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Abstract
We investigate the ground states of a dipolar Bose-Einstein condensate (BEC) subject to Raman laser induced spin-orbit coupling with mean-field theory. Owing to the interplay between spin-orbit coupling and atom-atom interactions, the BEC presents remarkable self-organization behavior and thus hosts various exotic phases including vortex with discrete rotational symmetry, stripe with spin helix, and chiral lattices with C4 symmetry. The peculiar chiral self-organized array of square lattice, which spontaneously breaks both U(1) and rotational symmetries, is observed when the contact interaction is considerable in comparison with the spin-orbit coupling. Moreover, we show that the Raman-induced spin-orbit coupling plays a crucial role in forming rich topological spin textures of the chiral self-organized phases by introducing a channel for atoms to turn on spin flipping between two components. The self-organization phenomena predicted here feature topology owing to spin-orbit coupling. In addition, we find long-lived metastable self-organized arrays with C6 symmetry in the case of strong spin-orbit coupling. We also present a proposal to observe these predicted phases in ultracold atomic dipolar gases with laser-induced spin-orbit coupling, which may stimulate broad theoretical as well as experimental interest.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Self-organization is a phenomenon of spontaneously ordered structure omnipresent in biological systems [1], synthetic materials [2], and atomic physics [3]. As the development of techniques in cold atoms and quantum optics, the light-atom interacting systems have been a promising platform to explore various self-organization phenomena, e.g., the light induced self-ordering regular pattern of atomic gas coupled to a cavity [4,5] is proposed as a typical route to realize a wealth of interesting effects of atoms [6] and optics [7,8]. These unexpected self-organized behaviors of atoms based on the exquisite control of light-atom interactions have been attracting broad interests to study spontaneous symmetry-breaking as well as phase transitions [9,10]. Especially, the system of Bose-Einstein condensate (BEC) coupled to the radiation field provides a new medium to realize exotic phases or study mechanisms of quantum optics, such as Dicke model [9,11], superradiant phase [12–15], multimode cavity quantum electrodynamics [16,17], etc. From the perspective of quantum optics, cold atomic systems can provide an ideal platform to develop all-optical control of symmetry breaking at the single-photon level [18], and achieve very low thresholds above which optomechanical instability and self-organizations can occur [19,20].
In cold atoms, the spontaneous emergence of spatial structures due to light-atom interaction has been predicted in optical cavities [21–24] or optical nonlinearity mediums [18,25]. Ultracold atomic gases confined in laser-induced traps are ideal systems to observe novel quantum phenomena owing to the highly controllable parameters, such as tunable contact interaction strength via Feshbach resonance, variable external potential via optical or magnetic trap, etc. Especially, Bose-Einstein condensates of atoms offer appealing settings to study such phenomena of self-organized quantum phases as the confinements and interactions between atoms can be accurately controlled by external fields or lasers [26]. The realizations of spin-orbit (SO) couplings [27,28] and dipole-dipole interactions (DDIs) [29] in BECs provide new freedoms to investigate exotic quantum phenomena related to gauge fields and long-range interactions in BECs, while the exploration of novel topological states of matter have been a long-thought goal in the research of quantum many-body physics.
Recent studies of SO coupling and dipolar interaction in ultracold atoms have shown a variety of topological quantum phases from the view of symmetry breaking and phase transition [30–33]. Topological properties of these phases, such as quantized topological charges [34] and topological bands [28], may play important roles in nontrivial topological materials which are appealing for potential applications in spintronics [35] and quantum computations [36,37]. Spin-orbit coupled BEC, owing to inversion symmetry breaking, presents a promising scenario for a variety of research in condensate physics and exotic materials [32]. Additionally, the long-range dipole-dipole interaction, e.g., of magnetic atoms, has the Legendre polynomial of second-order $P_2(\cos {\theta })$, i.e., d-wave angular symmetry, which may lead to unexpected properties of trapped spinor BEC. In the absence of an external magnetic field, the spinor BEC is rotationally invariant in spin space, while the DDI breaks this symmetry and thus induces novel quantum phases by tuning the effective strength of the DDI via a modification of the trapping geometry [29,38–48]. Previous works mainly focused on the crystallization of droplets solely induced by the dipole-dipole interactions in BECs [49]. With the new ingredient of SO coupling, an intriguing question is, does there exist self-organized pattern (array) with nontrivial topology in a quantum gas with dipole-dipole interaction? On one hand, it possesses topological structure which is different from the self-organization induced in usual BECs. On the other hand, it features superfluidity distinguished from the chiral self-organization realized in condensed matter systems. Here we propose a protocol to observe quantum topological self-organization in SO coupled dipolar BEC [50], which is feasible to implement experimentally. Remarkably, we can also access a long-lived metastable state, which exhibits distinct geometry and topology in comparison with the ground states, by tuning the strength of contact interaction.
