## Abstract

The fundamental mode of rotation in quantum fluids is given by a vortex whose quantized value yields the orbital angular momentum (OAM) per particle. If the vortex is displaced (off-centered) from the reference point for rotation, the angular momentum is reduced and becomes fractional. Such displaced vortices can further exhibit a peculiar dynamics in the presence of confining potentials or couplings to other fields. We study analytically a number of 2D systems where displaced vortices exhibit a noteworthy dynamics, including time-varying self-sustained oscillation of the OAM, complex reshaping of their morphology with possible creation of vortex–antivortex pairs, and peculiar trajectories for the vortex core with sequences of strong accelerations and decelerations that can even send the core to infinity and bring it back. Interestingly, these do not have to occur conjointly, with complex time dynamics of the vortex core and/or their wavepacket morphology possibly taking place without affecting the total OAM. Our results generalize to simple and fundamental systems a phenomenology recently reported with Rabi-coupled bosonic fields, showing their wider relevance and opening prospects for new types of control and structuring of the angular momentum of light and/or quantum fluids.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Topology is essential to different areas of physics, in particular to condensed matter physics [1,2]. This involves invariants under deformation from the ground state to a new configuration for which any path to the ground state is energetically too costly. As such, topological invariants powered by homotopy groups provide subtle tools to detect holes in a given space. Examples are topological defects in systems with continuous broken symmetry. A vortex is a typical such defect in systems with global $U(1)$ broken symmetry. It is characterized by a null density region and an integer (topological) charge associated with a singularity in the gradient of the phase of the order parameter, which defines the core of the vortex. In this way, the topological charge (TC) defines an intrinsic orbital angular momentum (OAM) embedded in the vortex state. Vortices are common in many media and systems, including light [3,4] and quantum matter, such as superconductors [5], superfluids [6], and atomic Bose–Einstein condensates [7], among others. Vortices also recently flourished in coupled light–matter, so-called “polaritons,” systems [8–14] that can interpolate between the purely optical and strongly interacting condensed matter cases.

A vortex beam (VB)—a vortex confined in a beam with a spiral wavefront—is special, as it can carry angular momentum [15], which can further be distinguished between one of two kinds: either with a non-uniform phase-varying wavefront, called an “anisotropic VB,” or with a uniform phase variation, referred to as an “isotropic VB.” In both cases, an interesting situation arises when the core of the vortex is displaced from the center of the beam, a situation that does not arise in infinite-sized systems. This leads to a reduced angular momentum, which is still fixed, and to a modified morphology for the phase of the field (its distribution in space [15]). For a static VB (not varying in time), either isotropic or anisotropic, it is known that the vortex angular momentum can be described by a combination of elementary vortices with integer TCs, which determines the overall morphology and associated angular momentum across the beam [16].

VBs can also be dynamic, i.e., their angular momentum can be time dependent. It is only recently, however, that VBs have entered this regime [17,18]. The earliest report, to the best of our knowledge, was by some of the present authors and collaborators [17] in the highly versatile polaritonic platform [19]. There, a time-varying angular momentum was shown to be self-sustained by the interplay between Rabi oscillations and VB morphologies in two coupled fields. This was achieved by preparing the system in a special topological initial condition imprinted in the microcavity polariton field with two delayed pulses of different TC. This produced a rich temporally and spatially structured dynamics of the off-centered core in the polariton fluid, resulting in oscillating linear and angular momenta of the emitted light. Such oscillations happen in the linear regime and can be described exactly. We study further this case in the last section of the present text. A similar scheme, with two retarded pulses but now partially overlapping in time, has also been used in the nonlinear process of high-harmonic generation, resulting in a VB with continuously time-varying angular momentum [18].

