# High-dimensional classically entangled light from a laser

Yijie Shen1,2,3, Isaac Nape1, Xilin Yang4, Xing Fu3,5, Mali Gong3,5, Darryl
Naidoo1,6, and Andrew Forbes1 1School of Physics, University of the
Witwatersrand, Private Bag 3, Wits 2050, South Africa
2State Key Laboratory of Precision Measurement Technology and Instruments,
Department of Precision Instrument, Tsinghua University, Beijing 100084, China
3Optoelectronics Research Centre, University of Southampton, Southampton SO17
1BJ, United Kingdom
4Electrical and Computer Engineering Department, University of California, Los
Angeles, CA 90095, USA
5Key Laboratory of Photonic Control Technology (Tsinghua University), Ministry
of Education, Beijing 100084, China
6CSIR National Laser Centre, PO Box 395, Pretoria 0001, South Africa

###### Abstract

Vectorially structured light has emerged as an enabling tool in many diverse
applications, from communication to imaging, exploiting quantum-like
correlations courtesy of a non-separable spatially varying polarization
structure. Creating these states at the source remains challenging and is
presently limited to two-dimensional vectorial states by customized lasers.
Here we invoke ray-wave duality in a simple laser cavity to produce
polarization marked multi-path modes that are non-separable in three degrees
of freedom and in eight dimensions. As a topical example, we use our laser to
produce the complete set of Greenberger-Horne-Zeilinger (GHZ) basis states,
mimicking high-dimensional multi-partite entanglement with classical light,
which we confirm by a new projection approach. We offer a complete theoretical
framework for our laser based on SU(2) symmetry groups, revealing a rich
parameter space for further exploitation. Our approach requires only a
conventional laser with no special optical elements, is easily scaleable to
higher dimensions, and offers a simple but elegant solution for at-the-source
creation of classically entangled states of structured light, opening new
applications in simulating and enhancing high-dimensional quantum systems.

††preprint: APS/123-QED

A quintessential property of quantum systems is the notion of entanglement
between particles, namely, that the measurement of some property of one
particle effects the measurement outcome of another. This has been
demonstrated with many systems, notably photonic states of light. The hallmark
of entangled states is non-separability, but non-separability is not unique to
quantum mechanics: it is found elsewhere in nature, and in particular it may
be engineered in the form of vectorial states of structured light,
controversially referred to as classically entangled light Spreeuw (1998);
Forbes  _et al._ (2019); Konrad and Forbes (2019); Toninelli  _et al._ (2019).
Here the entanglement is local, between internal degrees of freedom (DoFs) of
one classical light field, and so cannot capture the non-locality property of
quantum entangled particles. Yet the notion of classical entanglement is more
than simple mathematical machinery, with such states of light having violated
Bell-like inequalities Kagalwala  _et al._ (2013), used for classical
teleportation Diego  _et al._ (2016), enabled quantum walks with bright light
Sephton  _et al._ (2019); D?Errico  _et al._ (2020), used to realize real-time
quantum error correction in noisy channels Ndagano  _et al._ (2017), and
exploited for their classical vectorial properties in a myriad of applications
Rosales-Guzmán  _et al._ (2018), including metrology Töppel  _et al._ (2014);
D’ambrosio  _et al._ (2013), spin-to–orbital conversion in metamaterial Devlin
_et al._ (2017), multi-channel high-capacity communication Liu  _et al._
(2018); Willner (2018); Wang  _et al._ (2020), trapping Skelton  _et al._
(2013); Min  _et al._ (2013), super-resolution imaging Kozawa  _et al._
(2018), and kinematic sensing Berg-Johansen  _et al._ (2015).

Given the many applications that such classically entangled states have
fostered, much attention has gone into their creation. A myriad of tools exist
for creation external to the source Zhan (2009); Rubano  _et al._ (2019); Wang
_et al._ (2018), but with only limited success directly from lasers Forbes
(2019), with demonstrations limited to two-dimensional (qubit equivalent) non-
separable states by special intra-cavity elements in the form of liquid
crystal geometric phase elements Naidoo  _et al._ (2016a), metasurfaces Maguid
_et al._ (2018) as well as the use of optical fibre Mao  _et al._ (2017) and
on-chip solutions that exploit topological Cai  _et al._ (2012); Zambon  _et
al._ (2019); Miao  _et al._ (2016) or organic Stellinga  _et al._ (2018)
structures. In contrast, the ability to access arbitrarily engineered high-
dimensional state spaces with vectorial light would be highly beneficial,
opening the way to many exciting applications Aiello  _et al._ (2015); Mabena
and Roux (2017, 2020); Korolkova and Leuchs (2019); Forbes and Nape (2019);
Shen  _et al._ (2019).

Here we invoke complementary ray-wave duality in a simple laser cavity to
produce polarization marked multi-path modes for high-dimensional classically
entangled structured light directly from the source. In our cavity, the
oscillating modes appear as geometric ray orbits (patterns) along the length
and transverse to the cavity, each with a fixed shape and period determined by
the cavity length and pump light position. We introduce the concept of gain
control to realize polarization marking of the ray orbits without the need for
any internal polarizing elements. Although each exiting ray appears to follow
an independent path upon propagation, the output of all paths from the same
geometric shape is a single transverse mode with spatial coherence, courtesy
of the wave picture. Thus we use the DoFs of cavity ray direction, location
and polarization to realize eight dimensional classically entangled light from
a laser, which we go on to tailor in amplitude and phase, demonstrating the
first complete set of classically entangled Greenberger-Horne-Zeilinger (GHZ)
states Greenberger  _et al._ (1990). Our work overcomes the two-dimensional
(2D) limitation of existing laser solutions, creates new states of non-
separable structured light not proposed before, and does so in an otherwise
empty laser cavity without any special elements. The birth of high-dimensional
classical entanglement states from such a simple and compact laser paves the
way for exciting new applications of vectorially structured light.

Figure 1: Laser concept. (a) A 2D planar representation of the desired
geometric mode, where the mode evolves from wave-like fringes (at $z=0$ and
$\pm z_{R}$) to a ray-like trajectory. The means to create this mode in the
cavity is illustrated in (b), where completed oscillation orbits can be
described by the direction states $\left|+\right\rangle$ and
$\left|-\right\rangle$ (equivalently sub-orbits shown in black and pink,
respectively, in the dash-line boxes), as well as paths originating from ray
positions $\left|1\right\rangle$ and $\left|2\right\rangle$ (equivalently sub-
orbits shown in orange and green, respectively). (c) A 3D schematic of a
skewed (vortex) SU(2) geometric mode propagating in free space, with
experimental beam images shown at example propagation distances. The planar
mode is converted to the skewed version with a pair of cylindrical lenses.

## Results

Concept. Spatial mode control in lasers allows one to specify the desired
modal properties of amplitude, phase and polarization. This may be as vector
or scalar states of light, but pertinently, there are only two DoFs with which
to execute the control: the transverse mode and its polarization, conveniently
mapped to a sphere Milione  _et al._ (2011); Yi  _et al._ (2015). As such,
only analogues to qubits have been demonstrated from lasers Forbes (2019). The
challenge is to find more DoFs in a system seemingly constrained to just two,
a requirement in order to reach higher dimensions in classically entangled
light, but paradoxically, we wish to do so in an otherwise empty laser
configuration with no special elements. We will show that we can select our
new DoFs to be the oscillating direction, round trip location, and
polarization so that we can mimic multi-partite high-dimensional quantum
states, classically. Our concept utilizes the ability to produce an SU(2)
coherent state from the laser, treating the cavity as a quantum harmonic
oscillator with both wave-packet and classical ray trajectory interpretations
Shen  _et al._ (2018a, b); Chen  _et al._ (2019). We do so by engineering a
frequency-degenerate plano-concave cavity whose output mode can be shown to be
exactly the desired coherent state (see Supplementary Material). Importantly,
when the resonator operates in this frequency-degenerate state and the gain is
excited off-axis (at some localized position) the laser mode too becomes
localized on periodic trajectories that are controlled by the off-axis pump
position. While the output is a coherent state mode and is wave-like, the ray
trajectories define particle-like positions of localized intensity. This ray-
wave duality allows us to label the output in terms of the ray trajectories
rather than the modes themselves.

For instance, one such trajectory is shown in Fig. 1, with the forward
($\left|+\right\rangle$) and backward ($\left|-\right\rangle$) oscillating
states and the first ($\left|1\right\rangle$) and second
($\left|2\right\rangle$) roundtrip location state forming a completed
oscillation with the potential to reach dimension $d=4$ (see Supplementary
Material for the other geometries to reach higher dimensions). In Fig. 1 (a)
we show a 2D simulation of this ray-like propagation with marked trajectories.
At $z=0$ and $\pm z_{R}$ ($z_{R}$ is the Rayleigh range) the wave-like
behaviour is evident by the fringes in the beam as trajectories overlap. To
understand how to create this from a laser, we highlight the generation step
inside a laser in Fig. 1 (b). Here the output four rays originating from two
points ($\left|1\right\rangle,\left|2\right\rangle$) on the rear mirror
comprises two V-shaped locations, and the pair of rays in a certain location
state has two different directions
($\left|+\right\rangle,\left|-\right\rangle$), so that the output is spanned
by the basis states ${\cal
H}_{\text{ray}}\in\\{\left|+\right\rangle\left|1\right\rangle,\left|-\right\rangle\left|1\right\rangle,\left|+\right\rangle\left|2\right\rangle,\left|-\right\rangle\left|2\right\rangle\\}$,
a 4-dimensional Hilbert space, while in the wave picture the emitted output
from the laser is a single spatial mode. The planar mode can be converted by
simple cylindrical lenses to a skewed ray version shown in Fig. 1 (c), with
the physical implementation in the laboratory shown in the Supplementary
Material, section .10. Alongside the simulation is the measured beam profile
at selected propagation distances outside the cavity. We can further verify
the laser operation by imaging the internal cavity mode to planes external to
the laser for ease of measurement, shown for a period-4 trajectory example
(frequency-degenerate state $\left|\Omega=1/4\right\rangle$) experimentally in
Figure 2 (a). The results show the tell-tale signs of a wave-like behaviour,
as evident by the interference fringes with high visibility at $z=0$ and at
$z=z_{R}$, while appearing to be independent ray trajectories. The output is a
4-dimensional scalar state.

Figure 2: Vectorial ray-wave beams. (a) Experimental verification of an SU(2)
geometric mode exhibiting ray-wave duality, where the corresponding positions
of the flat and concave cavity mirrors are shown. The images show the
evolution of the output mode from one end of the cavity to the other as a
sequence of camera images. When the rays overlap, the wave-like nature is
evident, as seen by the fringes shown in the yellow box. (b) The polarization
structured of the rays can be adjusted by tuning the pumping, inducing a
change in fringe visibility with the pump power as the polarization state of
each ray is modified. The insets show the measured interference fringes at
$z=z_{R}$ for selected pump powers. The dashed boxes depict the corresponding
ray representation together with polarization change. (c) The notation of all
the elements in eight dimensions in an output vectorial ray-wave beam.

Since one of the DoFs is polarization, our scheme requires us to engineer the
cavity to have trajectory-dependent polarization control without any intra-
cavity elements. We achieve this by exploiting the fact that differing ray
trajectories will impinge on the laser crystal at angles other than normal, as
well as thermal effects as a function of pump power (see Supplementary
Materials, section .9). By deploying a c-cut crystal which exhibits angle
dependent birefringence at non-normal incident angles, we can mark our orbits
with polarization by simple adjustment of the pump light position and power,
with results shown in Fig. 2 (b). The visibility of the fringes decreases to
zero as the pump power is increased (for a particular angle of trajectory) as
a result of orthogonal ray polarization states. The power control of the gain
results in all rays evolving from an entirely linear horizontal state to a
vector state of diagonal ($\left|D\right\rangle$) and anti-diagonal
($\left|A\right\rangle$) polarizations. The results confirm that the cavity
can be forced into a ray-like mode and further, that the various orbits can be
individually modulated in polarization, resulting in a polarization structured
SU(2) vector beam. We derive a general analytical form for the output state
[see Eq. (S.55) in the Supplementary Material, section .6]; for the example
geometry of Figs. 1 and 2 it is a laser mode in an 8-dimensional Hilbert space
spanned by the basis states ${\cal
H}_{8}\in\\{\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle,\left|-\right\rangle\left|1\right\rangle\left|D\right\rangle,\left|+\right\rangle\left|2\right\rangle\left|D\right\rangle,$
$\left|-\right\rangle\left|2\right\rangle\left|D\right\rangle,\left|+\right\rangle\left|1\right\rangle\left|A\right\rangle,\left|-\right\rangle\left|1\right\rangle\left|A\right\rangle,\left|+\right\rangle\left|2\right\rangle\left|A\right\rangle,\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle\\}$.
We show this full state experimentally in Fig. 2 (c).

Figure 3: Classical GHZ states. The experimental setup used to generate
classical GHZ states from our laser, including the core steps of high-
dimensional state generation (laser), GHZ-state generation, and two
measurement steps to confirm the state properties. (a) shows the required path
and polarization transformations needed for each GHZ state, performed by an
iris (located at I1 or I2) and SLM phases ($3\pi/2$ and $\pi/2$),
respectively. In (b) and (c) this is unpacked graphically: the incoming four
lobes are reduced to two, and which are subsequently altered in polarization.
(d) shows results for the vector beam corresponding to the first maximally
entangled group $|\Phi^{\pm}\rangle$ both experimentally (Exp.) and simulated
(Sim.). The arrows depict the orientation of the polarizer in the measurement
stage of the OAM-state measure. In the tomography measurement (Bell-state
measure), each of the eight GHZ states can be inferred by just a polarizer and
a CCD camera. The CCD camera is moved to different locations and captures the
interferometric fringes for a visibility calculation. (OC: output coupler
mirror, DM: dichroic mirror, PBS: polarization splitting prism, QWP: quarter-
wave plate, HR: high-reflective mirror, PR: partial-reflective mirror, SLM:
spatial light modulator, CCD: charge-coupled device camera, P: Polarizer.)
Figure 4: Identification of each GHZ state. The experimental (Exp.) and
theoretical (Sim.) results showing interference patterns at the overlapped
region of the two orbits corresponding to each GHZ state. The plot on the left
for each group shows the location of the CCD camera for capturing the
interference fringes. $|\Phi^{+}\rangle$: no fringe for the original state,
center-bright fringes would be observed after projecting polarization on
$|H\rangle$ state i.e. for $\langle H|\Phi^{+}\rangle$, center-dark fringes
for $\langle V|\Phi^{+}\rangle$; $|\Phi^{-}\rangle$: center-dark fringes for
$\langle H|\Phi^{-}\rangle$, center-bright fringes for $\langle
V|\Phi^{-}\rangle$; $|\Psi_{1}^{+}\rangle$: center-bright fringes for $\langle
H|\Psi_{1}^{+}\rangle$, center-dark fringes for $\langle
V|\Psi_{1}^{+}\rangle$; $|\Psi_{1}^{-}\rangle$: center-dark fringes for
$\langle H|\Psi_{1}^{-}\rangle$, center-bright fringes for $\langle
V|\Psi_{1}^{-}\rangle$; $|\Psi_{2}^{+}\rangle$: center-bright fringes for
$\langle H|\Psi_{2}^{+}\rangle$, center-dark fringes for $\langle
V|\Psi_{2}^{+}\rangle$; $|\Psi_{2}^{-}\rangle$: center-dark fringes for
$\langle H|\Psi_{2}^{-}\rangle$, center-bright fringes for $\langle
V|\Psi_{2}^{-}\rangle$; $|\Psi_{3}^{+}\rangle$: center-bright fringes for
$\langle H|\Psi_{3}^{+}\rangle$, center-dark fringes for $\langle
V|\Psi_{3}^{+}\rangle$; $|\Psi_{3}^{-}\rangle$: center-dark fringes for
$\langle H|\Psi_{3}^{-}\rangle$, center-bright fringes for $\langle
V|\Psi_{3}^{-}\rangle$.

GHZ states from a laser. The results shown thus far confirm that this simple
laser cavity (with no special internal elements) is able to lase on an SU(2)
vector beam expressed in an 8-dimensional Hilbert space with three DoFs. In
our example results, this is achieved through oscillation on the frequency-
degenerate state $\left|\Omega=1/4\right\rangle$ and phase state
$|\phi=\pi\rangle$ but higher dimensions are possible by small adjustments to
the cavity length and pump position (see Supplementary Material). Now we show
that it is possible to controllably modulate this high-dimensional laser state
to some desired state; by way of example we create the first classically
entangled GHZ states, with the concept and setup shown in Fig. 3. In
particular we wish to create a complete set of GHZ states, namely,
$\left|\Phi^{\pm}\right\rangle$, $\left|\Psi^{\pm}_{1}\right\rangle$,
$\left|\Psi^{\pm}_{2}\right\rangle$, and $\left|\Psi^{\pm}_{3}\right\rangle$,
with full details given in the Supplementary Material, section .7.

