#  Coupled quantum vortex kinematics and Berry curvature in real space

Lorenzo Dominici  CNR NANOTEC, Istituto di Nanotecnologia, Via Monteroni,
73100 Lecce, Italy  Amir Rahmani  Institute of Physics, Polish Academy of
Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland  David Colas  Aix
Marseille Université, CNRS, Centrale Marseille, LMA UMR 7031 Marseille, France
Dario Ballarini  CNR NANOTEC, Istituto di Nanotecnologia, Via Monteroni, 73100
Lecce, Italy  Milena De Giorgi  CNR NANOTEC, Istituto di Nanotecnologia, Via
Monteroni, 73100 Lecce, Italy  Giuseppe Gigli  CNR NANOTEC, Istituto di
Nanotecnologia, Via Monteroni, 73100 Lecce, Italy  Dipartimento di Matematica
e Fisica E. De Giorgi, Università del Salento, Campus Ecotekne, via Monteroni,
73100 Lecce, Italy  Daniele Sanvitto  CNR NANOTEC, Istituto di Nanotecnologia,
Via Monteroni, 73100 Lecce, Italy  Fabrice P. Laussy  Faculty of Science and
Engineering, University of Wolverhampton, Wulfruna Street, WV1 1LY
Wolverhampton, UK  Russian Quantum Center, Skolkovo innovation city, 121205
Moscow, Russia  Nina Voronova  Russian Quantum Center, Skolkovo innovation
city, 121205 Moscow, Russia  National Research Nuclear University MEPhI
(Moscow Engineering Physics Institute), 115409 Moscow, Russia
*email<EMAIL_ADDRESS><EMAIL_ADDRESS>Cite this work as Commun. Phys. 6, 197 (2023).

###### Abstract

Abstract. The Berry curvature provides a powerful tool to unify several
branches of science through their geometrical aspect: topology, energy bands,
spin and vector fields. While quantum defects–phase vortices and
skyrmions–have been in the spotlight, as rotational entities in condensates,
superfluids and optics, their dynamics in multi-component fields remain little
explored. Here we use two-component microcavity polaritons to imprint a
dynamical pseudospin texture in the form of a double full Bloch beam, a
conformal continuous vortex beyond unitary skyrmions. The Berry curvature
plays a key role to link various quantum spaces available to describe such
textures. It explains for instance the ultrafast spiraling in real space of
two singular vortex cores, providing in particular a simple expression–also
involving the complex Rabi frequency–for their intricate velocity. Such Berry
connections open new perspectives for understanding and controlling highly-
structured quantum objects, including strongly asymmetric cases or even higher
multi-component fields.

INTRODUCTION
Quantized vortices embody the essence of circular motion in all fields and
fluids that can be described by an oscillating wavefunction, such as
superfluids, superconductors, atomic Bose-Einstein condensates (BECs) and
optical fields 1, 2, 3, 4, 5, 6, 7, 8. The complex-valued nature of the
wavefunction implies an integer number of phase windings, or wave crests,
around the core of such a vortex, contributing to the intrinsic orbital
angular momentum (OAM) 9, 10, 11. Unlike classical rigid rotations, the
rotating quantum fluid takes on greater momentum and velocity closer to the
core of rotation 12, as in a gravitational orbital motion. In an optical
vortex this applies to the group velocity, differently from the phase
velocity. An isolated Bose-Einstein condensate rotating around a single or
multiply charged central vortex can be thought of as a fundamental gyroscope
13, candidating them as sensitive detectors for gravitational waves when
implemented in an orbiting laboratory 14. At different levels, analogies were
drawn, both in the past and recently, to point-like atoms 15, 16 or other
elementary particles dressed with two quantum numbers and capable of tunable
pair-wise interactions 17, 18 in the nonlinear regime. Quantum vortices have
been mostly studied when dealing with nonlinear fluids and their phase
transitions, relevant in their observed macroscopic degree of coherence 19, in
quasi-two-dimensional (2D) as well as in three-dimensional (3D) fields, giving
rise to vortex tubes, networks, rings and knots 20, 21. These entities are
called wave dislocations 22 or topological defects, because the phase
singularity at their core and the order parameter winding around it are
features of all waves 23 at the basis of many structured objects of nontrivial
geometry. When implemented in photonics, vortices represent a further degree
of freedom 24, 25, for data multiplexing and for free-air transmission 26, 27,
28 as well as for many optical tweezers applications 29, 30 and structured
light schemes 31, 32. Further dressed with the spin angular momentum (SAM) or
polarization degree of freedom, they can create complex textures around their
core 33, 34, subtending 2D skyrmion (baby-skyrmion) and even 3D particle-like
optical skyrmions 35, 36, 37 eventually evolving in time 38 thanks to
different interactions with a material medium. These objects can be of
different type, such as Bloch-, Neel- and anti-skyrmions, or strained into a
combination of multiple subparticles known as merons 34. Recently, there is a
rising interest in the generation of optical skyrmion beams 35, 36, 37, 38,
39, 40, 41, 42, thanks to their tunability and possible use in data encoding
and transmission. This interest is further motivated by the fact that, both in
condensed matter and in optics, even the most fundamental skyrmion in the
linear regime can be viewed as a full Poincaré 43 or full Bloch beam 44,
highlighting their topological connections.

Full Poincaré or Bloch beams can be created using a combination of vortical
and vortex-free excitation beams, whose wavefunction simultaneously comprises
all the possible quantum states of the associated Hilbert space 44: a sphere
of polarizations or pseudospins. They belong to the class of continuous
vortices, also known as filled-core, bright core, coreless or non-singular
vortices in condensed matter and half-vortices in optics and polaritons 45.
The Mermin-Ho 46 and Anderson-Toulouse vortices 47 in superfluid
${}^{3}\text{He-A}$ are also examples of continuous vortices representing
half- and full-texture states, respectively. In that case, the pseudospin is
the local unit vector of relative orbital angular momentum of a Cooper pair.
In a ferromagnetic BEC, similar structures may exist 48, where the texture
refers to the unit vector parallel to the local magnetization 49. In these two
examples, the pseudospin vector represents a direction in real space. In
general, in any two-state system, such as, for example, two-component atomic
BECs or polarized optical fields, the associated Hilbert space is a four-
dimensional space 50. Apart from the degrees of freedom represented by the
total density and the total phase, it can be interesting to focus on the
remaning spinor order parameter (two-dimensional Hilbert space of
pseudospins). This expresses the relative amplitude and phase between the two
system components, that can be associated to the polar $\theta$ and azimuthal
$\varphi$ angles on a sphere, i.e., in terms of Stokes-like parameters. Their
quantum geometry can be conveniently described by mapping the sphere texture
onto a given parameter space, such as the real or momentum space. The density
of the mapped texture is represented by the so-called Mermin-Ho texture 46, 51
or Berry curvature 52, a quantity that, being integrated over the whole target
space, yields the net number of topological wraps around the sphere, or
skyrmion number 53, 54, 36. Compared to full Poincaré beam, the full Bloch
beam changes dynamically, because its texture is composed of two eigenmodes
with different energies. This results in the coherent oscillations of the two
components (e.g., the strongly coupled exciton and photon fields of
microcavity polaritons), and can be naturally associated with counter-
intuitive physical concepts such as time-varying OAM, a feature that has
recently attracted attention 55, 44. Indeed, the coherent exchange of energy
and momentum between the components also manifests in a dynamical offset of
the vortex cores from the centre in these composite beams. This provides a
nice visualisation of what happens to the wavefunction in the case of time-
varying OAM, implementing the fact that the mean OAM per particle is not
directly determined by the presence of a given number of vortices, but depends
on their positions, their motion and the surrounding density 56, 57, 58, 59,
60, 61. This could be used for the direct encoding of specific OAM waveforms.

Here, we study a double full Bloch beam, created in a polariton fluid, and
track its evolution that takes the form of spiraling vortices, driven by a
mechanism which we clarify in the following. Microcavity polaritons are hybrid
light-matter quasiparticles arising from the strong coupling of microcavity
(MC) photons and quantum well (QW) excitons. This makes them a two-component
system (even in the case of homogeneous polarization). Differently from a
polarization pseudospin, the polariton pseudospin represents the composition
of the full wavefunction in terms of the two normal modes of the system, upper
and lower polaritons (UP and LP). They inherit nonlinearity from the excitons,
interacting composite bosons 62, but show interesting aspects also in the
linear regime. Polaritons provide a powerful testbed for many quantum-fluids
phenomena 63 spanning from Bose-Einstein condensation, long range order
coherence, phase transitions 19 and superfluidity, as well as spontaneous or
imprinted vortex structures formation and dynamics 64. Their open-dissipative
nature allows for both the resonant excitation by means of continuous-wave or
pulsed laser light, as well as for detection of their state via the emitted
light.

In the present setting, the double vortex is imprinted by a photon pulse and
perturbed by a second pulse which is vortex-free. This splits the two initally
singular vortex into four coreless vortices (in the polariton pseudospin
texture), representing the initial state. Its evolution is imaged by means of
a digital off-axis holography scheme 65 allowing to retrieve both the
amplitude and phase maps of the emission in ultrafast time. The observed
dynamics happens in the linear regime and is enabled thanks to the natural
assets of polaritons, that is their Rabi oscillations 65 and radiative decay.
In fact, the motion of photonic vortices inside the composite vortex 66 is due
to the spatially varying relative phase and amplitude between the UP and LP
mode, and their drift in time due to the different eigenfrequencies. This is
possible upon associating the photonic vortex core to a specific pseudospin on
the equator of the Bloch sphere of polariton states, and mapping the
pseudospins by means of the Bloch sphere metric in real space. In the case of
a unitary full Bloch beam 44 (when the excitation pulse carries a unitary
vortex charge), the mapping is a conformal stereographic projection, and the
dynamics consists of reshaping Apollonian circles. In the case of a moving
packet 67 (when the excitation imprints a moving vortex upon oblique
incidence), the mapping is neither conformal nor stereographic, but involves
vortex pair creation and annihilation events. Here, we study the dynamical
texture of a double full Bloch beam of polaritons, and show that it is
conformal despite not being a stereographic projection. Consequently, the two
spiraling vortex cores are following reshaping orbits that are not circular.
This complicated motion can be described in terms of the two orthogonal
velocities on the sphere and in real space. We use this to reveal a general
property for the superposition of Laguerre-Gaussian beams (LGs).

Furthermore, our study is also the first general description, to the best of
our knowledge, of the real-space Berry curvature of a polariton state,
associated to their Bloch sphere pseudospin mapping. Its integral allows us to
show that the topological charge of the dynamical double full Bloch beam is
indeed two and that it is conserved in time, despite the metric spinning and
the Berry curvature itself constantly reshaping. Conceptually, this brings the
notion of a pseudospin texture manifesting as an observable point-like object
(the photonic vortex core) with a conserved topological number. Finally, we
derive a direct connection between the rate of change of the polariton state
on the Bloch sphere and the velocity of the pseudospins in real space. This
further simplifies into an elegant and intriguing expression for the
instantaneous speed of the observable vortex cores in real space, only
depending on the complex Rabi frequency and the local Berry curvature. Such a
link is inveresely proportional to the curvature, differently from other more
known cases.

Fig. 1: Double Rabi spiraling vortex. a,b, Experimental amplitude and phase
maps of the photonic emission from the polariton fluid, when exciting with a
double vortex ($l_{A}=2$) and after arrival of a plain Gaussian pulse
($l_{B}=0$) at the time $t_{AB}=2.04$ ps. The different frames correspond to
times $t=2.0,2.32,2.56$ and $2.80~{}\text{ps}$. The instantaneous phase
singularities are tracked as yellow dots in the amplitude maps, and the solid
lines represent the vortex trajectories drawn over two Rabi-oscillations
periods ($t=2.0-3.6~{}\text{ps}$, red and blue lines are used to distinguish
the two different cores in the photon emission). c, Vortex lines plotted as
$xyt$ curves (time range $t=0.5-10\text{ ps}$, step $\delta t=0.02\text{
ps}$). See also the Supplementary Movie 1.

