Spin-orbital-angular-momentum-coupled quantum gases

In the semiclassical theory, the atom-light interaction during the Raman process is described by the following Hamiltonian

$$\hat{H}_{al}=\left[\begin{array}[]{ccc}\hbar\omega_{\uparrow}&0&V_{\uparrow 3}\\ 0&\hbar\omega_{\downarrow}&V_{\downarrow 3}\\ V_{3\uparrow}&V_{3\downarrow}&\hbar\omega_{3}\end{array}\right]$$

(1)

in the bare hyperfine basis $\left[\left|\uparrow\right\rangle,\left|\downarrow\right\rangle,\left|3\right\rangle\right]^{T}$, where $\hbar\omega_{\sigma}$ ($\sigma=\uparrow,\downarrow,3$) is the bare energy of atoms in different hyperfine states, and $V_{\sigma\sigma^{\prime}}=-\left\langle\sigma\right|{\bf d}\cdot{\bf E}_{(\pm)}\left|\sigma^{\prime}\right\rangle$ characterizes the dipole interactions between the valence electron and light fields. Here, ${\bf d}$ is the electric moment of the valence electron and

$${\bf E}_{\pm}\left({\bf r},t\right)=\frac{1}{2}\hat{{\bf e}}_{\pm}\left[\mathcal{E}_{\pm}\left({\bf r}\right)e^{-i\omega_{\pm}t}+h.c.\right]$$

(2)

are the electric fields experienced by atoms, where $\hat{{\bf e}}_{\pm}$ denote the unit vectors of the polarization direction of light, $\omega_{\pm}$ are the angular frequencies of Raman beams, and $\mathcal{E}_{\pm}\left({\bf r}\right)$ are the spatial complex amplitudes. The wave function of atoms can also be written in the bare hyperfine basis as

$$\Psi\left(t\right)=\left[\begin{array}[]{c}c_{\uparrow}\left(t\right)e^{-i\eta_{\uparrow}t}\\ c_{\downarrow}\left(t\right)e^{-i\eta_{\downarrow}t}\\ c_{3}\left(t\right)e^{-i\eta_{3}t}\end{array}\right],$$

(3)

and $\eta_{\sigma}$ are arbitrary factors that are chosen for later convenience. Inserting Eqs. (1) and (3) into the Schrödinger equation $i\hbar\partial_{t}\Psi=\hat{H}_{al}\Psi$ and by choosing

	$\displaystyle\eta_{\uparrow}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(\omega_{\uparrow}+\omega_{\downarrow}-\omega_{+}+\omega_{-}\right),$		(4)
	$\displaystyle\eta_{\downarrow}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(\omega_{\uparrow}+\omega_{\downarrow}+\omega_{+}-\omega_{-}\right),$		(5)
	$\displaystyle\eta_{3}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(\omega_{\uparrow}+\omega_{\downarrow}+\omega_{+}+\omega_{-}\right),$		(6)

we easily obtain

	$\displaystyle i\hbar\frac{dc_{\uparrow}}{dt}$	$\displaystyle=$	$\displaystyle+\frac{\delta}{2}c_{\uparrow}+\frac{1}{2}\rho_{+}^{*}\mathcal{E}_{+}^{*}\left({\bf r}\right)c_{3},$		(7)
	$\displaystyle i\hbar\frac{dc_{\downarrow}}{dt}$	$\displaystyle=$	$\displaystyle-\frac{\delta}{2}c_{\downarrow}+\frac{1}{2}\rho_{-}^{*}\mathcal{E}_{-}^{*}\left({\bf r}\right)c_{3},$		(8)
	$\displaystyle i\hbar\frac{dc_{3}}{dt}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\rho_{+}\mathcal{E}_{+}\left({\bf r}\right)c_{\uparrow}+\frac{1}{2}\rho_{-}\mathcal{E}_{-}\left({\bf r}\right)c_{\downarrow}-\Delta c_{3}$		(9)

under the rotating-wave approximation (RWA) Scully2008Q , where $\rho_{+}\equiv\left\langle 3\right|-{\bf d}\cdot\hat{{\bf e}}_{+}\left|\uparrow\right\rangle$ and $\rho_{-}\equiv\left\langle 3\right|-{\bf d}\cdot\hat{{\bf e}}_{-}\left|\downarrow\right\rangle$ are the matrix elements of the electric dipole moments, and $\delta/\hbar\equiv\left(\omega_{+}-\omega_{-}\right)-\left(\omega_{\downarrow}-\omega_{\uparrow}\right)$ and $\Delta/\hbar\equiv\left(\omega_{+}+\omega_{-}\right)/2-\left[\omega_{3}-\left(\omega_{\uparrow}+\omega_{\downarrow}\right)/2\right]$ are respectively the two-photon and single-photon detunings. When the excited state $\left|3\right\rangle$ is far from the resonance, we may adiabatically eliminate it by setting $\partial_{t}c_{3}\left(t\right)=0$, and then Eq. (9) yields

$$c_{3}\approx\frac{\rho_{+}\mathcal{E}_{+}\left({\bf r}\right)}{2\Delta}c_{\uparrow}+\frac{\rho_{-}\mathcal{E}_{-}\left({\bf r}\right)}{2\Delta}c_{\downarrow}.$$

(10)

After inserting Eq. (10) into Eqs. (7) and (8), we arrive at

$$i\hbar\frac{d}{dt}\left[\begin{array}[]{c}c_{\uparrow}\\ c_{\downarrow}\end{array}\right]=\left[\begin{array}[]{cc}\delta/2+\chi_{+}\left({\bf r}\right)&\Omega\left({\bf r}\right)\\ \Omega^{*}\left({\bf r}\right)&-\delta/2+\chi_{-}\left({\bf r}\right)\end{array}\right]\left[\begin{array}[]{c}c_{\uparrow}\\ c_{\downarrow}\end{array}\right],$$

(11)

where

	$\displaystyle\chi_{\pm}\left({\bf r}\right)$	$\displaystyle\equiv$	$\displaystyle\frac{\left|\rho_{\pm}\mathcal{E}_{\pm}\left({\bf r}\right)\right|^{2}}{4\Delta},$		(12)
	$\displaystyle\Omega\left({\bf r}\right)$	$\displaystyle\equiv$	$\displaystyle\frac{\rho_{+}^{*}\rho_{-}\mathcal{E}_{+}^{*}\left({\bf r}\right)\mathcal{E}_{-}\left({\bf r}\right)}{4\Delta}$		(13)

are respectively the diagonal and off-diagonal ac-stark shifts. We find that the diagonal ac-stark shift $\chi_{\pm}\left({\bf r}\right)$ provides an effective trapping potential for atoms that could be removed by properly choosing a tune-out wavelength of LG beams in the experiment (Zhang2019G, ; Holmgren2012M, ; Herold2012P, ). The off-diagonal ac-stark shift $\Omega\left({\bf r}\right)$ leads to a space-dependent coupling between two hyperfine states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$, and then results in a SOAM coupling of atoms as we will see below.