The idea is to externally modulate the output state from the laser to engineer
the amplitude and phase of each term independently, producing the general
state

$\displaystyle\left|u_{8}\right\rangle$ $\displaystyle=$
$\displaystyle\alpha_{1}\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle+\alpha_{2}\left|-\right\rangle\left|1\right\rangle\left|D\right\rangle+\alpha_{3}\left|+\right\rangle\left|2\right\rangle\left|D\right\rangle$
$\displaystyle+\alpha_{4}\left|+\right\rangle\left|1\right\rangle\left|A\right\rangle+\alpha_{5}\left|-\right\rangle\left|2\right\rangle\left|D\right\rangle+\alpha_{6}\left|-\right\rangle\left|1\right\rangle\left|A\right\rangle$
$\displaystyle+\alpha_{7}\left|+\right\rangle\left|2\right\rangle\left|A\right\rangle+\alpha_{8}\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle,$

and converting it to each of the eight GHZ basis states. For example, the
transformation

$\left|u_{8}\right\rangle\rightarrow\left|\Phi^{\pm}\right\rangle=\frac{\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle\pm\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle}{\sqrt{2}}$

requires a modulation that sets all amplitudes to zero except
$|\alpha_{1}|=|\alpha_{8}|=\frac{1}{\sqrt{2}}$ and a relative phase shift
between the two decomposed ray modes. The general setup to achieve this (and
other modulations) is shown in Fig. 3. The main experimental arrangement
includes the laser for creating the initial high-dimensional state followed by
a tailoring step to convert it into specific desirable classes, here the GHZ
states are generated as an example. Finally, the states are directed to two
measurement devices: the vectorial nature of the prepared states are measured
by a polarizer and camera (OAM-state measure) and a Bell-state measurement
device, which we introduce here, is used for the tomographic projections. The
four kinds of vector beams corresponding to the four maximum entangled groups
of the eight GHZ states are illustrated by the graphical procedure as Fig. 3
(a), and unpacked in parts (b) and (c). Here we outline the process using the
$\left|\Phi^{\pm}\right\rangle$ state as an example, with full details for all
cases given in the Methods and Supplementary Material. We switch from a linear
polarization basis to circular with a quarter-wave plate (QWP), eliminate
light on the ray states $\left|+\right\rangle\left|2\right\rangle$ and
$\left|-\right\rangle\left|1\right\rangle$ by iris I1 and then modulate the
polarization of the $\left|+\right\rangle\left|1\right\rangle$ and
$\left|-\right\rangle\left|2\right\rangle$ paths into diagonal and anti-
diagonal states using programmed phases on a spatial light modulator. Similar
transformations allow us to generate all GHZ states in the complete family of
maximally entangled states. The controllable generation of all the GHZ states
provides further verification that the general SU(2) beam in our system is
indeed expressed in an 8-dimensional space, as GHZ states form a complete
basis in 8-dimensions.

The experimental images of skewed rays, together with the theoretical
predictions, are shown in Fig. 3 (d) for the $\left|\Phi^{\pm}\right\rangle$
states as a function of the orientation of the polarizer in the OAM-state
measure step, showing excellent agreement. The intensity distribution is a
two-lobed structure, consistent with the corresponding GHZ state, while the
evolution of the lobe intensities confirm the vectorial nature of the field. A
reconstruction of the GHZ-state mode’s propagation in free-space is shown in
Fig. 3 (e). Results for all other states are shown in the Supplementary
Material, Fig. S7.

To quantitatively infer the fidelity of our classically entangled GHZ states,
we introduce a new tomography based on Bell state projections, with the
results shown in Fig. 4. The two lobes of the GHZ state beam are overlapped at
the non-OAM planar state, projected onto polarization states, and the
visibility in fringes measured: a Bell state projection (see Supplementary
Material, section .8, for the theory of this). This allows the state
amplitudes and relative phases to be determined, and hence the GHZ states can
be clearly distinguished. From the visibility of each projection in the
tomographic measurement, a density matrix can be inferred and the fidelity of
each GHZ state quantitatively calculated. The results of this are shown in
Fig. 5 (a) for the $|\Phi^{+}\rangle$ state (as the lowest fidelity example)
with the measured fidelities of all the GHZ states shown in Figure 5 (b). All
other density matrix results can be found in the Supplementary Material, Fig.
S5. We find that the theoretical (inset) and measured density matrices are in
very good agreement, with the laser generating GHZ states with fidelities of
approximately $90\%$.

Figure 5: GHZ state tomography. (a) The theoretical (inset) and experimental
density matrix of the first GHZ state, $|\Phi^{+}\rangle$. From the density
matrix of each state, the fidelity was calculated, shown in (b). Error bars
show the standard error. Figure 6: Towards higher-dimensional classical
entanglement. Spatial SU(2) geometric beams can carry OAM and the OAM can be a
DoF manifesting additional dimensions in classical entanglement. (a) A
superposed trajectory on SU(2) state $|\Omega=1/4\rangle$ including the
positive-OAM and negative-OAM decomposed SU(2) trajectories, (b) which can
exactly fulfill a complete SU(2) oscillation in a degenerate cavity. (c) Based
on the ray-wave duality, the corresponding geometric mode can be represented
by the superposition of a positive-OAM and a negative-OAM vortex SU(2) beams
(the topological phase manifesting the OAM and intensity wave-packet are shown
in the inserts). This generalized beam can be expressed into 16-dimensional
state with 4 DoFs and 16 eigenstates noted in figure (d). (e) An exotic multi-
vortex SU(2)-structured mode is obtained by the superposition of two multi-
vortex SU(2) modes with opposite main-OAMs, and the sub-OAM carried by each
sub-ray mode can also play as a new DoF. This generalized beam can be
expressed into 32-dimensional state with 5 DoFs and 32 eigenstates noted in
figure (f).

Towards higher dimensions. So far we have demonstrated 8-dimensional 3-partite
GHZ states by SU(2) structured light. Nevertheless, the SU(2) structured light
has the potential to realize higher-dimensional multi-partite states. For
instance, we can use a superposed OAM state, as shown in the Fig. 6 (a), where
the superposed SU(2) trajectory includes the positive and negative twisted
decomposed SU(2) trajectories. More importantly, the superposed trajectory
fulfills the condition of a complete oscillation in a degenerate cavity, as
illustrated in Fig. 6 (b), so that it can be directly generated from the laser
(see Supplementary Material). Therefore, the corresponding ray-wave beam
introducing OAM as a new DoF shown in Fig. 6 (c), can be expressed as a
16-dimensional and 4-partite classically entangled state, with the eigenstates
shown in Fig. 6 (d). For exploring even more DoFs, we can replace the Gaussian
beams along the SU(2) orbits in a geometric mode based on ray-wave duality. In
this generalized beam, there is not only a main OAM state $|\pm\ell\rangle$
along the propagation axis, but also OAM $|\pm\rangle$ carried by the sub-
vortex beams along the SU(2) ray-orbits, as shown in Fig. 6 (e). This exotic
SU(2) beam also satisfies the conditions of an SU(2) coherent state, realizing
a 32-dimensional and 5-partite state, as noted in Fig. 6 (f). See
Supplementary Material, section .11, for more theoretical descriptions on
these states.

## Discussion and conclusion

Here we have demonstrated a simple laser that can produce exotic vectorial
states of light that are non-separable in high-dimensions, beyond the existing
laser limitation of two dimensions in polarization structured vector beams.
Our cavity has no internal elements and achieves mode control by simple
adjustments to the cavity (e.g., length and pump position/power). We have
introduced a new tomography approach for verifying high-dimensional states,
which we used to quantify the fidelity of our GHZ states. In our experimental
demonstration we have used a cavity operating at the degenerate state of
$|\Omega=1/4\rangle$ for 8 dimensions and simulating three partite
entanglement. By operating at a more general degenerate state, the general
SU(2) modes would be extended to a higher-dimensional space, which could be
extended even further by replacing the Gaussian-like orbital modes by more
exotic structures Lu  _et al._ (2011); Tuan  _et al._ (2018), including those
based on OAM (as described in the previous section). While we have tailored
our laser modes to create the first classically entangled GHZ states, other
states could be tailored such as $W$ Dür  _et al._ (2000), NOON Kok  _et al._
(2002) and CAT Liu  _et al._ (2019) states, or our high-dimensional laser
could be used as the input to alternative path modulation schemes Fickler  _et
al._ (2014); Pabón  _et al._ (2019).

Our work allows for a compact and simple generation scheme to produce new
states of structured light directly from the source. In particular, we
demonstrate classical analogues to multi-partite high-dimensional quantum
states, which have already been suggested for tasks such as quantum channel
error estimation and correction, super-resolution imaging, metrology and
sensing, optical communication, and quantum decoherence studies under easily
controlled conditions Aiello  _et al._ (2015); Mabena and Roux (2017, 2020);
Korolkova and Leuchs (2019); Forbes and Nape (2019). Finally, in addition to
the practical laser device we also offer a complete theoretical framework for
our vectorially structured light beams, providing a rich parameter space for
extending fundamental studies with structured light.

## Methods

Creating GHZ states. The GHZ generation step is shown as part of Fig. 3. The
relevant transformations for generating our GHZ states are executed in two
consecutive steps. First through intensity modulation then followed by dynamic
phase (SLM) modulation. A graphical representation of the scheme is
illustrated in Fig. 3 (a), showing the intensity modulation of the orbits
(rows) and polarization control (columns). Each panel describes the required
modulation to produce each of the desired GHZ states.

To achieve the path modulation, we use an iris strategically positioned at I1
or I2. We unpack this technique in Fig. 3 (b) showing an example for
modulation at location I1. The four quadrants represent the full orbits while
the grey regions represent the filtered orbits. Due to the special spatial
structure of the SU(2) geometric beam, the iris induces a path-dependent
intensity modulation: when an iris with an appropriate aperture size is placed
at the negative Rayleigh length position (I1 position), paths
$\left|-\right\rangle\left|1\right\rangle$ and
$\left|+\right\rangle\left|2\right\rangle$ can be blocked, resulting in a
diagonal intensity pattern in the corresponding vortex SU(2) beam [see Fig. 3
(b)]. When applied at the positive Rayleigh length position (I2 position),
$\left|+\right\rangle\left|1\right\rangle$ and
$\left|-\right\rangle\left|2\right\rangle$ are blocked, resulting in an anti-
diagonal intensity pattern of vortex SU(2) beam.

For polarization modulation, we first convert all rays to circular
polarization with a QWP and direct them towards the SLM with controlled
location. This is illustrated graphically in Fig. 3 (c), where the four orbits
are marked with (anti-)diagonal polarization states shows as arrows. The phase
mask of the SLM is split into two parts, one to modulate the states
$\left|1\right\rangle$ and the other to modulate states
$\left|2\right\rangle$. Since the beam is circularly polarized and the SLM is
only sensitive to horizontal component, the polarization of the
$\left|1\right\rangle$ and $\left|2\right\rangle$ states can be independently
controlled by encoding phase retardations on each section of the SLM. When the
encoded phase-step mask is set to $\pi/2$ and $3\pi/2$ for the split screen,
respectively, the states $\left|1\right\rangle\left|D\right\rangle$ and
$\left|2\right\rangle\left|A\right\rangle$ are produced. Likewise, when the
mask is flipped ($3\pi/2$ and $\pi/2$) the states
$\left|1\right\rangle\left|A\right\rangle$ and
$\left|2\right\rangle\left|D\right\rangle$ are produced. The combination of
intensity and polarization modulation results into the four kinds of vector
vortex beams corresponding to the four maximally entangled groups of three-
partite GHZ states. Finally, the inter-modal phase between states is easily
controlled with a thin BK7 plate that is partially inserted into the path of
one ray group to add a phase difference 0 or $\pi$.

Bell state projections. We describe the working principle of our Bell state
measurement approach, shown as part of Fig. 3, comprised of a $F=250$ mm
focusing lens, polarizer and CCD camera. The camera can be placed at different
regions within the focus of the lens.

Our measurement scheme exploits the properties of the multi-particle GHZ
states. A GHZ state will be reduced to a two particle Bell state after a
superposition measurement of one of the particles (See Supplementary
Materials, section .7). For classical GHZ states, superposition projections of
one of the DoFs leaves the remaining DoFs to collapse to maximally entangled
Bell states (See Supplementary Material, section .8, for the full
explanation). Here the polarization DoF of freedom is chosen as a candidate to
realize the Bell projection.

After projections onto $|H\rangle$ and $|V\rangle$ states,
$|\Phi^{\pm}\rangle$ or $|\Psi^{\pm}_{3}\rangle$ states would be reduced to
$|\psi^{\pm}\rangle$ and $|\psi^{\mp}\rangle$ states, $|\Psi^{\pm}_{1}\rangle$
or $|\Psi^{\pm}_{2}\rangle$ to $|\phi^{\pm}\rangle$ and $|\phi^{\mp}\rangle$
[refer to Eqs. (S.72)-(S.77), Supplementary Material]. The “$+$” and “$-$”
signals in the Bell states can be distinguished by the complementary
interferometric fringes of the corresponding phase difference of 0 and $\pi$
between two SU(2) ray-orbits. For measuring $|\Phi^{\pm}\rangle$ and
$|\Psi^{\pm}_{3}\rangle$, the CCD camera can be located at $z=-z_{R}$ position
where $|-1\rangle$ and $|+2\rangle$ orbits are overlapped. For
$|\Psi^{\pm}_{1}\rangle$ and $|\Psi^{\pm}_{2}\rangle$, the CCD camera should
be located at $z=z_{R}$ position where $|+1\rangle$ and $|-2\rangle$ orbits
are overlapped. Without the polarization projection, the pattern shows no
fringes since the light on the corresponding two orbits is incoherent. After
projection on $|H\rangle$ or $|V\rangle$ states, different interference
fringes would be observed for different reduced Bell states.

For the group $|\Phi^{\pm}\rangle$, the “$\pm$” cannot be distinguished by the
intensity patterns. If we project the polarization onto $|H\rangle$ state to
observe the pattern of $\langle H|\Phi^{\pm}\rangle$, pattern of original
state $|\Phi^{\pm}\rangle$ will be reduced into Bell states
$|\psi^{\pm}\rangle$ and two different patterns of complementary fringes will
be observed, center-bright fringes for $|\psi^{+}\rangle$ (the original state
should be $|\Phi^{+}\rangle$) and center-dark fringes for $|\psi^{-}\rangle$
(the original state should be $|\Phi^{-}\rangle$). We can also use projected
state $\langle V|\Phi^{\pm}\rangle$ to distinguish the “$\pm$” that $\langle
V|\Phi^{+}\rangle$ should be center-dark fringes corresponding to Bell state
$|\psi^{-}\rangle$ and $\langle V|\Phi^{-}\rangle$ should be center-bright
fringes corresponding to Bell state $|\psi^{+}\rangle$. In the experiment, we
can use a BK7 thin plate to cover one of the two orbits and rotate slightly to
control the phase difference between them to control a phase difference of
$\pi$, switching from “$+$” to “$-$” state. Other GHZ states can be generated
by the similar way fulfilling a completed set in 8-dimensional Hilbert space.

## Acknowledgements

National Key Research and Development Program of China (2017YFB1104500);
National Natural Science Foundation of China (61975087); Natural Science
Foundation of Beijing Municipality (4172030); Beijing Young Talents Support
Project (2017000020124G044).

## Authors’ contribution

Y.S. and A.F. proposed the idea and conceived the experiment, Y.S. performed
experiments and the theory, Y.S. and I.N. proposed the projection method of
GHZ states, Y.S and X.Y. performed the numerical simulations, D.N. helped with
the SLM modulation experiments, all authors contributed to data analysis and
writing the manuscript, and the project was supervised by A.F. and M.G.

## Competing financial interests

The authors declare no financial or competing interests.

## Materials and correspondence

Correspondence and requests for materials should be addressed to Y.S.
<EMAIL_ADDRESS>and A.F. (Andrew.Forbes@wits.ac.za).