RESULTS AND DISCUSSION
Spiraling of the polariton double vortex
By means of a $q$-plate carrying a double topological charge, we shape the
photonic pulse $A$ into a modified Laguerre-Gauss $\text{LG}_{02}$ state 18.
Its double winding is translated into two unitary co-winding vortices, whose
initial separation can be controlled by the tuning of the $q$-plate device. We
exploit this effect to realize a polariton fluid with two spatially separated
phase singularities, introducing the potentiality for an asymmetry factor with
respect to the following dynamics. After the first pulse, the vortex cores are
stably located in the inner region of the spot, split by about 12 $\mu$m along
an oblique direction (i.e., they are off-axis with respect to the center of
the beam), as shown in the bottom panels of Fig. 1a,b reporting the amplitude
(a) and phase (b) of the double vortex state at the time $t=2.0$ ps, before
the arrival of the second pulse $B$, which is a plain Gaussian,
$\text{LG}_{00}$. Upon the overlap with the pulse $B$, the vortices are
displaced even more off-axis in opposite directions and reach a maximum
distance of $\approx 40~{}\mu$m (at $t=2.32$ ps), starting a rotational
motion. We note that similarly to a precession, while the vortices move on the
ps time scale, the field is rotating around each core at the optical frequency
in the fs time scale. They reach a horizontal alignment ($t=2.56$ ps) before
coming back to the central region after one Rabi-oscillation period, which is
shown in the top panels ($t=2.80$ ps). Their trajectories, extended into the
next Rabi period ($t=2.0$ – 3.6 ps), are reported as superimposed solid lines
(red and blue). In summary, the two $xyt$ vortex lines start a damped
spiraling motion to end up at two new positions, and this can be clearly seen
when reporting them as a 3D perspective in Fig. 1c.

Fig. 2: Phase delay control of vortex lines. a, Polariton amplitude maps in
the phase delay experiment. The snapshots are taken at $t=3.7$ ps, with four
different phase delays $\varphi_{AB}$ spaced by $\uppi/4$. The phase
singularities have been marked with blue/red dots in the amplitude maps, and
the black/white line is the circle fitting their positions. b, The specific
$xyt$ vortex lines (at a given phase delay) of the two unitary vortices (blue
and red tubes) during one and half Rabi-oscillations period (time range
$t=2.5-3.7$ ps). The topological bottle surface described by the vortex cores
when sweeping the optical phase delay between the two pulses has been mapped
(by spheres) at 100 fs or 50 fs time intervals, spanning $\varphi_{AB}$ in a
$15\uppi$ range by successive $\lambda/8$ steps. c, Blue points represent the
positions of photonic vortex cores in real space at fixed times, when scanning
the phase delay as in panel b. At the times when the photon vortex cores are
more distant from the centre (e.g., $t=3.0$ ps), the phase scan makes their
positions rotate with central symmetry, overlapping the fitting circle (black
solid line). When the cores are closer to the centre (e.g., $t=2.6$ and $3.4$
ps), the phase scan makes them describe two ellipses not overlapping anymore
with the fitting circle, but composing a 8-shaped line.

From the photonic point of view, it is possible to perform a fine control of
the vortex trajectories upon tuning of the optical phase delay $\varphi_{AB}$
between the pulses $A$ and $B$ (which is directly controlled by the time-delay
$t_{AB}$ between the two pulses). Figure 2a shows the emitted photon density
at a fixed time of the dynamics ($t=3.7$ ps), but for varying $\varphi_{AB}$
spaced by equal intervals of $\uppi/4$ (corresponding to $\lambda/8$ steps in
the physical delay line for the second pulse). Both vortices describe the same
circle (black/white line) when sweeping the phase delay, exchanging their
positions when changing $\varphi_{AB}$ along a $\lambda/2$ length. Although
the distribution of the polariton fluid and the positions of the vortex cores
look invariant through such a change, nevertheless, the two vortices can be
identified by the continuity of their positions. The “topological bottle”
surface in the $xyt$ space described by the double vortex strings is shown in
Fig. 2b, in the time range of one and a half Rabi-oscillations period
($t=2.5-3.7$ ps). Here the solid spheres represent the position of the two
phase singularities, tracked at time intervals of $t=0.1$ ps ($t=0.05$ ps in
the last part) and sweeping $\varphi_{AB}$ in $\uppi/4$ steps. The red and
blue solid tubes are the specific vortex lines associated to a fixed phase
delay (that of the previous figure, sampled with $\delta t=0.02$ ps), climbing
on the surface. The surface apparently resembles a self-twisting double
cylinder, but in reality, two different topologies are observed, that can be
ascribed to the anisotropy factor given by the initial splitting of the two
vortices, when compared to the cores displacement induced by the pulse $B$. In
fact, there is a nodal string of the topological bottle (intersection with a
fixed $t$ plane), where the double concentric cylinder undergoes a
metamorphosis into two non-concentric and separated quasi-cylinders: the
projection of such $xy\varphi_{AB}$ line onto the $xy$-plane is a 8-shaped
line. The two different situations are represented in Fig. 2c, where the
circular symmetry is present at given times (e.g., $t=3.0$ ps), while the
8-shape is visible at $T_{\rm R}/2$ time interval from the former (e.g., at
$t=2.6$ ps and $3.4$ ps). These two situations therefore correspond to the
times of the Rabi cycle when the twin cores are further from and closer to the
centre, respectively.

Retrieval of the Bloch sphere metric in real space
In order to model and discuss the observed spiraling, we first retrieve the
topological texture, or quantum states metric, underlying the structured
vortex dynamics. Figure 3a shows that the time-oscillations of the photonic
density at a given spatial point can be fitted by the coupled oscillators
model (see Methods). The two normal modes (UP and LP fields) are decaying in
time differently, while also undergoing a phase shift with different
frequencies. This evolution can be described by introducing the complex
frequencies $\omega_{\rm U,L}+{\rm i}\gamma_{\rm U,L}$, so that $\psi_{\rm
U,L}(t)=\psi_{\rm U,L}(0)\exp({\rm i}\omega_{\rm U,L}t)\exp(-\gamma_{\rm
U,L}t)$. Because of the two different eigenfrequencies, the off-axis
displacement of the two vortices induced by the arrival of the pulse B appears
to different azimuthal positions (since $\varphi_{AB}$ for the UP and LP modes
is different for the same time delay $t_{AB}$). The fitting of the photonic
emission $\psi_{\rm C}$ as an interferometric sum of $\psi_{\rm U}$ and
$\psi_{\rm L}$ allows us to retrieve the spatial maps of the relative phase
and amplitude of the two normal modes (for more details, see Methods section).
Since the position of the phase singularities (after the two-pulse excitation)
in one of the normal modes is shifted with respect to the other component, the
spatial relative phase $\varphi_{\rm LU}(x,y)=\varphi_{\rm U}-\varphi_{\rm L}$
profile has the shape of a quadrupole, shown in Fig. 3b. While the displaced
UP and LP vortices are motionless, the two directly observable photon vortex
cores appear to be moving in round orbits around the singularities of
$\varphi_{\rm LU}$, in the specific experimental case around those
corresponding to the LP vortex cores (see Fig. 3b).

To track the position of any quantum state from the polariton Hilbert space
(UP, LP, photon, exciton or any other linear combination), one needs to
introduce the polariton Bloch sphere (shown in the inset of Fig. 3a). This
sphere has a metric which in turn defines a corresponding metric on the real
plane. In real space, we use the local polariton imbalance $s(x,y)=(|\psi_{\rm
U}|^{2}-|\psi_{\rm L}|^{2})/(|\psi_{\rm U}|^{2}+|\psi_{\rm
L}|^{2})=\cos{\theta}$ which, when shown on the Bloch sphere, is the analogue
of the $s_{3}$ Stokes parameter. Indeed, $\theta$ represents the polar angle
of the corresponding polariton state on the Bloch sphere, while $\varphi_{\rm
LU}$ is its azimuthal angle. Thus, in the spatial relative phase profile, the
isophase lines $\varphi_{\rm LU}=$ const (thin black lines in Fig. 3b)
correspond to the meridians on the Bloch sphere. The north and south poles of
the polariton Bloch sphere represent the pure UP and LP eigenstates
(analogously to the right and left circular polarizations on the Poincaré
sphere 68), whereas the equator represents all possible dynamical states of a
pure photon or exciton, which are continuously transforming into each other at
the Rabi frequency $\Omega_{\rm R}=\omega_{\rm U}-\omega_{\rm L}$. The
$s(x,y)$ map at $t=2.7$ ps is shown in Fig. 3c. The two photon vortex cores,
associated to zeros of the photon intensity, are pure excitonic states.
Furthermore, such zeros can be understood in terms of a destructive
interference between the UP and LP modes, and hence they have to move along
the orbit corresponding to $s(x,y)=0$. The orbits obtained from the fitting of
the experimental photon intensity (white loops in panel c) fail to precisely
retrieve the observed photon vortex cores trajectories (red and blue lines),
due to the high sensitivity of the fit in the weak density areas close to the
LP/UP cores, but the agreement is qualitatively good. While the photonic cores
move along the orbits due to the continuous dynamical drift of the relative
phase in time $\varphi_{\rm LU}=\varphi_{\rm LU}^{0}+\Omega_{\rm R}t$, the
orbits themselves shrink due to the differential decay between the normal
modes $\gamma_{\rm R}=\gamma_{\rm U}-\gamma_{\rm L}$. From the maps of
$s(x,y)$ and $\varphi_{\rm LU}(x,y)$ one can see that each state appearing as
one point on the Bloch sphere is mapped twice to the real plane.

Fig. 3:  Intensity oscillations fit and the Bloch metric maps. a, Photonic
oscillations at a specific point (experimental data) are fitted by the two
interfering LGs model (see Methods). The procedure allows to retrieve the
intensities of the two normal modes in time (solid purple and green lines for
the UP and LP modes, respectively) and their relative phase at each point and
moment of time. Repeating the procedure all along the spatial domain allows to
plot their 2D profiles. Inset: the polariton Bloch sphere with indicated UP,
LP states at the poles, photon (C) and exciton (X) states at the equator, and
the sphere metric consisting in the azimuthal angle $\varphi_{\rm LU}$
(relative phase) and $s=\cos{\theta}$ (local UP-LP content imbalance). An
example of area item $\text{d}A^{\rm sphere}=l_{\rm p}l_{\rm
m}=\sin{\theta}\text{d}\varphi\text{d}\theta$ is marked by the orange patch.
b,c, Relative phase map $\varphi_{\rm LU}(x,y)$ and local content imbalance
$s(x,y)$ (at $t=2.7$ ps). The superimposed lines (red, blue) represent the
experimental orbits of photonic vortex cores along the second Rabi cycle after
the pulse B arrival (time range $t=2.7-3.5$ ps). One clearly observes their
rotation around the two relative phase singularities corresponding to the two
LP-mode vortex cores. The asymmetry manifests itself both in the fact that the
UP mode cores are more distant from each other than the two LP cores, and that
the displacement lines of sight are mutually oblique. The superimposed orange
patches in b mark the real-space area elements $dA^{\rm real}$ corresponding
to a given element on the sphere. d, Module of the total density $|\Psi_{\rm
tot}(x,y)|$ (with $|\Psi_{\rm tot}|^{2}=|\psi_{\rm L}|^{2}+|\psi_{\rm
U}|^{2})$. The shape is a footprint mainly of the LP mode. e, Normalized Berry
curvature map in log scale, retrieved from the maps in panels b,c.