Here, we study analytically fundamental and simple physical systems that display similar nontrivial motion of a dynamical vortex. The results are quite remarkable given the complex and far-reaching phenomenology that is displayed by a simple theoretical model, which basically reduces to a first-year quantum mechanics textbook exposition. Given the ease of access of both the experimental scheme (combinations of pulsed excitations with various TCs) and the variety of possible platforms where to explore such physics, we expect the field of dynamic VBs to quickly take off in a variety of systems in the near future. What we find is that such displaced vortices sustain a rich dynamics with a morphology that is stretched or folded up in time to accompany the variation of the angular momentum. Interestingly, although one may expect a time-varying morphology to also result in a time-varying angular momentum, we show that this is not compulsorily the case. To this end, we consider a displaced vortex in several elementary systems, namely, an infinite (circular) quantum well potential (Section 2), a slightly anharmonic potential (Section 3), a spatially squeezed harmonic potential (Section 4), and the already discussed case of two coupled condensates (polaritons, in Section 5). Section 6 concludes and gives general remarks. It is worth noting that all the examples considered hereon are in the linear regime, where the self-interaction between particles is negligible. The phenomenology is however robust in the sense that it is maintained in the presence of nonlinearities or interactions, as we have checked numerically. In such cases, however, further complications arise with more complex reshaping of the vortices, proliferations of vortex–antivortex pairs, and other features that we leave for other studies.

## 2. INFINITE CIRCULAR QUANTUM WELL

Our first platform to consider the dynamics of a displaced vortex in a VB is the fully confined case with canonical circular geometry, namely, the infinite circular quantum well. This provides an example of interactions between vortices and the system’s boundaries. As we show below, interference between the rotating wavepacket and its reflection from the hard walls leads to a time-dependent morphology of the phase, which, nevertheless, leaves the expectation value for the total angular momentum invariant, as expected from the conservation rules. Manipulating the size of the initial packet introduces different content of radial momentum adding up to the rotational one. This leads to various regimes for the dynamics, ranging from a smooth rotation of the pattern to pair creation of vortices–antivortices. Such a case can be realized experimentally, for instance, with polaritons in etched micropillars [20–22]. The Hamiltonian for a circular quantum well of radius $R$ reads as

with where $r = \sqrt {{x^2} + {y^2}}$, and $\varphi = \arg (x + iy)$. The system is easily integrated exactly. Solving the equation ${H_{{\rm qw}}}{\chi _{n,l}} = {E_{n,l}}{\chi _{n,l}}$, one finds the eigenstates and eigenenergies as where ${J_{|l|}}$ is the Bessel function of integer order, ${N_{n,l}}$ is the normalization constant, $l = 0, \pm 1, \pm 2, \cdots$, and ${\beta _{n,l}} = {k_{n,l}}R$ is the $n$th zero of the $|l|$th Bessel function for which ${J_{|l|}}({\beta _{n,l}}) = 0$ as imposed by the boundary condition, namely, ${\chi _{n.l}}(R,\varphi) = 0$. From these stationary solutions ${\chi _{n,l}}$, one then solves Schrödinger’s equation $i\hbar {\partial _t}\psi = H_{\rm qw}\psi$ for any initial state ${\psi _0}(r,\varphi) \equiv \psi (r,\varphi ,0)$ as the linear superpositionTo describe the dynamics of a displaced vortex, we take for the initial condition a superposition of two Bessel–Gauss wavepackets, one with ${\rm TC} = 1$ and another with ${\rm TC} = 0$:

The dynamics depends strongly on the size $w$ of the packet. When $w$ is on the order of quantum well size $R$, the terms in Eq. (6) that dominate are those weighted by ${\alpha _{1,1}}$ and ${\alpha _{1,0}}$, which results in a smooth rotation of the wave packet, with simply a quantum beating between two states. However, when $w$ is smaller than $R$ by one order of magnitude or more, a larger number of ${\alpha _{n,i}}$ coefficients ($i = 0,1$) enters the dynamics, which acquires a markedly different character as higher $n$ amplitudes become significant. Examples are shown in Fig. 1. We can observe different stages in the dynamics. At first, we see an increase in the wavepacket size inside the well as a result of its diffusion, which also separates the minima and maxima of the density. However, since the wavepacket should remain zero at the boundary of the well, there is a point in time when diffusion stops and a second crest starts to form, with a half-moon shape, being thinner than the first one but larger in radius. At the same time, the vortex core itself keeps rotating [Fig. 1(g)], and the distortion of the phase excites new vortices, namely, a vortex–antivortex pair, to conserve angular momentum [Fig. 1(h)]. These are moving inside the fluid until their later recombination [Fig. 1(i)]. Shortly after that, another vortex–antivortex pair is formed locally [Fig. 1(j)]. Such processes are repeated during the full time evolution of the dynamics (see also Visualization 1 for better illustration of the dynamics). Despite the complex motion of the core inside this fluid bouncing back and forth as it rotates inside the well, the angular momentum remains steady. This is expected since the external potential is a purely confining one, being radially symmetric and azimuthally homogeneous. Considering the potential gradient and its symmetry, its boundaries act as both a local and net force on the fluid at any moment; however, there is no net torque acting on the fluid. Therefore, while the center of mass of the fluid and its net linear momentum keep changing due to the continuous bouncing of the fluid against the well boundaries, its net angular momentum remains constant. Yet, some peculiar morphology reshaping, involving also the TCs, happens during the evolution. It is interesting to check how this is self-consistently ensured by the formalism that produces correspondingly intricate dynamics of the fields:

Although the way the Bessel functions balance each other in space is not transparent, one can see indeed how Eq. (7) produces a time-independent average.

Another interesting quantity is the OAM per particle, which in the case above is less than one, as expected for an off-center vortex [23], namely, $\langle {\tilde L_z}\rangle \equiv \langle {L_z}\rangle /N \lt 1$, with $N$ the total number of particles [also, in the next sections, we refer to $\langle {\tilde L_z}\rangle$ as the mean (expectational) value of OAM per particle], given by

The condition $\langle {\tilde L_z}\rangle \lt 1$ comes from the initial field [Eq. (5)], where we have introduced two kinds of particles with only a fraction of them carrying (integer and equal to one) angular momentum. The ratio of particles that carry angular momentum is responsible for the position of the core, making it closer to the center (less displaced) as more particles have angular momentum (i.e., for smaller ${a_c}$). Similarly, increasing the number of particles without angular momentum (i.e., for larger ${a_c}$) pushes the core towards the boundary.

Although both the total and mean values of the angular momentum are constant for the whole wavepacket, one can consider a local OAM, which reveals time-varying features. In particular, we can define

*on*a cirle of radius $r$ is obtained as $d{\langle {L_z}\rangle _r}/dr$. Similarly, one can define the mean number of particles on a circle of radius $r$ as $d{N_r}/dr$ with ${N_r} = \int_0^r \int_0^{2\pi} \rho |\psi (\rho ,\varphi ,t{)|^2}{\rm d}\varphi {\rm d}\rho$, so that the chain rule yields the mean value of OAM per particle on a circle of radius $\rho$, which we call ${\langle {\tilde L_z}\rangle _r}$, as ${\langle {\tilde L_z}\rangle _r} = \frac{{d{{\langle {L_z}\rangle}_\rho}}}{{d\rho}}\frac{{d\rho}}{{d{N_\rho}}}$, which is thus given by

This quantity is shown in Figs. 2(d) and 2(e), where it is seen how the local OAM exhibits strikingly different behaviors for different cases. For large $w$, with a smooth rotation of the field, the local angular momentum is constant in time on any radial distance from the center of the quantum well. However, by decreasing the initial packet size, which induces a nontrivial dynamics of the vortex, one then finds strongly time-varying features of the angular momentum but only locally, since these cancel when averaging over the entire wavepacket. This is shown in (e). At a given radius $r$, a torque can be felt locally due to the internal dynamics of the field, which is absent in (d). One can see that variations are stronger in the central part rather than close to the boundaries ($r = 0,2$), where the density of the field has less dynamics. This is clear in Figs. 1(c) and (d), where a half-moon-shaped density near the boundary remains almost steady, while the central part has a rich dynamics of vortex rotation and pair excitation.