## References

  * Spreeuw (1998) R. J. Spreeuw, Foundations of physics 28, 361 (1998).
  * Forbes  _et al._ (2019) A. Forbes, A. Aiello, and B. Ndagano, Progress in Optics 64, 99 (2019).
  * Konrad and Forbes (2019) T. Konrad and A. Forbes, Contemporary Physics 60, 1 (2019).
  * Toninelli  _et al._ (2019) E. Toninelli, B. Ndagano, A. Vallés, B. Sephton, I. Nape, A. Ambrosio, F. Capasso, M. J. Padgett, and A. Forbes, Advances in Optics and Photonics 11, 67 (2019).
  * Kagalwala  _et al._ (2013) K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. Saleh, Nature Photonics 7, 72 (2013).
  * Diego  _et al._ (2016) G.-S. Diego, B. Robert, Z. Felix, V. Christian, G. Markus, H. Matthias, N. Stefan, D. Michael, A. Andrea, O. Marco, and S. Alexander, Laser & Photonics Reviews 10, 317 (2016).
  * Sephton  _et al._ (2019) B. Sephton, A. Dudley, G. Ruffato, F. Romanato, L. Marrucci, M. Padgett, S. Goyal, F. Roux, T. Konrad, and A. Forbes, PLOS ONE 14, e0214891 (2019).
  * D?Errico  _et al._ (2020) A. D?Errico, F. Cardano, M. Maffei, A. Dauphin, R. Barboza, C. Esposito, B. Piccirillo, M. Lewenstein, P. Massignan, and L. Marrucci, Optica 7, 108 (2020).
  * Ndagano  _et al._ (2017) B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, Nature Physics 13, 397 (2017).
  * Rosales-Guzmán  _et al._ (2018) C. Rosales-Guzmán, B. Ndagano, and A. Forbes, J. Opt. 20, 123001 (2018).
  * Töppel  _et al._ (2014) F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, New Journal of Physics 16 (2014), 10.1088/1367-2630/16/7/073019.
  * D’ambrosio  _et al._ (2013) V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, Nature Communications 4, 2432 (2013).
  * Devlin  _et al._ (2017) R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, Science 358, 896 (2017).
  * Liu  _et al._ (2018) J. Liu, S.-M. Li, L. Zhu, A.-D. Wang, S. Chen, C. Klitis, C. Du, Q. Mo, M. Sorel, S.-Y. Yu, _et al._ , Light: Science & Applications 7, 17148 (2018).
  * Willner (2018) A. E. Willner, Light: Science & Applications 7, 18002 (2018).
  * Wang  _et al._ (2020) E. Wang, J. Niu, Y. Liang, H. Li, Y. Hua, L. Shi, and C. Xie, Advanced Optical Materials , 1901674 (2020).
  * Skelton  _et al._ (2013) S. Skelton, M. Sergides, R. Saija, M. Iatì, O. Maragó, and P. Jones, Optics Letters 38, 28 (2013).
  * Min  _et al._ (2013) C. Min, Z. Shen, J. Shen, Y. Zhang, H. Fang, G. Yuan, L. Du, S. Zhu, T. Lei, and X. Yuan, Nature Communications 4, 2891 (2013).
  * Kozawa  _et al._ (2018) Y. Kozawa, D. Matsunaga, and S. Sato, Optica 5, 86 (2018).
  * Berg-Johansen  _et al._ (2015) S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, Optica 2, 864 (2015).
  * Zhan (2009) Q. Zhan, Advances in Optics and Photonics 1, 1 (2009).
  * Rubano  _et al._ (2019) A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, J. Opt. Soc. Am. B 36, D70 (2019).
  * Wang  _et al._ (2018) X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, Nanophotonics 7, 1533 (2018).
  * Forbes (2019) A. Forbes, Laser & Photonics Reviews 13, 1900140 (2019).
  * Naidoo  _et al._ (2016a) D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, Nature Photonics 10, 327 (2016a).
  * Maguid  _et al._ (2018) E. Maguid, R. Chriki, M. Yannai, V. Kleiner, E. Hasman, A. A. Friesem, and N. Davidson, ACS Photonics 5, 1817 (2018).
  * Mao  _et al._ (2017) D. Mao, T. Feng, W. Zhang, H. Lu, Y. Jiang, P. Li, B. Jiang, Z. Sun, and J. Zhao, Applied Physics Letters 110, 021107 (2017).
  * Cai  _et al._ (2012) X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science 338, 363 (2012).
  * Zambon  _et al._ (2019) N. C. Zambon, P. St-Jean, M. Milićević, A. Lemaître, A. Harouri, L. Le Gratiet, O. Bleu, D. Solnyshkov, G. Malpuech, I. Sagnes, _et al._ , Nature Photonics 13, 283 (2019).
  * Miao  _et al._ (2016) P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, Science 353, 464 (2016).
  * Stellinga  _et al._ (2018) D. Stellinga, M. E. Pietrzyk, J. M. Glackin, Y. Wang, A. K. Bansal, G. A. Turnbull, K. Dholakia, I. D. Samuel, and T. F. Krauss, ACS Nano 12, 2389 (2018).
  * Aiello  _et al._ (2015) A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, New Journal of Physics 17, 043024 (2015).
  * Mabena and Roux (2017) C. M. Mabena and F. S. Roux, Physical Review A 96, 053860 (2017).
  * Mabena and Roux (2020) C. M. Mabena and F. S. Roux, Physical Review A 101, 013807 (2020).
  * Korolkova and Leuchs (2019) N. Korolkova and G. Leuchs, Reports on Progress in Physics 82, 056001 (2019).
  * Forbes and Nape (2019) A. Forbes and I. Nape, AVS Quantum Science 1, 011701 (2019).
  * Shen  _et al._ (2019) Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, Light: Science & Applications 8, 1 (2019).
  * Greenberger  _et al._ (1990) D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, American Journal of Physics 58, 1131 (1990).
  * Milione  _et al._ (2011) G. Milione, H. Sztul, D. Nolan, and R. Alfano, Physical Review Letters 107, 053601 (2011).
  * Yi  _et al._ (2015) X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, Physical Review A 91, 023801 (2015).
  * Shen  _et al._ (2018a) Y. Shen, Z. Wan, Y. Meng, X. Fu, and M. Gong, IEEE Photonics Journal 10, 1 (2018a).
  * Shen  _et al._ (2018b) Y. Shen, X. Fu, and M. Gong, Optics Express 26, 25545 (2018b).
  * Chen  _et al._ (2019) Y. Chen, S. Li, Y. Hsieh, J. Tung, H. Liang, and K. Huang, Optics Letters 44, 2649 (2019).
  * Lu  _et al._ (2011) T.-H. Lu, Y. Lin, Y. Cexpresshen, and K. Huang, Applied Physics B 103, 991 (2011).
  * Tuan  _et al._ (2018) P. Tuan, Y. Hsieh, Y. Lai, K. Huang, and Y. Chen, Optics Express 26, 20481 (2018).
  * Dür  _et al._ (2000) W. Dür, G. Vidal, and J. I. Cirac, Physical Review A 62, 062314 (2000).
  * Kok  _et al._ (2002) P. Kok, H. Lee, and J. P. Dowling, Physical Review A 65, 052104 (2002).
  * Liu  _et al._ (2019) S.-L. Liu, Q. Zhou, S.-K. Liu, Y. Li, Y.-H. Li, Z.-Y. Zhou, G.-C. Guo, and B.-S. Shi, Communications Physics 2, 1 (2019).
  * Fickler  _et al._ (2014) R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, and A. Zeilinger, Nature communications 5, 1 (2014).
  * Pabón  _et al._ (2019) D. Pabón, S. Ledesma, and L. Rebón, arXiv preprint arXiv:1912.10153 (2019).

## Supplementary Information

###  .1 Schrödinger coherent state

A coherent state is a specific quantum state whose behavior most closely
resembles the classical state, and particularly adapted for studying the
quantum-to-classical transition, where the quantum probability wave-packet can
be coupled with classical movement Perelomov (2012). For a simple 1-D linear
oscillator, the Hamiltonian is given by:

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{1}{2m}p^{2}+\frac{1}{2}m\omega^{2}x^{2}$
$\displaystyle={\frac{1}{2}\left(a^{\dagger}{{a}}+{{a}}a^{\dagger}\right)\hbar{{\omega}}}={\left(a^{\dagger}{{a}}+\frac{1}{2}\right)\hbar{{\omega}}},$
(S.1)

where $m$ is the mass, $\omega$ is the unperturbed frequency, $p$ is the
momentum operator, $x$ is the coordinate operator, ${a}^{\dagger}$ and ${a}$
are the ladder (creation and annihilation) operators of photon, and $\hbar$ is
the reduced Planck constant. The eigenstate under coordinate representation is
expressed by Hermite function:

$|n\rangle=\sqrt{\frac{1}{\sqrt{\pi}2^{n}n!}}H_{n}(\xi)\text{e}^{-\xi^{2}/2},$
(S.2)

where $\xi=\sqrt{m\omega/\hbar}\cdot x$, with eigenvalue of:

$E_{n}=\left(n+\frac{1}{2}\right)\hbar{{\omega}},$ (S.3)

According to the definition of coherent state, the coherent state under
coordinate representation, i.e. Schrödinger coherent state Schrödinger (1926),
is given by:

$|\alpha\rangle={{\text{e}}^{-{{{\alpha}^{*}}\alpha}/{2}\;}}\sum\limits_{n=0}^{\infty}{\frac{{{\alpha}^{n}}}{\sqrt{n!}}}\left|n\right\rangle{{\text{e}}^{-{\text{i}{{E}_{n}}t}/{\hbar}\;}}.$
(S.4)

Substituting Eq. (S.2) and Eq. (S.3) into Eq. (S.4) and applying the
generating function of Hermite polynomials,
$\exp(2xt-t^{2})$=$\sum_{n=0}^{\infty}H_{n}(x)t^{n}/n!$, we get:

$\displaystyle\left|\alpha\right\rangle={{\text{e}}^{-\frac{{{\left|\alpha\right|}^{2}}}{\text{2}}}}\sum\limits_{n=0}^{\infty}{\frac{{{\left(\left|\alpha\right|{{\text{e}}^{\text{i}\delta}}\right)}^{n}}}{\sqrt{n!}}}{\frac{{H}_{n}\left(\xi\right)}{\sqrt{\sqrt{\pi}{{2}^{n}}n!}}}{{\text{e}}^{-\frac{{{\xi}^{2}}}{2}}}{{\text{e}}^{-\text{i}\left(n+\frac{\text{1}}{\text{2}}\right)\omega
t}}$
$\displaystyle=\frac{1}{\sqrt{\sqrt{\pi}}}{{\text{e}}^{-\frac{{{\left|\alpha\right|}^{2}}+{{\xi}^{2}}}{\text{2}}}}{{\text{e}}^{-\frac{\text{i}\omega
t}{\text{2}}}}\sum\limits_{n=0}^{\infty}{\frac{{{\left[\frac{\left|\alpha\right|{{\text{e}}^{-\text{i}\left(\omega
t-\delta\right)}}}{\sqrt{2}}\;\right]}^{n}}{{H}_{n}}\left(\xi\right)}{n!}}$
$\displaystyle=\frac{1}{\sqrt{\sqrt{\pi}}}{{\text{e}}^{-\frac{{{\left|\alpha\right|}^{2}}+{{\xi}^{2}}}{\text{2}}}}{{\text{e}}^{-\frac{\text{i}\omega
t}{\text{2}}}}{{\operatorname{e}}^{-\frac{{{\left|\alpha\right|}^{2}}{{\text{e}}^{-\text{i2}\left(\omega
t-\delta\right)}}}{2}+\sqrt{2}\left|\alpha\right|{{\text{e}}^{-\text{i}\left(\omega
t-\delta\right)}}\xi}},$ (S.5)

where $\delta$ is the argument of $\alpha$. Then, the probability wave-packet
of Schrödinger coherent state can be derived by:

$\displaystyle\langle\alpha|\alpha\rangle$
$\displaystyle=\frac{1}{\sqrt{\pi}}{{\text{e}}^{-{{\left|\alpha\right|}^{2}}-{{\xi}^{2}}}}{{\operatorname{e}}^{-{{\left|\alpha\right|}^{2}}\text{cos}\left[\text{2}\left(\omega
t-\delta\right)\right]+2\sqrt{2}\left|\alpha\right|\xi\cos\left(\omega
t-\delta\right)}}$
$\displaystyle=\frac{1}{\sqrt{\pi}}{{\operatorname{e}}^{-{{\xi}^{2}}-2{{\left|\alpha\right|}^{2}}\text{co}{{\text{s}}^{2}}\left(\omega
t-\delta\right)+2\sqrt{2}\left|\alpha\right|\xi\cos\left(\omega
t-\delta\right)}}$
$\displaystyle=\frac{1}{\sqrt{\pi}}{{\operatorname{e}}^{-{{\left[\xi-\sqrt{2}\left|\alpha\right|\cos\left(\omega
t-\delta\right)\right]}^{2}}}}.$ (S.6)

As shown in Fig. S1, the peak of wave-packet of is along the trajectory of
corresponding classical oscillator, i.e.
$\xi=\sqrt{2}\left|\alpha\right|\cos\left(\omega t-\delta\right)$, manifesting
the quantum-classical coupling.

Figure S1: Schrödinger coherent state. (a) Probability wave-packet of
eigenstate of 1-D linear oscillator ($n$=20). (b) Trajectory of the classical
movement of 1-D linear oscillator and (c) the probability wave-packet of
coherent state ($\alpha=1$, $\omega=1$, $\delta=0$).

###  .2 SU(2) coherent state

In quantum optics, the Hamiltonian for the 3-D linear harmonic oscillator is
given by:

$\mathcal{H}=\sum\limits_{j}{\frac{1}{2}\left(a_{j}^{\dagger}{{a}_{j}}+{{a}_{j}}a_{j}^{\dagger}\right)\hbar{{\omega}_{j}}}=\sum\limits_{j}{\left(a_{j}^{\dagger}{{a}_{j}}+\frac{1}{2}\right)\hbar{{\omega}_{j}}},$
(S.7)

where $j=x,y,z$. The generalized Hamiltonian related to SU(2) transformations
for the coupled oscillator systems can be modeled as:

$\mathcal{H}=\mathcal{H}_{0}+\sum\limits_{j}{\Omega_{j}J_{j}},$ (S.8)

where
$\mathcal{H}_{0}=(a_{x}a_{x}^{\dagger}+a_{y}a_{y}^{\dagger}+1)\hbar\omega_{0}$
is the Hamiltonian for the 2-D isotropic oscillator deciding the transverse
wave-packet on $(x,y)$ plane. The coupling parameters $\Omega_{j}$ are assumed
to be real constants, and the operators under Schwinger representation reveal
the SU(2)-Lie group accommodating two linear oscillators and an angular
momentum oscillator Bužek and Quang (1989); Wodkiewicz and Eberly (1985).

$\left\\{\begin{aligned}
{{J}_{x}}&=\frac{1}{2}\left(a_{x}^{\dagger}{{a}_{y}}+a_{y}^{\dagger}{{a}_{x}}\right)\\\
{{J}_{y}}&=\frac{-\text{i}}{2}\left(a_{x}^{\dagger}{{a}_{y}}-a_{y}^{\dagger}{{a}_{x}}\right)\\\
{{J}_{z}}&=\frac{1}{2}\left(a_{x}^{\dagger}{{a}_{x}}-a_{y}^{\dagger}{{a}_{y}}\right)\end{aligned}\right.$
(S.9)

Operators $J_{j}$ satisfy the SU(2)-Lie commutator algebra
$[J_{i},J_{j}]=\text{i}\varepsilon_{i,j,k}J_{k}$
($\\{i,j,k\\}=\\{1,2,3\\}=\\{x,y,z\\}$), where the Levi-Civita tensor
$\varepsilon_{i,j,k}$ is equal to $+1$ and $-1$ for even and odd permutations
of its indices, respectively, and zero otherwise. The Hamiltonian in Eq. (S.8)
can not only represent a host of entanglement mechanisms Sanders  _et al._
(1995); Kim  _et al._ (2002) but also be associated with astigmatism and
aberration in wave optics, relevant in high-order laser pattern formations
Agarwal (1999); Chen  _et al._ (2017). For the statistics of quantum number at
transverse oscillation, SU(2) coherent state is defined as:

$\left|\alpha\right\rangle=\exp\left(\alpha{{J}_{+}}-{{\alpha}^{*}}{{J}_{-}}\right)\left|j,-j\right\rangle,$
(S.10)

where ${{J}_{\pm}}={{J}_{x}}\pm\text{i}{{J}_{y}}$ are the ladder (creation and
annihilation) operators of angular momentum, and $j$ is a certain integer or
half-integer. Using the disentangling theorem for angular-momentum operators
Wodkiewicz and Eberly (1985), we can rewrite Eq. (S.10) in the following form:

$\left|\tau\right\rangle={{\left(1+{{\left|\tau\right|}^{2}}\right)}^{-j}}\exp\left(\tau{{J}_{+}}\right)\left|j,-j\right\rangle,$
(S.11)

Hereinafter, we express Eq. (S.11) into eigenstates representation via unitary
transformation. According to Taylor expansion, the exponential operator in Eq.
(S.11) can be expanded as:

$\exp\left(\tau{{J}_{+}}\right)=\sum\limits_{n=0}^{\infty}{\frac{{{\left(\tau{{J}_{+}}\right)}^{n}}}{n!}}.$
(S.12)

Substitute Eq. (S.12) into Eq. (S.11) and apply unitary transformation into
angular-momentum representation:

$\displaystyle\left|\tau\right\rangle$
$\displaystyle={{\left(1+{{\left|\tau\right|}^{2}}\right)}^{-j}}\sum\limits_{k=-j}^{j}{\left\langle
j,k\right|\exp\left(\tau{{J}_{+}}\right)\left|j,-j\right\rangle\left|j,k\right\rangle}$
$\displaystyle={{\left(1+{{\left|\tau\right|}^{2}}\right)}^{-j}}\sum\limits_{k=-j}^{j}{\left\langle
j,k\right|\sum\limits_{n=0}^{\infty}{\frac{{{\left(\tau{{J}_{+}}\right)}^{n}}}{n!}}\left|j,-j\right\rangle\left|j,k\right\rangle}$
$\displaystyle={{\left(1+{{\left|\tau\right|}^{2}}\right)}^{-j}}\sum\limits_{k=-j}^{j}{\left\langle
j,k\right|\frac{{{\left(\tau{{J}_{+}}\right)}^{j+k}}}{\left(j+k\right)!}\left|j,-j\right\rangle\left|j,k\right\rangle}.$
(S.13)

According to the property of ladder operators:

${{J}_{\pm}}\left|j,k\right\rangle=\sqrt{j\left(j+1\right)-k\left(k\pm
1\right)}\hbar\left|j,k\pm 1\right\rangle,$ (S.14)

Eq. S.13 can be rewritten as:

$\displaystyle\left|\tau\right\rangle$
$\displaystyle=\frac{\sum\limits_{k=-j}^{j}{\frac{{\underset{i=-j}{\overset{s-1}{\mathop{\prod}}}\sqrt{j\left(j+1\right)-i(i+1)}}}{\left(j+k\right)!}{{\tau}^{j+k}}\left|j,k\right\rangle}}{{\left(1+{{\left|\tau\right|}^{2}}\right)}^{j}}$
$\displaystyle={{\left(1+{{\left|\tau\right|}^{2}}\right)}^{-j}}\sum\limits_{k=-j}^{j}{{{\left(\begin{matrix}2j\\\
j+k\\\
\end{matrix}\right)}^{{1}/{2}\;}}{{\tau}^{j+k}}\left|j,k\right\rangle},$
(S.15)

After substituting $N=2j$ and $K=j+k$ ($N$ is a constant integer, $K$ is
integer yielded $0\leqslant K\leqslant N$), we get:

$\left|\tau\right\rangle={{\left(1+{{\left|\tau\right|}^{2}}\right)}^{-{N}/{2}\;}}\sum\limits_{K=0}^{N}{{{\left(\begin{matrix}N\\\
K\\\ \end{matrix}\right)}^{{1}/{2}}}{{\tau}^{K}}\left|K,N\right\rangle}.$
(S.16)

In another usually used form, $\tau$ is rewritten as the normalized argument
form $\tau={{\text{e}}^{\text{i}\phi}}$, and Eq. (S.16) is rewritten as the
phase state:

$\left|\phi\right\rangle=\frac{1}{{{2}^{{N}/{2}\;}}}\sum\limits_{K=0}^{N}{{{\left(\begin{matrix}N\\\
K\\\
\end{matrix}\right)}^{{1}/{2}\;}}{{\text{e}}^{\text{i}K\phi}}\left|K,N\right\rangle},$
(S.17)

where the eigenstates $|K,N\rangle$ should fulfill the orthogonality
$\left\langle K,N|L,N\right\rangle={\delta_{KL}}$, where $\delta_{i,j}$ is the
Kronecker delta, and the completeness
$a_{x}^{\dagger}{{a}_{x}}\left|K,N\right\rangle=K\left|K,N\right\rangle$,
$a_{y}^{\dagger}{{a}_{y}}\left|K,N\right\rangle=\left(N-K\right)\left|K,N\right\rangle$,
$\sum\limits_{K=0}^{N}{\left|K,N\right\rangle\left\langle K,N\right|}=1$.

Figure S2: Frequency-degenerate spectrum and the ray representation in laser
cavity. The frequency-degenerate spectrum
$(f_{n,m,l}-f_{n_{0},m_{0},l_{0}})/\Delta f_{L}$ of the ideal spherical cavity
as a function of the normalized cavity length $L/R$ for the range of
$|n-n_{0}|\leq 12$, $|m-m_{0}|\leq 12$, and $|l-l_{0}|\leq 12$, where some
degeneracy states $|\Omega=P/Q\rangle$ are marked at corresponding positions
with corresponding schematics of ray representation of SU(2) oscillation.

