Collectively induced many-vortices topology via rotatory Dicke quantum phase transition

Dicke superradiance (SR) is a fundamental effect that is the collective spontaneous emission of radiation by an ensemble of coherently excited atoms [1, 2]. The ensemble atoms are coupled together by a common electromagnetic field within the extent of the pump's coherence length, and are excited to phase synchronized multi-atom states. Cooperative rapid release of the stored energy leads to a so-called superradiant pulse with an intensity proportional to the square of the number of the atoms in the ensemble. The onset of SR is triggered by vacuum fluctuations in an empty photon mode, which stimulates strongly directional decay of atoms to this mode [3, 4] with a probability proportional to the exponential of the ensemble length along the emission direction [4].

In addition to dynamical SR, equilibrium SR can be studied as a first order quantum phase transition (PT) from a normal to cooperative radiation phase in finite systems [5]. Dicke SR quantum PT is proposed to occur in trapped, laser driven atomic Bose–Einstein condensates (BECs) in high finesse optical cavities [6]; and it is experimentally demonstrated [7]. Above a critical driving strength, SR PT happens simultaneously with the spatial fragmentation of the BEC into a periodical pattern. The collectivity in superradiant PT can be heralded by bi-partite quantum entanglement [8], single-mode non-classicality [9], and atomic (spin) squeezing [10] that can induce N-particle entanglement [11]. Collisions between atoms in BECs can also contribute to quantum entanglement and its interplay with the SR PT.

When an N atom BEC is stirred, it does not rotate until a total of $N{\hslash }$ orbital angular momentum (OAM) is transferred to it [12]. Instead of stirring, one can also control atomic center of motion to transfer OAM to a BEC using optical fields [13–16]. Such schemes operate at optical frequencies so that single atom recoil energy (${\hslash }{\omega }_{{\rm{R}}}$) exceeds the mean atom–atom interaction energy (${U}_{\mathrm{int}}/N$). Though they cannot transfer the condensate to a rotatory state collectively, they can induce partial rotation on a BEC and yield vortex nucleation. In order to transfer sufficient OAM to generate vortices in a BEC, regular optical pumping schemes need two or more laser beams [13–18]. A BEC, which is pumped by a Laguerre–Gaussian (LG) beam with OAM, radiates by stimulated emission into a second laser in Gaussian (G) mode without any OAM [15]. Without using the G beam, high probability of emission into the pump beam relative to emission of zero OAM photons results in negligible OAM transfer into the BEC, which leads to weakly populated vortex side-mode.

In the present manuscript, we consider only a single LG pump beam, but with an intensity (or the BEC-field coupling) above a critical threshold at which SR with zero OAM occurs. Our idea is a generalization of the physical mechanisms behind Rayleigh [19] and Raman superradiance [20–22] from an off-resonant driven BEC in free space, which are the creation of density and spin polarization gratings, respectively [4, 23–25]. In our case recoiled condensate atoms form a ring grating, due to the conservation of linear momentum and interference of the recoiled matter waves. In addition, due to the OAM transfer, the grating is not frozen but rotating. The grating coherently overlaps with the condensate at rest. The contrast gets deeper with the more scattering, and in turn the scattered intensity is proportional to the square of the density contrast or to the square of the number of active atoms. Using large amount of OAM transfer at the threshold of 'rotatory' SR PT, weakly populated rotational symmetry broken field and vortex side modes can be coherently amplified which leads to significant nucleation of a 'vortex molecule' at the edge of the grating. The collective spontaneous emission character of the SR makes it fundamentally different than the known optical pumping schemes to transfer OAM to BEC, as the intensity of emission scales quadratically with the number of atoms inside the BEC, leading to larger amount of OAM transfer. From practical perspective SR scheme is also different, as it only uses a single pump laser beam. Besides, SR scheme can lead to complete transfer of BEC to rotational state by depleting the zero momentum condensate mode, in contrast to partial transfer achieved by regular two pump optical schemes.