In this work, we consider a pseudospin-$1/2$ SO coupled system with both contact and dipolar interactions. Such setting can be implemented with laser-coupling states of highly magnetic (or electric) atoms such as Dy, Er, or Cr, BECs of which have been realized recently [51]. By employing mean-field theory, we explore the ground state phases of such 2D harmonically trapped BEC and find three types of quantum phases. Depending on the magnetic properties of each phase, the ground states appear distinct spin textures as well as topological characteristics.
2. Theoretical model
The spin-orbit coupled dipolar atoms can be realized by engineering an ensemble of $^{52}\textrm {Cr}$ atoms illuminated by a pair of Raman beams which propagate along two directions with a fixed angle (see Fig. 1) [48]. In this setting, the condensate is tightly confined along its polarization direction and thus can be considered as a quasi-2D system. Here we focus on the Rashba-type SO coupling, which can be induced and tuned by varying the relative angle of Raman beams [52].
The ultracold atomic gas of $^{7}S_3$ ground-state manifold is trapped in a crossed optical dipole trap [48,53,54]. In this setup, a laser-induced quadratic Zeeman shift $\Delta E_{QZS}=\alpha m_J^2$ [55] is employed to extract two spin states $|1\rangle =|J=3, m_J=-1\rangle$, and $|2\rangle =|3, 0\rangle$ with quadratic Zeeman strength $\alpha =p$ ($p$ is the strength of linear Zeeman shift [56]) when the external bias magnetic field $B_0=7$ $\mathrm{mG}$ [55]. In the crossed dipole trap with a waist of $30$ $ {\mathrm{\mu} \mathrm{m}}$ formed by a $20$ $\mathrm{W}$ fiber laser at $1064$ $\mathrm{nm}$, the homogeneous magnetic field $B_0$ can generate a total Zeeman shift $\Delta \omega _z/2\pi \approx 39.2$ $\mathrm{kHz}$ between $|3, 0\rangle$ and $|3, 1\rangle$ [55,57]. Hence, one can prepare the cold atoms on these two lowest levels and then achieve a pseudospin-$1/2$ SO coupled system [27] with tunable Rashba SO coupling $\mathcal {H}_I=\kappa (p_x\hat {\sigma }_x+p_y\hat {\sigma }_y)$.
Fig. 1. Schematic of energy levels of $^{52}$Cr for creating 2D SO coupling. (a) Hyperfine energy levels of $^{52}$Cr and the corresponding couplings with optical fields relevant for cooling and optical pumping [53]. Three $\sigma ^+$-$\sigma ^-$ combined polarized lights (colored lines with arrows) are used to drive the atoms to the lowest two states and circumvent dipolar relaxation [53,54]. Here, different colored lines represent lights with different frequencies, and we suppose that the laser linewidth are narrow enough to distinguish different transitions from states of $^{7}S_3$ to $^{7}P_3$ (e.g., the green colored laser can only drive the transition from state $|J=3, m_J=-3\rangle$ or $|J=3, m_J=2\rangle$ of $^{7}S_3$, to $|J=3, m_J=-2\rangle$ or $|J=3, m_J=1\rangle$ of $^{7}P_3$, respectively). (b) Details of the optical pumping. A bias magnetic field $B_0\mathbf {e}_z$ induces linear Zeeman shifts between the sublevels of $^{52}$Cr atoms. The quadratic Zeeman shift ($\Delta E_{QZS}=\alpha m_J^2$) is induced by the light field of the optical trap. Two lowest spin states $|1\rangle =|J=3, m_J=-1\rangle$, $|2\rangle =|3, 0\rangle$ are selected as the two spin components of the binary BECs considered in this work by tuning the strength of linear and quadratic Zeeman shifts to be balanced (i.e., $\alpha =p$). The $\sigma ^-$ polarized light (blue line with the arrow) is used to drive the atoms to the lowest states and circumvent dipolar relaxation [53].
