psi }_0{|}^2$, and (c) $|{\psi }_{-1}{|}^2$ components of the condensate. The coupling strengths are $k_L=5.0$ and $\mathrm{\Omega }=0.0$ with interaction parameters $c_0=0.5$ and $c_2=-0.1$. Throughout the dynamics, the unpolarized superstripe wave holds its behavior for $t<30$. Further, it breaks into two domains and oscillates on either side of the condensates of $\pm 1$ components. The zeroth component disappears, where $\pm 1$ components gain density, which shows the change in the density profile confirms the dynamical instability.

Figure 16

Variation of the band gap ${\mathrm{\Delta }}_g$ between negative and positive branches of the spectrum for (a i) the fixed value SO coupling strength upon variation of the Rabi-coupling strength and (b i) vice versa. (a i) The green dash-dotted line and the red dashed line represent gap calculations for $k_L=1.14$ and $3.10$, respectively. (b i) The blue dash-dotted line and the black dashed line represent gap calculations for $\mathrm{\Omega }=1.3$ and $3.1$, respectively. The interaction parameter strengths are $c_0=0.5$ and $c_2=-0.1$. (a ii) and (b ii) Logarithmic scale plots corresponding to (a i) and (b i), respectively. In (a ii) and (b ii) the maroon (orange) dotted lines represent the fitting curve ${\mathrm{\Delta }}_g=a{\mathrm{\Omega }}^b$ with the parameters $a=0.9811$ and $b=0.9974$ ($a=1.0825$ and $b=1.0258$) and $a=126.2992$ and $b=-4.9568$ ($a=46.4438$ and $b=-2.2146$), respectively.

Figure 17

Phase diagram in the $k_L\text{−}\mathrm{\Omega }$ plane for interaction parameters (a) $c_0=5.0$ and $c_2=-0.1$ and (b) $c_0=885.72$ and $c_2=-4.09$. Based on the different characteristics of the eigenspectrum and ground state, the phase diagram is divided into regions I, IIa, IIb, and III. Region I is stable, while regions IIa and IIb are unstable. The line $\mathrm{\Omega }=k_L^2$ is drawn, separating region I from II. Blue dots separate regions IIa and IIb. The horizontal line with $\mathrm{\Omega }\sim 0$ denotes region III. The behavior of all the regions remains the same as for $c_0=0.5$ and $c_2=-0.1$.

Figure 18

Spin-magnetization density components $(m_x,m_y,m_z)$ for the ground state at initial time $t=0$ (solid line) and later time $t=1000$ (dashed line) in the dynamically unstable region IIa in the $k_L\text{−}\mathrm{\Omega }$ plane for different sets of coupling parameters of the condensate with ferromagnetic interactions $c_0=0.5$ and $c_2=-0.1$: (a i), (b i), and (c i) $k_L=2.0$ and $\mathrm{\Omega }=2.0$ and (a ii), (b ii), and (c ii) $k_L=2.35$ and $\mathrm{\Omega }=4.0$. Owing to the dynamical instability, the spin texture deviates from its initial state during the time evolution.

Figure 19

Spatial profile of the spin-magnetization density components $(m_x,m_y,m_z)$ for the ground state at initial time $t=0$ (solid line) and later time $t=1000$ (dashed line) in the dynamically unstable region IIb in the $k_L\text{−}\mathrm{\Omega }$ plane for different sets of coupling parameters of the condensate with ferromagnetic interactions $c_0=0.5$ and $c_2=-0.1$: (a i), (b i), and (c i) $k_L=3.1$ and $\mathrm{\Omega }=1.17$ and (a ii), (b ii), and (c ii) $k_L=5.0$ and $\mathrm{\Omega }=2.5$. Owing to the dynamical instability, spin texture shows a deviation from the initial behavior during the time evolution.