Present contribution generalizes the case of superradiant generation of a single axial vortex along a cigar shaped BEC [26] to the superradiant generation of multiple vortices in a disk geometry and relaxes the condition of frozen spatial mode for the radiation field. Both the normal SR (without OAM transfer) and rotatory SR (with OAM transfer) can exhibit distinct behaviors in the small and large recoil regimes [26, 27]. When a cigar shaped BEC is pumped with an LG beam along its long axis, it is found that normal SR can either precede the rotatory SR in the small recoil regime or it can be completely suppressed by the rotatory SR in the large recoil regime when the LG pump width larger than the width of vortex side mode [26]. As the normal SR cannot lead to nucleation of vortices, we focus here only on rotatory SR. For brevity we use the term SR threshold for rotatory SR. Below the SR threshold, our scenario falls into the same case of single LG pump where emission of G-mode radiation, hence OAM transfer or vortex nucleation, is negligibly small; while above the SR threshold, cooperative emission of BEC atoms leads to enhancement of OAM transfer and vortex nucleation becomes significant. The rotatory SR threshold can be taken as an 'effective' vortex nucleation threshold in our scheme; though SR threshold and the critical threshold of vortex nucleation are different in principle and SR does not change the critical rotation threshold for vortex nucleation.

Our physical system is depicted in figure 1, where a pancake BEC is driven by an LG beam with winding number ℓ. Dynamical simulations of the model system show that single-charge vortices at the arms of the ℓ-fold symmetric structures becomes significant in the density profile of the BEC after SR threshold. We determine the intensity profile of the superradiant scattered pulse as well. Reorganization of the condensate into ℓ-fold rotational symmetric structures resembles SR induced spatial ordering of driven BEC in high finesse optical cavity [6, 7, 28–30], where the ordering arises due to the collective transfer of linear momentum via Dicke SR. Here, ℓ-fold rotational ordering and quantized vorticity emerge due to the OAM transfer to the BEC which is enhanced via rotatory SR. Breaking of the cylindrical symmetry into the ℓ-fold symmetry and OAM transfer to the BEC become significant at the same critical pump strength. Quantized rotations (single-charge vortices) emerge at the edges of the ℓ-fold rotational ordered form. We tested the SR nature of the scattered radiation by verifying that the temporal width of the scattered pulse peak is inversely proportional to the number of atoms [31, 32]. In addition, peak intensity scales quadratically with the number of condensate atoms. Furthermore, we compared the dependence of the SR threshold on the radiation frequency and atom-field coupling to the case of SR of BEC in optical cavity [28] and find that they behave similarly.

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This paper is organized as follows. In section 2 we present our model system, where we introduce and discuss the corresponding Hamiltonian and the equations of motion (EOM) in two subsections. Main results of the numerical simulations and their discussions are presented in section 3. Critical pump strength is discussed in more detail in section 4. We conclude in section 5.

We examine the dynamics of a pancake BEC of N atoms [33], as shown in figure 1, which is illuminated with a strong LG laser of optical frequency ${\omega }_{{\rm{L}}}$ along the symmetry axis z. The pump beam is well collimated with radial width w_L and propagates along the z-axis with a wave-vector k_L. LG pumpmode is assumed to have a normalized spatial profile [34]

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{L}}}({\bf{r}})=\displaystyle \frac{1}{\sqrt{{\ell }!\pi {V}_{{\rm{L}}}}}{(r/{w}_{{\rm{L}}})}^{{\ell }}{{\rm{e}}}^{-{r}^{2}/2{w}_{{\rm{L}}}^{2}}{{\rm{e}}}^{{\rm{i}}{\ell }\phi }{{\rm{e}}}^{{{ik}}_{{\rm{L}}}z}\end{eqnarray} \tag{ 1 }$$