Berry curvature link between the two spaces
We now show that the real-space density of the relative phase and local
imbalance isolines, that comprises the metric of quantum states on the Bloch
sphere, is described by the Berry curvature—the imaginary part of the quantum
geometric tensor 52—that can be defined for any given target space. In
condensed matter, the role of momentum-space and real-space Berry curvatures
of magnetization (or in general, of other vector textures) was recently
highlighted in the anomalous and topological Hall effect (AHE and THE),
respectively 69, 70, affecting the motion of an electron wavepacket. For the
2D target spaces, such as the one we are dealing with here ($xy$-plane), the
generally tensorial Berry curvature becomes a vector with only one non-zero
component $B_{z}$. In microcavity polaritons, such a curvature has been
recently used to express the change of polarization pseudospin (i.e.,
polarization of only one of the polariton modes), thus mapping the density of
the Poincaré coordinates in the reciprocal space 71, 72. That was related to
the AHE of an accelerating wavepacket 73. Here instead, we use the Berry
curvature to express the change of the polariton pseudospin (i.e., composition
of the full wavefunction in terms of both the polariton modes), mapping the
density of the Bloch sphere coordinates to real space:

$B_{z}=\frac{1}{2}\sin\theta(\partial_{x}\theta\partial_{y}\varphi_{\rm
LU}-\partial_{y}\theta\partial_{x}\varphi_{\rm LU}).$ (1)

Such an expression can be shown to be equivalent to:

$B_{z}=\frac{1}{2}\textbf{S}\cdot(\partial_{x}\textbf{S}\times\partial_{y}\textbf{S}),$
(2)

where S is the pseudospin or Bloch vector with components
$(\sin{\theta}\cos{\varphi_{\rm LU}},\sin{\theta}\sin{\varphi_{\rm
LU}},\cos{\theta})$ on the sphere. The functional form in Eq. (2) was brought
into attention in the Mermin-Ho relations 46 and was used to express the
topological charge density, or real-space Berry curvature, of different order
parameter vectors 70, 54, 34, 36, 69. Since Eq. (1) also embeds a cross
product of two gradients in real space,
$B_{z}=\text{\textonehalf}\sin\theta~{}|\bm{\nabla}\theta\times\bm{\nabla}\varphi_{\rm
LU}|$, it is straightforward to see (at least in the case of a conformal
mapping between the metrics) that the Berry curvature describes how much area
of the Bloch sphere is covered when a given area element is spanned in real
space. In other terms, the local density of the sphere surface itself on the
real plane can be written as $\text{d}A^{\rm sphere}/\text{d}A^{\rm
real}=2B_{z}$. We note that in the present case, each area element on the
sphere has a multiplicity $C=2$ in real space (see the color filled areas in
both panels Fig. 3a,b). The Berry curvature therefore plays the role of
linking the full-wavefunction density $|\Psi_{\rm tot}|^{2}=|\psi_{\rm
L}|^{2}+|\psi_{\rm U}|^{2}$ between the two spaces. In both cases, the meaning
of the full wavefunction is to quantify the total number of particles in a
small area element: $|\Psi_{\rm tot}|^{2}=\text{d}N/\text{d}A$. Equating the
number of particles in a given real-space area to the corresponding Bloch
sphere surface area, $dN=C|\Psi^{\rm real}_{\rm tot}(x,y)|^{2}\text{d}A^{\rm
real}=|\Psi^{\rm sphere}_{\rm tot}(\varphi,\theta)|^{2}\text{d}A^{\rm
sphere}$, one gets $|\Psi_{\rm tot}^{\rm sphere}|^{2}=C|\Psi_{\rm tot}^{\rm
real}|^{2}/(2B_{z})$. In this derivation, we have assumed a perfect symmetry
of the real-space distributions which is not observed in the experiments, but
present in the model (see below). For the objects considered in the
experiments (i.e., superpositions of LG00 and LG02 beams), the total density
has a central maximum and two dips, see Fig. 3d, being a footprint of the UP
and LP states at short and long times, respectively, and rotated at
intermediate times. The Berry curvature assumes the shape of a double peak
(centered in the intensity minima) on top of a less intense surrounding ring,
with a neat central hole corresponding to a saddle point of the isophase
lines, as shown in the experimentally-reconstructed Fig. 3e. Recalling the
previous connection of the total densities between the two spaces, it is
therefore expected that the most populated state on the Bloch sphere is the
one for which the pseudospin is associated with the central point of the
imprinted LGs in real space (at any given time).

The integral of the Berry curvature over the surface limited by a closed loop
in real space represents a solid angle on the sphere. The integral of $B_{z}$
over the whole plane defines the topological charge, the number of times the
texture wraps over the sphere, also known as the Chern number 53 or skyrmion
number 54, 36:

$C=\frac{1}{4\uppi}\int\textbf{S}\cdot(\partial_{x}\textbf{S}\times\partial_{y}\textbf{S})\text{d}x\text{d}y=\frac{1}{4\uppi}\int
2B_{z}\text{d}x\text{d}y.$ (3)

Remarkable examples of Chern numbers can be found for a number of textures in
condensed matter. In the case of superfluid 3He-A 46, 70, the integrand in Eq.
(3) represents the topological density of the texture of the unit vector
$\bm{\hat{\textbf{l}}}$, that is the relative orbital angular momentum of
Cooper pairs in the chiral superfluid. In that case, the Mermin–Ho relations
establish that the local vorticity of the continuous vortex is represented by
the texture of such a unit vector. In other terms, they link the circulation
of the superfluid velocity to the solid angle (i.e., area on the sphere) that
is contained inside the real-space loop. The same can be applied to the
current in ferromagnetic BECs where the integrand instead refers to the
density of magnetization texture in real space. In those cases also the
pseudospins represent physical directions in real space. In our case, the
pseudospin refers to a direction on the auxiliary Bloch sphere of the
polariton state. We have verified that for both the experimental and the model
cases described below, the entire space integral of the Berry curvature is
4$\uppi$, associated to the twofold mapping of the sphere, and remains such at
any time despite the reshaping of the mapping. This is indeed consistent with
the integral of Eq. (3) over the whole parameter space giving a net number of
two wraps on the sphere. In the case of stereographic mapping from the sphere
to the real plane 43, 44 (i.e., for the single full Bloch beam), such a
wrapping is unitary. In the current realization, instead, the twofold mapping
and the associated Chern number are reflected in the term double full Bloch
beam (or, equivalently, dynamical double skyrmion) that we use for such
states.

LGs model of the dynamical double full Bloch beam
The essence of the dynamics can be reproduced by means of a model which starts
from the overlapping of the two initial LG beams in the normal modes and lets
them evolve due to the differential decay and the Rabi oscillations.
Theoretically, the quadrupole relative-phase profile formed of four poles
(after the second pulse arrival) can be that of a square, a rectangle, a
rhombus or a parallelogram, depending on the time-delay $t_{AB}$ which defines
the difference in azimuthal displacement of the UP and LP vortex cores.

Experimentally, there are two effects to be taken into account. One is that
the initial $\text{LG}_{02}$ pulse has always a small $\text{LG}_{00}$
component (due to imperfect tuning of the $q$-plate or other residual
intensity) and the first pulse then imprints in the polariton fluid two
vortices which are not ideally overlapped and centered, but have a slight
preliminary offset in opposite directions with respect to the center of the
beam. Then, the arrival of the second pulse induces a further displacement
which may happen along the same or different direction, and this displacement
can also differ for the LP and UP vortex cores due to the difference in
$\varphi_{AB}$ for the two modes. A second factor is that upon arrival of the
second pulse, the differential decay of the modes will have altered the
amplitudes ratio between the two eigenmodes with respect to the one imprinted
by the pulse $A$. The lesser content is that of the UP (as this mode decays
faster), which makes this mode more sensitive to the displacement induced by
the second pulse. As a matter of fact, it can be seen in Fig. 3b that the UP
cores are more offset than the LP ones with respect to the center of the spot.
The minimal set to account for the initial situation after the two pulses is
the four-LG model, one $\text{LG}_{02}$ and one $\text{LG}_{00}$ in both of
the normal modes, with the same center and widths $\sigma$ but with different
amplitudes and phases (see Methods). The initial displacement of the vortex
cores inside each of the normal modes is then symmetric, but the direction and
amount of such displacement is different between the modes.

Fig. 4:  Model of a spiraling double vortex. a, Spatial map of the relative
phase $\varphi_{\rm LU}$ when starting with a larger UP content. The black
lines are phase isolines ($\varphi_{\rm LU}$ every 30∘), the white lines are
isolines of the local imbalance ($s$ from 0.8 to -0.8 with step 0.2). The
superimposed orange patches are an example of real-space area elements
corresponding to a given sphere area element. The two arrows (not is scale)
depict the velocity components in the motion of a given pseudospin. b, Spatial
maps of the local UP-LP imbalance $s(x,y)$ at four different times, with its
reshaping due to the differential decay. The instantaneous orbits of the
photon vortex cores (red and blue circles) follow the white contour lines
$s=0$. Two initial orbits expand till they join at infinity, then become an
eight-shaped line until they finally recoil into two separated petals again.
c, Spatial map of the amplitude of velocity of the moving vortex cores (as
well as of any other pseudospin state). The trajectory of the vortex cores in
the photonic field is tracked (blue and red open circles) at regular time
intervals. d, Perspective views of the isocontent $s=0$ line evolving in time,
drawing a 3D surface with two valleys and two hills. The early (bottom) cones
open out from the position of the UP vortex cores, the later (top) cones wrap
into the position of the LP cores. The overlapped red and blue circles mark
the photon vortex cores trajectory. e, Absolute value of the full wavefunction
$|\Psi_{\rm tot}(x,y)|=\sqrt{|\psi_{\rm L}|^{2}+|\psi_{\rm U}|^{2}}$ at the
moments of time corresponding to panel b. f, Normalized Berry curvature map in
log scale for the same times as in panels b,e. The two dashed black lines mark
the UP and LP vortex pairs line of sight. The scale bar is equal to the width
of the beams $\sigma=16~{}\upmu\text{m}$ in all the panels, the Rabi period
$T_{\rm R}=0.785$ ps. See also Methods and the Supplementary Movie 2.

A typical (corresponding to experiment) scenario is reproduced in Fig. 4a
where the more asymmetric condition of the four poles makes a parallelogram.
The vortex quadrupole is shown together with the relative phase (black) and
local imbalance (white) isolines. It is worth to note the electric-like
(conservative) shape of the isophase lines and the magnetic-like (solenoidal)
shape of the isocontent lines. Interestingly, even while the isolines
$s(x,y)=$ const are not circles, they are still perpendicular to the relative
phase isolines. We express this by using the condition on the gradients of the
Bloch sphere coordinates in real space,
$\bm{\nabla}\theta~{}\perp~{}\bm{\nabla}\varphi_{\rm LU}$. In other terms, the
Bloch sphere’s parallels and meridians that are by definition orthogonal on
the sphere surface, remain mutually orthogonal in real space, and likewise any
other angle relation is preserved, meaning that the mapping between the Bloch
sphere of polariton states and the real space is fully conformal. However, it
cannot be described simply by a stereographic projection or a homeomorphism as
in a fundamental full Poincaré43 or single full Bloch beam case 44, since the
one-to-one link is lost (as already mentioned, each pseudospin is mapped twice
to the real space).

The theoretically modeled evolution of the UP-LP local imbalance map $s(x,y)$
is shown in Fig. 4b for four selected moments in time. The initially closed
orbits ($s=0$, white loop regions) expand until when, approximately at the
moment of the global populations equality, they become two diagonal edge lines
connecting to each other at an infinite distance 74, after which they start to
shrink again, forming a single 8-shaped line. Finally, they separate in the
central point leaving two closed orbits again. The experimentally-retrieved
panel Fig. 3c taken at $t=2.7$ ps corresponds to the final panel in Fig. 4b.
In essence, the motion of the cores (as well as of any other pseudospin state)
has two orthogonal components, and its velocity can be expressed as
$\textbf{v}_{\rm ps}^{\rm real}=\textbf{v}_{\theta}+\textbf{v}_{\varphi}$,
where $\textbf{v}_{\theta}=(\gamma_{\rm
R}\sin{\theta}/|\bm{\nabla}{\theta}|)\bm{\hat{\nabla}}{\theta}$ and
$\textbf{v}_{\varphi}=(\Omega_{\rm R}/|\bm{\nabla}{\varphi_{\rm
LU}}|)\bm{\hat{\nabla}}{\varphi}_{\rm LU}$ (with
$\bm{\hat{\nabla}}{\theta}=\bm{\nabla}{\theta}/|\bm{\nabla}{\theta}|$ and
$\bm{\hat{\nabla}}{\varphi}_{\rm LU}=\bm{\nabla}{\varphi}_{\rm
LU}/|\bm{\nabla}{\varphi}_{\rm LU}|$ the unit vectors along the direction of
the two gradients, see also Methods and the blue arrows in Fig. 4a for their
schematic representation, not in scale). Such two components are conservative
and solenoidal, respectively. The velocity and the trajectory of a given
pseudospin state are further characterized as following a constant angle
$\alpha$ with respect to the imbalance isolines, defined as
$\tan{\alpha}=\gamma_{\rm R}/\Omega_{\rm R}$. This is derived from the
pseudospin of a fixed point in real space evolving on the Bloch sphere along
loxodromes, the curves forming a constant angle with its parallels, and from
the full conformal mapping. The amplitude of the velocity in real space is
shown in a log scale map in Fig. 4c. The four darker spots represent the four
poles of larger gradients, where the photon vortex cores move slower (in the
same panel we superimposed also the vortices cores trajectories). The moving
cores can end up their spiral in one of the two final orbits, depending on
which side they are immediately before the orbits’ separation (which is set by
the initial conditions and $\gamma_{\rm R}/\Omega_{\rm R}$ ratio), or
eventually cross in the very center. This crossing possibility is also
expressed by the central point being a saddle point 75, which means that the
relative phase is defined, but its isolines have a singularity of their
direction. Due to the conformality of the mapping, the same is valid for the
local imbalance isolines. A full view of the orbit line evolution in time is
presented in Fig. 4d, depicted as two different perspectives for the sake of
clarity, with the central point visible as a 3D saddle point also of the $s=0$
surface in the $xyt$ domain. The steeper slope of the surface at earlier and
later times correspond to the smaller velocity of the swirling photon vortices
cores close to the LP and UP cores positions.