Without loss of generality, we consider a two-dimensional (2D) configuration for simplicity in that atoms are confined in the $z=0$ plane, and the pair of Raman LG beams copropagate along the $-{\bf\it{z}}$ direction with the spatial complex amplitude

$$\mathcal{E}_{\pm}\left({\bf r}\right)=\sqrt{2I_{0}}e^{il_{\pm}\varphi}\left(\frac{r}{w}\right)^{\left|l_{\pm}\right|}e^{-r^{2}/w^{2}},$$

(14)

where $I_{0}$ and $w$ are respectively the intensity and waist of the beams, and $l_{\pm}$ are their winding numbers. Here, we have adopted a polar coordinate ${\bf r}=\left(r,\varphi\right)$. Consequently, the single-atom Hamiltonian in the presence of Raman LG beams takes an effective form of

$$\hat{H}_{0}=-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{ext}\left(r\right)+\hat{H}_{so}$$

(15)

with

$$\hat{H}_{so}=\left[\begin{array}[]{cc}\delta/2+\chi\left(r\right)&\Omega\left(r\right)e^{-i2n\varphi}\\ \Omega\left(r\right)e^{+i2n\varphi}&-\delta/2+\chi\left(r\right)\end{array}\right],$$

(16)

where $V_{ext}\left(r\right)=m\omega^{2}r^{2}/2$ is the external harmonic potential, $n=\left(l_{+}-l_{-}\right)/2$ is the angular momentum transferred to atoms, and $\Omega\left(r\right)\equiv\Omega_{R}\left(r/w\right)^{\left|l_{+}\right|+\left|l_{-}\right|}e^{-2r^{2}/w^{2}}$, with the coupling strength $\Omega_{R}=\left|\rho_{+}^{*}\rho_{-}\right|I_{0}/2\Delta$, is the radial Rabi frequency. Here, we have assumed $\chi_{+}\left(r\right)=\chi_{-}\left(r\right)\equiv\chi\left(r\right)$ for simplicity, which is only dependent on $\left|{\bf r}\right|=r$. Besides, we also discard the internal phase difference between the matrix elements of electric dipole moments $\rho_{+}$ and $\rho_{-}$ that is irrelevant to the problem.

Let us then analyse the possible symmetries of the single-atom Hamiltonian (15). It is obvious that the rotational symmetry with respect to the $z$ axis is broken by the dependence of Raman coupling $\hat{H}_{so}$ on the azimuthal angle $\varphi$. The orbital angular momentum of the atom is thus no longer conserved, since it follows a change of orbital angular momentum accompanied by the spin flip in the Raman process DeMarco2015A ; Chen2020A ; Duan2020S . By introducing a unitary transformation $\hat{U}=\exp\left(in\varphi\hat{\sigma}_{z}\right)$, the single-atom Hamiltonian becomes $\hat{\mathcal{H}}_{0}=\hat{U}\hat{H}_{0}\hat{U}^{\dagger}$, i.e.,

$$\hat{\mathcal{H}}_{0}=-\frac{\hbar^{2}}{2mr}\frac{\partial}{\partial r}r\frac{\partial}{\partial r}+\frac{\left(\hat{L}_{z}-n\hbar\hat{\sigma}_{z}\right)^{2}}{2mr^{2}}\\ +V_{ext}\left(r\right)+\chi\left(r\right)+\Omega\left(r\right)\hat{\sigma}_{x}+\frac{\delta}{2}\hat{\sigma}_{z},$$

(17)

where $\hat{L}_{z}=-i\hbar\partial/\partial\varphi$, and $\hat{\sigma}_{x,z}$ are Pauli matrices. We find that the angular momentum $\hat{L}_{z}$ commutes with the single-atom Hamiltonian $\hat{\mathcal{H}}_{0}$ under the unitary transformation $\hat{U}$, and thus $\hat{L}_{z}$ is conserved. Therefore, the original single-atom Hamiltonian $\hat{H}_{0}$ demonstrates the symmetry under the redefined rotational transformation $\hat{\mathcal{R}}\left(\varphi\right)\equiv\hat{U}^{\dagger}e^{-i\hat{L}_{z}\varphi/\hbar}\hat{U}$. Then $\hat{L}_{z}$ may be regarded as the operator of quasi-angular momentum (QAM), and each eigenstate of the system possesses definite values of QAM characterized by the corresponding quantum number $l_{z}$. It is related to the angular momentum of each spin component in the laboratory frame by $l_{\uparrow,\downarrow}=l_{z}\mp n$. Here, we find that the term of $\hat{L}_{z}\hat{\sigma}_{z}$ appears, which characterizes the crucial SOAM-coupling effect that is well familiar to us in atomic physics.

At the resonance of Raman coupling with $\delta=0$, the system demonstrates an additional symmetry, namely the time-reversal symmetry. The single-atom Hamiltonian $\hat{H}_{0}$ commutes with the time-reversal operator $\hat{T}=\hat{\sigma}_{x}\hat{K}$, and thus is invariant under the time-reversal transformation, where $\hat{K}$ denotes the operator of complex conjugation DeMarco2015A ; Duan2020S . This symmetry may be translated into a QAM frame as $\hat{\mathcal{T}}=\hat{U}\hat{T}\hat{U}^{\dagger}$, and then we have $\left[\hat{\mathcal{T}},\hat{\mathcal{H}}_{0}\right]=0$. The time-reversal symmetry guarantees that the spectrum of a single atom is symmetric about the QAM $l_{z}=n$.