with zero azimuthal mode number and positive radial mode (winding) number ${\ell }\gt 0$. LG beam carries ${\ell }{\hslash }$ OAM per photon. Here, ${V}_{{\rm{L}}}={w}_{{\rm{L}}}^{2}{z}_{{\rm{L}}}$, with z_L being the mode quantization length. The electric field operator of the LG beam is given by

$$\begin{eqnarray}&&{\hat{{\bf{E}}}}_{{\rm{L}}}({\bf{r}},t)=-{\rm{i}}{{\bf{e}}}_{{\rm{L}}}{{ \mathcal E }}_{{\rm{L}}}{{\rm{\Phi }}}_{{\rm{L}}}(r,\phi ,z){\hat{a}}_{{\rm{L}}}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}t}+{\rm{h}}.{\rm{c}}\mathrm{.},\end{eqnarray} \tag{ 2 }$$

where ${{\bf{e}}}_{x}$ is the unit polarization vector of the laser, ${{ \mathcal E }}_{{\rm{L}}}=\sqrt{{\hslash }{\omega }_{{\rm{L}}}/2{\epsilon }_{0}}$, with ${\epsilon }_{0}$ being the vacuum permittivity, and ${\hat{a}}_{{\rm{L}}}$ is the annihilation operator for the pump laser photons.

Similar to the end-fire modes emitted out of a pencil shaped condensate [4, 35], scattered radiation off a pancake BEC would consist of radial edge-fire modes, in the x–y plane shown in figure 1. End-fire modes are emitted along the radial directions since the BEC is tightly confined along the z-axis. When the x–y profile of the scattered radiation has a smaller winding number than the pump, OAM is transferred to the BEC.In the case of cooperative (SR) scattering, OAM is transferred to the BEC at once through an exponentially fast and collectively enhanced spontaneous emission [4]. We express the electric field operator of the scattered radiation as

$$\begin{eqnarray}&&{\hat{{\bf{E}}}}_{{\rm{s}}}({\bf{r}},t)=-{\rm{i}}{{\bf{e}}}_{{\rm{s}}}{{ \mathcal E }}_{{\rm{s}}}\hat{{\rm{\Phi }}}(r,\phi ,z;t){{\rm{e}}}^{{\rm{i}}{\bf{k}}\cdot {\bf{r}}}+{\rm{h}}.{\rm{c}}\mathrm{.},\end{eqnarray} \tag{ 3 }$$

where ${{\bf{e}}}_{{\rm{s}}}$ is the unit polarization vector of the scattered field; ${{ \mathcal E }}_{{\rm{s}}}=\sqrt{{\hslash }{\omega }_{{\rm{e}}}/2{\epsilon }_{0}};$ ${\omega }_{{\rm{e}}}$ is the carrying end-fire mode frequency and $\hat{{\rm{\Phi }}}({\bf{r}})$ is the annihilation field operator for the scattered radiation. We include the factor $\exp (-{\rm{i}}{\omega }_{{\rm{e}}}t)$ into the field operator $\hat{{\rm{\Phi }}}({\bf{r}})$.

2.1. Hamiltonian

Assuming the LG beam is far off resonant from the atomic transition frequency, excited state can be adiabatically eliminated to describe the many body dynamics using an effective Hamiltonian [36, 37], $\hat{{ \mathcal H }}$, which can be expressed as

$$\begin{eqnarray}&&\hat{{ \mathcal H }}={\hat{{ \mathcal H }}}_{0}+{\hat{{ \mathcal H }}}_{\mathrm{col}}+{\hat{{ \mathcal H }}}_{f}+{\hat{{ \mathcal H }}}_{{af}}+{\hat{{ \mathcal H }}}_{{af}}^{(2)},\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{0}=\int {{\rm{d}}}^{3}{\bf{r}}{\hat{{\rm{\Psi }}}}^{\dagger }({\bf{r}}){\hat{H}}_{g}\hat{{\rm{\Psi }}}({\bf{r}}),\end{eqnarray} \tag{ 5 }$$

is the Hamiltonian of the external (motional) states of the BEC;