###  .3 Frequency-degenerate state

For realizing SU(2) coherent state in a laser cavity, the eigenstates should
be the eigenmodes of the resonator and fulfill the coherent-superposition
condition of SU(2) wave-packet. Without loss of generality, we consider a
plano-concave cavity with the length of $L$, formed by a gain medium, a
concave spherical mirror with the radius of curvature of $R$ as the output
coupler, and a plane mirror high-reflective for laser. The eigenmodes
${\psi}_{n,m,l}$ ($n,m$ are the indices of transverse mode and $l$ is the
index of longitudinal mode) and the eigenvalues ${k}_{n,m,l}$ for a laser
cavity can be solved from the Helmholtz equation:

$\left({{\nabla}^{2}}\text{+}k_{n,m,l}^{2}\right){{\psi}_{n,m,l}}\left(x,y,z\right)=0.$
(S.18)

Under the paraxial approximation, the eigenmodes being separable in Cartesian
coordinate can be expressed as Hermite-Gaussian (HG) modes:

$\displaystyle\psi_{n,m,l}^{\left(\text{HG}\right)}\left(x,y,z\right)=$
$\displaystyle\frac{1}{\sqrt{{{2}^{m+n-1}}\pi
m!n!}}\frac{1}{w\left(z\right)}\text{e}^{-\frac{{{x}^{2}}+{{y}^{2}}}{{{w}^{2}}\left(z\right)}}$
$\displaystyle\times
H_{n}\left[\frac{\sqrt{2}x}{w\left(z\right)}\right]H_{m}\left[\frac{\sqrt{2}y}{w\left(z\right)}\right]$
$\displaystyle\times\text{e}^{\text{i}{{k}_{n,m,l}}\widetilde{z}-\text{i}\left(m+n+1\right){{\vartheta}}\left(z\right)},$
(S.19)

where $\vartheta(z)=\tan^{-1}(z/z_{R})$ is the Gouy phase, $H_{n}(\cdot)$
represents the Hermite polynomials of $n$-th order, $k_{n,m,l}=2\pi
f_{n,m,l}/c$, $f_{n,m,l}$ is the eigenmode frequency, $c$ is the speed of
light, $\widetilde{z}=z+(x^{2}+y^{2})z/[2(z^{2}+z_{R}^{2})]$,
$w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}}$, $w_{0}=\sqrt{(\lambda z_{R})/\pi}$ is the
beam radius parameter, and $\lambda$ is the emission wavelength. The eigenmode
frequency of resonator is given by Chen  _et al._ (2013); Tung  _et al._
(2016):

$\displaystyle f_{n,m,l}$ $\displaystyle=l\Delta f_{L}+(n+\frac{1}{2})\Delta
f_{x}+(m+\frac{1}{2})\Delta f_{y}$
$\displaystyle=\left[l+(n+m+1)\Omega\right]\Delta f_{L},$ (S.20)

where the longitudinal mode spacing $\Delta f_{L}=c/(2L)$, here the minor
disparity between the physical length and the geometric length is neglected.
Without consideration of symmetry breaking, the transverse mode spacing should
be $\Delta f_{x}+\Delta f_{y}=\Delta f_{T}=\Delta f_{L}\vartheta(L)/\pi$. The
mode-spacing ratio $\Omega=P/Q=(1/\pi)\cos^{-1}(\sqrt{1-L/R})$ reveals the
degeneracy, which is varied in the range between 0 and 1/2 by changing the
cavity length as $0<L<R$. The frequency difference in the neighborhood of the
indices $(n_{0},m_{0},l_{0})$ is given by
$(f_{n,m,l}-f_{n_{0},m_{0},l_{0}})/\Delta f_{L}$, which can illustrate the
various degeneracy states distribution as topological joints in the fractal
spectrum Tung  _et al._ (2016). Fig. S2 depicts a diagram of the frequency-
degenerate spectrum where some degeneracy states $|\Omega=P/Q\rangle$ are
marked at corresponding positions. In order to fulfill the condition of
coherent superposition, the frequency of every decomposed eigenmodes should be
a constant, which requires a coupling effect between transverse and
longitudinal modes. If the transverse mode at $x$-axis and longitudinal mode
are coupled, i.e. $n+m+l=N$, we can choose a frequency-degenerate family of HG
modes as the complete set of orthogonal bases:

$\left|K,N\right\rangle=\psi_{{{n}_{0}}+QK,m_{0},{l_{0}}-PK}^{\left(\text{HG}\right)}\left(x,y,z\right),$
(S.21)

where $n_{0}$, $m_{0}$, and $l_{0}$ are constants with $n_{0}+m_{0}+l_{0}=N$,
thus the constant frequency should be:

$\displaystyle f_{n,m,l}$
$\displaystyle=\left[{{l}_{0}}-PK+({{n}_{0}}+QK+m_{0}+1)\Omega\right]\Delta
f_{L}$ $\displaystyle=\left[l_{0}+(n_{0}+m_{0}+1)\Omega\right]\Delta f_{L}$
$\displaystyle=f_{n_{0},m_{0},l_{0}},$ (S.22)

and the corresponding laser wave-packet of SU(2) coherent state is given by:

$\displaystyle\Psi_{{n_{0},m_{0}}}^{N,\Omega,\phi}\left(x,y,z\right)=$
$\displaystyle\frac{1}{{{2}^{N/{2}}}}\sum\limits_{K=0}^{N}{{\left(\begin{matrix}N\\\
K\\\ \end{matrix}\right)}^{{1}/{2}}}{{\text{e}}^{\text{i}K{\phi}}}$
$\displaystyle\times\psi_{{{n}_{0}}+QK,m_{0},{{l}_{0}}-PK}^{\left(\text{HG}\right)}\left(x,y,z\right),$
(S.23)

sharing the same form of Eq. (S.17). Here we already proved that frequency-
degenerate state of laser cavity fulfills the condition for generating a laser
wave-packet as SU(2) coherent state. We can also use Laguerre-Gaussian (LG)
modes being separable in circular coordinate as the eigenmodes to generate
SU(2) vortex beams:

$\displaystyle\Phi_{{n_{0},m_{0}}}^{N,\Omega,\phi}\left(x,y,z\right)=$
$\displaystyle\frac{1}{{{2}^{N/{2}}}}\sum\limits_{K=0}^{N}{{\left(\begin{matrix}N\\\
K\\\ \end{matrix}\right)}^{{1}/{2}}}{{\text{e}}^{\text{i}K{\phi}}}$
$\displaystyle\times\varphi_{{{n}_{0}}+QK,m_{0},{{l}_{0}}-PK}^{\left(\text{LG}\right)}\left(x,y,z\right),$
(S.24)

where the LG modes are given by:

$\displaystyle\varphi_{n,m,l}^{\left(\text{LG}\right)}\left(\rho,\theta,z\right)=$
$\displaystyle\sqrt{\frac{2p!}{\pi\left(p+\left|\ell\right|\right)!}}\frac{1}{w\left(z\right)}{{\left[\frac{\sqrt{2}r}{w\left(z\right)}\right]}^{\left|\ell\right|}}\text{e}^{-\frac{{{r}^{2}}}{{{w}^{2}}\left(z\right)}}\text{e}^{\text{i}\ell\theta}$
$\displaystyle\times
L_{p}^{\left|\ell\right|}\left[\frac{2{{r}^{2}}}{{{w}^{2}}\left(z\right)}\right]\text{e}^{\text{i}{{k}_{n,m,l}}\widetilde{z}-\text{i}\left(m+n+1\right){{\vartheta}}\left(z\right)},$
(S.25)

where $p=\min\left(m,n\right)$, $\ell=\pm\left(m-n\right)$, and
$L_{p}^{\ell}\left(\cdot\right)$ represents the associated Laguerre polynomial
with radial and azimuthal indices of $p$ and $\ell$. For $m_{0}=0$, Fig. 6(c)
in the main text shows the 3-D simulation of SU(2) vortex beams with positive,
negative, and superposed OAMs, and the inserts show the topological phases of
the SU(2) vortex beams. For $m_{0}\geq 1$, the SU(2) beams manifest the multi-
LG vortex beams Lu  _et al._ (2011); Tuan  _et al._ (2018), as shown in Fig.
6(e) in the main text for 3-D simulation together with topological phase,
where the main OAM at the center is decided by the index $n_{0}$ and the sub-
OAM carried by sub-LG beams is decided by the index $m_{0}$. For constituting
a completed oscillation in cavity, the positive and negative oscillations
should be superposed together forming a standing wave mode, the phase state
expression of which is $|+\rangle+|-\rangle=|\phi\rangle+|2\pi-\phi\rangle$
Chen  _et al._ (2004). Figure S3 shows the ray presentation (a) intensity
wave-packet (b-d) of intracavity geometric modes versus the coherent state
phase. From larger $n_{0}$ to smaller one, the wave-packets perform from ray-
like cases to wave-like cases. Figure S4 shows the transverse patterns of
planar and vortex geometric modes and the topological phase of vortex
geometric modes at SU(2) coherent state $|\Omega=1/4\rangle|\phi=\pi\rangle$
with parameters as $z=z_{R}$, $M=20$, and various $n_{0}$ and $m_{0}$, where
some cases perform multi-spot shape while some wave fringes unravel the
interference among lights on the sub-orbits, which is manifested by the
property of ray-wave duality.

Figure S3: Phase states of SU(2) oscillation with ray-wave duality.
Intracavity planar SU(2) geometric mode $|\phi\rangle+|2\pi-\phi\rangle$
oscillating at $(x,z)$-plane at degenerate state $|\Omega=1/4\rangle$: (a) The
ray representations and (b-d) intensity wave-packets with $n_{0}$ from larger
to smaller ($n_{0}=30,20,10$) for various $\phi$. The patterns of wave
representations from (b) to (d) change from the case of more ray-like
properties to that of more wave-like properties.

There are also other ways to realize frequency degeneracy in order to fulfill
the coherent superposition of SU(2) wave-packet. For instance:

$\left|K,N\right\rangle=\psi_{{{n}_{0}}+pK,m_{0}+qK,{l_{0}}-PK}^{\left(\text{HG}\right)}\left(x,y,z\right),$
(S.26)

where the integers $p$ and $q$ yield $p+q=Q$, thus the eigenmodes also
constitute a frequency-degenerate family with frequency
$f_{n_{0},m_{0},l_{0}}$. Using the more general Eq. (S.26) as the bases of
SU(2) coherent state, we can obtain more exotic structured light beams Chen
_et al._ (2017); Tung  _et al._ (2016).

Figure S4: Planar and vortex SU(2) geometric modes. The theoretical intensity
transverse patterns of planar and vortex geometric modes and the topological
phase of vortex geometric modes at SU(2) coherent state
$|\Omega=1/4\rangle|\phi=\pi\rangle$ with parameters as $z=z_{R}$, $M=20$, and
various $n_{0}$ and $m_{0}$.

###  .4 Ray-wave duality

When an optical resonator is operating close to frequency-degenerate state,
named frequency-degenerate cavity, its laser mode and intensity would undergo
dramatic changes with the principle that laser modes have a preference to be
localized on the periodic ray trajectories under off-axis pumping, which is
called the ray-wave duality. Like the Schrödinger coherent state coupled with
the trajectory of classical oscillator, the SU(2) coherent state can also be
coupled with the periodic oscillating trajectories in frequency-degenerate
cavity.

Based on geometrical optics, the ABCD matrix is used to characterize the
propagation property of the optical ray trajectories inside a stable plano-
concave cavity. Since the cavity length satisfies $L/R=\sin^{2}(\Omega\pi)$
under degeneracy state $|\Omega=P/Q\rangle$, the corresponding ABCD matrix of
the frequency-degenerate cavity is given by:

$\mathbf{A}=\left[\begin{matrix}1-\frac{2L}{R}&2L\left(1-\frac{L}{R}\right)\\\
-\frac{2}{R}&1-\frac{2L}{R}\\\
\end{matrix}\right]=\left[\begin{matrix}\cos\left(2\Omega\pi\right)&\frac{R}{2}{{\sin}^{2}}\left(2\Omega\pi\right)\\\
-\frac{2}{R}&\cos\left(2\Omega\pi\right)\\\ \end{matrix}\right].$ (S.27)

After $n$ times of round trips in the frequency-degenerate cavity, the matrix
is derived as:

${{\mathbf{A}}^{n}}=\left[\begin{matrix}\cos\left(2n\Omega\pi\right)&\frac{R}{2}{{\sin}^{2}}\left(2n\Omega\pi\right)\\\
-\frac{2}{R}\frac{\sin\left(2n\Omega\pi\right)}{\sin\left(2\Omega\pi\right)}&\cos\left(2n\Omega\pi\right)\\\
\end{matrix}\right].$ (S.28)

Because $Q\Omega=P$ is an integer, $\cos\left(2Q\Omega\pi\right)=1$,
$\sin\left(2Q\Omega\pi\right)=0$, and $Q$-th power of $\mathbf{A}$ is an unit
matrix:

${{\mathbf{A}}^{Q}}=\left[\begin{matrix}\cos\left(2Q\Omega\pi\right)&\frac{R}{2}{{\sin}^{2}}\left(2Q\Omega\pi\right)\\\
-\frac{2}{R}\frac{\sin\left(2Q\Omega\pi\right)}{\sin\left(2\Omega\pi\right)}&\cos\left(2Q\Omega\pi\right)\\\
\end{matrix}\right]=\mathbf{I}.$ (S.29)

Equation (S.29) reveals that an optical ray oscillating at an arbitrary
position within the cavity would coincide exactly with the initial state after
$Q$ times of round trips. Therefore, it is proved that the lasing modes have a
preference to be localized on the periodic ray trajectories in a frequency-
degenerate cavity. The schematics of classical oscillating trajectories at
various states $|\Omega=P/Q\rangle$ are shown in Fig. S2. Manifested by the
ray matrix, the parametric equation for each periodic orbit in SU(2)
oscillation can be derived. For the planar geometric modes, the orbits can be
derived as:

$\displaystyle x_{s}^{\pm}(z)$
$\displaystyle=\sqrt{N}{{w}_{0}}\left[\cos\left({{\theta}_{s}}+{{\phi}_{x}}\right)\mp\left(z/{{z}_{R}}\right)\sin\left({{\theta}_{s}}+{{\phi}_{x}}\right)\right]$
$\displaystyle=\sqrt{N}w(z)\cos\left[{{\theta}_{s}}+{{\phi}_{x}}\pm\vartheta(z)\right]$
(S.30)

where ${{\theta}_{s}}=(P/Q)2\pi s,s=0,1,2,\cdots,Q-1$ is the running index for
the different rays, ${\phi}_{x}$ is the phase factor related to the initial
position and direction, and $+$ and $-$ in the symbol of $\pm$ indicate the
backward and forward rays, respectively. Defining the dimensionless variable
$\widetilde{x}=\sqrt{2}x/w(z)$, the expression for the ray equation can be
expressed as $\widetilde{x}(z)=\operatorname{Re}[\sqrt{2}{u}_{s}^{\pm}(z)]$
with:

$u_{s}^{\pm}(z)=\sqrt{{N}}\text{e}^{-\text{i}[{{\theta}_{s}}+{{\phi}_{x}}\pm{{\vartheta}}(z)]}.$
(S.31)

when $\phi_{x}=n\pi/Q$ ($n\in\mathbb{Z}$), the forward and backward rays would
be coincidently overlapped, and the bouncing orbits with positive and negative
transverse directions share the same location. In this case, the trajectories
in frequency-degenerate cavities at various degenerate states
$|\Omega=P/Q\rangle$ are depicted in Fig. S2. For example at
$|\Omega=1/4\rangle$ ($\phi_{x}=\pi/4$), positive and negative W-shaped
trajectories coincide exactly, i.e.
$|+\rangle=|u_{0}^{\pm}\rangle|u_{1}^{\pm}\rangle$ and
$|-\rangle=|u_{2}^{\pm}\rangle|u_{3}^{\pm}\rangle$, as shown in Fig. 1 in the
main text. Additionally, in the four periodic round-trips, two share the first
bouncing location and other two share the second one, i.e.
$|1\rangle=|u_{0}^{\pm}\rangle|u_{3}^{\pm}\rangle$ and
$|2\rangle=|u_{1}^{\pm}\rangle|u_{2}^{\pm}\rangle$. For the general spatial
geometric mode, the ray equations for the 3-D periodic orbits can be written
as:

$\left\\{\begin{aligned}
u_{s}^{\pm}(z)&=\sqrt{{N_{x}}}\text{e}^{-\text{i}[{{\theta}_{s}}+{{\phi}_{x}}\pm{{\vartheta}}(z)]}\\\
v_{s}^{\pm}(z)&=\sqrt{{N_{y}}}\text{e}^{-i[{{\theta}_{s}}+{{\phi}_{y}}\pm{{\vartheta}}(z)]}\end{aligned}\right.$
(S.32)

where the dimensionless variable in the $y$-direction is similarly defined as
$\widetilde{y}=\sqrt{2}y/w(z)$ with the ray equation
$\widetilde{y}(z)=\operatorname{Re}[\sqrt{2}{v}_{s}^{\pm}(z)]$, here $+$ and
$-$ in the symbol of $\pm$ indicate the positive and negative OAM states
$|\pm\ell\rangle$, they together constitute a completed oscillation in cavity,
as shown in Figs. 6(a) and (b) in the main text. For constituting a completed
oscillation with both OAM states in a cavity, the phase factors yield
$|\phi_{x}-\phi_{y}|=\pi/2$.