Consequently, the single particle and interaction Hamiltonian of this system $\hat {\mathcal {H}}=\hat {\mathcal {H}}_s+\hat {\mathcal {H}}_{int}$ reads
Next, we implement the mean-field approximation for the binary BEC, in which the ground-state of the condensate is described by two-component order parameter $\Psi =(\Psi _1,\Psi _2)^T$ subject to the normalization condition $\int \left (|\Psi _1|^2+|\Psi _2|^2\right )d\mathbf {r_\perp }=1$. For a cloud of $^{52}\textrm {Cr}$ atoms polarized along the $z$-axis by a strong magnetic field whose relative position between two atoms is $r=|\mathbf {r}-\mathbf {r}'|$, the mean-field dipole-dipole interaction energy within the Hilbert space spanned by $\{|\Psi _1\rangle, |\Psi _2\rangle \}$ is [40,59–61]
Here we consider a cloud of magnetic $^{52}\textrm {Cr}$ gases with Rashba SO coupling confined in a two-dimensional harmonic trap, and assume the strength of intra-component contact interaction $g_{11}=g_{22}=g$ and the atoms of the second component have vanishing dipolar interaction, thus the ground state can be obtained through minimization of the following mean-field energy [26]
At $g=g_{12}$, the symmetry (either parity symmetry or combined parity-time-reversal symmetry) of the ground state, which is obtained through minimizing the energy in Eq. (5), can not be determined within mean-field theory in the absence of dipolar interaction ($\varepsilon _{dd}=0$) [64,65]. However, we will prove that it can be determined by the dipolar interaction term when $\varepsilon _{dd}\neq 0$. Let us consider an eigenstate of parity operator of $\mathcal {P}$ with wave function $\Psi _P=[\Psi _1,\Psi _2]^T$, where $\mathcal {P}=\hat {\sigma }_z\mathcal {I}$ with $\mathcal {I}$ the spatial inversion operator [64]. The corresponding eigenstate of $\mathcal {PT}$ operator is assumed to be $\Psi _{PT}=\left (\Psi _P+\mathcal {T}\Psi _P\right )/\sqrt {2}$, where $\mathcal {T}=i\hat {\sigma }_y\mathcal {C}$ with $\mathcal {C}$ the complex conjugate operator. The mean-field energy difference between these two states is given by calculating Eq. (5) as
where $\Delta _d=\mathrm{Re}\{\Psi _1\Psi _2\}-|\Psi _1|^2/2$. Generally speaking, since $\Delta _d\neq 0$, the ground state will be a $\mathcal {P}$ eigenstate if $\Delta E>0$, or a $\mathcal {PT}$ eigenstate if $\Delta E<0$. In most cases, a stable ground state pattern with a determined symmetry will be observed when the contact interaction varies. Nevertheless, our numerical simulation also finds some nearly-degenerate metastable states at some points in parameter space that have similar energies compared to that of the ground states when $\Delta E\approx 0$, as discussed below. In these cases, the metastable states can be eliminated and the wave functions obtained through minimizing the energy in Eq. (5) converge back to the real ground states by imposing small perturbations. For the metastable states, wave functions converge back to the real stable ground states which have lower energy when we prepare proper initial wave functions.3. Results and discussion
By numerically minimizing the energy $\mathcal {E}[\Psi _1,\Psi _2]$ in Eq. (5), we will show the ground-states and phase transitions in the $^{52}\textrm {Cr}$ BEC when $g=g_{12}$ as well as the ratio between dipolar and contact interaction $\varepsilon _{dd}$ is fixed for the cases of weak and strong SO couplings, respectively.