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{\mathrm{col}}={g}_{{\rm{s}}}\int {{\rm{d}}}^{3}{\bf{r}}{\hat{{\rm{\Psi }}}}^{\dagger }({\bf{r}}){\hat{{\rm{\Psi }}}}^{\dagger }({\bf{r}})\hat{{\rm{\Psi }}}({\bf{r}})\hat{{\rm{\Psi }}}({\bf{r}}),\end{eqnarray} \tag{ 6 }$$

is the interatomic collision Hamiltonian with g_s being the s-wave scattering strength;

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{{f}}={\hslash }{\omega }_{{\rm{e}}}\int {{\rm{d}}}^{3}{\bf{r}}{\hat{{\rm{\Phi }}}}^{\dagger }({\bf{r}})\hat{{\rm{\Phi }}}({\bf{r}}),\end{eqnarray} \tag{ 7 }$$

is the Hamiltonian of the scattered field;

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{{af}}={\hslash }{g}_{a}\int {{\rm{d}}}^{3}{\bf{r}}{\hat{{\rm{\Psi }}}}^{\dagger }({\bf{r}}){\hat{{\rm{\Phi }}}}^{\dagger }({\bf{r}}){\alpha }_{{\rm{L}}}{{\rm{\Phi }}}_{{\rm{L}}}({\bf{r}})\hat{{\rm{\Psi }}}({\bf{r}})+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 8 }$$

is the Hamiltonian of the scattering of the pump beam off the atoms with ${\alpha }_{{\rm{L}}}$ being the c-number parametric approximation for the ${\hat{a}}_{{\rm{L}}}$; and

$$\begin{eqnarray}&&{\hat{{ \mathcal H }}}_{{af}}^{(2)}=2{\hslash }{g}_{a}\int {{\rm{d}}}^{3}{\bf{r}}{\hat{{\rm{\Psi }}}}^{\dagger }({\bf{r}}){\hat{{\rm{\Phi }}}}^{\dagger }({\bf{r}})\hat{{\rm{\Phi }}}({\bf{r}})\hat{{\rm{\Psi }}}({\bf{r}}),\end{eqnarray} \tag{ 9 }$$

is the Hamiltonian describing the interaction of the scattered field with the atoms. The first-quantized Hamiltonian of the BEC in a harmonic potential V(r), with radial and axial trapping frequencies ${\omega }_{r}$ and ${\omega }_{z}$, respectively, is given by ${\hat{H}}_{g}=-{{\hslash }}^{2}{{\rm{\nabla }}}^{2}/2m+V(r)$. The effective strength of the atom-field interaction is denoted by g_a.