The above is the ray representation of geometric modes in frequency-degenerate
cavity. Hereinafter, we derive the wave representation coupled with the
geometric modes and prove that it fulfills the SU(2) coherent state. The
Gaussian wave packet with the central peak moving along the path
$x=\operatorname{Re}(\sqrt{2}u)=\sqrt{2N}\cos(\omega t+{{\varphi}_{0}})$ can
be derived as Chen  _et al._ (2019):

${{\pi}^{-1/4}}{{\text{e}}^{-{{x}^{2}}/2}}F(x,u)=\sum\limits_{n=0}^{\infty}{{{a}_{n}}\psi_{n}(x)}{{\text{e}}^{-\text{i}n(\omega
t+{{\varphi}_{0}})}},$ (S.33)

where coefficients ${{a}_{n}}={N^{n/2}}{{e}^{-N/2}}/\sqrt{n!}=\sqrt{P(n,N)}$,
where $P(n,N)$ is the Poisson distribution, $F(x,u)$ is the wave function of
Schrödinger coherent state and $\psi_{n}(x)$ is the Hermite–Gaussian function:

$F(x,u)={{\operatorname{e}}^{-\left({{u}^{2}}+|u{{|}^{2}}-2\sqrt{2}ux\right)/2}},$
(S.34)
${\psi}_{n}(x)={{\left({{2}^{n}}n!\right)}^{-1/2}}{{\pi}^{-1/4}}{{H}_{n}}(x){{e}^{-{{x}^{2}}/2}}.$
(S.35)

The wave representation of a Gaussian wave packet moving along the $s$-th ray
in a spatial geometric mode can be given by:

$\Phi(\mathbf{r},u_{s}^{\pm},v_{s}^{\pm})=G(\mathbf{r})F(\widetilde{x},u_{s}^{\pm})F(\widetilde{y},v_{s}^{\pm}).$
(S.36)

where
$G(\mathbf{r})={{\pi}^{-1/2}}{\text{e}^{-({{{\widetilde{x}}}^{2}}+{{{\widetilde{y}}}^{2}})(1+\text{i}\widetilde{z})/2}}{\text{e}^{\mp\text{i}{{\vartheta}}(\widetilde{z})}}$
represents the fundamental mode Gaussian beam and $\mathbf{r}=(x,y,z)$ the
Cartesian coordinates. In terms of Eq. (S.36), the resonant mode for the
forward and backward components of a complete period is given by:

$\Psi_{N_{x},N_{y}}^{\pm}(\mathbf{r})=\frac{1}{Q}\sum\limits_{s=0}^{Q-1}{\Phi\left(\mathbf{r},u_{s}^{\pm},v_{s}^{\pm}\right){\text{e}^{\text{i}\left(N_{x}+N_{y}\right)\theta_{s}}}},$
(S.37)

where the phase term $\text{e}^{\text{i}\left(N_{x}+N_{y}\right)\theta_{s}}$
is associated with the transverse frequency. For $N_{x}=0$ or $N_{y}=0$, Eq.
(S.37) represents the planar geometric modes with ray structure on $(x,z)$ or
$(y,z)$ plane; for $N_{x}=N_{y}\neq 0$, circular vortex geometric modes; for
$0\neq N_{x}\neq N_{y}\neq 0$, elliptical vortex geometric modes. In the above
description, $N_{x}$ or $N_{y}$ should be large enough to stimulate more ray-
like properties, otherwise the pattern will be nearly a certain eigenmode.
Hereinafter, we demonstrate the wave representation fulfills the form of SU(2)
coherent state. Using planar trajectory $N_{x}=N$ and $N_{y}=0$ for
convenience, it can be obtained that $v_{s}^{\pm}=0$ and
$F(\widetilde{y},0)=1$, and the planar geometric mode is given by:

$\Psi_{N}^{\pm}(\mathbf{r})=\frac{1}{Q}\sum\limits_{s=0}^{Q-1}G(\mathbf{r}){F}\left(\widetilde{x},u_{s}^{\pm}\right){{\text{e}}^{\text{i}N{{\theta}_{s}}}},$
(S.38)

Substituting Eq. (S.33) into Eq. (S.38) and ignoring the constant coefficient,
we get:

$\displaystyle\Psi_{{N}}^{\pm}(\mathbf{r})$
$\displaystyle\propto\sum\limits_{s=0}^{Q-1}{G(\mathbf{r}){{\text{e}}^{\frac{{\widetilde{x}}^{2}}{2}}}\sum\limits_{n=0}^{\infty}{{{a}_{n}}{{\psi}_{n}}(\widetilde{x}){{\text{e}}^{-\text{i}n\left[{{\theta}_{s}}+{{\phi}_{x}}\pm\vartheta(z)\right]}}}}{{\text{e}}^{\text{i}{{N}}{{\theta}_{s}}}}$
$\displaystyle=\sum\limits_{s=0}^{Q-1}{\sum\limits_{n=0}^{\infty}{{{a}_{n}}{{\psi}_{n}}(\widetilde{x}){{\text{e}}^{\frac{{{\widetilde{x}}^{2}}}{2}}}G(\mathbf{r}){{\text{e}}^{\mp\text{i}n\vartheta(z)-\text{i}n\left({{\theta}_{s}}+{{\phi}_{x}}\right)+\text{i}N\theta_{s}}}}}$
$\displaystyle\propto\sum\limits_{s=0}^{Q-1}{\sum\limits_{n=0}^{\infty}{{{a}_{n}}\psi_{n,0,l}^{(\text{HG})}(x,y,\pm
z){{\text{e}}^{-\text{i}n{{\phi}_{x}}}}{{\text{e}}^{\text{i}\left(N-n\right){{\theta}_{s}}}}}}$
$\displaystyle=\sum\limits_{n=0}^{\infty}{{{a}_{n}}\psi_{n,0,l}^{(\text{HG})}(x,y,\pm
z){{\text{e}}^{-\text{i}n{{\phi}_{x}}}}\sum\limits_{s=0}^{Q-1}{{{\text{e}}^{\text{i}\left(N-n\right){{\theta}_{s}}}}}},$
(S.39)

where the indices of HG modes should also fulfill the frequency-degenerate
condition. Setting $n^{\prime}=N-n$, the last term in the external summation
notation of Eq. (S.39) can be written as:

$\sum\limits_{s=0}^{Q-1}{{{\text{e}}^{\text{i}n^{\prime}{{\theta}_{s}}}}}=\sum\limits_{s=0}^{Q-1}\text{e}^{\text{i}n^{\prime}\frac{P}{Q}2\pi
s}.$ (S.40)

When $n^{\prime}=KQ$ ($K\in\mathbb{Z}$),
$\text{e}^{\text{i}n^{\prime}\frac{P}{Q}2\pi s}=\text{e}^{\text{i}2\pi KPs}=1$
and Eq. (S.40) is equal to a constant $Q$; when $n^{\prime}\neq KQ$, Eq.
(S.40) is always a sum of the complex numbers uniformly distributed on the
unit circle of the complex plane, thus it should be zero. Then Eq. (S.39) can
be reduced as:

$\displaystyle\Psi_{{N}}^{\pm}(\mathbf{r})$
$\displaystyle\propto\sum\limits_{K=0}^{N}{{{a}_{n_{0}+KQ}}{\text{e}}^{-\text{i}KQ\phi_{x}}\psi_{n_{0}+KQ,0,l_{0}-KP}^{(\text{HG})}(x,y,\pm
z)}$
$\displaystyle=\sum\limits_{K=0}^{N}\sqrt{P(n_{0}+KQ,N)}{\text{e}}^{-\text{i}KQ\phi_{x}}|K,N\rangle$
$\displaystyle\propto\sum\limits_{K=0}^{N}\sqrt{B(K;N,\frac{1}{2})}{\text{e}}^{-\text{i}KQ\phi_{x}}|K,N\rangle$
$\displaystyle=\frac{\text{1}}{{{\text{2}}^{N/2}}}\sum\limits_{K=0}^{N}{{{\left(\begin{matrix}N\\\
K\\\ \end{matrix}\right)}^{1/2}}{\text{e}}^{\text{i}K\phi}|K,N\rangle},$
(S.41)

where the coherent state phase $\phi=Q\phi_{x}$, and the Poisson distribution
$P(n_{0}+KQ,N)$ is approximated by Binomial distribution $B(K;N,1/2)$ when
$N=4Q^{2}n_{0}$ is large enough, where $B(k;n,p)=\left(\begin{matrix}n\\\ k\\\
\end{matrix}\right){{p}^{k}}{{\left(1-p\right)}^{n-k}}$ is Binomial
distribution, according to the central-limit theorem. Then the laser mode Eq.
(S.41) shares the same form of SU(2) coherent state as Eq. (S.17).

Therefore, the SU(2) wave-packet in frequency-degenerate cavity has the
property of ray-wave duality, the laser mode can be not only characterized by
the wave function representation but also coupled with classical oscillating
trajectory given by ray representation. The preponderance of wave-like or ray-
like property can be actually controlled by the parameters in SU(2) wave-
packet.

###  .5 Generation of SU(2) geometric beams

In section .3, we demonstrated that the frequency-degenerate cavity fulfills
the condition of generating SU(2) wave-packet. Hereinafter, we will
demonstrate the actual method to generate SU(2) planar geometric beams in a
frequency-degenerate cavity by off-axis pumping. Considering the gain
distribution $f(x,y,z)$ , the resonant modes of the laser system pumped by a
localized source can be solved from the inhomogeneous Helmholtz equation:

$\left({{\nabla}^{2}}+{{\widetilde{k}}^{\text{2}}}\right)\Psi(x,y,z)={{\eta}_{c}}f(x,y,z),$
(S.42)

where $\widetilde{k}=k_{0}+\text{i}\alpha$, $k_{0}=2\pi/\lambda_{0}$ is the
given wave number of the laser with given wavelength $\lambda_{0}$, the factor
$\eta_{c}$ represents the conversion efficiency for the excitation source, and
$\alpha$ is a small loss parameter including losses from the scattering, the
absorption, and the output coupling. Based on the orthogonality and
completeness of HG modes, the lasing mode and the source distribution can be
expressed as the superposition of HG eigenmodes:

$\Psi(x,y,z)=\sum\limits_{n,m,l}{{{a}_{n,m,l}}\psi_{n,m,l}^{(\text{HG})}(x,y,z)}$
(S.43)

and

$\eta_{c}F(x,y,z)=\sum\limits_{n,m,l}{{b_{n,m,l}}\psi_{n,m,l}^{(\text{HG})}(x,y,z)}$
(S.44)

Substituting Eqs. (S.43) and (S.44) into the wave equation Eq. (S.42), the
relationship between the coefficients $a_{n,m,l}$ and $b_{n,m,l}$ can be
found:

${{a}_{n,m,l}}={{b}_{n,m,l}}/({{\widetilde{k}}^{2}}-k_{n,m,l}^{2}).$ (S.45)

Considering the condition of $\alpha\ll k_{0}$ and substituting Eq. (S.45)
into Eq. (S.43), the eigenmode expansion of the resonant mode is given by:

$\Psi(x,y,z)=\sum\limits_{n,m,l}{\frac{{{b}_{n,m,l}}}{({{k}^{2}_{0}}-k_{n,m,l}^{2})+2\text{i}\alpha
k_{0}}}\psi_{n,m,l}^{(\text{HG})}(x,y,z).$ (S.46)

With the orthonormal property of eigenmodes, the coefficient $b_{n,m,l}$ is
given by:

${{b}_{n,m,l}}={{\eta}_{c}}\iiint{\psi_{n,m,l}^{(\text{HG})}(x,y,z)f(x,y,z)\text{d}x\text{d}y\text{d}z}.$
(S.47)

Supposing the pump light is a Gaussian beam along $z$-axis with focused waist
at $z=z_{c}$ position and an off-axis displacement $\Delta x$ at
$x$-direction, which is larger than pump spot size, $\Delta x\gg w_{0}$, also
supposing that the gain medium center is located at the pump waist with the
thickness much less than the cavity length, i.e. $L_{c}\ll L$, the pump source
distribution can be written as:

$f(x,y,z)=\frac{2}{\pi{{w}^{2}}(z-z_{c}){{L}_{c}}}{{\operatorname{e}}^{-\frac{{{(x-\Delta
x)}^{2}}+{{y}^{2}}}{{{w}^{2}}(z-z_{c})}}},$ (S.48)

for $|z-z_{c}|\leq L_{c}/2$, here $w(z)$ represents the pump beam radius
distribution. Since the longitudinal distribution of the pump source is nearly
uniform, the coefficient $b_{n,m,l}$ related to the source term $f(x,y,z)$ can
be considered to be independent of the index $l$ in the neighborhood of the
central index $l_{0}$. Consequently, the integral in Eq. (S.47) can be
approximately reduced as:

${{b}_{n,m,l}}=\frac{2\eta}{\pi
w_{0}^{2}{{L}_{c}}}\iint{\psi_{n,m,l}^{(\text{HG})}(x,y,0)}{{\operatorname{e}}^{-\frac{{{(x-\Delta
x)}^{2}}+{{y}^{2}}}{w_{0}^{2}}}}\text{d}x\text{d}y,$ (S.49)

where $\eta$ is a constant that includes the effective conversion efficiency
$\eta_{c}$ and the overlap integral in the longitudinal direction, here
$w_{0}$ represents the pump beam radius at its waist. Substituting Eqs. (S.19)
and (S.48) into Eq. (S.49), setting $X=\sqrt{2}x/{{w}_{0}}$,
$Y=\sqrt{2}y/{{w}_{0}}$, ${{n}_{x}}={{\left(\Delta x/{{w}_{0}}\right)}^{2}}$,
and applying generating function of the Hermite polynomials, $b_{n,m,l}$ can
be derived as:

$\displaystyle{{b}_{n,m,l}}$ $\displaystyle=\frac{2\eta}{\pi
w_{0}^{2}{{L}_{c}}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\frac{1}{\sqrt{{{2}^{m+n-1}}\pi
m!n!}}\frac{1}{{{w}_{0}}}{{H}_{n}}\left(\frac{\sqrt{2}x}{{{w}_{0}}}\right){{H}_{m}}\left(\frac{\sqrt{2}y}{{{w}_{0}}}\right){{\operatorname{e}}^{-\frac{{{x}^{2}}+{{y}^{2}}}{w_{0}^{2}}}}{{\operatorname{e}}^{-\frac{{{\left(x-\Delta
x\right)}^{2}}+{{y}^{2}}}{w_{0}^{2}}}}\text{d}x\text{d}y}$
$\displaystyle=\frac{\eta}{\pi{{w}_{0}}{{L}_{c}}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\frac{1}{\sqrt{{{2}^{m+n-1}}\pi
m!n!}}{{H}_{n}}\left(X\right){{H}_{m}}\left(Y\right){{\operatorname{e}}^{-\frac{{{X}^{2}}+{{Y}^{2}}}{2}}}{{\operatorname{e}}^{-{{\left(\frac{X}{\sqrt{2}}-{{n}_{x}}\right)}^{2}}-{{\left(\frac{Y}{\sqrt{2}}\right)}^{2}}}}\text{d}X\text{d}Y}$
$\displaystyle=\frac{\eta}{\pi{{w}_{0}}{{L}_{c}}}{{\delta}_{m,0}}\int_{-\infty}^{\infty}{\frac{1}{\sqrt{{{2}^{n-1}}\pi
n!}}{{H}_{n}}\left(X\right){{\operatorname{e}}^{-\frac{{{X}^{2}}}{2}}}{{\operatorname{e}}^{-\frac{{{X}^{2}}}{2}+\sqrt{2{{n}_{x}}}X-{{n}_{x}}}}\text{d}X}$
$\displaystyle=\frac{\eta}{\pi{{w}_{0}}{{L}_{c}}}\frac{1}{\sqrt{{{2}^{n-1}}\pi
n!}}{{\operatorname{e}}^{-\frac{{{n}_{x}}}{2}}}{{\delta}_{m,0}}\int_{-\infty}^{\infty}{{{H}_{n}}\left(X\right){{\operatorname{e}}^{-{{X}^{2}}}}{{\operatorname{e}}^{\sqrt{2{{n}_{x}}}X-\frac{{{n}_{x}}}{2}}}\text{d}X}$
$\displaystyle=\frac{\eta}{\pi{{w}_{0}}{{L}_{c}}}\frac{1}{\sqrt{{{2}^{n-1}}\pi
n!}}{{\operatorname{e}}^{-\frac{{{n}_{x}}}{2}}}{{\delta}_{m,0}}\int_{-\infty}^{\infty}{{{H}_{n}}\left(X\right){{\operatorname{e}}^{-{{X}^{2}}}}\left[\sum\limits_{p=0}^{\infty}{\frac{{{H}_{p}}\left(X\right){{\left(\sqrt{\frac{{n}_{x}}{2}}\right)}^{p}}}{p!}}\right]\text{d}X}$
$\displaystyle=\frac{\eta}{\pi{{w}_{0}}{{L}_{c}}}\frac{1}{\sqrt{{{2}^{n-1}}\pi
n!}}\frac{{{\left({{n}_{x}}\right)}^{\frac{n}{2}}}}{\sqrt{{{2}^{n}}}}\frac{1}{n!}{{\operatorname{e}}^{-\frac{{{n}_{x}}}{2}}}{{\delta}_{m,0}}\int_{-\infty}^{\infty}{{{H}_{n}}\left(X\right){{H}_{n}}\left(X\right){{\operatorname{e}}^{-{{X}^{2}}}}\text{d}X}$
$\displaystyle=\frac{\eta{{\delta}_{m,0}}}{\pi{{w}_{0}}{{L}_{c}}}\frac{1}{\sqrt{{{2}^{n-1}}\pi
n!}}\frac{{{\left({{n}_{x}}\right)}^{\frac{n}{2}}}}{\sqrt{{{2}^{n}}}}\frac{1}{n!}{{\operatorname{e}}^{-\frac{{{n}_{x}}}{2}}}{{\delta}_{m,0}}\left(\sqrt{\pi}{{2}^{n}}n!\right)$
$\displaystyle=\frac{\sqrt{2}\eta}{\pi{{w}_{0}}{{L}_{c}}}\frac{{{\left({{n}_{x}}\right)}^{\frac{n}{2}}}}{\sqrt{n!}}{{\operatorname{e}}^{-\frac{{{n}_{x}}}{2}}}{{\delta}_{m,0}}=\frac{\sqrt{2}\eta}{\pi{{w}_{0}}{{L}_{c}}}\sqrt{P\left(n,{{n}_{x}}\right)}{{\delta}_{m,0}},$
(S.50)

where $\delta_{i,j}$ is the Kronecker delta. Because $\Delta x\gg w_{0}$ and
similarly $n_{x}\gg\sqrt{2}$, there should be no overlap between the pump spot
and the high-order mode at $y$-direction, we can use $\delta_{m,0}$ to
simplify the formula. We also use the orthogonality with respect to the weight
function of Hermite polynomials,
$\int_{-\infty}^{\infty}H_{m}(x)H_{n}(x)\text{e}^{-x^{2}}\text{d}x=\sqrt{\pi}2^{n}n!\delta_{n,m}$,
to reduce the integral in Eq. (S.50). Based on Eq. (S.20) and relationship
${{k}_{n,m,l}}=2\pi{{f}_{n,m,l}}/c$, and no high-order mode at $y$-axis here,
the expressions of center and variable wave numbers are derived as
${{k}_{0}}=\pi\left[{{l}_{0}}+({{n}_{0}}+1)\Omega\right]/L$, and
${{k}_{n,m,l}}=\pi\left[{{l}}+({{n}}+1)\Omega\right]/L$. Substituting them and
Eq. (S.50) into Eq. (S.46), we can derive the expression of laser mode:

$\displaystyle\Psi(x,y,z)$
$\displaystyle=\frac{\sqrt{2}\eta}{\pi{{w}_{0}}{{L}_{c}}}{{\delta}_{m,0}}\sum\limits_{n,m,l}{\frac{\sqrt{P({n,n_{x}})}}{(k+{{k}_{n,m,l}})(k-{{k}_{n,m,l}})+2\text{i}\alpha
k}}\psi_{n,m,l}^{(\text{HG})}(x,y,z)$
$\displaystyle=\frac{\sqrt{2}\eta}{\pi{{w}_{0}}{{L}_{c}}}\sum\limits_{n}{\sum\limits_{l}{\frac{\sqrt{P({n,n_{x}})}}{{{\pi}^{2}}/{{L}^{2}}\\{[l+{{l}_{0}}+(n+{{n}_{0}}+2)\Omega][{{l}_{0}}-l+({{n}_{0}}-n)\Omega]+2\text{i}\gamma[{{l}_{0}}+({{n}_{0}}+1)\Omega]\\}}}\psi_{n,0,l}^{(\text{HG})}(x,y,z)}$
$\displaystyle=\frac{\sqrt{2}\eta}{\pi{{w}_{0}}{{L}_{c}}}\frac{{{L}^{2}}}{{{\pi}^{2}}}\sum\limits_{n}{\sum\limits_{l}{\frac{\sqrt{P({n,n_{x}})}}{[l+{{l}_{0}}+(n+{{n}_{0}}+2)\Omega][{{l}_{0}}-l+({{n}_{0}}-n)\Omega]+2\text{i}\gamma[{{l}_{0}}+({{n}_{0}}+1)\Omega]}}\psi_{n,0,l}^{(\text{HG})}(x,y,z)}$
$\displaystyle=\frac{\sqrt{2}\eta{{L}^{2}}}{{{\pi}^{3}}{{w}_{0}}{{L}_{c}}}\sum\limits_{n}{\sum\limits_{l}{\frac{\sqrt{P({n,n_{x}})}}{\beta[({{l}_{0}}-l)+({{n}_{0}}-n)\Omega]+\text{i}\gamma}}\psi_{n,0,l}^{(\text{HG})}(x,y,z)},$
(S.51)

where $\gamma=\alpha L/\pi$ and
$\beta=\frac{l+{{l}_{0}}+(n+{{n}_{0}}+2)\Omega}{2[{{l}_{0}}+({{n}_{0}}+1)\Omega]}=\frac{k+{{k}_{n,m,l}}}{2k}\approx
1$. Note that the value of the parameter $n_{x}$ signifies the magnitude of
the off-axis displacement, also indicates that the maximum contribution in the
resonant mode comes from the eigenmode with the transverse index $n$ to be
closest to the value $n_{0}$. Therefore, we can take the parameter
$n_{x}=n_{0}$ for convenience. For a Poisson distribution with large
parameter, it can be approximated using a Gaussian distribution of an
effective range near its mean, then the effective range of mode index $n$ can
be limited as $|n-n_{0}|\leq n^{\prime}$ ($n^{\prime}=2\sqrt{{{n}_{0}}}$) for
the sum in Eq. (S.51):

$\Psi(x,y,z)=\frac{\sqrt{2}\eta{{L}^{2}}}{{{\pi}^{3}}{{w}_{0}}{{L}_{c}}}\sum\limits_{l={{l}_{0}}-l^{\prime}}^{{{l}_{0}}+l^{\prime}}{\sum\limits_{n={{n}_{0}}-n^{\prime}}^{{{n}_{0}}+n^{\prime}}{\frac{\sqrt{P({n,n_{0}})}}{\beta[({{l}_{0}}-l)+({{n}_{0}}-n)\Omega]+\text{i}\gamma}}}\psi_{n,m,l}^{(\text{HG})}(x,y,z).$
(S.52)

Considering $\gamma=\alpha L/\pi$ where $\alpha\sim{{10}^{-6}}$m and
$\Omega=P/Q$ is a simple fraction number, when
$\left|({{l}_{0}}-l)+({{n}_{0}}-n)\Omega\right|\neq 0$, there should be
$\left|({{l}_{0}}-l)+({{n}_{0}}-n)\Omega\right|\gg\left|\text{i}\gamma\right|$,
so that we can neglect the terms of corresponding eigenstates and only
consider the HG modes that satisfy
$\left|({{l}_{0}}-l)+({{n}_{0}}-n)\Omega\right|=0$, which can also be written
as $-\left({{l}_{0}}-l\right)/\left({{n}_{0}}-n\right)=\Omega=P/Q$. Defining
$K=(l-l_{0})/P=-(n_{0}-n)/Q$, Eq. (S.52) can be simplified with only one
accumulative symbol:

$\displaystyle\Psi(x,y,z)=\frac{\sqrt{2}\eta{{L}^{2}}}{{{\pi}^{3}}{{w}_{0}}{{L}_{c}}}\sum\limits_{K=-n^{\prime}/Q}^{n^{\prime}/Q}{\frac{\sqrt{P({n_{0}-KQ,n_{0}})}}{\text{i}\alpha
L/\pi}}\psi_{{{n}_{0}}+KQ,0,{{l}_{0}}-KP}^{(\text{HG})}(x,y,z).$ (S.53)

Setting $\bar{n}_{0}=n_{0}-n^{\prime}$, $\bar{l}_{0}=l_{0}-l^{\prime}$, and
$N=4Q^{2}n_{0}$, and according to the central-limit theorem in maximum
likelihood estimation, the Poission distribution $P(n_{0}-KQ,n_{0})$ can be
approximated by Binomial distribution $B(K;N,1/2)$ when $N$ is large enough.
Then the laser mode Eq. (S.53) can be derived into the form of SU(2) coherent
state:

$\displaystyle\Psi(x,y,z)$
$\displaystyle\propto\sum\limits_{K=0}^{N}{\sqrt{B(K;N,\frac{1}{2})}\psi_{{\bar{n}_{0}}+KQ\,,0,\,{\bar{l}_{0}}-KP}^{(\text{HG})}}(x,y,z)$
$\displaystyle=\frac{\text{1}}{{{\text{2}}^{N/2}}}\sum\limits_{K=0}^{N}{{{\left(\begin{matrix}N\\\
K\\\
\end{matrix}\right)}^{1/2}}\psi_{{\bar{n}_{0}}+KQ,0,\bar{l}_{0}-PK}^{(\text{HG})}(x,y,z)},$
(S.54)

sharing the same form of Eq. (S.17). If we stretch the pump size along
$y$-axis to stimulate the high-order mode at $y$-direction and control the
astigmatism in pumping light to lock additive phase among eigenmodes, the
laser mode can be controlled as more general SU(2) coherent state as Eq.
(S.23). The laser modes can also be transformed into vortex geometric modes
Eq. (S.24) by astigmatic mode converter that transforms the HG bases into
corresponding LG bases. Hereto, we have proved that the frequency-degenerate
cavity with off-axis pumping can emit the laser mode of SU(2) coherent state.

Figure S5: State tomography. The theoretical and experimental results of the
tomography of density matrices for various GHZ states .

###  .6 General SU(2) vector beams

By elaborate external modulation, a normal SU(2) geometric beam can be
modulated as a general SU(2) vector beam, where arbitrary amplitude, phase,
and polarization for each orbit can be modulated. By modifying Eq. (S.37), a
general SU(2) vector beam can be given by:

$\mathbf{\Psi}_{N_{x},N_{y}}^{\pm}(\mathbf{r})=\frac{1}{Q}\sum\limits_{s=0}^{Q-1}A_{s}\text{e}^{\phi_{s}}\mathbf{J}_{s}\Phi_{N_{x},N_{y}}^{(s)}\left(\mathbf{r}\right),$
(S.55)

where we set
$\Phi_{N_{x},N_{y}}^{(s)}=\Phi\left(\mathbf{r},u_{s}^{\pm},v_{s}^{\pm}\right){\text{e}}^{\text{i}\left(N_{x}+N_{y}\right)\theta_{s}}$
for convenience; $A_{s}$, $\phi_{s}$, and $\mathbf{J}_{s}$ are the amplitude,
phase, and polarization Jones vector of light at the $s$-th orbit. In
degenerate state $|\Omega=1/4\rangle$, we can firstly control the SU(2) beam
into ray-like state and then make special modulations to realize various
classical GHZ states [see Eqs. (S.68) to (S.71) in the next section .7] as
specific cases:

  * •

$|{\Phi}^{+}\rangle$: $A_{0}=A_{2}=\frac{1}{\sqrt{2}}$, $A_{1}=A_{3}=0$,
$\phi_{0}=\phi_{2}$, $\mathbf{J}_{0}=|D\rangle$ and
$\mathbf{J}_{2}=|A\rangle$;

  * •

$|{\Phi}^{-}\rangle$: $A_{0}=A_{2}=\frac{1}{\sqrt{2}}$, $A_{1}=A_{3}=0$,
$\phi_{0}=\phi_{2}+\pi$, $\mathbf{J}_{0}=|D\rangle$ and
$\mathbf{J}_{2}=|A\rangle$;

  * •

$|\Psi_{1}^{+}\rangle$: $A_{1}=A_{3}=\frac{1}{\sqrt{2}}$, $A_{0}=A_{2}=0$,
$\phi_{1}=\phi_{3}$, $\mathbf{J}_{1}=|D\rangle$ and
$\mathbf{J}_{3}=|A\rangle$;

  * •

$|\Psi_{1}^{-}\rangle$: $A_{1}=A_{3}=\frac{1}{\sqrt{2}}$, $A_{0}=A_{2}=0$,
$\phi_{1}=\phi_{3}+\pi$, $\mathbf{J}_{1}=|D\rangle$ and
$\mathbf{J}_{3}=|A\rangle$;

  * •

$|\Psi_{2}^{+}\rangle$: $A_{0}=A_{3}=\frac{1}{\sqrt{2}}$, $A_{1}=A_{2}=0$,
$\phi_{0}=\phi_{3}$, $\mathbf{J}_{0}=|D\rangle$ and
$\mathbf{J}_{3}=|A\rangle$;

  * •

$|\Psi_{2}^{-}\rangle$: $A_{0}=A_{3}=\frac{1}{\sqrt{2}}$, $A_{1}=A_{2}=0$,
$\phi_{0}=\phi_{3}+\pi$, $\mathbf{J}_{0}=|D\rangle$ and
$\mathbf{J}_{3}=|A\rangle$;

  * •

$|\Psi_{3}^{+}\rangle$: $A_{0}=A_{2}=\frac{1}{\sqrt{2}}$, $A_{1}=A_{3}=0$,
$\phi_{0}=\phi_{2}$, $\mathbf{J}_{0}=|A\rangle$ and
$\mathbf{J}_{2}=|D\rangle$;

  * •

$|\Psi_{3}^{-}\rangle$: $A_{0}=A_{2}=\frac{1}{\sqrt{2}}$, $A_{1}=A_{3}=0$,
$\phi_{0}=\phi_{2}+\pi$, $\mathbf{J}_{0}=|A\rangle$ and
$\mathbf{J}_{2}=|D\rangle$;

Figure S6: Fidelity. The experimental results of fidelities for various GHZ
states.

When an SU(2) vector beam was controlled into ray-like state with large enough
$N_{x}$ and $N_{y}$ without interference among lights on sub-orbits, the
wavefunction along different orbits are independent without interference to
each other, thus intensity pattern can be simplified as:

$\left\|\mathbf{\Psi}_{N_{x},N_{y}}^{\pm}(\mathbf{r})\right\|^{2}=\frac{1}{Q^{2}}\sum\limits_{s=0}^{Q-1}A^{2}_{s}\left|\Phi_{N_{x},N_{y}}^{(s)}\left(\mathbf{r}\right)\right|^{2}.$
(S.56)

where $\left\|\mathbf{\Psi}\right\|$ is the Frobenius norm of vector
$\mathbf{\Psi}$. Hereinafter, we derive the expression for the polarization
projection states. The Jones matrix for a linear polarizer with a inclined
angle of $\theta_{P}$ is
${{\mathbf{J}}_{P}}=\left[\begin{matrix}\cos{{\theta}_{P}}&0\\\
0&\sin{{\theta}_{P}}\\\ \end{matrix}\right]$, which can project the light into
the linear polarization state with inclined angle of $\theta_{P}$. the SU(2)
vector beam after projection can be given by:

$\widetilde{\mathbf{\Psi}}_{N_{x},N_{y}}^{\pm}(\mathbf{r})=\frac{1}{Q}\sum\limits_{s=0}^{Q-1}A_{s}\text{e}^{\phi_{s}}\mathbf{J}_{P}\mathbf{J}_{s}\Phi_{N_{x},N_{y}}^{(s)}\left(\mathbf{r}\right).$
(S.57)

When $N_{x}$ and $N_{y}$ are both large enough, the lights on various orbits
cannot make interference to each other, the intensity pattern can be given by.

$\left\|\widetilde{\mathbf{\Psi}}_{N_{x},N_{y}}^{\pm}(\mathbf{r})\right\|^{2}=\frac{1}{Q^{2}}\sum\limits_{s=0}^{Q-1}A^{2}_{s}\left\|\mathbf{J}_{P}\mathbf{J}_{s}\right\|^{2}\left|\Phi_{N_{x},N_{y}}^{(s)}\left(\mathbf{r}\right)\right|^{2},$
(S.58)

The 3-D intensity patterns of various GHZ states with various polarization
projection states ($\theta_{P}=0$, $\pi/2$, $\pi$, and $3\pi/2$) are shown in
Fig. (S7), where we set $N_{x}=N_{y}$ in simulation. When the vortex geometric
mode is reduced into planar geometric mode, i.e. $N_{x}$ or $N_{y}=0$, we can
directly use $\langle\Psi_{N_{x},N_{y}}^{\pm}|\Psi_{N_{x},N_{y}}^{\pm}\rangle$
to observe the interference effect among sub-orbits, as shown in Fig. 4 for
various GHZ states. By experimentally measuring the amplitude and polarization
of each orbit, we can evaluate the experimental state:

$\displaystyle\left|\psi_{exp}\right\rangle=$ $\displaystyle\text{
}{\alpha_{1}}\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle+{{\alpha}_{2}}\left|-\right\rangle\left|1\right\rangle\left|D\right\rangle+{{\alpha}_{3}}\left|+\right\rangle\left|2\right\rangle\left|D\right\rangle$
$\displaystyle+{{\alpha}_{4}}\left|+\right\rangle\left|1\right\rangle\left|A\right\rangle+{{\alpha}_{5}}\left|-\right\rangle\left|2\right\rangle\left|D\right\rangle+{{\alpha}_{6}}\left|-\right\rangle\left|1\right\rangle\left|A\right\rangle$
$\displaystyle+{{\alpha}_{7}}\left|+\right\rangle\left|2\right\rangle\left|A\right\rangle+{{\alpha}_{8}}\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle,$
(S.59)

reconstitute the density matrix
$\hat{\rho}_{exp}=|\psi_{exp}\rangle\langle\psi_{exp}|$ for each GHZ state,
and calculate the fidelities
$F=\langle\Phi^{\pm}|\hat{\rho}_{exp}|\Phi^{\pm}\rangle$ and
$F=\langle\Psi^{\pm}_{i}|\hat{\rho}_{exp}|\Psi^{\pm}_{i}\rangle$ ($i=1,2,3$)
comparing with the theoretical density matrices of GHZ states. The theoretical
and experimental tomography of density matrices and fidelities for various GHZ
states are shown in Fig. S5 and Fig. S6.