Regarding the rotational symmetry of the single-atom Hamiltonian (17), each eigenstate of a single atom should possess a definite QAM $l_{z}$, whose wave function may be written as

$$\Psi_{l_{z}}\left({\bf r}\right)=\left[\begin{array}[]{c}\psi_{\uparrow}\left(r\right)\\ \psi_{\downarrow}\left(r\right)\end{array}\right]\frac{e^{il_{z}\varphi}}{\sqrt{2\pi}}.$$

(18)

Then the Schrödinger equation $\hat{\mathcal{H}}_{0}\Psi_{l_{z}}\left({\bf r}\right)=E\Psi_{l_{z}}\left({\bf r}\right)$ reduces to two coupled one-dimensional radial equations of $\psi_{\uparrow,\downarrow}\left(r\right)$, which are easily solved by using the finite-difference method DeMarco2015A . The single-particle dispersion relations between the energy $E$ and QAM $l_{z}$ are shown in Fig. 2 for three typical coupling strengths. Here, the winding numbers of two Raman beams are respectively chosen to be $l_{+}=-2$ and $l_{-}=0$, and we consider the situation at the two-photon resonance with $\delta=0$. The symmetry of the single-particle dispersion about $l_{z}=0$ is guaranteed by the time-reversal symmetry $\hat{\mathcal{T}}$ discussed above. This means that the Schrödinger equation is invariant under the exchange of the spin indices, which sends $n\rightarrow-n$ as well.

In the absence of SOAM coupling, i.e., $\Omega_{R}=0$, the energy band structure is simply that of the spinor harmonic oscillator with the excitation interval $\hbar\omega$. The ground state is characterized by the QAM $l_{z}=\pm 1$, which is doubly degenerate. The angular momentum for each spin component in the laboratory frame is $\left(l_{\uparrow},l_{\downarrow}\right)=\left(0,-2\right)$ and $\left(l_{\uparrow},l_{\downarrow}\right)=\left(2,0\right)$ corresponding to the QAM $l_{z}=-1$ and $1$, respectively. As the SOAM-coupling strength $\Omega_{R}$ gradually increases, the spin is no longer a good quantum number, and small amounts of atoms are transferred into the previously vacant spin component, while the ground state still has the two-fold degeneracy with definite QAM. By further increasing the coupling strength, the system finally jumps into the QAM $l_{z}=0$ ground state, which gives rise to a first-order phase transition to a spin-balanced population (Zhang2019G, ). This is due to the unique property of the quantized angular momentum in SOAM-coupled systems. In SLM-coupled systems, such phase transition is a continuous type, where the doubly degenerate ground states finally merge into each other as the Raman-coupling strength increases Li2012Q ; Martone2012A ; Chen2017Q ; Chen2018Q .

Let us look closely at the evolution of the lowest-energy band as presented in Fig. 3 at different coupling strengths. Its symmetry about $l_{z}=0$ is preserved by the time-reversal symmetry at the two-photon resonance $\delta=0$ as shown in Fig. 3(b). We demonstrate how the ground state evolves from the doubly degenerate states ($l_{z}=\pm 1$) into a single one ($l_{z}=0$) as the coupling strength increases. However, away from the two-photon resonance with $\delta\neq 0$, we find that the degeneracy of the ground states at weak coupling strength is lifted, since the time-reversal symmetry is broken by the two-photon detuning (as shown in Fig. 3(a) and (c)). The ground state is located either at $l_{z}=-1$ or $1$ determined by the sign of the two-photon detuning. This gives rise to an additional first-order phase transition by continuously varying the two-photon detuning Zhang2019G . At strong coupling strength, the system again jumps into the QAM $l_{z}=0$ ground state as we discussed above.

The SO coupling gives rise to intriguing spin textures. For example, it has been found in studies of 2D Rashba SO-coupled BECs Hu2012S ; Ramachandhran2012H and BECs exposed to LG beams (Leslie2009C, ) that the spin texture contains a topological knot known as a 2D skyrmion. It is one kind of topological defects protected by their topological nontriviality. Regarding the spin texture of SOAM-coupled systems, it is useful to define a spin vector DeMarco2015A ; Hu2012S ; Ramachandhran2012H

$${\bf S}\left({\bf r}\right)=\frac{\Psi^{\dagger}\left({\bf r}\right)\hat{{\bf s}}\Psi\left({\bf r}\right)}{\left|\Psi\left({\bf r}\right)\right|^{2}},$$

(19)

with the spin operator $\hat{{\bf s}}=\left(\hbar/2\right)\left(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\right)$. The skyrmion number defined as

$$n_{\text{skyrmion}}=\frac{1}{4\pi}\int{\bf S}\cdot\left(\partial_{x}{\bf S}\times\partial_{y}{\bf S}\right)d{\bf r}$$

(20)

is a measure of the winding of the spin texture, which distinguishes a skyrmion texture from that of the vacuum. If it equals $1$ or $-1$, a topological knot exists in the spin texture Binz2008C ; Muhlbauer2009S . The ground-state spin texture at four typical coupling strengths is presented in Fig. 4. At weak coupling strength, the two-fold degenerate ground state gives rise to a spin texture corresponding to a half skyrmion. As the coupling strength increases, the ground state finally jumps to the QAM $l_{z}=0$ state. The system reaches a spin-balanced state, which does not support a skyrmion texture. The local spin becomes planar and lies in the $x-y$ plane, and thus the skyrmion number vanishes.

In the following, we are going to demonstrate how the artificial (synthetic) gauge field may emerge for neutral atoms in the presence of atom-light interactions. It is one of the essential ingredients for the simulation of charged particles moving in the electromagnetic field by using cold atoms Dum1996G ; Visser1998G ; Lin2009S ; Lin2009B ; Dalibard2011A , which gives rise to a series of intriguing phenomena, such as the spin Hall effect Zhu2006S ; Liu2007O ; Beeler2013T . To this end, we may simply rewrite the atom-light Hamiltonian (16) in an explicit form of

$$\hat{H}_{so}={\bf Z}\left({\bf r}\right)\cdot{\bf s}$$

(21)

with the effective Zeeman field

$${\bf Z}\left({\bf r}\right)=\frac{2}{\hbar}\left(\Omega\left(r\right)\cos 2n\varphi,\Omega\left(r\right)\sin 2n\varphi,\frac{\delta}{2}\right).$$

(22)