In general, the interaction strengths in equations (8) and (9) are different. Instead of g_a, we could write g_b in equation (9). They are given by ${g}_{a}={g}_{{\rm{L}}}{g}_{{\rm{e}}}/{{\rm{\Delta }}}_{a}$, and ${g}_{b}={g}_{{\rm{e}}}^{2}/{{\rm{\Delta }}}_{a}$ [7, 28, 36, 37], where ${{\rm{\Delta }}}_{a}={\omega }_{a}-{\omega }_{{\rm{L}}}$ is the detuning of the pump frequency from the atomic transition frequency ${\omega }_{a};$ ${g}_{{\rm{L}}}={d}_{{\rm{L}}}{{ \mathcal E }}_{{\rm{L}}}/{\hslash }$ and ${g}_{{\rm{e}}}={d}_{{\rm{e}}}{{ \mathcal E }}_{{\rm{s}}}/{\hslash }$ are the single photon Rabi frequencies for the laser and scattered field, respectively. For simplicity we take ${{\bf{e}}}_{{\rm{L}}}={{\bf{e}}}_{{\rm{s}}}$. Different polarizations, including the radial one, can be synthesized using linearly polarized LG beams [38]. Accordingly, we can take d_L = d_e, where d_L and d_e are the matrix elements of the atomic dipole moment operator components along ${{\bf{e}}}_{{\rm{L}}}$ and ${{\bf{e}}}_{{\rm{s}}}$, respectively. Due to the energy conservation in Rayleigh scattering, and small atomic side mode energies relative to the pump laser energy, pump and scattered field are in quasi-resonance, ${\omega }_{{\rm{L}}}\approx {\omega }_{{\rm{e}}}$. In fact, frequency dependence of the coupling constants is neglected over wider range of field detuning ${\rm{\Delta }}={\omega }_{{\rm{e}}}-{\omega }_{{\rm{L}}}$, which is the case in SR of BEC in optical cavity [7, 28, 36, 37]. Under these conditions describing our physical system, we find that g_a = g_b. We scale the BEC wave-function ${\rm{\Psi }}({\bf{r}})$ and the scattered field ${\rm{\Phi }}({\bf{r}})$ with $\sqrt{N}$ [28]. Parameters $\eta ={g}_{a}\sqrt{N}| {\alpha }_{{\rm{L}}}|$ and ${U}_{0}={g}_{a}N$ reflect the strength of the ${\hat{{ \mathcal H }}}_{{af}}$ and ${\hat{{ \mathcal H }}}_{{af}}^{(2)}$ atom-field couplings, respectively on the dynamics of scaled BEC and scattered field profiles. The physical condition of g_a = g_b hence has no restrictive effect on the control of the dynamics in terms of independent parameters η and U₀.

The dynamics of the field operator for the scattered pulse $\hat{{\rm{\Phi }}}({\bf{r}})$ and the field operator of the BEC $\hat{{\rm{\Psi }}}({\bf{r}})$ will be determined by the EOM.

2.2. Equations of motion

We evolve $F(x,y)$ and ${\rm{\Psi }}({\bf{r}})=\psi (x,y)Z(z)$ using the EOM given in the appendix in dimensionless time scaled by ${\omega }_{{\rm{R}}}$. In figure 2, macroscopic OAM transfer to the BEC follows the emission peak (at $t\simeq 1.45/{\omega }_{{\rm{R}}}$). In figure 2(a), the $\langle {\hat{L}}_{z}\rangle$ can have values different than pump LG photon's OAM due to collective nature of the superradiant scattering. We calculated the OAM of the scattered field $F(x,y)$. We find that it always remains close to zero (∼10^–5). The difference between the BEC sample OAM and the pump LG photon OAM is not due to the scattered field OAM. BEC atoms act as a single object and scatter pump photons collectively. Each absorbed pump photon transfers ${\ell }{\hslash }$ amount of OAM to the BEC. BEC OAM is determined by the total contribution of a fraction of pump photons. The OAM that can be received collectively by the N-atom BEC must be an integer multiple of ℏ per atom so that we can express it as ${nN}{\hslash }$, where n is an integer. The fraction of absorbed pump photons is then given by ${nN}/{\ell }$. Accordingly the number of scattered photons per atom, n/ℓ, can be different than one in the collective scattering.

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The temporal width ${\rm{\Delta }}\tau$ of the peak in figure 2(b) exhibits the characteristic superradiant behavior [31, 32]. When we increase the number of atoms by n times, temporal width of the peak shrinks to ${\rm{\Delta }}\tau /n$, as can be seen in figure 3. In addition, the peak intensitygrows quadratically with the number of BEC atoms, ∝N², which is another signature of SR [31, 32]. Hence, we conclude that the scattering displays the superradiant character.