The time dynamics of the dips in the total density and the peaks of the Berry
curvature consists in drifting from the initial positions on top of the two UP
vortex cores to the positions of the two LP cores, as shown in Fig. 4e,f,
respectively. This reshaping is only due to the differential decay, since the
Rabi rotation is homogeneously swapping the sphere meridians, without altering
the metric density in real space. In more symmetric cases, the Berry curvature
can as well assume the shape of a perfect ring, due to orthogonal
displacements between the UP and LP vortex pairs (achievable upon proper phase
delay). The two maxima can also become diverging curvature points when the
global population imbalance is mainly in one of the two modes, i.e., when
initializing the system at the UP mode energy, or at long times, when the
differential decay leads to the mainly LP content. Such points can also be
thought of as the limit of the projected sphere contracting to a zero radius,
representing two monopoles. In other terms, this is equivalent to the
continuous vortices becoming two singular vortices (as the system loses its
spinorial degree of freedom and becomes a one-component wavefunction). As
anticipated, using the space integral of the Berry curvature in Eq. (3) gives
a skyrmion number equal to two, which remains the same during the Rabi
spinning and reshaping in time, indicating the topological conservation of the
textures.

Speed of the photon vortex cores in real space
As a very interesting property of dynamical full Bloch beams, we derived also
the speed of the polariton pseudospins in real space, revealing their link to
velocities of the polariton states spiraling on the Bloch sphere, again via
the Berry curvature connection of the metrics (see Methods). Such a relation
can be written as

$v_{\rm ps}^{\rm real}=\frac{v_{\rm ps}^{\rm
sphere}}{\sqrt{2B_{z}}}=\frac{\sin\theta}{\sqrt{2B_{z}}}\sqrt{\gamma_{\rm
R}^{2}+\Omega_{\rm R}^{2}}.$ (4)

Here, the spatial map of the velocities remains constant in time despite
$B_{z}(x,y)$ is reshaping, compensated by the simultaneous reshaping of the
polar angle $\theta$, whereas the second multiplier in (4) can be viewed as
the absolute value of the complex-valued frequency $\tilde{\Omega}_{\rm
R}=\Omega_{\rm R}+i\gamma_{\rm R}$.

The connection between real-space Berry curvature of a topological texture and
some physical properties has been known for different systems. In superfluid
${}^{3}\text{He-A}$, the celebrated Mermin-Ho relations link the vorticity
(curl) of the superfluid velocity $\textbf{v}_{\rm s}$ to the texture density
of the unit vector $\bm{\hat{\textbf{l}}}$ 46 (or of the magnetization in
ferromagnetic multicomponent BECs 49). For simplicity, in the case of a 2D
$xy$-dependence of the order parameter, that is, a vorticity parallel to $z$
(as also in the case of 2D polariton fluids), these relations assume the
following form 76

$\bm{\nabla}\times\textbf{v}_{\rm
s}\propto\bm{\hat{\textbf{l}}}\cdot(\partial_{x}\bm{\hat{\textbf{l}}}\times\partial_{y}\bm{\hat{\textbf{l}}}),$
(5)

where the right-hand side is the $B_{z}$ associated with
$\bm{\hat{\textbf{l}}}$. We now point out one main difference between the two
expressions. In Eq. (5), the curl of the total superfluid velocity is directly
proportional to the real-space Berry curvature of the unit vector of the order
parameter. In the case of Eq. (4), the velocity of the pseudospin states (not
its curl) is, instead, inversely proportional to the square root of the real-
space Berry curvature of the polariton Bloch sphere. Here it is useful to
recall that in the case of a homogenenous $\bm{\hat{l}}$ texture, or of a
single-component fluid or condensate, the superfluid velocity is proportional
to the gradient of the phase of the wavefunction. In the language of optics or
exciton-polaritons, it is therefore equivalent to the group velocity
($|\textbf{v}_{\rm g}|=\partial{\omega}/\partial{k}\propto
k=|\bm{\nabla}\varphi|$, in the parabolic approximation for one component). In
these cases, such a velocity is irrotational and its curl is zero everywhere
apart from the diverging point of a singular vortex core (where, conversely,
the density needs to be zero). In two-component wavefunctions, the concept of
phase needs to be modified, leading to the notion of total superfluid velocity
49, 70, and the vorticity may be diffused (i.e., a finite curl of velocity in
an extended area where the total density doesn’t need to be zero). This is the
case of $v_{s}$ appearing in Eq. (5). However, in our case and the situation
described by Eq. (4), we are not looking at the total phase. The associated
total group velocity plays a role, as it is known, in the motion of the total
density, whose reshaping is further affected in two-component cases by the
relative decay term, or differential radiative losses between the two
components. While the total density is still plotted in our representation, we
are not focusing on its driving terms. Here, we are rather dealing with the
velocity of the pseudospin states themselves moving on top of the total
density. To this end, since the motion of the pseudospin can be thought of in
terms of interference between the normal modes, their speed is more analogue
to a phase velocity ($|\textbf{v}_{\rm ph}|=\omega/k\propto
k^{-1}=|\bm{\nabla}\varphi|^{-1}$). This intuitively explains that the
velocity of the pseudospins is inversely proportional to the Berry curvature
(if one thinks of the latter as the gradient of a 2D-valued phase). Another
difference is that the overall velocity in Eq. (4), as mentioned before, has
both the conservative and the solenoidal terms linked to the Berry curvature.
Furthemore, when projecting the full wavefunction onto one specific component
(such as the emitted photon field), one can study the motion of the (singular)
vortex cores themselves inside that component (rather than looking at the
group velocities, or fluid velocities, around the cores, for a fixed vortex
configuration, be it singular or continuous).

At the same time, the velocity described in Eq. (4) is also different from the
terms involved in the AHE and THE, that represent velocity and force terms
that are directly proportional to the momentum- and real-space Berry
curvatures, respectively 69. In the AHE, an accelerated wavepacket aquires a
transverse velocity term too, proportional to the momentum-space Berry
curvature (of magnetization vector in condensed matter, and of polarization
for polaritons 73). On the other hand, the real-space and mixed spatio-
temporal Berry curvatures of the magnetization vector result in the emergent
magnetic and electric fields, respectively 77. In the former case, also known
as THE, an electron wavepacket acquires an extra force proportional to the
real-space skyrmion texture 54. In the latter case, known as spinmotive force
(and requiring a time derivative of the relevant vector, e.g., local
magnetization), it is the moving skyrmion that generates a force perpendicular
to its moving direction 78. Here we are not dealing with velocities obtained
by (or forces exerted on) external objects, but with the self-evolution of the
pseudospin texture in a two-component wavefunction (externally generated and
then driven by an effective Rabi field, as indicated below).

It is now interesting to focus on the velocity of the observable photon vortex
cores, which represent pure excitonic states that lie on the equator of the
Bloch sphere, i.e., $\sin\theta=1$. In this case, Eq. (4) reduces to

$v_{\rm core}^{\rm real}=\frac{|\tilde{\Omega}_{\rm R}|}{\sqrt{2B_{z}}},$ (6)

where the vortices cores’ velocity only depends on the complex Rabi frequency
and is inversely proportional to the (square root of the) local and
instantaneous Berry curvature, changing during the motion. It can also be seen
that there is no mass or any other prefactor in the right-hand side of Eq. (6)
(while this was omitted in the case of Eq. (5)). This is a general relation
that is assumed to hold also in the case of superposition of two optical LG
(or even differently shaped) beams with different frequencies (when looking at
the dynamics on a plane orthogonal to the propagation direction).

CONCLUSIONS
When dealing with a two-component fluid, its full wavefunction can be
expressed in terms of a total density and total phase, plus a further
pseudospin unit vector or 2D valued phase. Such a pseudospin is representable
on a Poincaré or Bloch sphere, and can be mapped by the sphere texture (or
metric) to the real space. In a fundamental unitary skyrmion (or single full
Bloch beam 44 and, analogously, Poincaré beam 43) the one-to-one mapping is
possible via a stereographic projection, a conformal homeomorphism between the
sphere metric and real space, while in more complex cases even the conformal
feature is not maintained 67. Our experiments implement for the first time
dynamical pseudospin textures going beyond the unitary skyrmion case in a
polariton fluid, based on multiple-vortex configurations. Such textures are
set into a spinning and reshaping dynamics thanks to the polariton Rabi
oscillations and the non-Hermitian feature imparted by the decay. The Rabi
frequency as an operator, similarly to other Hamiltonians, can be thought of
as an effective magnetic field, as for example was proposed in the optical
Maxwell-Schrödinger formalism 79, 80 for the dynamics of the polarization
pseudospin on the Poincaré sphere. For the polariton state, the Rabi field is
along the vertical axis $\bm{\hat{s}}_{3}$ of the Bloch sphere, resulting in
the precession of the polariton pseudospin around such axis (i.e., along a
parallel)

$\frac{d\textbf{S}}{dt}=\Omega_{\rm
R}\bm{\hat{s}}_{3}\times\textbf{S}+(\gamma_{\rm
R}\bm{\hat{s}}_{3}\times\textbf{S})\times\textbf{S},$ (7)

while the second term adds the decay-induced drift along a vertical plane
(i.e., along a meridian). The two terms result in the solenoidal and
conservative velocities of the pseudospin states in real space, respectively,
as a feature directly inherited from the associated parallels and meridians on
the sphere (and mediated by the conformal mapping). The form of Eq. 7 is
equivalent to the Landau-Lifshitz (LL) 81, 82 equation that describes the
Larmor precession of the magnetic moment of an object (or the magnetization
vector) around the external (or effective) magnetic field (that is, on the
other hand, also analogue to the precession of a tilted mechanical gyroscope
in an external, torque-exerting gravitational field). From a physical point of
view, the LL equation has a term that corresponds to the torque exerted by the
effective field on the magnetic moment forcing it to precession, and a second
damping (relaxation) term that represents another torque that pushes the
magnetization in the direction of the field. Also in the case of
${}^{3}\text{He-A}$, there are equivalent and even more complex equations 2,
describing how the vector $\bm{\hat{\textbf{l}}}$ varies in time due to the
external rotation of the system, the magnetic field in case of locked spin-
orbital configurations, and other torques due to incoherent fermionic
excitations. Here, the two main terms are represented by the Rabi frequency
(responsible of a cyclic conversion of photons into excitons and so on) and
the differential decays of polaritons (responsible for the unidirectional
evolution from the upper polariton to the lower polariton mode). Furthermore,
in the LL case, both the magnetic moment and the magnetic field are directions
in real space. Here, the effective Rabi field as well as the decay term, are
directions on the pseudospin sphere. The field and decay are assumed to be
uniform, and the pseudospin texture varying in real space (e.g., creating a
unitary or multiply-charged skyrmion). It is this effective field and
associated decay that rules the dynamics of the pseudospin and reshapes the
texture. This results in the observable photonic vortex cores (being
associated to a specific state on the equator of the sphere) following a
peculiar spiraling motion. Such an effective field preserves the initial
conformal feature of the texture mapping at all times, despite being non-
stereographic and non-homeomorphic. Conversely, it is also possible to show
that, since the composing beams only change in intensity and phase, the
transformation maintains the initial ingredients of the LGs superposition. We
empirically devise the conformal feature as a general property of concentric
LG beams of the same width bearing cowinding charges (of different absolute
value).