###  .7 Classical GHZ states

Maximally entangled states of two particles are called Bell states, acting as
4 eigenstates of 4-D Hilbert space, and those of $N$ ($N\geq 3$) particles are
called GHZ states, acting as $2^{N}$ eigenstates of $2^{N}$-D Hilbert space
Pan and Zeilinger (1998). The 4 Bell states in 2 maximumly entangled groups
for polarization-entangled photon pair are given by:

$\left|{{\phi}^{\pm}}\right\rangle=\frac{\left|H\right\rangle\left|V\right\rangle\pm\left|V\right\rangle\left|H\right\rangle}{\sqrt{2}},$
(S.60)
$\left|{{\psi}^{\pm}}\right\rangle=\frac{\left|H\right\rangle\left|H\right\rangle\pm\left|V\right\rangle\left|V\right\rangle}{\sqrt{2}}.$
(S.61)

They can linearly express a general 4-D polarization-entangled state:

$\left|\psi\right\rangle=\alpha\left|H\right\rangle\left|H\right\rangle+\beta\left|H\right\rangle\left|V\right\rangle+\gamma\left|V\right\rangle\left|V\right\rangle+\delta\left|V\right\rangle\left|V\right\rangle.$
(S.62)

The 8 GHZ states in 4 maximumly entangled groups for three-photon
polarization-entanglement are given by Pan and Zeilinger (1998):

$\left|{{\Phi}^{\pm}}\right\rangle=\frac{\left|H\right\rangle\left|H\right\rangle\left|H\right\rangle\pm\left|V\right\rangle\left|V\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.63)
$\left|\Psi_{1}^{\pm}\right\rangle=\frac{\left|V\right\rangle\left|H\right\rangle\left|H\right\rangle\pm\left|H\right\rangle\left|V\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.64)
$\left|\Psi_{2}^{\pm}\right\rangle=\frac{\left|H\right\rangle\left|V\right\rangle\left|H\right\rangle\pm\left|V\right\rangle\left|H\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.65)
$\left|\Psi_{3}^{\pm}\right\rangle=\frac{\left|H\right\rangle\left|H\right\rangle\left|V\right\rangle\pm\left|V\right\rangle\left|V\right\rangle\left|H\right\rangle}{\sqrt{2}}.$
(S.66)

They can linearly express a general 8-D polarization-entangled state:

$\displaystyle\left|\psi\right\rangle=$ $\displaystyle\text{
}{\alpha_{1}}\left|H\right\rangle\left|H\right\rangle\left|H\right\rangle+{{\alpha}_{2}}\left|V\right\rangle\left|H\right\rangle\left|H\right\rangle+{{\alpha}_{3}}\left|H\right\rangle\left|V\right\rangle\left|H\right\rangle$
$\displaystyle+{{\alpha}_{4}}\left|H\right\rangle\left|H\right\rangle\left|V\right\rangle+{{\alpha}_{5}}\left|V\right\rangle\left|V\right\rangle\left|H\right\rangle+{{\alpha}_{6}}\left|V\right\rangle\left|H\right\rangle\left|V\right\rangle$
$\displaystyle+{{\alpha}_{7}}\left|H\right\rangle\left|V\right\rangle\left|V\right\rangle+{{\alpha}_{8}}\left|V\right\rangle\left|V\right\rangle\left|V\right\rangle.$
(S.67)

By comparing the representation of vector beam
($|\psi\rangle=|\ell_{1}\rangle|H\rangle+|\ell_{2}\rangle|V\rangle$) and Eqs.
(S.60) and (S.61), the vector beams show an excellent classical picture
corresponding to the quantum Bell states, because the non-separability of
transverse mode and polarization DoFs in vector beam is just corresponding to
the non-separability of two polarization states in entangled photon pair. Thus
this exotic non-separability in vector beams is also called classical
entanglement. By manipulating the spatial modes $|u_{1}\rangle$ and
$|u_{2}\rangle$, a completed qubit can be revealed by specially designed
vector beams such as the higher-order Poincaré sphere Naidoo  _et al._ (2016b)
and the hybrid-order Poincaré sphere Liu  _et al._ (2017). Moreover, quantum
measure and tomography methods can be referred to measure degree of non-
separability in classical entanglement McLaren  _et al._ (2015). However, this
previous works are constrained within the 2-D Bell states. In order to
transfer GHZ states into classical entanglement, we should find more DoFs
beyond transverse mode and polarization. It seems difficult to find more DoFs
because the transverse mode and polarization are the basic elements
constituting a common laser beam. In contrast to the common beams, the
geometric beams in SU(2) coherent states have a coupling effect between the
transverse and longitudinal modes induced by the frequency degeneracy where
the spatial patterns vary with the longitudinal distance Chen  _et al._
(2017). In an SU(2) coherent state, the actual 3-D wave-packet pattern can be
located at some periodic oscillating orbits, manifesting the ray-wave duality
Barré  _et al._ (2017); Shen  _et al._ (2018c). We can elaborately modulate
the intensity and polarization of each periodic orbit to manipulate three DoFs
and realize GHZ states alternatively: the first DoF is the oscillating
direction, the second is the location of periodic orbits, the third is the
polarization, then the classical GHZ states yield:

$\left|{{\Phi}^{\pm}}\right\rangle=\frac{\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle\pm\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle}{\sqrt{2}},$
(S.68)
$\left|\Psi_{1}^{\pm}\right\rangle=\frac{\left|-\right\rangle\left|1\right\rangle\left|D\right\rangle\pm\left|+\right\rangle\left|2\right\rangle\left|A\right\rangle}{\sqrt{2}},$
(S.69)
$\left|\Psi_{2}^{\pm}\right\rangle=\frac{\left|+\right\rangle\left|2\right\rangle\left|D\right\rangle\pm\left|-\right\rangle\left|1\right\rangle\left|A\right\rangle}{\sqrt{2}},$
(S.70)
$\left|\Psi_{3}^{\pm}\right\rangle=\frac{\left|+\right\rangle\left|1\right\rangle\left|A\right\rangle\pm\left|-\right\rangle\left|2\right\rangle\left|D\right\rangle}{\sqrt{2}}.$
(S.71)

where $|+\rangle$ and $|-\rangle$ represent the positive and negative SU(2)
oscillating states, $|1\rangle$ and $|2\rangle$ are the first and second
round-trip locations of periodic orbits, here we change the polarization
states from horizontal and vertical linear-polarization states $|H\rangle$ and
$|V\rangle$ into diagonal and antidiagonal linear-polarization state
$|D\rangle=\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$ and
$|A\rangle=\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle)$ only because it is
corresponding to our experimental description more conveniently.

Figure S7: Control of four maximumly entangled groups in GHZ states. The
experimental and theoretical results of four general SU(2) vector beams
corresponding to the four maximumly entangled groups in classical GHZ states.
The white arrow means the allowed polarization orientation of the polarizer.
The plot on the left for each group shows the theoretical spatial wave-packet
of the corresponding SU(2)-like vector beams. $|\Phi^{\pm}\rangle$: the
SU(2)-like vector beam with diagonal intensity pattern where $|+1\rangle$
orbit with diagonal polarization $|-2\rangle$ orbit with anti-diagonal
polarization; $|\Psi^{\pm}_{1}\rangle$: anti-diagonal intensity pattern,
$|-1\rangle$ orbit with diagonal polarization $|+2\rangle$ orbit with anti-
diagonal polarization; $|\Psi^{\pm}_{2}\rangle$: anti-diagonal intensity
pattern, $|+2\rangle$ orbit with diagonal polarization $|-1\rangle$ orbit with
anti-diagonal polarization; $|\Psi^{\pm}_{3}\rangle$: diagonal intensity
pattern, $|-2\rangle$ orbit with diagonal polarization $|+1\rangle$ orbit with
anti-diagonal polarization.

###  .8 Projected Bell states from GHZ states

Via intensity modulation by iris on I1 or I2 and SLM with “$\pi/2|3\pi/2$” or
“$3\pi/2|\pi/2$” phase-step mask, four orthogonal vector beams corresponding
to the four maximumly entangled groups $|\Phi^{\pm}\rangle$,
$|\Psi^{\pm}_{1}\rangle$, $|\Psi^{\pm}_{2}\rangle$, and
$|\Psi^{\pm}_{3}\rangle$ would be obtained. By using the polarizer, we
measured the vector properties in these beams. Figure S7 shows the theoretical
and experimental results of the intensity pattern of the four maximumly
entangled groups before and after polarizer measure. For $|\Phi^{\pm}\rangle$
and $|\Psi^{\pm}_{3}\rangle$, the intensity pattern is diagonal, and
polarizations are orthogonal at the corresponding orbit location, i.e.
$|1\rangle|D\rangle$ with $|2\rangle|A\rangle$ and $|1\rangle|A\rangle$ with
$|2\rangle|D\rangle$. For $|\Psi^{\pm}_{1}\rangle$ and
$|\Psi^{\pm}_{2}\rangle$, the intensity pattern is antidiagonal, polarization
distribution are $|1\rangle|D\rangle$ with $|2\rangle|A\rangle$ and
$|1\rangle|A\rangle$ with $|2\rangle|D\rangle$. So far, we cannot say we
obtain the vector beams completely expressed in 8-D space because we cannot
distinguish the “$\pm$” signals and get the complete tomography of the 8 GHZ
states.

Hereinafter, we demonstrate how to distinguish the “$\pm$” states in the four
groups of maximum entanglement. Actually, the “$+$” and “$-$” states are
corresponding to the classical orbits with phase differences of 0 and $\pi$.
The phase difference of two classical light can be revealed by observation of
interference fringes. In order to observe interference fringes of orbits, we
should project the vector beams into a certain polarization for producing
coherence. According to the properties of GHZ states, a GHZ state will be
reduced into a Bell state after polarization projection. For classical GHZ
states, different Bell states will also be obtained after polarization
projection, yielded by:

$\displaystyle\left|{{\Phi}^{\pm}}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}{\left(\left|+\right\rangle\left|1\right\rangle\frac{\left|H\right\rangle+\left|V\right\rangle}{\sqrt{2}}\pm\left|-\right\rangle\left|2\right\rangle\frac{\left|H\right\rangle-\left|V\right\rangle}{\sqrt{2}}\right)}$
$\displaystyle=\frac{\left|+\right\rangle\left|1\right\rangle\pm\left|-\right\rangle\left|2\right\rangle}{2}\left|H\right\rangle+\frac{\left|+\right\rangle\left|1\right\rangle\mp\left|-\right\rangle\left|2\right\rangle}{2}\left|V\right\rangle$
$\displaystyle=\frac{\left|{{\psi}^{\pm}}\right\rangle\left|H\right\rangle+\left|{{\psi}^{\mp}}\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.72) $\displaystyle\left|\Psi_{1}^{\pm}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|-\right\rangle\left|1\right\rangle\frac{\left|H\right\rangle+\left|V\right\rangle}{\sqrt{2}}\pm\left|+\right\rangle\left|2\right\rangle\frac{\left|H\right\rangle-\left|V\right\rangle}{\sqrt{2}}\right)$
$\displaystyle=\frac{\left|-\right\rangle\left|1\right\rangle\pm\left|+\right\rangle\left|2\right\rangle}{2}\left|H\right\rangle+\frac{\left|-\right\rangle\left|1\right\rangle\mp\left|+\right\rangle\left|2\right\rangle}{2}\left|V\right\rangle$
$\displaystyle=\frac{\pm\left|{{\phi}^{\pm}}\right\rangle\left|H\right\rangle\mp\left|{{\phi}^{\mp}}\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.73) $\displaystyle\left|\Psi_{2}^{\pm}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|+\right\rangle\left|2\right\rangle\frac{\left|H\right\rangle+\left|V\right\rangle}{\sqrt{2}}\pm\left|-\right\rangle\left|1\right\rangle\frac{\left|H\right\rangle-\left|V\right\rangle}{\sqrt{2}}\right)$
$\displaystyle=\frac{\left|+\right\rangle\left|2\right\rangle\pm\left|-\right\rangle\left|1\right\rangle}{2}\left|H\right\rangle+\frac{\left|+\right\rangle\left|2\right\rangle\mp\left|-\right\rangle\left|1\right\rangle}{2}\left|V\right\rangle$
$\displaystyle=\frac{\left|{{\phi}^{\pm}}\right\rangle\left|H\right\rangle+\left|{{\phi}^{\mp}}\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.74) $\displaystyle\left|\Psi_{3}^{\pm}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|+\right\rangle\left|1\right\rangle\frac{\left|H\right\rangle-\left|V\right\rangle}{\sqrt{2}}\pm\left|-\right\rangle\left|2\right\rangle\frac{\left|H\right\rangle+\left|V\right\rangle}{\sqrt{2}}\right)$
$\displaystyle=\frac{\left|+\right\rangle\left|1\right\rangle\pm\left|-\right\rangle\left|2\right\rangle}{2}\left|H\right\rangle-\frac{\left|+\right\rangle\left|1\right\rangle\mp\left|-\right\rangle\left|2\right\rangle}{2}\left|V\right\rangle$
$\displaystyle=\frac{\left|{{\psi}^{\pm}}\right\rangle\left|H\right\rangle-\left|{{\psi}^{\mp}}\right\rangle\left|V\right\rangle}{\sqrt{2}},$
(S.75)

where the classical Bell states should be:

$\left|{{\phi}^{\pm}}\right\rangle=\frac{\left|+\right\rangle\left|2\right\rangle\pm\left|-\right\rangle\left|1\right\rangle}{\sqrt{2}},$
(S.76)
$\left|{{\psi}^{\pm}}\right\rangle=\frac{\left|+\right\rangle\left|1\right\rangle\pm\left|-\right\rangle\left|2\right\rangle}{\sqrt{2}}.$
(S.77)

After projection onto $|H\rangle$ and $|V\rangle$ states, $|\Phi^{\pm}\rangle$
or $|\Psi^{\pm}_{3}\rangle$ state would be reduced to $|\psi^{\pm}\rangle$ and
$|\psi^{\mp}\rangle$ states, $|\Psi^{\pm}_{1}\rangle$ or
$|\Psi^{\pm}_{2}\rangle$ to $|\phi^{\pm}\rangle$ and $|\phi^{\mp}\rangle$. The
“$+$” and “$-$” in Bell states can be distinguished by the complementary
interferometric fringes of the corresponding phase difference of 0 and $\pi$
between two SU(2) orbits. For measuring $|\Phi^{\pm}\rangle$ and
$|\Psi^{\pm}_{3}\rangle$, the CCD camera should be located at $z=-z_{R}$
position where $|-1\rangle$ and $|+2\rangle$ orbits are overlapped. For
$|\Psi^{\pm}_{1}\rangle$ and $|\Psi^{\pm}_{2}\rangle$, the CCD camera should
be located at $z=z_{R}$ position where $|+1\rangle$ and $|-2\rangle$ orbits
are overlapped. Without polarization projection, the pattern shows no fringes
because the light on the corresponding two orbits are incoherent. After
projection on $|H\rangle$ or $|V\rangle$ states, different interference
fringes would be observed for different reduced Bell states. For the group
$|\Phi^{\pm}\rangle$, the “$\pm$” cannot be distinguished by the intensity
patterns. However if we project the polarization onto $|H\rangle$ state to
observe the pattern of $\langle H|\Phi^{\pm}\rangle$, pattern of original
state $|\Phi^{\pm}\rangle$ will be reduced into Bell states
$|\psi^{\pm}\rangle$ and two different patterns of complementary fringes will
be observed, center-bright fringes for $|\psi^{+}\rangle$ (the original state
should be $|\Phi^{+}\rangle$) and center-dark fringes for $|\psi^{-}\rangle$
(the original state should be $|\Phi^{-}\rangle$). We can also use projected
state $\langle V|\Phi^{\pm}\rangle$ to distinguish the “$\pm$” that $\langle
V|\Phi^{+}\rangle$ should be center-dark fringes corresponding to Bell state
$|\psi^{-}\rangle$, and $\langle V|\Phi^{-}\rangle$ should be center-bright
fringes corresponding to Bell state $|\psi^{+}\rangle$. In experiment, we can
use a BK7 thin plate to cover one of the two orbits and rotate slightly to
control the phase difference between them to control a phase difference of
$\pi$, switching from “$+$” to “$-$” state. Other GHZ states can be generated
by the similar way fulfilling a completed set in 8-D Hilbert space.

Figure S8: Picture of experimental setup. The actual picture is in accord with
the schematic of experimental setup of Fig. 3, where one insert (left) shows
the details of the arrangement of c-cut Nd:YVO4, another insert (right) shows
the two cylindrical lenses playing as astigmatic mode converter with the ray-
represent of the transformation from a planar geometric mode to a vortex
geometric mode. The trajectory of SU(2) geometric mode is depicted in the
cavity. The index ellipsoide is depicted for determining the effective
refraction index and angle of a geometric beam in crystal. OC: output coupler;
DM: dichroic mirror; PBS: polarization splitting prism; QWP: quarter-wave
plate; HR, high-reflective mirror; SLM, spatial light modulator; P: polarizer;
CL: cylindrical lens; CCD: charge-coupled device. Figure S9: Experimental
generation of general SU(2) vector beams. (a) The experimental measure of
transverse patterns versus the propagation distance of an SU(2) geometric mode
in state $|\Omega=1/4\rangle|\phi=\pi\rangle$, where different orbits show
different intensities and polarizations. (b) The experimentally observed
evolution of the transverse pattern at $z=z_{R}$ plane with the adjustment of
the pumping power and displacement.

###  .9 Generating general high-dimensional classical entanglement from a
laser

The general SU(2) vector beams with general high-dimensional entanglement can
be directly generated in a cavity. In other words, a degenerate cavity with
off-axis displacement should commonly generate the general SU(2) vector beams
while normal SU(2) scalar beams are just specific cases. Actually, considering
the anisotropism in gain medium induced by crystal cutting geometry and
nonuniform thermal effect by asymmetric pumping, the geometric beams can
undergo complex amplitude and polarization modulations in cavity and be output
as vector fields Lu and Lin (2012); Lu and He (2015). In our experiment, we
used a c-cut Nd:YVO4 as gain medium to realize intracavity complex amplitude
and polarization modulations. Nd:YVO4 is a positive uniaxial crystal with
anisotropic refractive indices and stimulated absorption and emission cross-
sections Guo  _et al._ (2019). The detailed schematic for arranging the
crystal in our experiment is depicted in Fig. S8. In c-cut Nd:YVO4, the
principal c-axis is located on $z$-axis and other two a-axes on the $(x,y)$
transverse plane. For a light beam with normal incidence, there is no
birefrigent effect; for a beam with a incident angle $\theta_{\text{in}}$,
there is a birefrigent modulation where the vertical linear polarized
component undergoes an ordinary refractive index $n_{\text{o}}$ and angle
$\theta_{\text{o}}$ while the horizontal linear polarized component undergoes
an effective refractive index $n_{\text{eff}}$ and angle $\theta_{\text{eff}}$
involved in the ordinary and extraordinary refractive indices $n_{\text{o}}$
and $n_{\text{e}}$:

$n_{\text{eff}}=\frac{n_{\text{o}}n_{\text{e}}}{\sqrt{n_{\text{e}}^{2}\cos^{2}{\theta_{\text{in}}}+n_{\text{o}}^{2}\sin^{2}{\theta_{\text{in}}}}}.$
(S.78)

The difference of the two refractive indices leads to the phase retardation
between the orthogonal polarized components, which can be represented as

$\Delta=\frac{2\pi
d}{\lambda_{0}}\left(\frac{n_{\text{eff}}}{\cos{\theta_{\text{eff}}}}-\frac{n_{\text{o}}}{\cos{\theta_{\text{o}}}}\right),$
(S.79)

where $d$ is the thickness of the crystal. According to Eq. (S.79), when the
laser crystal is given ($d$ and $\lambda_{0}$ is determined), the polarization
control of geometric mode can be experimentally realized by two methods of:
(1) modulating the refractive indices; (2) modulating the incidence angle,
i.e. the included angle of classical orbits of geometric mode. The method-(1)
can be realized by control of pump power, and the method-(2) can be realized
by control of off-axis displacement.

Modulation by pump power. It has been proved that the temperature-dependent
thermal effects in Nd:YVO4 solid-state laser is highly related to the pump
power, while the refractive indices, stimulated absorption and emission cross-
sections are all related to the thermal effect Shen  _et al._ (2017a, b,
2018d). Thus the different powers corresponding to different thermal effects
and then to different refractive indices. In SU(2) geometric mode, the pump
spot is off-axis, thus nonuniform thermal effect is nonuniform and the
polarization modulation is also nonuniform, resulting into the output of SU(2)
vector beams.