The vortex–antivortex pair formation, evolution, and later recombination, which all occur without affecting the total angular momentum but can be detected locally, can be understood from the expanding cloud reaching the boundary and being reflected, producing a circular ripple of lower density and with a larger phase gradient [visible starting from Figs. 1(b) and 1(g) on]. This interference between the outward diffusing and the inward reflected waves is the origin of the secondary vortex–antivortex pair, which is nucleated starting from this loop of locally low density. The pair creation can be modeled by using a toy model that interferes two waves, both with a Laguerre–Gaussian type envelope of the same amplitude and propagating in the directions ${{\textbf{k}}_1}$ and ${{\textbf{k}}_2} \ne {{\textbf{k}}_1}$ as ${A_1} = {e^{- i{{\textbf{k}}_1} \cdot {\textbf{r}}}}{e^{- {r^2}/(2{w^2})}}$ and ${\rm TC} = 0$, ${A_2} = {e^{- i{{\textbf{k}}_2} \cdot {\textbf{r}}}}{e^{- {r^2}/(2{w^2})}}(r{e^{{i\varphi}}} - {r_c}{e^{i{\varphi _c}}})$, with a TC. If, for simplicity, ${A_1}$ travels along the $x$ direction (${{\textbf{k}}_1} = k\hat \imath$) and ${A_2}$ does not propagate (${{\textbf{k}}_2} = 0$), then the two interfering waves yield a total density $I \propto |{e^{\textit{ikx}}} + r{e^{{i\varphi}}} - {r_c}{e^{i{\varphi _c}}}|^2$. The condition for a vanishing density is given as ${-}\cos (kx) = x - {x_c}$ and ${-}\sin (kx) = y - {y_c}$. Depending on the wavenumber $k$, such a density can have either one or more roots. The former case corresponds to the initial ${\rm TC} \le 1$, while when there are three roots, a vortex–antivortex pair is added to the initial TC. An example for this behavior of the phase is shown in Figs. 2(a)–2(c), where we take ${x_c} = 1\,\,\unicode{x00B5}{\rm m}$ and ${y_c} = 0$ (obtained for the polar parameters ${r_c} = 1\,\,\unicode{x00B5}{\rm m}$ and ${\varphi _c} = 0$). One can see how, upon increasing $k$, the phase gets distorted to the point of creating a vortex pair. This condition also depends on the relative amplitude of the two waves, which we did not consider to highlight the importance of the momentum. Coming back to the displaced vortex in the circular quantum well, the excitation of the pair comes into play when the reflected wave from the hard boundary interferes with the field inside the well, matching the toy model above, where a vortex–antivortex pair is created and annihilated repeatedly. One could also consider higher-energy initial conditions with a larger number of vortex–antivortex pairs. The maximum number of vortices (MNV) observed during the dynamics as a function of the initial packet size $w$ is shown in Fig. 2(f). It is clear that by decreasing $w$, MNV is growing in a nonlinear fashion. Interestingly, also configurations of even number of vortices are possible, where a single vortex or antivortex can be excited from the boundary of the quantum well. In fact, since the distance of a vortex fractionalizes its contribution to the OAM, and a pair of vortex–antivortex immediately drifts apart with different distances, it is clear that the field itself must locally accommodate for the corresponding changing angular momentum, and it is thus not surprising that, would a vortex come from the boundary of the well, it is allowed to drift alone with no anti-counterpart without affecting the total OAM, which remains constant. These additional vortices and antivortices thus also display an intricate dynamics that also affects the morphology, which has to maintain a constant total along with the mean angular momentum. While we do not consider it here, nonlinearities also result in further vortex–antivortex dynamics with added complexity to the overall phenomenology.

## 3. HARMONIC POTENTIAL

Next, we consider the displaced vortex in a harmonic potential, still with a confinement but that now grows quadratically with the distance from the center of the potential and thus with the possibility for the core to stray arbitrarily far from it. We will see that, surprisingly, such an opportunity is actually seized. Schrödinger’s equation $i\hbar {\partial _t}\psi = {H_{{\rm ho}}}\psi$ now has Hamiltonian

This gives the orbit described by the core, which is a circle of radius ${r_c}(0)$. Its motion proceeds with a constant speed ${v_c}(0) = {r_c}(0){\omega _{{\rm ho}}}$. Although the core oscillates in time, as previously and for the same reason, the total angular momentum content remains constant,