We find easily that the Raman-induced atom-light interaction effectively provides a spin-magnetic interaction equivalent to that for a spin-half charged particle in a space-dependent Zeeman field ${\bf Z}\left({\bf r}\right)$. The diagonalization of $\hat{H}_{so}$ simply gives the dressed spin states Dalibard2011A , i.e.,

$$\left|\zeta_{+}\right\rangle=\left[\begin{array}[]{c}\cos\left(\theta/2\right)\\ e^{i\phi}\sin\left(\theta/2\right)\end{array}\right],\;\left|\zeta_{-}\right\rangle=\left[\begin{array}[]{c}-e^{-i\phi}\sin\left(\theta/2\right)\\ \cos\left(\theta/2\right)\end{array}\right]$$

(23)

with eigenvalues $\pm\hbar\left|{\bf Z}\left({\bf r}\right)\right|/2$, respectively, and $\tan\phi=\tan 2n\varphi$ and $\tan\theta=2\Omega\left(r\right)/\delta$, which determine the orientation of the effective Zeeman field. It is obvious that $\left|\zeta_{\pm}\right\rangle$ denotes the states that the pseudo-spin of the atom aligns along or inversely to the direction of the local Zeeman field. In the following, let us consider the atom moves slowly in the space-dependent Zeeman field ${\bf Z}\left({\bf r}\right)$, and its pseudo-spin follows adiabatically one of the eigenstates of $\hat{H}_{so}$, namely for example $\left|\zeta_{+}\right\rangle$. Then the full wave function of the atom can be written as $\left|\Psi\left({\bf r},t\right)\right\rangle=\psi_{+}\left({\bf r},t\right)\left|\zeta_{+}\right\rangle$, the evolution of which is governed by the Schrödinger equation $i\hbar\partial_{t}\left|\Psi\left({\bf r},t\right)\right\rangle=\hat{H}_{0}\left|\Psi\left({\bf r},t\right)\right\rangle$ under the Hamiltonian (15). Here, $\psi_{+}\left({\bf r},t\right)$ is the spatial wave function characterizing the center-of-mass motion of the atom in the internal dressed state $\left|\zeta_{+}\right\rangle$. By projecting the Schrödinger equation onto the internal dressed state $\left|\zeta_{+}\right\rangle$, we obtain easily

$$i\hbar\frac{\partial\psi_{+}}{\partial t}=\left[\frac{\left(\hat{{\bf P}}-{\bf A}\right)^{2}}{2m}+V_{ext}+W+\frac{\hbar\left|{\bf Z}\right|}{2}\right]\psi_{+}.$$

(24)

Two geometric potentials ${\bf A}$ and $W$ emerge in addition to the external trapping potential $V_{ext}$ and Zeeman energy $\hbar\left|{\bf Z}\right|/2$, which are respectively the vector potential Dalibard2011A

$${\bf A}\left({\bf r}\right)\equiv\left\langle\zeta_{+}\right|i\hbar\nabla\left|\zeta_{+}\right\rangle=\frac{\hbar}{2}\left(\cos\theta-1\right)\nabla\phi,$$

(25)

and the scalar potential

$$W\left({\bf r}\right)\equiv\frac{\hbar^{2}}{2m}\left|\left\langle\zeta_{-}\right|\nabla\left|\zeta_{+}\right\rangle\right|^{2}=\frac{\hbar^{2}}{8m}\left[\left(\nabla\theta\right)^{2}+\sin^{2}\theta\left(\nabla\phi\right)^{2}\right].$$

(26)

It is obvious that Eq. (24) takes the same form as that for a spin-half particle with a unit charge ($q=1$) (Landau2007Q, ). The effective magnetic field ${\bf B}\left({\bf r}\right)$ associated with ${\bf A}\left({\bf r}\right)$ is

$${\bf B}\left({\bf r}\right)=\nabla\times{\bf A}\left({\bf r}\right)=\frac{\hbar}{2}\nabla\left(\cos\theta\right)\times\nabla\phi.$$

(27)

Here, we should pay special attention to the difference between the Zeeman field ${\bf Z}\left({\bf r}\right)$ and the effective magnetic field ${\bf B}\left({\bf r}\right)$: the Zeeman field ${\bf Z}\left({\bf r}\right)$ only lifts the degeneracy of the bare spin states and does not affect the center-of-mass motion of the atom, while ${\bf B}\left({\bf r}\right)$ is the magnetic field experienced by the center-of-mass motion of the atom when it moves in the presence of the atom-light interaction and keeps staying in the internal dressed state $\left|\zeta_{+}\right\rangle$. In the presence of Raman LG beams, the effective magnetic field ${\bf B}\left({\bf r}\right)$ takes the explicit form of

$${\bf B}\left({\bf r}\right)=\frac{n\hbar}{r}\left[\frac{\partial}{\partial r}\frac{\delta/2}{\sqrt{\Omega^{2}\left(r\right)+\delta^{2}/4}}\right]\hat{{\bf e}}_{z},$$

(28)

which is perpendicular to the $x-y$ plane and along the $z$ axis.

When the interaction is taken into account, the nonlinearity may spontaneously break the rotational symmetry of the system. We can no longer assume that the condensate possesses the definite QAM $l_{z}$. The ground-state solution of the GP equation is first attempted by using a split-step imaginary time evolution DeMarco2015A . Three kinds of quantum phases are identified: two of them still preserve the rotational symmetry as the many-body analogs of the single-particle ground state, while the third one is a newly emergent angular stripe phase, which does not possess a definite QAM and breaks the rotational symmetry.

Therefore, apart from the angular stripe phase, one may take the states with a definite QAM for the condensates as a good starting point when taking into account interactions. If the system or the condensate wave function preserves the axial symmetry, one may consider the QAM $l_{z}$ as a good quantum number and adopt the Ansatz for the condensate Chen2020A

$$\psi_{{}_{l_{z}}}\left({\bf{r}}\right)=\begin{bmatrix}f_{{}_{l_{z}\uparrow}}(r)\\ f_{{}_{l_{z}\downarrow}}(r)\end{bmatrix}\frac{e^{il_{z}\varphi}}{\sqrt{2\pi}},$$

(30)

with the radial wave function $f_{{}_{l_{z}\sigma}}$, and substitute it into the GP equation,

$$\begin{bmatrix}\hat{\mathcal{H}}_{0}(\hat{L}_{z})+\mathrm{diag}(\mathcal{L}_{\uparrow},\mathcal{L}_{\downarrow})\end{bmatrix}\psi_{{}_{l_{z}}}=\mu\psi_{{}_{l_{z}}},$$

(31)

where $\mu$ is the chemical potential, and the diagonal element takes the form of $\mathcal{L}_{\sigma}\equiv g_{\sigma\sigma}n_{\sigma}+g_{{}_{\uparrow\downarrow}}n_{\bar{\sigma}}$ (spin index $\sigma\neq\bar{\sigma}$). Thus, one can then determine the ground-state wave function $\psi_{{}_{l_{z}}}$ of the rotationally symmetric phases at zero temperature by solving self-consistently this GP equation.