Furthermore, we apply for the first time the Berry curvature concept to the
polariton Bloch-sphere mapping to real space, specifically describing it for
dynamical double skyrmions. The Berry curvature here represents the density of
the Bloch-sphere metric (the variation of the pseudospin) in the target space
(real space). As such, its integral over the whole parameter space represents
the number of times the texture wraps around the Bloch sphere, in our case
equal to two (in both experiments and model cases) and conserved during time,
despite the spinning and reshaping of the texture and of the curvature itself.
One could think of the initial state in the model as composed by two skyrmions
centered in the positions of the UP mode vortex cores. The decay strains the
two initial skyrmions into four merons and then transforms them back again
into two skyrmions, centered in the LP vortex cores at later times, while the
Rabi oscillations transform them from Bloch- to Neel-type continuously. This
is consistent with the topological charge equal to two at all times. It would
be possible to use counter-polarized optical pulses to achieve a four-
component fluid and build a one-to-one mapping of the Bloch and Poincaré
spheres, realizing a dynamical full Poincaré beam. Each polarization would be
present at a given time in a different position, and any point of space would
see a sequence of polarizations in time, too. In another configuration, having
a different (asymmetric content between the two pulses) superposition (such as
for the simplest full Bloch beam possible, a LG01 in the UP mode and a LG00 in
the LP mode), would lead to the observation of an initial vortex in the
photonic component (emission), spiraling out of the polariton cloud (and vice
versa, spiraling in, when using the opposite scheme). Similar reasonings
pertain to any generic LGs superposition of two (or more) optical beams with
different (complex) frequencies. These structured beams could be used to drive
THE and spinmotive forces when applied to optically anisotropic particles.
Experimentally, it is interesting to see how a pseudospin texture can manifest
itself into an observable point-like object, here the photonic vortex cores.
Their motion depends on how the pseudospin trajectories are linked between the
two spaces. Along the sphere, these directions are loxodromes with an angle
directly linked to the ratio of the two components of the complex Rabi
frequency. On the other hand, we show that the same velocity is projected onto
the real plane with a speed defined by the Berry curvature. Intuitively, the
pseudospin gradient in real space can be further thought of as a
pseudomomentum, similarly to the gradient of a standard phase, which is now a
2D-valued phase. One of the meanings of the Berry curvature is hence to
describe the amplitude of the pseudomomentum vector. The real-space velocity
of the pseudospins’ motion is inversely linked to such a pseudomomentum
(similar to standard phase velocity being inversely proportional to the
momentum in one-component fields). This allows us to derive a compact equation
for the real-space velocity of the observable photonic vortex cores, which
ultimately depends only on the complex Rabi frequency and is inversely
proportional to the square root of the local Berry curvature.

METHODS
Experimental methods. The polariton device used here is fabricated by means of
MBE technique and consists in an AlGaAs $2\lambda$ MC with three
In0.04Ga0.96As quantum wells of 8 nm placed at the antinodes of the cavity
mode field. The cavity is placed inside two distributed Bragg reflectors made
of 21 and 24 pairs of alternated $\lambda/4$ AlAs and GaAs layers. The
polariton modes are at 836.2 nm and 833.2 nm (LP and UP mode, respectively),
their splitting is 3 nm (5.4 meV) at zero momentum which converts into a Rabi
period of $T_{\rm R}=0.780$ ps. The lifetime of the normal modes is $\tau_{\rm
L}=1/\gamma_{\rm L}\sim 10$ ps and $\tau_{\rm U}=1/\gamma_{\rm U}\sim 2$ ps
for the lower and upper modes, respectively. The experiments are performed in
a region of the sample clean from defects. The device is kept at a temperature
of 10 K inside a closed-loop He cryostat. The resonant beam is a 130 fs pulse
laser with 80 MHz repetition rate and a 8 nm bandwidth. The central energy is
tuned at approximately 835 nm in order to overlap both the LP and UP branch
energies. The photonic beam is passed through a $q$-plate device 83, 84 with
winding charge 2, in order to obtain a double optical vortex which is sent
onto the microcavity sample at normal incidence. Once imprinted, the polariton
vortices just decay and oscillate in intensity in the photonic component
keeping a fixed position. The second pulse is sent by means of a small
Michelson-Morley (MM) scheme interferometer in the excitation side, with
controllable time delay. The second pulse is derived from the unconverted
portion of the beam crossing the $q$-plate, through polarization filtering.
The $q$-plate is a patterned Liquid Crystal (LC) device that is driven by a
oscillating voltage in the kHz range. The applied voltage controls the degree
of LC tilting along the vertical direction, hence the anisotropy degree along
the patterned directions and ultimately the $q$-plate conversion efficiency.
An impinging $\text{LG}_{00}$ beam with right or left circular polarization is
totally or partially converted into the opposite circular polarization with a
wavefront shaped into an $\text{LG}_{02}$ mode. So there can be a tunable
amount of residual original $\text{LG}_{00}$ beam. Upon using a different
polarization filtering scheme in the small MM interferometer arms, such a
residual portion can be both left in the pulse A and used for pulse B. In
summary, pulse A will consist of a $\text{LG}_{02}$ and $\text{LG}_{00}$
beams, whose relative amplitudes control the initial two cores displacement
(the same in the two normal modes), while pulse B will only consist of an
$\text{LG}_{00}$ pulse used to further and differentially split the two cores
(i.e., along different directions in the normal modes). An ultrafast time
resolved detection scheme is based on the off-axis digital holography, where
the emission from the sample plane is focused on an imaging camera together
with a reference beam. The reference pulse is derived by the resonant laser
beam in a Mach-Zehnder scheme interferometer scheme and is not focused but
passed through a small iris, so that it becomes a homogeneous and flat front
when arriving on the camera. The digital fast Fourier transform (FFT)
filtering is applied to the interferograms and this procedure allows to
reconstruct the amplitude and phase of the emitted photonic wavefunction. The
FFT filtering can be applied directly in the laboratory in order to aid for
the setting of the measurements. Further details on the polariton sample or
digital holography technique can be found in previous works 65, 85, 44, 59 and
descriptions therein.

Nonlinear regime effects. We checked the changes added in the experiments when
deviating from the linear regime, in this and previous works configurations
and add here a brief note on those. When increasing the power we effectively
see two main effects. First, a differential shift of the oscillations
frequency between spatial regions of low and small density, that we ascribe to
the nonlinear energy increase of the exciton line, mainly reflected in the LP
mode blueshift. On the other hand, the UP mode energy acts for this sample
mainly in a self-compensating way, due to the exciton blueshift and the
partial loss of coupling. Second, increasing further the power we observe the
damping of the oscillations amplitude, due to an increased linewidth of the UP
mode, associated to its faster decay and/or dephasing. This damping of the
oscillations converts, in the current vortex experiments, into a faster
damping of the swirling cores orbits.

Interference-fitting procedure for two coupled oscillators. In the linear
regime, it is possible to express the evolution of the observable photon
density at each point in space as a mean value plus a modulation term, i.e.,
in terms of interference between the LP and UP eigenmodes:

$|\psi_{\rm C}|^{2}=\frac{|\psi_{\rm L}|^{2}+|\psi_{\rm U}|^{2}}{2}+|\psi_{\rm
L}||\psi_{\rm U}|\cos(\varphi_{\rm LU}^{0}+\Omega_{\rm R}t).$

The numerator defines also the total density, expressed as $|\Psi_{\rm
tot}|^{2}=|\psi_{\rm L}|^{2}+|\psi_{\rm U}|^{2}$. The UP and LP modes decay
are included upon adding $|\psi_{\rm U,L}(t)|=|\psi_{\rm
U,L}(0)|\exp(-\gamma_{\rm U,L}t)$. Imposing fixed values for $\gamma_{\rm U}$,
$\gamma_{\rm L}$, and $\Omega_{\rm R}$, and treating $|\psi_{\rm L}({\bf
r},0)|$, $|\psi_{\rm U}({\bf r},0)|$, and $\varphi_{\rm LU}^{0}({\bf r})$ as
fitting parameters, we can fit the time traces at each point $(x,y)$ as in
Fig. 3a starting from a time taken after the setting sequence of both pulses
$A$ and $B$, and retrieve the profiles of the UP and LP polariton fields
(which are not directly observable), as well as their relative phase spatial
profile shown in Fig. 3b. The fitting is performed along seven Rabi periods
(from 2.6 ps to 8.2 ps). The experimental maps in Fig. 3c-e are retrieved by
using the fitted values in the expressions for the local imbalance parameter
$s$, amplitude of the total density $|\Psi_{\rm tot}|$ and Berry curvature
$B_{z}$, respectively.

Interfering LGs model. The initial condition and its evolution are reproduced
by use of the four LGs packets model, representing the vortex and Gaussian
beam imprinted in the upper and lower polariton fields. Assuming for
simplicity a zero detuning between the cavity photon to exciton mode, the bare
modes are hence obtained as linear combinations of the normal modes $\psi_{\rm
C,X}=(\psi_{\rm U}\pm\psi_{\rm L})/\sqrt{2}$. We write the spatial profiles of
the fields in complex polar coordinates $(r,\phi)$, where $r=|x+{\rm i}y|$ and
$\phi=\arg(x+{\rm i}y)$. The initial state can be written as a combination of
the four LGs,

$\psi_{\rm U,L}(0)=A_{\rm U,L}\text{LG}_{02}+B_{\rm U,L}\text{LG}_{00}.\\\ $

Expanding the LGs forms and adding the time evolution on the right side, it
can be written

$\psi_{\rm U,L}={\rm e}^{-r^{2}/2\sigma^{2}}\left[A_{\rm
U,L}\frac{r}{\sigma}{\rm e}^{{\rm i}2\phi}+B_{\rm U,L}{\rm e}^{{\rm
i}\varphi_{AB}^{\rm U,L}}\right]{\rm e}^{({\rm i}\omega_{\rm U,L}-\gamma_{\rm
U,L})t}.\\\ $

We have used the same width $\sigma$ and same center in the origin for all the
four components. The four coefficients (real positive) represent the amplitude
of the UP and LP modes set by the two pulses $A$ and $B$. Despite the two
laser pulses have the same frequencies content, we use $B_{\rm U}/B_{\rm
L}>A_{\rm U}/A_{\rm L}$. In fact, at the time arrival of second pulse $B$
(about 2 ps time delay), the UP/LP content set by pulse $A$ is already
decreased due to the differential decay. The pulse delay also sets a phase
delay between the plain Gaussian and the vortex packet that is in general
different in each of the two modes, $\varphi_{AB}^{\rm U}\neq\varphi_{AB}^{\rm
L}$. These phase delays control the direction of the displacement of the
vortex cores in each respective (UP or LP) mode. Upon changing time delay on a
few wavelengths scale, the relative displacement directions of the UP and LP
remain the same, meaning they are rotated of the same amount. For such reason
we use in the previous sections the simplified notation $\varphi_{AB}$. This
simple model gives the same results than would be derived with a full coupled
Schrödinger equations (cSE) model, apart from neglecting the dispersion and
diffraction effects and the transients from the incoming photon pulses being
injected into the polariton components. We are not interested in the sub-ps
transient and the beams are wide enough in order to neglect any
dispersive/diffraction effects in the 10 ps time scale range. The cSE model
was used in order to verify the behavior of the Rabi-oscillating vortices
against the LGs model in previous configurations 44, 67. The parameter values
used in the model are: $A_{\rm U}=6$, $A_{\rm L}=2$, $B_{\rm U}=10$, $B_{\rm
L}=2$, $\varphi_{AB}^{\rm U}=\uppi/2$, $\varphi_{AB}^{\rm L}=0$ and
$\sigma=16~{}\upmu\text{m}$. The used relative frequency and decay are:
$\Omega_{\rm R}=\omega_{\rm U}-\omega_{\rm L}=8~{}\text{ps}^{-1}$ and
$\gamma_{\rm R}=\gamma_{\rm U}-\gamma_{\rm L}=0.8~{}\text{ps}^{-1}$, resulting
in a ratio $\gamma_{\rm R}/\Omega_{\rm R}=0.1$. The time frames shown in Fig.
4 are $t_{1,2,3,4}=0.89,1.38,1.58~{}\text{and}~{}2.11~{}\text{ps}$,
corresponding to a global content imbalance $S_{1,2,3,4}=0.5$, $0.16$, $0.0$
and $-0.4$, respectively. The global imbalance is defined as
$S=\frac{\int|\psi_{\rm U}|^{2}d{\bf r}-\int|\psi_{\rm L}|^{2}d{\bf
r}}{\int|\psi_{\rm U}|^{2}d{\bf r}+\int|\psi_{\rm L}|^{2}d{\bf r}}$.