Modulation by off-axis displacement. According to the ray-wave duality, the
SU(2) geometric modes with different orders have different incident angles for
various orbits and the ray trajectory is coupled with the pumping spot.
Therefore, the control of pumping spot is related to the included angles of
incidence rays in the classical trajectory. The different off-axis
displacements corresponding to different transverse-orders of geometric mode
result into different incidence angles. Note that the nonuniform thermal
distribution and transverse-order comprehensively impact on the polarization
of geometric mode.

Therefore, there should be asymmetrical complex amplitude and polarization
modulations by general pumping control. For instance, Fig. S9(a) shows an
experimental result of an SU(2) geometric mode in state
$|\Omega=1/4\rangle|\phi=\pi\rangle$, where different orbits show different
intensities and polarizations. For each orbit, the left- and right-handed
circular polarized components are measured by a polarization grating, which
uses geometric phase to diffract light into two beams in the $+1$ and $-1$
orders such that the two output beams have opposite circular polarizations,
and the horizontal and vertical linear polarized components are measured by a
rotating polarizer. Through the polarization components measurement, we can
evaluate the actual polarizations of light on various orbits. The experimental
results show that the SU(2) beam is a general vector state, i.e. the lights on
different orbits have different polarization and intensity.

The actual vector property can also be revealed by the interference fringes
emerged at $z=0$ and $z=\pm z_{R}$ position. By adjusting the pumping power
and displacement, we can modulate the vector field in the output general SU(2)
vector beam, modulating the polarizations on $|+1\rangle$ and $|-2\rangle$
orbits from coherent to orthogonal. The evolution of the transverse pattern at
$z=z_{R}$ shows the coherence between $|+1\rangle$ and $|-2\rangle$ orbits
changing from weak to strong with the interference fringes from vague to
clear, unraveling the polarizations on the two orbits changing into orthogonal
states. This experimentally observed evolution is shown in Fig. S9(b).

In summary, using off-axis pumping in a frequency-degenerate cavity with c-cut
Nd:YVO4 as gain medium, general SU(2) vector beams with high-dimensional
entanglement property can be generated.

###  .10 Experimental setup details

The main setup is depicted in Fig. 3 in the main text and Fig. S8 for real
picture. The frontend is a laser oscillator with cavity precisely adjusted
into frequency-degenerate state $|\Omega=1/4\rangle$ and phase state
$|\phi=\pi\rangle$. Through controlling the off-axis pumping, a ray-like SU(2)
coherent state wave-packet can be generated and coupled with a W-shape
classical geometric trajectory. The positive and negative oscillating states
$|+\rangle$ and $|-\rangle$ share the overlapped trajectory. Here the signal
is defined by the projection of photon propagating direction on $x$-axis.
There are two round-trip orbits bouncing at flat mirror, i.e. the two V-shape
orbits, noted as $|1\rangle$ and $|2\rangle$, for the coherent state
$|\Omega=1/4\rangle|\phi=\pi\rangle$ Shen  _et al._ (2018c). The intracavity
geometric trajectory is constituted by four periodic oscillating orbits
$|+1\rangle$, $|+2\rangle$, $|-1\rangle$, and $|-2\rangle$, each of them
contributes a ray-like orbit in the actual output SU(2) geometric mode. The
planar geometric mode can be astigmatically converted into vortex geometric
mode with OAM where the corresponding states $|\pm\rangle$, $|1\rangle$ and
$|2\rangle$ are located at a hyperbolic ruled surface with orthogonal SU(2)
coordinates as illustrated in the introduction of concept in the main text.

A 808 nm fiber-coupled laser diode (LD) (FOCUSLIGHT, FL-FCSE08-7-808-200) was
used as the pump source. With a telescope system with magnification about 1:1
constituted by two identical anti-reflective (AR) coated lenses (focal length
$F=25$ mm), the pump light was focused into a c-cut Nd:YVO4 slice-like crystal
with dopant of 0.5 at.% and thickness of 5 mm, which was wrapped in a copper
heat sink and conductively water cooled at 18∘C. The outside surface of
crystal was coated AR at 808 nm and high-reflective (HR) at 1064 nm and the
inner surface AR at 1064 nm. A plano-concave mirror was used as the output
coupler, where the radius of curvature is 100 mm and the transmittance is 10%
at 1064 nm for inner surface and AR for outside surface.

The laser was firstly passed through a dichroic mirror (DM, 45∘ incidence, HR
at pump light and AR at laser) for filtering residual pumping light. After
lens focusing and passing through a polarization splitting prism (PBS) and
quarter-wave plate (QWP), we can control the SU(2) beam as right-hand circular
polarization state. Using a high-reflective mirror and lens, we can illuminate
the SU(2) beam on a spatial light modulator (SLM) with beam waist location
overlapped with the phase mask and incidence angle less than 5∘. A filter was
used before the illumination on SLM for avoiding over-power damage. The phase
modulation is only sensitive to horizontal linear polarization component, thus
it can be used to modulate polarization and generate SU(2) structured vector
beams by phase mask design. For a input light with right-hand circular
polarization, if the phase mask plays a $\pi/2$ constant phase, the output
will be diagonal linear polarization, and if $3\pi/2$, anti-diagonal linear
polarization. Here we equally divided the phase mask into two parts, one for
modulating the polarization on $|1\rangle$ and another for $|2\rangle$. When
the SLM add divided phase “$\pi/2|3\pi/2$” on the mask, $|1\rangle|D\rangle$
and $|2\rangle|A\rangle$ states can be produced, and when “$3\pi/2|\pi/2$” on
phase mask, $|1\rangle|A\rangle$ and $|2\rangle|D\rangle$ states are produced.

For introducing more structure control, we used an iris to make on-demand
intensity modulation. Due to the special spatial structure of the SU(2) beam,
different effects would be introduced for different location of iris
application. When the iris with a proper aperture size was applied at negative
Rayleigh length position (I1 position), the orbits $|-1\rangle$ and
$|+2\rangle$ could be blocked, resulting in a diagonal intensity pattern in
the corresponding vortex SU(2) beam. When applied at positive Rayleigh length
position (I2 position), $|+1\rangle$ and $|-2\rangle$ could be blocked,
resulting in anti-diagonal intensity pattern vortex SU(2) beam. A BK7 thin
plate can be placed partially at the beam waist position for adding a phase
difference for $|1\rangle$ or $|2\rangle$ states. After intensity modulation,
the structured light can no longer be a normal SU(2) geometric mode, we here
call the new structured light as general SU(2) beams and the beam after
polarization modulation as general SU(2) vector beams. Because we used a c-cut
Nd:YVO4 providing anisotropic intracavity polarization modulation on geometric
modes Guo  _et al._ (2019); Lu and Lin (2012), our laser output should be a
general SU(2) vector beam. Hereinafter, we will demonstrate that the general
SU(2) vector beams can be measured completely by 3-DoF bases in 8-D Hilbert
space and a complete set of GHZ states ($|\Phi^{\pm}\rangle$,
$|\Psi^{\pm}_{1}\rangle$, $|\Psi^{\pm}_{2}\rangle$, and
$|\Psi^{\pm}_{3}\rangle$) can be experimentally generated and controlled.

After the SLM modulation, the tomography measure system was designed. A
astigmatic mode convertor (AMC) including two 45∘-inclined AR-coated
cylindrical lenses was used to convert a planar general SU(2) beam into vortex
beam with disjoint geometric orbits. After that, a charge coupled device (CCD)
camera was used to identify the diagonal and anti-diagonal intensity patterns.
A rotatable polarizer should be inset before the CCD to measure the vector
properties and identify the diagonal and anti-diagonal linear polarization
states for each orbit. The combination of this measure results in the
tomography of for vector beams corresponding to the four maximumly entangled
groups of $|\Phi^{\pm}\rangle$, $|\Psi^{\pm}_{1}\rangle$,
$|\Psi^{\pm}_{2}\rangle$, and $|\Psi^{\pm}_{3}\rangle$. The “$\pm$” states are
revealed by the phase difference on a certain orbit, which can be identified
by the interference fringes of corresponding two orbits. For the SU(2)
coherent state $|\Omega=1/4\rangle|\phi=\pi\rangle$, the interference of
$|+\rangle$ and $|-\rangle$ orbits occurred at the beam waist, and the
interference of $|1\rangle$ and $|2\rangle$ orbits occurred at the Rayleigh
length $z_{R}$ away from the beam waist. For identifying “$\pm$” in GHZ
states, the measure of the interference fringes is required. Before the
measure of the interference fringes, a polarizer was used to project the
polarization on a unified $|H\rangle$ or $|V\rangle$ state, in order to
provide coherence, just corresponding to the process of Bell state projection.
When the CCD was located at $z=-z_{R}$, the interference for $|-1\rangle$ and
$|+2\rangle$ can be detected, and when $z=-z_{R}$, the interference for
$|+1\rangle$ and $|-2\rangle$ can be detected.

###  .11 Towards higher-dimensional classically entangled state

Hereto, we can already generate a complete set of GHZ states from a laser,
sharing the same form of three-photon quantum entanglement. The general SU(2)
modes also have potential to be used to generate even higher-dimensional
entanglement states with the form of $N$-photon ($N>3$) quantum entanglement.
To this end, we should find more DoFs to extend the dimension in classical
entanglement such as OAM. As shown in Fig. 6(a-d) in the main text, two SU(2)
modes with opposite OAM can be superposed together fulfilling a complete
oscillating trajectory in a cavity, which can also be experimentally realized
Lu and He (2015). After inhomogeneous intensity and polarization modulation,
this kind of general SU(2) modes should be represented in 16-D Hilbert space
like the 4-photon quantum entanglement:

$\displaystyle\left|\psi\right\rangle=$ $\displaystyle\text{
}{{\alpha}_{1}}\left|+\ell\right\rangle\left|+\right\rangle\left|1\right\rangle\left|H\right\rangle+{{\alpha}_{2}}\left|-\ell\right\rangle\left|+\right\rangle\left|1\right\rangle\left|H\right\rangle$
$\displaystyle+{{\alpha}_{3}}\left|+\ell\right\rangle\left|+\right\rangle\left|1\right\rangle\left|V\right\rangle+{{\alpha}_{4}}\left|-\ell\right\rangle\left|+\right\rangle\left|1\right\rangle\left|V\right\rangle$
$\displaystyle+{{\alpha}_{5}}\left|+\ell\right\rangle\left|-\right\rangle\left|1\right\rangle\left|H\right\rangle+{{\alpha}_{6}}\left|-\ell\right\rangle\left|-\right\rangle\left|1\right\rangle\left|H\right\rangle$
$\displaystyle+{{\alpha}_{7}}\left|+\ell\right\rangle\left|-\right\rangle\left|1\right\rangle\left|V\right\rangle+{{\alpha}_{8}}\left|-\ell\right\rangle\left|-\right\rangle\left|1\right\rangle\left|V\right\rangle$
$\displaystyle+{{\alpha}_{9}}\left|+\ell\right\rangle\left|+\right\rangle\left|2\right\rangle\left|H\right\rangle+{{\alpha}_{10}}\left|-\ell\right\rangle\left|+\right\rangle\left|2\right\rangle\left|H\right\rangle$
$\displaystyle+{{\alpha}_{11}}\left|+\ell\right\rangle\left|+\right\rangle\left|2\right\rangle\left|V\right\rangle+{{\alpha}_{12}}\left|-\ell\right\rangle\left|+\right\rangle\left|2\right\rangle\left|V\right\rangle$
$\displaystyle+{{\alpha}_{13}}\left|+\ell\right\rangle\left|-\right\rangle\left|2\right\rangle\left|H\right\rangle+{{\alpha}_{14}}\left|-\ell\right\rangle\left|-\right\rangle\left|2\right\rangle\left|H\right\rangle$
$\displaystyle+{{\alpha}_{15}}\left|+\ell\right\rangle\left|-\right\rangle\left|2\right\rangle\left|V\right\rangle+{{\alpha}_{16}}\left|-\ell\right\rangle\left|-\right\rangle\left|2\right\rangle\left|V\right\rangle,$
(S.80)

with 16 (4 partite and 8 maximumly entangled group) GHZ states as eigenstates
(just show the first maximumly entangled group here):

$\left|{{\Phi}^{\pm}}\right\rangle=\frac{\left|+\ell\right\rangle\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle\pm\left|-\ell\right\rangle\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle}{\sqrt{2}}.$
(S.81)

For exploring more DoFs to extend the dimension in classical entanglement, we
can utilize multi-LG SU(2) beams Lu  _et al._ (2011); Tuan  _et al._ (2018),
where LG beams replace the Gaussian beams along SU(2) orbits in a geometric
mode based on ray-wave duality. Thus, there is a main OAM state
$|+\ell\rangle$ along the propagation axis, but also OAM $|+m\rangle$ along an
SU(2) orbit. The sub-OAM $|+m\rangle$ has potential to be a DoF to reach
higher-dimensional entanglement. As shown in Fig. 6(e,f) in the main text,
exotic beams can be obtained by the superposition of two multi-LG SU(2) beams
with opposite main OAM, realizing the 32-D entangled stated with 5-partite
16-group GHZ states as eigenstates (just show the first maximumly entangled
group here):

$\left|{{\Phi}^{\pm}}\right\rangle=\frac{\left|+m\right\rangle\left|+\ell\right\rangle\left|+\right\rangle\left|1\right\rangle\left|D\right\rangle\pm\left|-m\right\rangle\left|-\ell\right\rangle\left|-\right\rangle\left|2\right\rangle\left|A\right\rangle}{\sqrt{2}}.$
(S.82)

Besides finding more DoFs, increasing the number of orbits can also extend the
dimension. The above demonstrations are all at degenerate state
$|\Omega=1/4\rangle$. If we can control higher-order degenerate state
$|\Omega=P/Q\rangle$ ($P$ and $Q$ are co-prime integers, $Q$ is even), the
general SU(2) modes would be extended into $4Q$-dimensional space:

$\left|\psi\right\rangle=\sum\limits_{i=\pm\ell}{\sum\limits_{j=\pm}{\sum\limits_{k=1}^{{Q}/{2}}{\left|i\right\rangle\left|j\right\rangle\left|k\right\rangle\left({{\alpha}_{ijk}}\left|H\right\rangle+{{\beta}_{ijk}}\left|V\right\rangle\right)}}}.$
(S.83)

Involving the sub-OAM in multi-LG SU(2) beams, that can be further extended
into $8Q$-dimensional space:

$\displaystyle\left|\psi\right\rangle=\sum\limits_{h=\pm
m}\sum\limits_{i=\pm\ell}\sum\limits_{j=\pm}$
$\displaystyle\sum\limits_{k=1}^{{Q}/{2}}\left|h\right\rangle\left|i\right\rangle\left|j\right\rangle\left|k\right\rangle$
$\displaystyle\times\left({{\alpha}_{hijk}}\left|H\right\rangle+{{\beta}_{hijk}}\left|V\right\rangle\right).$
(S.84)

The realization of high-dimensional classical entanglement can pave the way
for developing a myriad of novel applications of quantum mechanism using
classical light.

## References

  * Perelomov (2012) A. Perelomov, Generalized coherent states and their applications (Springer, 2012).
  * Schrödinger (1926) E. Schrödinger, Naturwissenschaften 14, 664 (1926).
  * Bužek and Quang (1989) V. Bužek and T. Quang, JOSA B 6, 2447 (1989).
  * Wodkiewicz and Eberly (1985) K. Wodkiewicz and J. Eberly, JOSA B 2, 458 (1985).
  * Sanders  _et al._ (1995) B. Sanders, K. Lee, and M. Kim, Physical Review A 52, 735 (1995).
  * Kim  _et al._ (2002) M. Kim, W. Son, V. Bužek, and P. Knight, Physical Review A 65, 032323 (2002).
  * Agarwal (1999) G. Agarwal, JOSA A 16, 2914 (1999).
  * Chen  _et al._ (2017) Y. Chen, J. Tung, P. Tuan, and K. Huang, Annalen der Physik 529, 1600253 (2017).
  * Chen  _et al._ (2013) Y. Chen, J. Tung, P. Chiang, H. Liang, and K. Huang, Physical Review A 88, 013827 (2013).
  * Tung  _et al._ (2016) J. Tung, P. Tuan, H. Liang, K. Huang, and Y. Chen, Physical Review A 94, 023811 (2016).
  * Chen  _et al._ (2004) Y. Chen, C. Jiang, Y. Lan, and K. Huang, Physical Review A 69, 053807 (2004).
  * Pan and Zeilinger (1998) J.-W. Pan and A. Zeilinger, Physical Review A 57, 2208 (1998).
  * Naidoo  _et al._ (2016b) D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, Nature Photonics 10, 327 (2016b).
  * Liu  _et al._ (2017) Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, Photonics Research 5, 15 (2017).
  * McLaren  _et al._ (2015) M. McLaren, T. Konrad, and A. Forbes, Physical Review A 92, 023833 (2015).
  * Ndagano  _et al._ (2016) B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, Optics Letters 41, 3407 (2016).
  * Barré  _et al._ (2017) N. Barré, M. Romanelli, M. Lebental, and M. Brunel, European Journal of Physics 38, 034010 (2017).
  * Shen  _et al._ (2018c) Y. Shen, X. Yang, X. Fu, and M. Gong, Applied Optics 57, 9543 (2018c).
  * Lu and Lin (2012) T.-H. Lu and L. Lin, Applied Physics B 106, 863 (2012).
  * Lu and He (2015) T.-H. Lu and C. He, Optics Express 23, 20876 (2015).
  * Guo  _et al._ (2019) R. Guo, Y. Shen, Y. Meng, and M. Gong, Chinese Physics B 28, 044204 (2019).
  * Shen  _et al._ (2017a) Y. Shen, W. Zhang, M. Gong, Y. Meng, Y. Wang, and X. Fu, Applied Sciences 7, 470 (2017a).
  * Shen  _et al._ (2017b) Y. Shen, M. Gong, E. Ji, X. Fu, and L. Sun, Optics Communications 383, 430 (2017b).
  * Shen  _et al._ (2018d) Y. Shen, M. Gong, and X. Fu, Applied Physics B 124, 85 (2018d).