3.1 Vortex with discrete rotational symmetry and stripe with spin helix
In the absence of dipolar interaction ($\varepsilon _{dd}=0$), the phase transition between half-quantum vortex and hexagonal Skyrmion lattice occurs when the SO coupling is strengthened for an imbalanced contact interacting condensate ($g_{ij}\neq g_{ii}$ with $i,j=1,2$) [64]. However, if $g_{ij}=g_{ii}$ the symmetry of the ground state phases are ordered by dipole-dipole interactions, as discussed above. As a result, novel phases with different topological properties and rotational symmetry emerge in this scenario.
The left three columns in Fig. 2 show the typical density and phase distributions of the two components for different repulsive contact interaction. For a two-component system with fixed SO coupling $\kappa =5$ as shown in Fig. 2, the repulsive interaction between the two species breaks the rotational symmetry in real space [58]. Consequently, atoms in these two components feel repulsion from each other and a density stripe is predicted with phase separation between the spin-up and spin-down components [66]. As shown in Fig. 2, a spin stripe pattern with finite-pitch spin helix [see Fig. 2(c6)] is obtained when $g=20$. For a trapped 2D BEC with SO coupling, the aspect ratio of the condensate profile and the contact interaction is subject to the following constraint [67]
where $\chi ^2\equiv \langle y^2\rangle /\langle x^2\rangle$. For a large contact interaction, e.g., $g=20$ considered here, the aspect ratio is increased to $\chi =0.89$. For a small contact interaction, e.g., $g=3$, the aspect ratio drops to $\chi =0.65$ [67]. If the contact interaction goes down further, the SO coupling will dominate the system. In this case, e.g., $g=1$, quantized vortex phase with different quantum number emerges in the two-component system [64,67,68]. Surprisingly, we find an exotic vortex phase with discrete $C_3$ rotational symmetry in both of the two components when $g=1$ [see Fig. 2(a)]. Intriguingly, the winding number (vorticity) [69] of the first and second components is $\mathcal {N}=1$ and $\mathcal {N}=2$, respectively. The corresponding spin texture of such vortex phase reveals the topological nature of the vortex [see Fig. 2(a6)].Fig. 2. (1), (3) Real-space density, (2), (4) phase, (5) first-component $k$-space density distributions and (6) spin textures of the quantum phases in the (a) vortex with discrete rotational symmetry, (b) stripe with elongation along modulation, and (c) stripe with spin helix, respectively. The density $\rho _1,\rho _2$ is indicated by a color map from low (blue) to high (yellow). The phases of the first and second components are $\phi _1=\rm {arg}(\Psi _1)$ and $\phi _2=\rm {arg}(\Psi _2)$, respectively. The phase ranges from $-\pi$ (blue) to $\pi$ (red). The values of the pseudospin density $S_z$ are from $-1$ (blue) to $1$ (red). Here $g\equiv \tilde {g}N/\left (\hbar ^2/M\right )$ is the strength of contact interaction. The strength of SO coupling is $\kappa =5\ \hbar /Ml_0$. The length unit is $l_0=\sqrt {\hbar /M\omega _\perp }$.
3.2 Chiral soliton and self-organized array with $C_4$ symmetry
In the following, we investigate the ground state phases of the system ($^{52}\textrm {Cr}$ condensate) in the presence of magnetic dipole-dipole interaction. To simplify the numerical calculations and focus on the effects of the dipole-dipole interaction of the first component, we consider a non-dipolar case of the second component while fixing the polarization direction along $z$-axis and the strength of DDI $\varepsilon _{dd}=1.88$ in the first component. For this case in which the dipolar interaction is written by Eq. (3), dipole-induced spin flipping between two components is suppressed while the spin textures of the ground state phases are governed by the SO coupling. Meanwhile, the d-wave angular symmetric term DDI, which can be expressed in terms of spherical harmonics $Y_{20}$, produces a saddle-like mean-field dipolar potential [29]. This kind of mean-field dipolar potential ultimately separates the stripe phase [preserving $C_2$ symmetry in the absence of DDI, see Fig. 2(b5)] and leads the density profile to be self-organized array distribution preserving $C_4$ symmetry in repulsive contact interaction region.