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In figure 4, we depict the time evolution of the spatial profile for both the BEC and the scattered field. In the close proximity of the scattering peak, figure 2(b), azimuthal symmetric BEC profile reorganizes to 3-fold rotational symmetry, figure 4(b). This happens even though both BEC and LG pump intensity have cylindrical symmetry initially. The winding number of the laser is transformed to the 3-fold rotational symmetry. Three single-charge vortices appear at the arms of the 3-fold rotational symmetric density profile. By enclosing small spatial regions around the three vortex positions, we calculate the expectation value of the ${\hat{L}}_{z}$ operator and find that they carry a single charge. In figure 4(d), we observe the expansion of the three vortices when the laser field isturned off. Remaining OAM $({m}_{\mathrm{tot}}-{\ell }){\hslash }$, per atom, is distributed to the body of the BEC, where ${m}_{\mathrm{tot}}=\langle {\rm{\Psi }}({\bf{r}})| {\hat{L}}_{z}| {\rm{\Psi }}({\bf{r}})\rangle$ is the total OAM of the BEC (per atom, due to the scaling of the ${\rm{\Psi }}({\bf{r}})$.

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In our simulations, ℓ-fold rotational symmetric structures are significantin a BEC which is pumped with an LG laser of winding number ℓ only when $\eta \gt {\eta }_{{\rm{c}}}$, as exemplified in figure 5. Single-charge vortices emerge at the ends of ℓ-fold rotational symmetric structures,and shown in figure 5 when the pump is switched off. When the OAM transferred to BEC is smaller than the ℓ-fold symmetry, ${m}_{\mathrm{tot}}\lt {\ell }$, ℓ-fold symmetric structures emerge again. However, in this case no single-charge vortex appear at the arms The whole OAM is distributed to the body of the BEC. No quantized vorticity can be identified in the body of the BEC. In our case of pure rotatory SR scattered field OAM cannot be used to probe BEC OAM, as it has no OAM. On the other hand, it could be used to probe cylindrical symmetry breaking. Scattered field intensity profile carries information on the cylindrical symmetry breaking of BEC as figures 4 and 5 demonstrate. In our treatment there is no stimulating secondary Gaussian beam that would impose a specific profile on the scattered field. Accordingly, we do not make frozen spatial profile or few mode expansion approximations but we let the spatial profiles of the scattered field and the BEC evolve consistently by the coupled EOM. The interplay in their mutual evolution allows us to use scattered field profile as a probe for BEC profile. The intensity profile of the scattered field exhibits the similar dynamical development as symmetry breaking in BEC. The scattered field does not show ℓ-fold ordering patterns if $\eta \lt {\eta }_{{\rm{c}}}$. Hence it can be used as an alternative probe for the dynamics of rotational self-organization and symmetry breaking in BEC.

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In figure 6, we observe that OAM transfer to the BEC can be achievedby using a single LG pump only above a critical pump strength ${\eta }_{{\rm{c}}}$ depending on the values of the parameters U₀, g_s and κ, similar to the Dicke SR of BEC in optical cavity [28]. This behavior can be compared with figure 7. The 3-fold ordering in the BEC profile start to grow with η visibly after ${\eta }_{{\rm{c}}}$ as can be seen in figure 7, in parallel with the figure 6, showing increasing OAM transfer to the BEC above the threshold ${\eta }_{{\rm{c}}}$ to macroscopically large values³ .

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In the absence of damping, $\kappa =0$, we clearly observe the staircase of plateaus at $\langle {\hat{L}}_{z}\rangle =1$ and $\langle {\hat{L}}_{z}\rangle =2$ horizontal lines. This resistance against the OAM transfer is due to the irrotational nature—unless integer multiplies of $N{\hslash }$ OAM is gained [12]—of the BEC. When we introduce damping, these plateaus tend to disappear.

In figure 6, we observe a smooth increase in the OAM transferred to the BEC.Let us note that when we say OAM transfer to the BEC with non-integer $\langle {\hat{L}}_{z}\rangle$ values, we mean that BEC can have transient OAM values in real time dynamics only for a short time unlike quantized vorticity. We emphasize that, for $\eta \lt {\eta }_{c}$, BEC does not have OAM even for a short time. Despite the smooth increase in $\langle {\hat{L}}_{z}\rangle$, the visibility of the ℓ-fold rotational ordering appears quite sharply just after $\eta \gt {\eta }_{{\rm{c}}}\simeq 2.25$. In figure 7, we observe that between $\eta =1.50\mbox{--}2.25$, visibility of the 3-fold ordering remains almost constant and low, while for $\eta \gt {\eta }_{{\rm{c}}}\simeq 2.25$, visibility of 3-fold ordering grows towards $\eta \sim 5$, in parallel with the behavior in figure 6. Therefore, onset of the OAM transfer to the BECdetermines when 3-fold rotational symmetric ordering becomes significant (see footnote 3).