Pseudospin evolution on the sphere. We have defined the pseudospin using the
polar and azimuthal angles $\theta,\varphi$, that define the latitude and
longitude on the Bloch sphere of polaritons, respectively. The polar angle can
also be written as a Stokes-like parameter, the local imbalance
$s=\frac{|\psi_{\rm U}|^{2}-|\psi_{\rm L}|^{2}}{|\psi_{\rm U}|^{2}+|\psi_{\rm
L}|^{2}}=\cos{\theta}$ while the azimuthal angle is represented by the
relative phase $\varphi_{\rm LU}=\varphi_{\rm U}-\varphi_{\rm L}$. A point in
real space has a given pseudospin at time $t_{0}$. Such state will change due
to the complex relative (i.e., Rabi) frequency. The azimuthal angle of the
pseudospin on the sphere will change according to $\varphi_{\rm
LU}(t)=\varphi_{\rm LU}(0)+\Omega_{\rm R}t$. Its arc line speed along a
parallel is hence $\text{d}l_{\rm p}/\text{d}t=\Omega_{\rm R}\sin{\theta}$.
The polar angle will change due to the differential decay term. Combining the
expression of the two decays, $\frac{|\psi_{\rm U}|}{|\psi_{\rm
L}|}(t)=\frac{|\psi_{\rm U}|}{|\psi_{\rm L}|}(0)e^{-\gamma_{\rm R}t}$, that
can be written in terms of the Stokes parameter $s(t)=\tanh\\{-\gamma_{\rm
R}t-\text{atanh}[s(0)]\\}$. The initial $s(0)$ is arbitrary, here it depends
both on the initialization conditions and on the specific point in space. The
speed of variation of the parameter is $\text{d}s/\text{d}t=\frac{-\gamma_{\rm
R}}{\text{cosh}^{2}\\{-\gamma_{\rm R}t-\text{atanh}[s(0)]\\}}=-\gamma_{\rm
R}(1-s^{2})=-\gamma_{\rm R}\sin^{2}{\theta}$ (where we used the hyperbolic
derivative and the $\text{cosh}^{2}=\frac{1}{1-\text{tanh}^{2}}$ relation).
Recalling that $\text{d}s=-\sin{\theta}\text{d}\theta$, the arc length speed
along a meridian is $\text{d}l_{\rm
m}/\text{d}t=\text{d}\theta/\text{d}t=\frac{1}{-\sin{\theta}}\text{d}s/\text{d}t=\gamma_{\rm
R}\sin{\theta}$. The pseudospin evolves on the sphere moving at an angle
$\alpha$ with respect to the parallels such that
$\tan{\alpha}=\frac{\text{d}l_{\rm m}/\text{d}t}{\text{d}l_{\rm
p}/\text{d}t}=\gamma_{\rm R}/\Omega_{\rm R}$. The angle is hence constant,
describing a so called loxodrome (we used this in a previous work 44, without
showing the current demonstration). The total velocity of the polariton state
evolution on the sphere can be expressed as $v_{\rm ps}^{s\rm
phere}=\text{d}l/\text{d}t=\sin{\theta}\sqrt{\gamma_{\rm R}^{2}+\Omega_{\rm
R}^{2}}$. The term in the squared root can be thought as the module of a
complex Rabi frequency $\tilde{\Omega}_{\rm R}=\Omega_{\rm R}+i\gamma_{\rm
R}$. The evolution holds for the pseudospin state in any point of real space,
regardless of the specific mapping of the Bloch sphere to real space.

Conformal mapping. We have verified that the superposition of LGs with the
same width, center and sign of all the winding charges among the two modes
always convert into a conformal mapping of the pseudospin sphere to real space
(regardless of the packets’ total number, amplitudes, relative phases and
absolute winding charges). Naturally, linearly dependent combinations of
coefficients between the two modes result in a 0D mapping on the sphere, that
is the same pseudospin on the whole space. Conformal mapping means that
parallels and meridians of the Bloch sphere are mapped to two families of
curves mutually orthogonal everywhere on the real plane. Since we didn’t find
a complete reference treating this property, we point to more specialized
bibliography or to a future work. In our multifrequency LGs models we neglect
diffusive and dispersion effects, the composing LGs only change their relative
amplitudes and phase, hence the conformal feature remains at all times. This
feature can be expressed in terms of the real-space gradients of the two
pseudospin angles, $\bm{\nabla}{\theta}~{}\perp~{}\bm{\nabla}{\varphi}_{\rm
LU}$. By application of the chain rule, we know that the motion of a
pseudospin in real space due to, e.g., the solely Rabi effect happens along
the direction of the gradient of $\varphi_{\rm LU}$ with speed
$v_{\varphi}=\frac{\partial d}{\partial t}=\frac{\partial
d}{\partial\varphi}\cdot\frac{\partial\varphi}{\partial
t}=(\frac{\partial\varphi}{\partial
d})^{-1}\cdot\frac{\partial\varphi}{\partial t}=\Omega_{\rm
R}/|\bm{\nabla}\varphi_{\rm LU}|$ (where we used $d$ for the distance along
such a direction, and the compact notation $\varphi=\varphi_{\rm LU}$ for
simplicity). Applying the same on the other component, and from the evolution
speed of the polar angle, $\frac{\partial\theta}{\partial t}=\gamma_{\rm
R}\sin{\theta}$, we end up with the total vector velocity $\bf{v}_{\rm
ps}^{\rm real}=\bf{v}_{\theta}+\bf{v}_{\varphi}$ being
$\bf{v}_{\theta}=(\gamma_{\rm
R}\sin{\theta}/|\bm{\nabla}{\theta}|)\bm{\hat{\nabla}}{\theta}$ and
$\bf{v}_{\varphi}=(\Omega_{\rm R}/|\bm{\nabla}\varphi_{\rm
LU}|)\bm{\hat{\nabla}}{\varphi}_{\rm LU}$. The second term does not change in
time, because the relative phase only changes by a homogeneous dynamical term
due to the Rabi frequency, so its gradient is constant. The first term does
not change as well, because our specific mapping is fully conformal, such that
$|\bm{\nabla}{\theta}|=\sin{\theta}|\bm{\nabla}\varphi_{\rm LU}|$, so that
$v_{\theta}/v_{\varphi}=\gamma_{\rm R}/\Omega_{\rm R}$, so the trajectory of a
given pseudospin is a loxodrome in real space too (with respect to the
$\varphi_{\rm LU},\theta$ curvilinear coordinates). Looking at the expression
of the Berry curvature in Eq. (1), it contains the vector product of the two
pseudospin angles gradients in real space,
$B_{z}=\text{\textonehalf}\sin\theta~{}|\bm{\nabla}{\theta}\times\bm{\nabla}{\varphi}_{\rm
LU}|$. Such a product represents the sphere areal density on real space. Since
the two gradients are here orthogonal, it is also
$B_{z}=\text{\textonehalf}\sin\theta~{}|\bm{\nabla}{\theta}||\bm{\nabla}{\varphi}_{\rm
LU}|=\text{\textonehalf}~{}|\bm{\nabla}{\theta}|^{2}=\text{\textonehalf}\sin^{2}\theta~{}|\bm{\nabla}{\varphi}_{\rm
LU}|^{2}$. The two velocities can then be written as
$v_{\theta}=\frac{1}{\sqrt{2B_{z}}}\gamma_{\rm R}\sin{\theta}$ and
$v_{\varphi}=\frac{1}{\sqrt{2B_{z}}}\Omega_{\rm R}\sin{\theta}$, while for the
module of the total velocity $v_{\rm ps}^{\rm
real}=\frac{1}{\sqrt{2B_{z}}}\sin{\theta}\sqrt{\gamma_{\rm R}^{2}+\Omega_{\rm
R}^{2}}$. It is possible to see that the Berry curvature is linking the
velocities between the two spaces (real plane and Bloch sphere), recalling
that $v_{\rm ps}^{\rm sphere}=\sin{\theta}\sqrt{\gamma_{\rm R}^{2}+\Omega_{\rm
R}^{2}}$. Hence, the speed of the spatial motion of a pseudospin and of the
evolution on the sphere of a spatial point’s pseudospin are linked by $v_{\rm
ps}^{\rm real}=\frac{1}{\sqrt{2B_{z}}}v_{\rm ps}^{\rm sphere}$. On the other
hand, since the total density represents the number of total particles in a
given area, the Berry curvature also plays a role in linking the total density
between the two spaces, $|\Psi_{\rm tot}^{\rm
sphere}|^{2}=\frac{C}{2B_{z}}|\Psi_{\rm tot}^{\rm real}|^{2}$, where $C$ is
the topological multiplicity factor discussed in the main text.

Total phase and continuity equation. For the sake of clarity and completeness,
we derive the continuity equation for the total density of particles $n_{\rm
tot}=n_{\rm U}+n_{\rm L}$, where $n_{\rm U}=|\psi_{\rm U}|^{2}$ and $n_{\rm
L}=|\psi_{\rm L}|^{2}$ are the local densities of particles in the two
components. With this notation the local imbalance reads $s=\frac{n_{\rm
U}-n_{\rm L}}{n_{\rm U}+n_{\rm L}}=\frac{n_{\rm U}-n_{\rm L}}{n_{\rm tot}}$.
It is also possible to consider the total phase $\varphi_{\rm
tot}=\varphi_{\rm U}+\varphi_{\rm L}$ in parallel with the relative phase
$\varphi_{\rm LU}=\varphi_{\rm U}-\varphi_{\rm L}$. For convenience, we also
introduce the total decay rate $\gamma_{\rm tot}=\gamma_{\rm U}+\gamma_{\rm
L}$ in addition to the already discussed relative decay $\gamma_{\rm
R}=\gamma_{\rm U}-\gamma_{\rm L}$, where $2\gamma_{\rm U}$ and $2\gamma_{\rm
L}$ are the particles decay rates in each component. The continuity equation
can be written as $\text{d}n_{\rm
tot}/\text{d}t=-\bm{\nabla}\cdot\textbf{j}_{\rm U}-2\gamma_{\rm U}n_{\rm
U}-\bm{\nabla}\cdot\textbf{j}_{\rm L}-2\gamma_{\rm L}n_{\rm L}$, where
$\textbf{j}_{\rm U,L}=n_{\rm U,L}\textbf{v}_{\rm U,L}$ are the density flow in
each component and $\textbf{v}_{\rm U,L}=\frac{\hbar}{m_{\rm
U,L}}\bm{\nabla}\varphi_{\rm U,L}$ the superfluid velocities, associated to
the respective effective masses $m_{\rm U,L}$ and phase gradients
$\bm{\nabla}\varphi_{\rm U,L}$. The total flow of particles can be written in
terms of the total density and velocity, $\textbf{j}_{\rm tot}=\textbf{j}_{\rm
U}+\textbf{j}_{\rm L}=n_{\rm tot}\textbf{v}_{\rm tot}$, from which the total
superfluid velocity is defined as $\textbf{v}_{\rm tot}=\frac{n_{\rm
U}\textbf{v}_{\rm U}+n_{\rm L}\textbf{v}_{\rm L}}{n_{\rm U}+n_{\rm L}}$. The
continuity equation can then be written in terms of the total and relative
quantities, rather than with respect to the single components,

${\text{d}n_{\rm tot}}/{\text{d}t}=-\bm{\nabla}\cdot(n_{\rm
tot}\textbf{v}_{\rm tot})-(\gamma_{\rm tot}+s\gamma_{\rm R})n_{\rm tot}.$

Also the total superfluid velocity contained in such expression can be written
in terms of the total and relative quantities,

$\textbf{v}_{\rm tot}=\frac{\hbar}{2m}(\bm{\nabla}\varphi_{\rm
tot}+s\bm{\nabla}\varphi_{\rm LU}),$

where we used the approximation of equal effective masses $m=m_{\rm U}=m_{\rm
L}$ (that is not strictly true for polaritons but valid at small momenta and
for small photon-exciton detuning). It is hence possible to see that the total
phase and its gradient is in general involved in the motion and reshaping of
the total density profile, similarly to what happens in one-component cases.
However, the total density is in principle also affected by the content
imbalance, if there is a difference between the two superfluid velocities
(i.e., if $\textbf{v}_{\rm U}\neq\textbf{v}_{\rm L}$). Finally, while in one-
component cases, a uniform decay rate is only affecting the absolute density,
not its shape, in two-component cases the total density is also reshaping due
to the different decay rates of the components. The dynamics of the total
density in Fig. 4e is in fact only due to this effect (while the model
neglects the density redistribution due to the group velocities, that is a
valid approximation, as said, for the used beams width and time scales). Apart
from the role on the total density, the total superfluid velocity satisfies
the Mermin-Ho relation in Eq. (5), upon replacing the $\bm{\hat{\textbf{l}}}$
vector with the polariton pseudospin S. It is possible to note that this is
due to the second term in the $\textbf{v}_{\rm tot}$ expression, that has a
nonzero curl, thanks to the $s$ factor that appears before the gradient of the
relative phase, while for the first term, its curl is zero. The circulation of
the total velocity has hence a discrete part in $2\uppi$ multiples, depending
on the total-phase singularities enclosed by the path, and a continuous part
that depends on the relative phase, whose gradient is weighted by the $s$
local imbalance.