Nonetheless, to capture the exotic angular stripe phase that does not possess a definite QAM, a variational Ansatz with various angular momenta is adopted for the condensate wave function in determining the ground state with an energy-minimizing method Chen2020A .

In the realistic configuration of recent experiments Chen2018S ; Zhang2019G , the orbital angular momentum transferred by the LG lasers is $n=\pm 1$, giving rise to minima with the QAM located at $l_{z}=\pm 1$, or $0$ in the single-particle dispersion. Thus, the adopted variational Ansatz could be taken as

$$\Psi\left({\bf r}\right)=\alpha e^{i\theta_{\alpha}}\psi_{{}_{-1}}+\beta e^{i\theta_{\beta}}\psi_{{}_{0}}+\gamma e^{i\theta_{\gamma}}\psi_{{}_{1}},$$

(32)

with real weighting coefficients $\alpha,\beta,\gamma$ and the associated phases $\theta_{\alpha}$, $\theta_{\beta}$, and $\theta_{\gamma}$. In general, $\psi_{{}_{l_{z}}}$ is the state with a definite QAM $l_{z}$, which can either be the single-particle state or the one obtained by self-consistently solving the GP equation (31) when the interaction is taken into account. Nonetheless, the latter is adopted here which contributes a lower mean-field energy than the one from the single-particle states particularly at low Rabi frequencies or at relatively strong interactions Chen2020A . Therefore, the ground-state wave function $\Psi\left({\bf r}\right)$ is then determined from the minimization of the mean-field energy in Eq. (29) with respect to the variational parameters. It’s worth mentioning that, three definite-QAM states are merely sufficient here, regarding the recent experimental configuration with the transferred orbital angular momentum $n=\pm 1$ Zhang2019G . One may need more definite-QAM states in the variational Ansatz for larger $n$.

After solving self-consistently the GP equation in Eq. (31) and minimizing the total energy in Eq. (29) with a constructed variational Ansatz, the ground-state wave function $\Psi({\bf r})$ can then be calculated and the phase diagram at zero temperature is conveniently depicted for a weakly-interacting SOAM-coupled Bose gas Chen2020A . In general, four distinct phases can be identified in the parameter space of $g_{\sigma\sigma^{\prime}}$ and $\Omega$ at a vanishing two-photon detuning $\delta=0$. The first is the vortex-antivortex pair phase which behaves as a vortex, as shown by the first column in Fig. 5. It locates at zero QAM $l_{z}=0$ (i.e., $\beta=1,\alpha=\gamma=0$) with zero magnetization $\langle\sigma_{z}\rangle=0$ and $\langle\sigma_{x}\rangle=-1$. The second one is the half-skyrmion phase with two spin-component densities being a Thomas-Fermi-like distribution and a vortex (see the second column in Fig. 5). This phase is magnetic, i.e., $\langle\sigma_{z}\rangle\neq 0$ and $\langle\sigma_{x}\rangle\neq 0$, with a definite QAM at $l_{z}=-1$ or $1$ (i.e., $\alpha=1,\beta=\gamma=0$ or $\gamma=1,\alpha=\beta=0$). The third and fourth ones are called angular stripe phases shown by the peanut-like and halo-like density profiles in the last two columns of Fig. 5. Both of them have no definite $l_{z}$ with $\alpha=\gamma=1/\sqrt{2},\beta=0$ or $\alpha=\gamma\neq 0,\beta\neq 0$ and they have no magnetization $\langle\sigma_{z}\rangle=0$ and $\langle\sigma_{x}\rangle\neq 0$. In addition, one can further investigate the elementary excitation spectrum to distinguish these phases and then characterize the phase transitions. The spectrum of a SOAM-coupled Bose gas is discrete due to the quantized angular momentum. In detail, the vortex-antivortex pair phase possesses a symmetric spectrum with a single phonon mode, see Fig. 7(a). However, the half-skyrmion phase breaks spontaneously the double degeneracy in the single-particle dispersion and thus has an excitation spectrum with a discrete roton-maxon structure Chen2020A , as shown in Fig. 7(b). Most interestingly, owing to the spontaneous breaking of both the U$(1)$ gauge symmetry and the continuous rotational symmetry, the excitation spectrum in the angular stripe phase exhibits two gapless Goldstone modes Chen2020Ground , see Fig. 7(c). A summary of the properties of these typical phases can be found in Table 1. It should be noted that these phases described above are based on recent experimental configurations with small transferred angular momentum $n$, and we use them for a brief illustration. It is possible to find more nontrivial phases such as other angular stripe phases with superposition of large-angular-momentum states under the condition of larger $n$ Chen2020Ground ; Chiu2020V or complex vortex molecule states Duan2020S . Among these typical ground-state phases, the most intriguing one is the angular stripe phase, which attracts intensive research attention and will be discussed in detail in the following.

In the past decade, an exotic phase of matter has attracted tremendous attention in ultracold quantum gases, i.e., namely the stripe phase, which breaks spontaneously U$(1)$ gauge symmetry and spatial translational symmetry wang2010spin ; Wu2011U ; ho2011bose ; Li2012Q . As a consequence, quantum gases will exhibit a superfluid behaviour and meanwhile show a periodic modulation in the density distribution. These counterintuitive features associate closely this nontrivial phase to the long-sought supersolid phase in solid Helium since the 1960s Boninsegni2012C , i.e., a rigid, spatially ordered solid that flows like a fluid without friction. Whether such a superfluid state can exist remains unclear for more than 50 years until the seminal breakthroughs using ultracold atoms. In 2017, two groups from ETH Zurich and MIT created successfully an ultracold quantum gas featuring supersolid properties using Bose-Einstein condensates with optical cavities or spin-orbit coupling leonard2017supersolid ; li2017stripe . Later in 2019, three independent groups from Stuttgart, Florence, and Innsbruck observed supersolidity using dipolar quantum gases of lanthanide atoms (i.e., ¹⁶²Dy, ¹⁶⁴Dy, and ¹⁶⁶Er) without any external optical lattice tanzi2019observation ; bottcher2019transient ; chomaz2019long , and very recently one of them extended successfully into two dimensions norcia2021two .