Figure 3 illustrates typical chiral soliton and self-organized arrays for different strengths of contact interaction in the weak SO coupling regime ($\kappa =5$). Chirality of the soliton is justified by the Skyrmion-type spin texture which is similar to the one of each cell shown in Fig. 5(d). A single chiral soliton with Skyrmion-type spin texture turns up when the contact interaction is weak ($g=1$), as shown in the left column of Fig. 3. The formation of Skyrmion-type chiral soliton is a combination effect of the spin-orbit coupling and the dipolar interaction. On one hand, the mean-field dipolar potential has the shape of a saddle with minima located on the $z$-axis, therefore the atoms in the dipolar component are energetically favorable to form a peak near the origin point of the transverse plane. Furthermore, the repulsive contact interaction leads to phase separation between two components and exerts force on the atoms in the non-dipolar component to form a toroidal structure, located in the energetic minima formed by the dipolar component. On the other hand, as discussed in Ref. [70], the system has velocity field $\mathbf {v}=\boldsymbol {\omega }_{rot}\times \mathbf {r}$ associated with SO coupling, which drives the condensate to rotate like a classical rigid body. As a result, the rotational nature of the superfluid combined with the spin rotation property both induced by SO coupling leads to the topological spin structure of the singly Skyrmion [48]. In this phase, spatial rotational symmetry is preserved both in real and momentum spaces. The nonzero phase of the wave function spontaneously breaks $U(1)$ gauge symmetry.
Fig. 3. Real-space density $\rho _1,\rho _2$ and the corresponding phase distributions in a spin-1/2 dipolar condensate for the first and second components, respectively. The phases $\phi _1$ and $\phi _2$ range from $-\pi$ (blue) to $\pi$ (red). The chiral soliton and self-organized arrays exist in the condensate for SO coupling $\kappa =5\ \hbar /Ml_0$ when the strength of contact interaction $\tilde {g}N=(1, 3, 7, 30)\ \hbar ^2/M$, respectively. The ratio between dipolar and contact interaction is $\varepsilon _{dd}=1.88$. The strength of SO coupling is $\kappa =5\ \hbar /Ml_0$. The length unit is $l_0=\sqrt {\hbar /M\omega _\perp }$.
As the contact interaction increases, the repulsion between atoms tend to separate them into several solitons in each component. For example, a chiral self-organized array holding four Skyrmions with $C_4$ symmetry is observed when $g=3$ (see the second column in Fig. 3). In stark contrast to stripe phase with $C_2$ symmetry in the absence of DDI, the morphogenesis of self-organized array with $C_4$ symmetry is owing to the system can minimize its energy by locally increasing density with the crystallization of self-organization [71] in the presence of DDI. This density modulation of crystallization due to DDI has been studied and explained by softening of roton excitation spectrum in a dipolar quantum gas [72–75]. By increasing the contact interaction further, more and more solitons emerge and form self-organized arrays of a square lattice. The third and fourth columns in Fig. 3 provide examples of self-organized arrays composed of dozens of Skyrmions when $g=7$ and $30$, respectively.
3.3 Topological charge and long-lived metastable self-organized array with $C_6$ symmetry
To further understand the topological property in chiral self-organization, we calculate the topological charge defined by [34]
In the strong SO coupling regime, e.g., $\kappa =10$ considered here, the ground-state phases in most cases converge to a chiral self-organized array with $C_4$ symmetry in the numerical imaginary time evolution, which is similar to the case when $\kappa =5$ as discussed above. Additionally, our simulation also finds some long-lived metastable chiral states with self-organized $C_6$ symmetry for cases when $\Delta E\approx 0$, as discussed above. However, the metastable states finally evolve to the correspondingly real lower-energy ground states with $C_4$ symmetry in the presence of perturbations. Fig. 5 presents a typical metastable chiral self-organized states with $C_6$ symmetry for a dipolar condensate with SO coupling $\kappa =10$. In this kind of chiral self-organized array, we find that the strong repulsive contact interaction ($\mathrm{g}=23$) exerts force on the condensate and leads to its phase separation, while SO coupling breaks spatial rotation symmetry [66]. The density distributions of both component appear periodic modulations of hexagonal lattices. Moreover, the rotational effects of the dipole-dipole interaction as well as SO coupling drive the atoms to rotate both in spatial and spin spaces and then leads to quantized vortex lattice with hexagonal structure in the second component.