Data Availability: All data needed to evaluate conclusions are present in the
paper. Original datasets and additional data related to this paper may be
requested from the corresponding author on reasonable request. Supplementary
Movies 1 and 2 contain the full dynamics of experimental and model double full
Bloch beams. Code Availability: The code used to analyze data or implement the
model may be requested from the corresponding author on reasonable request.

## References

  * Salomaa and Volovik [1987] M. M. Salomaa and G. E. Volovik,  _Quantized vortices in superfluid He 3,_Rev. Mod. Phys. 59, 533 (1987).
  * Vollhardt and Wölfle [1990] D. Vollhardt and P. Wölfle,  _The superfluid phases of helium 3_ (Taylor & Francis, London - New York, 1990).
  * Lounasmaa and Thuneberg [1999] O. V. Lounasmaa and E. Thuneberg,  _Vortices in rotating superfluid ${}^{\textrm{3}}$ He, _Proc. Natl. Acad. Sci. 96, 7760 (1999).
  * Volovik [2009] G. E. Volovik,  _The universe in a helium droplet_ , International series of monographs on physics No. 117 (Oxford University, Oxford, 2009).
  * Leggett [1999] A. J. Leggett,  _Superfluidity,_Rev. Mod. Phys. 71, S318 (1999).
  * Aranson and Kramer [2002] I. S. Aranson and L. Kramer,  _The world of the complex Ginzburg-Landau equation,_Rev. Mod. Phys. 74, 99 (2002).
  * Simula [2019] T. Simula,  _Quantised Vortices_ (IOP Publishing, 2019).
  * Shen  _et al._ [2019] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan,  _Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,_Light Sci. Appl. 8, 90 (2019).
  * Allen  _et al._ [1992] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman,  _Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,_Phys. Rev. A 45, 8185 (1992).
  * Soskin  _et al._ [1997] M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg,  _Topological charge and angular momentum of light beams carrying optical vortices,_Phys. Rev. A 56, 4064 (1997).
  * Milonni and Boyd [2010] P. W. Milonni and R. W. Boyd,  _Momentum of Light in a Dielectric Medium,_Adv. Opt. Photonics 2, 519 (2010).
  * Pitaevskii and Stringari [2016] L. Pitaevskii and S. Stringari,  _Bose-Einstein Condensation and Superfluidity_, 1st ed. (Oxford University Press, Oxford, 2016).
  * Hodby  _et al._ [2003] E. Hodby, S. A. Hopkins, G. Hechenblaikner, N. L. Smith, and C. J. Foot,  _Experimental Observation of a Superfluid Gyroscope in a Dilute Bose-Einstein Condensate,_Phys. Rev. Lett. 91, 090403 (2003).
  * Aveline  _et al._ [2020] D. C. Aveline, J. R. Williams, E. R. Elliott, C. Dutenhoffer, J. R. Kellogg, J. M. Kohel, N. E. Lay, K. Oudrhiri, R. F. Shotwell, N. Yu, and R. J. Thompson,  _Observation of Bose–Einstein condensates in an Earth-orbiting research lab,_Nature 582, 193 (2020).
  * Thomson [1869] W. Thomson,  _4\. On Vortex Atoms,_Proc. R. Soc. Edinburgh 6, 94 (1869).
  * Karch and Tong [2016] A. Karch and D. Tong,  _Particle-Vortex Duality from 3D Bosonization,_Phys. Rev. X 6, 031043 (2016).
  * Zhao  _et al._ [2017] H. J. Zhao, V. R. Misko, J. Tempere, and F. Nori,  _Pattern formation in vortex matter with pinning and frustrated intervortex interactions,_Phys. Rev. B 95, 104519 (2017).
  * Dominici  _et al._ [2018] L. Dominici, R. Carretero-González, A. Gianfrate, J. Cuevas-Maraver, A. S. Rodrigues, D. J. Frantzeskakis, G. Lerario, D. Ballarini, M. De Giorgi, G. Gigli, P. G. Kevrekidis, and D. Sanvitto,  _Interactions and scattering of quantum vortices in a polariton fluid,_Nat. Commun. 9, 1467 (2018).
  * Dagvadorj  _et al._ [2015] G. Dagvadorj, J. Fellows, S. Matyjaśkiewicz, F. Marchetti, I. Carusotto, and M. Szymańska,  _Nonequilibrium Phase Transition in a Two-Dimensional Driven Open Quantum System,_Phys. Rev. X 5, 041028 (2015).
  * Leach  _et al._ [2005] J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett,  _Vortex knots in light,_New J. Phys. 7, 55 (2005).
  * Dennis  _et al._ [2010] M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett,  _Isolated optical vortex knots,_Nat. Phys. 6, 118 (2010).
  * Nye and Berry [1974] J. F. Nye and M. V. Berry,  _Dislocations in Wave Trains,_Proc. Roy. Soc. A 336, 165 (1974).
  * Berry [2000] M. Berry,  _Making waves in physics,_Nature 403, 21 (2000).
  * Molina-Terriza  _et al._ [2007] G. Molina-Terriza, J. P. Torres, and L. Torner,  _Twisted photons,_Nat. Phys. 3, 305 (2007).
  * Yao and Padgett [2011] A. M. Yao and M. J. Padgett,  _Orbital angular momentum: origins, behavior and applications,_Adv. Opt. Photonics 3, 161 (2011).
  * Krenn  _et al._ [2016] M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger,  _Twisted light transmission over 143 km,_Proc. Natl. Acad. Sci. 113, 13648 (2016).
  * Yan  _et al._ [2014] Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner,  _High-capacity millimetre-wave communications with orbital angular momentum multiplexing,_Nat. Commun. 5, 4876 (2014).
  * Gibson  _et al._ [2004] G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold,  _Free-space information transfer using light beams carrying orbital angular momentum,_Opt. Express 12, 5448 (2004).
  * Padgett and Bowman [2011] M. Padgett and R. Bowman,  _Tweezers with a twist,_Nat. Photon. 5, 343 (2011).
  * He  _et al._ [1995] H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop,  _Direct Observation of Transfer of Angular Momentum to Absorptive Particles from a Laser Beam with a Phase Singularity,_Phys. Rev. Lett. 75, 826 (1995).
  * Forbes  _et al._ [2021] A. Forbes, M. de Oliveira, and M. R. Dennis,  _Structured light,_Nat. Photon. 15, 253 (2021).
  * Secor  _et al._ [2016] J. Secor, R. Alfano, and S. Ashrafi,  _Complex Light_ (IOP Publishing, 2016).
  * Lopez-Mago [2019] D. Lopez-Mago,  _On the overall polarisation properties of Poincaré beams,_J. Opt. 21, 115605 (2019).
  * Göbel  _et al._ [2021] B. Göbel, I. Mertig, and O. A. Tretiakov,  _Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles,_Phys. Rep. 895, 1 (2021).
  * Sugic  _et al._ [2021] D. Sugic, R. Droop, E. Otte, D. Ehrmanntraut, F. Nori, J. Ruostekoski, C. Denz, and M. R. Dennis,  _Particle-like topologies in light,_Nat. Commun. 12, 6785 (2021).
  * Parmee  _et al._ [2022] C. D. Parmee, M. R. Dennis, and J. Ruostekoski,  _Optical excitations of Skyrmions, knotted solitons, and defects in atoms,_Commun. Phys 5, 54 (2022).
  * Shen  _et al._ [2023] Y. Shen, B. Yu, H. Wu, C. Li, Z. Zhu, and A. V. Zayats,  _Topological transformation and free-space transport of photonic hopfions,_Adv. Photonics 5 (2023), 10.1117/1.AP.5.1.015001.
  * Donati  _et al._ [2016] S. Donati, L. Dominici, G. Dagvadorj, D. Ballarini, M. De Giorgi, A. Bramati, G. Gigli, Y. G. Rubo, M. H. Szymańska, and D. Sanvitto,  _Twist of generalized skyrmions and spin vortices in a polariton superfluid,_Proc. Natl. Acad. Sci. 113, 14926 (2016).
  * Gutiérrez-Cuevas and Pisanty [2021] R. Gutiérrez-Cuevas and E. Pisanty,  _Optical polarization skyrmionic fields in free space,_J. Opt. 23, 024004 (2021).
  * Shen  _et al._ [2022] Y. Shen, E. C. Martínez, and C. Rosales-Guzmán,  _Generation of Optical Skyrmions with Tunable Topological Textures,_ACS Photon. 9, 296 (2022).
  * Lin  _et al._ [2021] W. Lin, Y. Ota, Y. Arakawa, and S. Iwamoto,  _Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers,_Phys. Rev. Res. 3, 023055 (2021).
  * Liu  _et al._ [2022] C. Liu, S. Zhang, S. A. Maier, and H. Ren,  _Disorder-Induced Topological State Transition in the Optical Skyrmion Family,_Phys. Rev. Lett. 129, 267401 (2022).
  * Beckley  _et al._ [2010] A. M. Beckley, T. G. Brown, and M. A. Alonso,  _Full Poincaré beams,_Opt. Express 18, 10777 (2010).
  * Dominici  _et al._ [2021a] L. Dominici, D. Colas, A. Gianfrate, A. Rahmani, V. Ardizzone, D. Ballarini, M. De Giorgi, G. Gigli, F. P. Laussy, D. Sanvitto, and N. Voronova,  _Full-Bloch beams and ultrafast Rabi-rotating vortices,_Phys. Rev. Res. 3, 013007 (2021a).
  * Rubo [2007] Y. Rubo,  _Half vortices in exciton polariton condensates,_Phys. Rev. Lett. 99, 106401 (2007).
  * Mermin and Ho [1976] N. D. Mermin and T.-L. Ho,  _Circulation and Angular Momentum in the A Phase of Superfluid Helium-3,_Phys. Rev. Lett. 36, 594 (1976).
  * Anderson and Toulouse [1977] P. W. Anderson and G. Toulouse,  _Phase Slippage without Vortex Cores: Vortex Textures in Superfluid He 3,_Phys. Rev. Lett. 38, 508 (1977).
  * Mizushima  _et al._ [2002] T. Mizushima, K. Machida, and T. Kita,  _Mermin-Ho Vortex in Ferromagnetic Spinor Bose-Einstein Condensates,_Phys. Rev. Lett. 89, 030401 (2002).
  * Yukawa [2023] E. Yukawa,  _su(N) Mermin-Ho Relation,_J. Phys. Soc. Jpn. 92, 044601 (2023).
  * Dandoloff  _et al._ [1992] R. Dandoloff, R. Balakrishnan, and A. R. Bishop,  _Two-level systems: space curve formalism, Berry’s phase and Gauss-Bonnet theorem,_J. Phys. A.: Math. Gen. 25, L1105 (1992).
  * Mermin [1979] N. D. Mermin,  _The topological theory of defects in ordered media,_Rev. Mod. Phys. 51, 591 (1979).
  * Berry [1989] M. Berry,  _in Geometric Phases in Physics_ , edited by F. Wilczek and A. Shapere (World Scientific, Singapore, 1989) p. 7.
  * Xiao  _et al._ [2010] D. Xiao, M.-C. Chang, and Q. Niu,  _Berry phase effects on electronic properties,_Rev. Mod. Phys. 82, 1959 (2010).
  * Everschor-Sitte and Sitte [2014] K. Everschor-Sitte and M. Sitte,  _Real-space Berry phases: Skyrmion soccer (invited),_J. Appl. Phys. 115, 172602 (2014).
  * Rego  _et al._ [2019] L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C.-T. Liao, J. San Román, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernández-García,  _Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,_Science 364, eaaw9486 (2019); only at the time of publishing we became aware of this recent work on the time-varying OAM, J. Zhang, P. Li, R. C. C. Cheung, A. M. H. Wong, and J. Li, _Generation of time-varying orbital angular momentum beams with space-time-coding digital metasurface,_ Adv. Photonics 5, 036001 (2023) .
  * Chevy  _et al._ [2000] F. Chevy, K. W. Madison, and J. Dalibard,  _Measurement of the Angular Momentum of a Rotating Bose-Einstein Condensate,_Phys. Rev. Lett. 85, 2223 (2000).
  * Madison  _et al._ [2000] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard,  _Vortex Formation in a Stirred Bose-Einstein Condensate,_Phys. Rev. Lett. 84, 806 (2000).
  * Berry [1998] M. V. Berry, in  _International Conference on Singular Optics_, Vol. 3487, edited by M. S. Soskin, International Society for Optics and Photonics (SPIE, 1998) pp. 6–11.
  * Maji  _et al._ [2019] S. Maji, P. Jacob, and M. M. Brundavanam,  _Geometric phase and intensity-controlled extrinsic orbital angular momentum of off-axis vortex beams,_Phys. Rev. Appl. 12, 054053 (2019).
  * Hosseini  _et al._ [2020] F. Hosseini, M. A. Sadeghzadeh, A. Rahmani, F. P. Laussy, and L. Dominici,  _Temporal shaping and time-varying orbital angular momentum of displaced vortices,_Optica 7, 1359 (2020).
  * Choi  _et al._ [2022] D. Choi, M. Park, B. Y. Oh, M.-S. Kwon, S. I. Park, S. Kang, J. D. Song, D. Ko, M. Sun, I. G. Savenko, Y.-H. Cho, and H. Choi,  _Observation of a single quantized vortex vanishment in exciton-polariton superfluids,_Phys. Rev. B 105, L060502 (2022).
  * Combescot [2015] M. Combescot,  _Excitons and Cooper pairs: two composite bosons in many-body physics_ , 1st ed. (Oxford University Press, New York, NY, 2015).
  * Byrnes  _et al._ [2014] T. Byrnes, N. Y. Kim, and Y. Yamamoto,  _Exciton–polariton condensates,_Nat. Phys. 10, 803 (2014).
  * Dominici  _et al._ [2015] L. Dominici, G. Dagvadorj, J. M. Fellows, D. Ballarini, M. D. Giorgi, F. M. Marchetti, B. Piccirillo, L. Marrucci, A. Bramati, G. Gigli, M. H. Szymańska, and D. Sanvitto,  _Vortex and half-vortex dynamics in a nonlinear spinor quantum fluid,_Sci. Adv. 1, e1500807 (2015).
  * Dominici  _et al._ [2014] L. Dominici, D. Colas, S. Donati, J. P. Restrepo Cuartas, M. De Giorgi, D. Ballarini, G. Guirales, J. C. López Carreño, A. Bramati, G. Gigli, E. del Valle, F. P. Laussy, and D. Sanvitto,  _Ultrafast control and Rabi oscillations of polaritons,_Phys. Rev. Lett. 113, 226401 (2014).
  * Maleev and Swartzlander [2003] I. D. Maleev and G. A. Swartzlander, Jr.,  _Composite optical vortices,_J. Opt. Soc. Am. B 20, 1169 (2003).
  * Dominici  _et al._ [2021b] L. Dominici, N. Voronova, D. Colas, A. Gianfrate, A. Rahmani, V. Ardizzone, D. Ballarini, M. De Giorgi, G. Gigli, F. P. Laussy, and D. Sanvitto,  _Shaping the topology of light with a moving Rabi-oscillating vortex,_Opt. Express 29, 37262 (2021b).
  * Bliokh  _et al._ [2019] K. Y. Bliokh, M. A. Alonso, and M. R. Dennis,  _Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects,_Rep. Prog. Phys. 82, 122401 (2019).
  * Verma  _et al._ [2022] N. Verma, Z. Addison, and M. Randeria,  _Unified theory of the anomalous and topological Hall effects with phase-space Berry curvatures,_Sci. Adv. 8, eabq2765 (2022).
  * Volovik [2019] G. E. Volovik,  _Topological Superfluids,_J. Exp. Th. Phys. 129, 618 (2019).
  * Bleu  _et al._ [2018] O. Bleu, D. D. Solnyshkov, and G. Malpuech,  _Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems,_Phys. Rev. B 97 (2018), 10.1103/PhysRevB.97.195422.
  * Polimeno  _et al._ [2021] L. Polimeno, G. Lerario, M. De Giorgi, L. De Marco, L. Dominici, F. Todisco, A. Coriolano, V. Ardizzone, M. Pugliese, C. T. Prontera, V. Maiorano, A. Moliterni, C. Giannini, V. Olieric, G. Gigli, D. Ballarini, Q. Xiong, A. Fieramosca, D. D. Solnyshkov, G. Malpuech, and D. Sanvitto,  _Tuning of the Berry curvature in 2D perovskite polaritons,_Nat. Nanotech. 16, 1349 (2021).
  * Gianfrate  _et al._ [2020] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech,  _Measurement of the quantum geometric tensor and of the anomalous Hall drift,_Nature 578, 381 (2020).
  * Molina-Terriza  _et al._ [2001] G. Molina-Terriza, L. Torner, E. M. Wright, J. J. García-Ripoll, and V. M. Pérez-García,  _Vortex revivals with trapped light,_Opt. Lett. 26, 1601 (2001).
  * Berry [2001] M. V. Berry, in  _Second International Conference on Singular Optics (Optical Vortices): Fundamentals and Applications_, Vol. 4403, edited by M. S. Soskin and M. V. Vasnetsov, International Society for Optics and Photonics (SPIE, 2001) pp. 1 – 12.
  * Parts  _et al._ [1995] U. Parts, J. M. Karimäki, J. H. Koivuniemi, M. Krusius, V. M. H. Ruutu, E. V. Thuneberg, and G. E. Volovik,  _Phase Diagram of Vortices in Superfluid 3He-A,_Phys. Rev. Lett. 75, 3320 (1995).
  * Volovik [1987] G. E. Volovik,  _Linear momentum in ferromagnets,_J. Phys. C: Solid State Phys. 20, L83 (1987).
  * Matsuki and Mochizuki [2023] J. Matsuki and M. Mochizuki,  _Thermoelectric effect of a skyrmion crystal confined in a magnetic disk,_Phys. Rev. B 107, L100408 (2023).
  * Kuratsuji and Kakigi [1998] H. Kuratsuji and S. Kakigi,  _Maxwell-Schroedinger Equation for Polarized Light and Evolution of the Stokes Parameters,_Phys. Rev. Lett. 80, 1888 (1998).
  * Singh  _et al._ [2023] K. Singh, P. Ornelas, A. Dudley, and A. Forbes,  _Synthetic spin dynamics with Bessel-Gaussian optical skyrmions,_Opt. Express 31, 15289 (2023).
  * Landau and Lifshitz [1992] L. Landau and E. Lifshitz, in  _Perspectives in Theoretical Physics_ (Elsevier, 1992) pp. 51–65.
  * García-Palacios [2007] J. L. García-Palacios, in  _Advances in Chemical Physics_, edited by I. Prigogine and S. A. Rice (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2007) pp. 1–210.
  * Marrucci  _et al._ [2006] L. Marrucci, C. Manzo, and D. Paparo,  _Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,_Phys. Rev. Lett. 96, 163905 (2006).
  * Karimi  _et al._ [2009] E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato,  _Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,_Appl. Phys. Lett. 94, 231124 (2009).
  * Colas  _et al._ [2015] D. Colas, L. Dominici, S. Donati, A. A. Pervishko, T. C. Liew, I. A. Shelykh, D. Ballarini, M. de Giorgi, A. Bramati, G. Gigli, E. d. Valle, F. P. Laussy, A. V. Kavokin, and D. Sanvitto,  _Polarization shaping of Poincaré beams by polariton oscillations,_Light Sci. Appl. 4, e350 (2015).