Even though the complexity of the functions determining the superradiant emission and the recoiled BEC prevents us from establishing a simple analytical expression for the threshold ${\eta }_{{\rm{c}}}$, we can make numerical comparison with the known behavior of the SR threshold with system parameters [6, 7, 28]. In figure 8, we present logarithmic plots showing behaviors of ${\eta }_{{\rm{c}}}$ with U₀ and Δ. We observe that ${\eta }_{{\rm{c}}}\sim {{\rm{\Delta }}}^{1/2}$ and ${\eta }_{{\rm{c}}}\sim {U}_{0}^{1/2}$, for non-interacting BEC, as expected behaviors by the analytical predictions of SR threshold [28]. According to [28], ${\eta }_{{\rm{c}}}\sim {U}_{0}^{1/2}$ behavior should be observed for the large values of U₀ and hence we consider a domain of U₀ in figure 8 so that ${\rm{\Delta }}=1$ can be neglected in this regime.

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In summary, we numerically observe that when a pancake shaped BEC is pumped with a single LG beam of winding number ℓ, it significantly reorganizes to ℓ-fold rotational symmetry above the critical pump strength for rotatory SR, ${\eta }_{{\rm{c}}}$. For $\eta \lt {\eta }_{{\rm{c}}}$, OAM transfer to the BEC is negligible. When $\eta \gt {\eta }_{c}$, both OAM transfer (see footnote 3) and ℓ-fold rotational symmetry become significant simultaneously. For excess OAM transfer, ℓ number of single-charge vortices appear at the arms of the ℓ-fold symmetric structures. In the time evolution, these structures develop visibly after a sharp SR scattering peak. The temporal width and the maximum of the peak conform to the SR characteristics [31, 32].

In this contribution, we limit our discussion to free space SR and do not consider the case of trapping the SR scattered pulse in a cavity in the x–y directions. In order to consider confined fields configuration, one can envision a system where a high finesse optical cavity is placed around the BEC [7]. In such a case one can imagine that the onset of the SR phase could also induce structural PT in translational degrees of freedom leading to a crystalline structure in the BEC, or a 'supersolid'. In other words, before SR induced rotational PT, there could be a supersolid phase. On the other hand, one can also imagine that, due to different thresholds for rotational and translational symmetry breaking, depending on system parameters a rotational transition can happen first. In addition, translational symmetry breaking could accompany the rotational ordering, too. In our opinion such a rich optical and structural phase diagram requires a separate study, including detailed examination of the phase boundaries and the onset of normal and rotatory SR regimes with second quantized approaches. Different mechanisms to generate supersolid vortex crystals are proposed in literature such as in Rydberg-Dressed BECs where supersolid and vortex lattice phases are found [44]. Our choice of free space SR case is only for clarity and simplicity to eliminate translational symmetry breaking to focus on generation of multiple vortices topologies.

Our results can be significant to probe vortex nucleation and rotational symmetry breaking dynamics using the scattered radiation, and as an alternative single optical pumping scheme for rapid and large OAM transfer to BEC. We hope our scheme can inspire further investigation directions such as cavity confinement effects [7], supersolidity and bosonic vortex molecules and crystals [44], and collective excitations [45].

The authors thank L You and L Deng for illuminating discussions and acknowledge the financial support from TUBITAK Projects Grant No. 112T927 and 114F170. ÖEM acknowledge support from TUBITAK Project Grant No. 112T974.