In the presence of SOAM coupling, several kinds of angular stripe phases are theoretically predicted as introduced above, which take the analogous behavior as that of supersolid. Here, we have discussed two kinds of angular stripe phases: phase (III), a superposition state of two orbital-angular-momentum states ($l_{z}=\pm 1$), and phase (IV), a superposition state of three orbital-angular-momentum states ($l_{z}=0,\pm 1$). Although experimentalists have achieved successfully most of the ground-state quantum phases of a SOAM-coupled Bose gas, the nontrivial angular stripe phase remains elusive in the laboratory. There are two main obstacles. Firstly, the typical energy scale of the SOAM coupling is about $E_{L}\sim 1/R^{2}$ with the atomic cloud size $R$. To enhance the SOAM-coupling effect to reach the regime of the angular stripe phase, one needs to reduce the atomic size $R$ which decreases the perimeter of the cloud and instead presents fewer periods of density order in the angular direction. Secondly, recent experimental attempts are made in a ⁸⁷Rb BEC. The critical Rabi frequency of the angular stripe phase is thus small due to the tiny difference between the intra- and inter-species interactions. As a consequence, the period of stripes becomes so small that the stripes show too low contrast and visibility, which makes them hard to be detected.

The interatomic interaction is one of the crucial factors to affect the parameter space of angular stripe phases as well as the visibility. This is quite similar to the situation when concerning the stripe phase in a SO-coupled Bose gas Li2012Q . The stripe phase prefers the region where the inter-species interaction is pretty smaller than the intra-species interaction. To this end, ⁴¹K atomic gases provide a promising candidate for exploring angular stripe phases with tunable interactions according to the Feshbach resonance centered at the magnetic field $B_{0}=51.95$G Chen2020A . Near the Feshbach resonance, the intraspecies scattering lengths are approximately constant, and the interspecies one can be tuned in a wide range Lysebo2010F ; Tanzi2018F . By setting typical realistic parameters, the phase diagram is determined and the angular stripe phases can occupy a relatively large parameter space, as shown in Fig. 6. An angular density-density correlation function $g_{i}^{(2)}(\theta)\equiv\int_{0}^{2\pi}n_{i}(\varphi)n_{i}(\varphi+\theta)\mathrm{d}\varphi/\int_{0}^{2\pi}n_{i}^{2}(\varphi)\mathrm{d}\varphi$ can be introduced to estimate the visibility in density profile, with the angular density $n_{i}(\varphi)=\int_{0}^{\infty}r\mathrm{d}rn_{i}(r,\varphi)$, and the label $i=\uparrow,\downarrow$ for each spin component and null for the total density. In contrast to other phases, the novel angular stripe phases break the rotational symmetry and exhibit spatial density modulation in the angular direction. Two angular stripe phases feature considerable spatial modulation and contrast in the angular density-density correlation for both spin components as well as the total one, as shown in Figs. 5(c) and 5(d). This hallmark feature might be useful in directly probing the existence of the angular stripes in future experiments with ultracold atoms.

There are also other theoretical attempts and developments emphasizing the angular stripe phases in SOAM-coupled Bose gases. For example, by utilizing LG beams with higher-order orbital angular momenta, it is shown that angular stripe phases can be achieved in a wide window of experimentally accessible parameters with high visibility contrast Chen2020G ; Chiu2020V . Besides, A number of distinct quantum phases are identified according to the symmetry analysis, and a complex vortex molecule state is discovered, which plays an important role in the continuous phase transitions Duan2020S . For a ring-shaped BEC in the presence of SOAM coupling, the fine structures of angular stripe phases are explored Bidasyuk2022F .

It is well known that a two-component spin-balance Fermi gas becomes unstable in the presence of an arbitrary small attractive interaction. The resulting instability gives rise to a Bardeen-Cooper-Schrieffer (BCS) ground state with zero center-of-mass momentum. It is also well accepted that in a Fermi superfluid with spin-imbalance or Zeeman fields, exotic pairing states emerge, for instance, Fulde-Ferrell and Larkin-Ovchinnikov states FF ; LO , where the former pairing state carries finite center-of-mass momentum and the latter oscillates both in coordinate and momentum spaces. Whereas, these novel states are only stable in a very small parameter region, which is one of the reasons for the exclusiveness of their experimental observations. However, the experimental achievement of SO coupling in cold atoms offers a new platform to pursue these long sought-after states. For instance, the interplay of SO coupling, Zeeman fields and interactions provides an alternative mechanism to induce Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states Dong2013F ; Shenoy2013F ; Wu2013U ; Qu2013T ; Zhang2013T ; Chen2013I ; Liu2013T . Besides SO-coupling-induced FFLO states, topological band structures become accessible under SO coupling Huang2016E ; Wu2016R ; Zhang-16 ; xjliu1D ; xjliu2D , which lays the fertile ground for topological superfluid–for instance, aided by the Zeeman fields and interactions, a topological superfluid emerges in a 2D Fermi gas under the Rashba-type SO coupling Kane-08 ; tsfsolid1 ; tsfsolid2 ; tsfsolid3 ; tsfsolid4 ; soc1 ; Yi-11 .

Intuitively, SOAM coupling as an angular analog of the conventional SO coupling, two novel pairing states can be expected. One is the SOAM-coupling-induced pairing state with finite quantized angular momenta, which is nothing but a vortex state. The other is the SOAM-coupling-induced angular topological superfluid state, the analog of the topological superfluid state in a one-dimensional lattice gas under a one-dimensional SO coupling, where quantized angular momenta in the former play the role of discretized linear momenta of the latter. In this review, whether the above two expectational pairing states, i.e., SOAM-coupling-induced vortex and topological superfluid states, could be stabilized is clarified. In the following, the former issue will be addressed in Sec. V.2 and the latter in Sec. V.3.