Acknowledgements: We acknowledge Romuald Houdré and Alberto Bramati for the
microcavity sample, Lorenzo Marrucci and Bruno Piccirillo for the $q$-plate
devices, Paolo Cazzato for technical support, Charlie Leblanc, Miguel Alonso,
Giovanni I. Martone, Grigory E. Volovik and sir Michael V. Berry for
interesting comments. Funding: We thank for funding the Italian Ministry of
University (MUR) PRIN project “Interacting Photons in Polariton Circuits”–
INPhoPOL (grant 2017P9FJBS); the PNRR MUR project “National Quantum Science
and Technology Institute” - NQSTI (PE0000023); the PNRR MUR project
“Integrated Infrastructure Initiative in Photonic and Quantum Sciences” -
I-PHOQS (IR0000016); the Russian Foundation for Basic Research (CNR-RFBR Joint
Bilateral Agreement Triennal Program 2021–2023 and CNR-RFBR Joint Project No.
20–52–7816). N.V. acknowledges the financial support of MEPhI Priority 2030
Program. A.R. acknowledges support from National Science Center, Poland (PL),
Grant No. 2016/22/E/ST3/00045. Author Contributions: L.D. proposed the effect
and realized the experiments, with the help of M.D.G., D.B., D.S. and G.G. in
realizing the set-up; L.D. analysed the data, proposed the model and
implemented numerical simulations, in coordination with N.V., A.R. and D.C.;
L.D., A.R., N.V and F.P.L. discussed the theoretical aspects; L.D. and N.V.
drafted the manuscript, with the supervision of F.P.L., and all the authors
discussed the results and the final manuscript editing. Competing interests:
No competing interests.

SUPPLEMENTARY MATERIALS.

Supplementary Movie  S1:  Experimental Rabi-oscillating double vortex seen in
the photonic component as in Fig. 1. Photonic amplitude and phase in a
$100\times 100~{}\mu\text{m}^{2}$ area, with 20 fs time step over a 4.20 ps
time span. The second photonic pulse is arriving at around
$t=2.4~{}\text{ps}$. The vortex cores positions are marked with yellow dots
and circles in the amplitude and phase maps. [Video SM1].  Supplementary Movie
S2:  The LGs model for the double Rabi-oscillating vortex, comprising the maps
and dynamics of four relevant quantities. The four panels show the local
imbalance $s(\bm{r},t)$ (purple-green map) and the relative phase
$\varphi_{LU}(\bm{r},t)$ (black and white map) on the top row, as in Fig.
4a,b, together with the photon and exciton amplitudes
$|\psi_{\mathrm{C,X}}(\bm{r},t)|$ (blue and red maps, respectively) in the
second row. The positions of the displaced UP/LP cores are corresponding to
the poles of the vortex quadrupole in the relative phase map. The spiraling
motion of the photon and exciton cores, seen as holes in their respective
fields, are tracked by the crossing points between the $s=0$ isocontent
(white) and the $\varphi_{LU}=0$ and $\varphi_{LU}=\pi$ isophase lines (black
lines). [Video SM2].  Supplementary Movie  S3:  Photonic amplitude
$|\psi_{\mathrm{C}}(\bm{r},t)|$ seen as a 3D image. The amplitude is plotted
in an arbitrary false colour bar on the $z$ axis, in the same $xy$ and time
domain as the previous model case. [Video SM3].