Fig. 4. Topological charge of the self-organized states obtained by Eq. (8) is plotted by blue curve. Red square and yellow hexagram present the stable and metastable self-organized arrays with $C_4$ and $C_6$ rotational symmetries, respectively. The ratio between dipolar and contact interaction is $\varepsilon _{dd}=1.88$. The strength of SO coupling is $\kappa =10\ \hbar /Ml_0$. The length unit is $l_0=\sqrt {\hbar /M\omega _\perp }$. Inset: Density distribution in k-space illustrates that the rotational symmetry breaks into 4- or 6-fold discrete symmetries for stable or metastable self-organized arrays when $\tilde {\mathrm{g}}\equiv \mathrm{g}N/\left (\hbar ^2/M\right )=2,3,7,23$, respectively. $\mathbf {k}$ is the reciprocal lattice vector.
Fig. 5. An example of metastable state composed of self-organized arrays with $C_6$ symmetry. Real-space density distributions illustrate (a) self-organized and (b) vortex lattices in the dipolar and non-dipolar components, respectively. (c) is the momentum-space density profiles corresponding to (a). (d) The spin textures of the self-organized array in which the pseudospin density $S_z$ is described by colors from $-1$ (blue) to $1$ (red). Here the strength of contact interaction $\tilde {\mathrm{g}}\equiv \mathrm{g}N/\left (\hbar ^2/M\right )=23$. The ratio between dipolar and contact interaction is $\varepsilon _{dd}=1.88$. The strength of SO coupling is $\kappa =10\ \hbar /Ml_0$. The length unit is $l_0=\sqrt {\hbar /M\omega _\perp }$.
4. Conclusions
We perform a numerical study of the ground states of a dipolar Bose-Einstein condensate subject to Raman laser induced spin-orbit coupling, and demonstrate that the competition between SO coupling, contact and dipolar interactions gives rise to a rich variety of topological ground state phenomena, such as vortex with discrete rotational symmetry, stripe with spin helix, and chiral self-organized array with $C_4$ symmetry. Comparing to the previous dipolar condensate without SO coupling, we observe a stable chiral self-organized pattern of square lattice with topology as well as a long-lived metastable self-organized array with $C_6$ symmetry in the case of strong repulsive contact interaction.
In the absence of dipole-dipole interaction, spin-orbit coupled BECs hosts vortex phase with discrete rotational symmetry in the case of weak contact interaction. The winding number of the vortex in the first and second components is $\mathcal {N}=1$ and $\mathcal {N}=2$, respectively. For strong contact interaction, elongated stripe phase with spin helix is induced. While the dipole-dipole interaction is switched on, chiral self-organization emerge as square-lattice arrays with $C_4$ symmetry, whose topological charge is found to increase with the contact interaction. We also find the long-lived metastable state in which the dipolar component forms self-organized array, while for non-dipolar component the inter-component repulsive contact interaction leads the atoms located in the space not occupied by the first component to become hexagonal lattice composed of quantized vortices with $C_6$ symmetry. The pattern formation mechanism of the metastable self-organized state obtained here is similar to the optical pattern reported recently in a cold Rydberg atomic gas, which is controlled by nonlocal Kerr nonlinearity and initial atomic population [76]. Due to the high degrees of control over the relevant parameters, such as laser-induced harmonic trap and spin-orbit coupling, the quantum phases with topology predicted in this paper are within the reach of current experiment capability.
Funding
National Natural Science Foundation of China (11847237, 12104359); Xi'an Jiaotong University Basic Research Funding (xtr042021012); National Key Research and Development Program of China (2021YFA1401700).
Acknowledgments
We thank H. Saito, L. Vernac, and L. Huang for helpful discussions. This work was supported by National Key Research and Development Program of China (2021YFA1401700) and NSF of China under Grants No. 11847237 and No. 12104359. Y.C.Z. acknowledges the support of Xi’an Jiaotong University through the “Young Top Talents Support Plan” and Basic Research Funding (Grant No. xtr042021012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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