As illustrated in Fig. 8, a two-component Fermi gas is confined in the $x-y$ plane, where the two ground states are labeled by $\uparrow$ and $\downarrow$, respectively. The two-photon Raman process is driven by two copropagating Raman beams carrying different orbital angular momenta $-l_{1}\hbar$ and $-l_{2}\hbar$ [see Figs. 8(a), 8(b)], and is characterized by the inhomogeneous Raman coupling $\Omega(r)$, two-photon detuning $\delta$ and a phase winding $e^{-2il\varphi}$. Here, the polar coordinate ${\bf r}=(r,\varphi)$ is taken. After a unitary transformation, the effective single-particle Hamiltonian again reduces to Eq. (17). As we can see that the Raman coupling $\Omega(r)$ and two-photon detuning $\delta$ serve as effective transverse and longitudinal Zeeman fields, respectively, which play crucial roles in stabilizing vortices. Different from the LG Raman beams used in previous experiments  Chen2018S ; Zhang2019G , Chen et al. propose Raman coupling as $\Omega(r)=\Omega_{0}e^{-2r^{2}/w^{2}}$ Chen2020G , with $\Omega_{0}$ the peak intensity and $w$ the beam waist, which can be experimentally realized Rumala-17 . Such a choice of the Raman beams is mainly on the basis that the LG Raman beams used in current experiments are suppressed over a considerable region near $r=0$, giving rise to an almost vanishing spin mixing effect in the vicinity, which is unfavorable for vortex formation essentially. Here, Raman lasers are assumed to operate at the tune-out wavelength Holmgren2012M ; Herold2012P ; Zhang2019G , leading to a vanishing diagonal ac-stark potential that is consistent with the experiment Zhang2019G . The external potential $V_{ext}(r)$ is chosen as an isotropic hard-wall box trap with a radius $R$, which offers a natural boundary.

It is helpful for understanding the mechanism of vortex formation by analysis of the single-particle properties before moving to the many-body calculation. Figs. 8(c) and (d) illustrate the impact of the two-photon detuning $\delta$ on the single-particle spectrum. At the resonance with $\delta=0$, the time-reversal symmetry gives rise to $E_{m,n}=E_{-m,n}$. Whereas away from the resonance at finite $\delta$, the breaking of the time-reversal symmetry results in a deformation of the Fermi surface. In the many-body setting when the attractive interaction is taken into account, pairing predominantly occurs between unlike spins with the same radial quantum number $n$ to maximize the overlap of radial wave functions. Thus, for the symmetric eigenspectrum under $\delta=0$, it is more favorable for two fermions with opposite angular quantum numbers ($m$ and $-m$), forming a Cooper pair with a zero total angular momenta. In contrast, under a finite $\delta$ with asymmetric eigenspectrum, the two fermions in a Cooper pair may possess different values of $|m|$, leading to a pairing state with a nonzero quantized angular momentum, which is the so-called vortex state. Such a mechanism for the vortex formation is analogous to that of the SO-coupling-induced Fulde-Ferrell states, where the interplay between SO coupling and Zeeman fields gives rise to the deformation of Fermi surfaces with broken time-reversal symmetry in the momentum space Wu2013U ; Qu2013T ; Zhang2013T ; Chen2013I ; Liu2013T .

Chen et al. confirm the above analysis by solving the many-body problem under the Bogoliubov–de Gennes (BdG) formalism Chen2020G . Specifically, the order parameter is written in the form of $\Delta({\bf r})=\Delta(r)e^{i\kappa\varphi}$, where the vorticity $\kappa=0$ ($\kappa\neq 0$) indicates the superfluid (vortex) state. With different $\kappa\in\mathbb{Z}$, the BdG equation and $\Delta({\bf r})$ are solved self-consistently with a fixed particle number. The ground state is then determined by comparing the free energies of vortex states with different $\kappa$, the usual superfluid state ($\kappa=0$), and the normal state ($\Delta=0$).

As shown in Fig. 9(a), Chen et al. give a typical phase diagram of the system with SOAM coupling in the $\Omega_{0}$–$\delta$ plane. For $\delta>0$ ($\delta<0$), vortex states with $\kappa=-1$ ($\kappa=1$) emerge, with the phase boundaries unchanged to the sign of $\delta$. At small $\Omega_{0}$ and $\delta$, the ground state is a usual superfluid (SF) with a zero vorticity $\kappa=0$. Under sufficiently large $\Omega_{0}$ and/or $\delta$, the free-energy difference between the SF and normal ($N$) states becomes vanishingly small, and hence the system enters the normal state. Remarkably, two vortex states emerge between SF and $N$ states. For example, with a fixed $\Omega_{0}/E_{F}=1.5$ [see Fig. 9(a)], the ground state is in the SF state under small detunings $\delta$, and becomes a fully-gapped vortex state ($V1$) beyond a critical value of $\delta$. Further increase of $\delta$ gives rise to a gapless vortex state ($V2$), whose bulk excitation gap closes. In Figs. 9(b), 9(c), free energies of different states are compared, as the phase diagram is traversed. Especially, for the case with $\Omega_{0}=0.5E_{F}$, the ground state remains vortexless for finite $\delta$, despite the deformation of the Fermi surface under SOAM coupling and effective Zeeman fields. This originates from the quantized nature of the angular momentum, and is on the sharp contrary to the SO-coupling-induced Fulde-Ferrell state which carries a nonzero, continuously varying center-of-mass momentum in the presence of SO coupling and Zeeman fields Dong2013F ; Shenoy2013F ; Wu2013U ; Qu2013T ; Zhang2013T ; Chen2013I ; Liu2013T .

Remarkably, as illustrated in Fig. 10, the SOAM-induced vortex state features a giant and tunable vortex-core size, similar to that in a BEC interacting with a microwave field Qin2016S . The vortex-core size, characterized by variations of the order parameter, is comparable to the waist $w$ of LG beams. Compared to the conventional vortex states in atomic Fermi superfluids, where changes in the vortex-core structure predominantly take place within a short length scale set by the interatomic separation Sensarma2006V ; Chien2006G , these SOAM-coupling-induced vortex states, with tunable size and core structure, provide unprecedented experimental access to topological defects in Fermi superfluids.

Different from Chen et al.’s configuration, Wang et al. investigate a similar vortex-forming scheme under SOAM coupling Wang2021E . Most strikingly, they predict that an unprecedented vortex state, which is an angular analog of SO-coupling-induced Larkin-Ovchinnikov state, to occur.