EOMs transforms to the following in the scaled form, where time and energy (frequency) scale is the recoil frequency, which is taken to be the same with the radial trapping frequency ${\omega }_{{r}}$, determining the spatial scaling

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{{\rm{d}}F(x,y)}{{\rm{d}}t}=(-{\rm{i}}\kappa +{\rm{\Delta }})F(x,y)+\eta | \psi (x,y){| }^{2}{F}_{{\rm{L}}}(x,y)\int | Z(z){| }^{2}{\rm{d}}z+2{U}_{0}F(x,y)| \psi (x,y){| }^{2}\int | Z(z){| }^{2}{\rm{d}}z\end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}\psi (x,y)}{{\rm{d}}t} & = & \eta ({F}_{{\rm{L}}}(x,y){F}^{* }(x,y)+{F}_{{\rm{L}}}^{* }(x,y)F(x,y))\psi (x,y)\displaystyle \int | Z(z){| }^{2}{\rm{d}}z\\ & & +2{U}_{0}\psi (x,y)| F(x,y){| }^{2}\displaystyle \int | Z(z){| }^{2}{\rm{d}}z+{\bar{g}}_{{\rm{s}}}| \psi (x,y){| }^{2}\psi (x,y)\displaystyle \int | Z(z){| }^{4}{\rm{d}}z\\ & & -\displaystyle \frac{1}{2}[({\partial }_{x}^{2}+{\partial }_{y}^{2})\psi (x,y)]\displaystyle \int | Z(z){| }^{2}{\rm{d}}z-\displaystyle \frac{1}{2}\psi (x,y)\displaystyle \int {Z}^{* }(z){\partial }_{z}^{2}Z(z){\rm{d}}z\\ & & +\displaystyle \frac{1}{2}({x}^{2}+{y}^{2})\psi (x,y)\displaystyle \int | Z(z){| }^{2}{\rm{d}}z+\displaystyle \frac{1}{2}{\left(\tfrac{{\omega }_{z}}{{\omega }_{{r}}}\right)}^{2}\psi (x,y)\displaystyle \int {z}^{2}| Z(z){| }^{2}{\rm{d}}z,\end{eqnarray} \tag{ A2 }$$

$$\begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}{Z}(z)}{{\rm{d}}t} & = & \eta Z(z)\displaystyle \int (| \psi (x,y){| }^{2}{F}_{{\rm{L}}}(x,y){F}^{* }(x,y)+| \psi (x,y){| }^{2}{F}_{{\rm{L}}}^{* }(x,y)F(x,y)){{\rm{d}}}^{2}r\\ & & +2{U}_{0}Z(z)\displaystyle \int | \psi (x,y){| }^{2}| F(x,y){| }^{2}{{\rm{d}}}^{2}r+{\bar{g}}_{{\rm{s}}}| Z(z){| }^{2}Z(z)\displaystyle \int | \psi (x,y){| }^{4}{{\rm{d}}}^{2}r\\ & & -\displaystyle \frac{1}{2}Z(z)\displaystyle \int {\psi }^{* }(x,y)({\partial }_{x}^{2}+{\partial }_{y}^{2})\psi (x,y){{\rm{d}}}^{2}r-\displaystyle \frac{1}{2}{\partial }_{z}^{2}Z(z)\displaystyle \int | \psi (x,y){| }^{2}{{\rm{d}}}^{2}r\\ & & +\displaystyle \frac{1}{2}Z(z)\displaystyle \int ({x}^{2}+{y}^{2})| \psi (x,y){| }^{2}{{\rm{d}}}^{2}r\\ & & +\displaystyle \frac{1}{2}{\left(\tfrac{{\omega }_{z}}{{\omega }_{r}}\right)}^{2}{z}^{2}Z(z)\displaystyle \int | \psi (x,y){| }^{2}{{\rm{d}}}^{2}r.\end{eqnarray} \tag{ A3 }$$