Nevertheless, for the inevitable heating introduced by the Raman process, it is difficult to cool a realistic Fermi gas with SOAM coupling below the superfluid temperature heating , which leads to the exclusiveness of SOAM-coupling-induced novel states. Instead, concerning the persistence of dressed molecules above the critical temperature in Fermi gases with SO coupling Williams2013R ; Fu2014P , it is reasonable to expect that molecular states in a SOAM-coupled Fermi gas should be readily accessible under typical experimental conditions. Based on the above consideration, Han et al. studied the two-body bound states in a SOAM-coupled quantum gas of fermions very recently Han2022M . They identify the condition for the emergence of molecular states with finite total angular momenta and propose to detect the molecules according to the radio-frequency spectroscopy. As the molecular states can form above the superfluid transition temperature, Han et al. offer an experimentally more accessible route toward the study of the underlying pairing mechanism under SOAM coupling.

As discussed in Sec. V.1, it is expected that SOAM coupling can induce a topological superfluid with the help of Zeeman fields and interactions. While, implemention of SOAM coupling relies on the spatial dependence of the LG beams, which gives rise to the dimensionality of atomic gases must higher than one; whereas, it is also realized that the Fermi superfluid becomes gapless and losses its topological features when its spatial dimensionality higher than that of SO coupling Wu2013U ; yisoc . Such that the one-dimensional nature of the SOAM coupling (coupling only occurs along the azimuthal direction) imposes a stringent constraint on the stability of an angular topological superfluid. Whether such a topological superfluid can also be stabilized under SOAM coupling should be verified. It is shown that the stability of a fully gapped angular topological superfluid survives the constraint above, provided that the radial motion of atoms is sufficiently suppressed and then the topological gap is not closed Chen-22 .

The survives of the angular topological superfluid can be understood through the following analysis. As shown in Eq. (17), the Raman coupling and the diagonal ac-stark potential can be written as $\Omega(r)=\Omega_{0}I(r)$ and $\chi(r)=\chi_{0}I(r)$, consistent with configurations of current experiments on SOAM coupling Chen2018S ; Zhang2019G . Here, $\Omega_{0}$ is the effective SOAM-coupling strength, $\chi_{0}$ is the trapping strength of the diagonal ac-stark potential, and $I(r)=(\sqrt{2}r/w)^{2l}e^{-2r^{2}/w^{2}}$ is the spatial intensity profile of LG lasers, with $w$ the beam waist. Different from the SOAM-induced vortex state in Sec. V.2, here, $\chi(r)$ provides an extra confinement potential, which plays an important role in stabilizing the topological superfluid. For a confinement that is sufficiently tight along the radial direction, the radial degrees of freedom of the atoms are frozen, and the remaining quantized angular motion is then well-captured by an effective one-dimensional model with discretized modes. Intuitively, such a scenario occurs when the trap depth $\chi_{0}$ is so large that the radial excitation energy ($\sim\sqrt{|\chi_{0}|\hbar^{2}/(mw^{2})}$) becomes much larger than any other relevant energy scales of the system. The atoms are then localized near $r_{0}=\sqrt{l/2}w$ in the radial direction and the system reduces to an effective one-dimensional model along the angular direction.

The above analysis is confirmed through the BdG approach Chen-22 . As illustrated in Fig. 11, Chen et al. show the Bogoliubov quasiparticle spectrum in a sufficiently deep ac-stark potential with $\chi_{0}/\epsilon_{0}=-8$ in the unit of energy $\epsilon_{0}=\pi^{2}\hbar^{2}/(2Mr^{2}_{0})$. For the case with $\delta=0$, the ground state always emerges at $\kappa=0$. Here, consistent with Sec. V.2, the vorticity $\kappa=0$ ($\kappa\neq 0$) denotes the superfluid (vortex) state. By increasing the coupling strength $\Omega_{0}$, the Bogoliubov spectrum undergoes a gap-closing and re-opening process, reminiscent of that of a topological phase transition. Specifically, the Bogoliubov quasiparticle excitation is fully gapped under small $\Omega_{0}$ [Fig. 11(a)], and then becomes gapless at a critical $\Omega^{c}_{0}/\epsilon_{0}\approx 0.18$ [Fig. 11(b)], and finally is again fully gapped for further increasing $\Omega_{0}$ [Fig. 11(c)]. The process of gap closing and re-opening is further demonstrated as a topological phase transition by calculating the Zak phase of the one-dimensional effective Hamiltonian Chen-22 . Besides, it is also demonstrated that the angular topological superfluid is stabilized when the ac-stark potential becomes sufficiently large.

Building upon the topological superfluid state above, an exotic topological vortex state can be induced by taking the two-photon detuning $\delta$ into account, which deforms the Fermi surface. As shown in Figs. 12(a)–12(c), the free energy is generically asymmetric with respect to $\kappa=0$ at a finite $\delta$. The asymmetry becomes more apparent with increasing $\Omega_{0}$, until the ground-state order parameter eventually acquires a finite phase with $\kappa\neq 0$. Intriguingly, such a transition into the vortex state is topological. As demonstrated in Chen-22 , while the Fermi-surface deformation is manifested as the asymmetric spectral shape with respect to $m=0$, the closing and re-opening of the energy gap persist. Indeed, after the gap is reopened, the angular momentum of the ground state jumps from $\kappa=0$ to $\kappa=1$. The ground state simultaneously becomes topological, which is confirmed by the Zak-phase calculation. Conceptually, such an exotic topological vortex state is the angular version of the topological Fulde-Ferrell state under the conventional SO coupling Qu2013T ; Zhang2013T ; Chen2013I ; Liu2013T .

In addition, the topological vortex leaves a direct signature in the angular-momentum-space density profile, as illustrated in Figs. 12(d)–(f). The density profile of the minority spin species exhibits a dip close to $\kappa/2$ only in the topological vortex state [Fig. 12(f)], since the spin polarization in the vortex state is a direct result of the two-photon detuning, which plays the role of an effective Zeeman field. Similar signatures have been identified in the topological Fulde-Ferrell state under SO coupling.

The origin of angular topological superfluid comes from the interplay of interactions, Zeeman fields and SOAM couplings. As topological superfluids are believed to host Majorana zero modes at boundaries, once a boundary is created, for instance, by shining a strong laser beam to break the ring geometry of the ac stark potential, Majorana zero modes should be observed.