This connection between quantum and classical pictures through the vortex itself also holds when the trajectories are not on equipotential lines but involve energy transfers between the kinetic and potential terms. This occurs when the initial condition is not spatially equipotential in the trap, i.e., with $w \ne \beta$. The solutions are obtained as before:

where ${H_{{\rm ho}}}{\phi _{n,m}} = {E_{n,m}}{\phi _{n,m}}$, and ${\alpha _{n,m}}(t) = \int {e^{- it{E_{n.m}}/\hbar}}{\psi _0}{\phi _{n,m}}{\rm d}x{\rm d}y$, with $n,m = 0,1,2, \cdots$. The time evolution of the density is shown in Fig. 3 for $w = 0.7\beta$ and for the core located initially at $(- \beta ,0)$. The wavepacket first shrinks in size, while the core moves but now on an ellipse ${({x_c}(t)/w)^2} + {(w{y_c}(t)/{\beta ^2})^2} = (\beta /w{)^2}$ rather than on a circle, with ${{\textbf{r}}_c} = ({x_c},{y_c})$ and ${{\textbf{v}}_c} = d{{\textbf{r}}_c}/dt$ obtained asNote as well that the speed is now non-uniform. The wavepacket recovers its initial size as the core completes one period of its motion. Such oscillations are repeated during the time evolution of the dynamics. Apart from the peculiar motion of the vortex core, with accelerations and decelerations along an ellipse, and with also an intricate time-varying morphology of the phase, the quantum average of angular momentum is still constant since in general, and for any initial condition, the expectation value of angular momentum reads as

To exhibit such a case, we now introduce a perturbation that triggers a time dynamics of the angular momentum. Namely, we add a spatial anharmonicity in the form of a quartic contribution added to the harmonic oscillator potential. Indeed, a fluid in a power law trap demonstrated rich vortex states including crossover from a vortex lattice to a giant vortex [28]. We do not consider such transitions at large rotational velocities here, as we rather focus on the onset of a time-varying angular momentum due to the anharmonicity, which could be naturally present in any vibrating system. Here, for simplicity, we consider an anisotropic anharmonic term ${H_{{\rm an}}} \equiv \lambda {x^4}$, with $\lambda \gt 0$. In this case, coefficients ${\alpha _{n,m}}$ are given by

Examples of the dynamics are shown in Fig. 4. The vortex core (white dot) moves on two disconnected boomerang trajectories (shown in cyan) as it shapes the morphology of the vortex phase, which is shown with the red contours. The most striking departure brought by this case is that the core now can escape to infinity [Figs. 4(a)–4(c) or 4(f)–4(g)], and reappears on another branch [Figs. 4(d)–4(e) or 4(h)–4(j)]. When the vortex is infinitely far, the phase morphology is uniform and the total angular momentum is zero. This thus also corresponds to the point where angular momentum changes sign. This is a nice illustration of a complex topological inversion imparted by even a possibly weak anharmonicity. Such topological inversions have been reported before with free linear propagation [29,30] and we hereby provide a counterpart rooted in rotation (and with a potential, in Section 5 we also provide a case in free space). While the reversal of the total angular momentum in itself is not a wave feature of the model, as it is also produced by a classical oscillator, the behavior of the core, which is also more striking, is peculiar to the topological character of the VB. The cycles are also repeated forever. Moreover, one can observe an edge dislocation, i.e., when ${t_{{\rm cr}}} \equiv {\cal N}\pi /[2({\gamma _1} - {\gamma _0})]$, with ${\cal N}$ an integer. Indeed, the overall movement of the fluid consists of an additional left and right oscillation due to the anisotropic term along this direction, in addition to also moving up and down due to the harmonic trap. This causes dark interference lines along the diagonal directions, which are reminiscent of the crossing lines between the two terms of the potential. Notably, these diagonal edge dislocations are the most pronounced when the vortex core is at infinity. As pointed out earlier, for a core that exhibits such an unfamiliar dynamics as far as any mechanical point-like object is concerned, relationships such as Eq. (16) do not hold anymore, with no effective mass being able to account simultaneously for all the Newtonian expressions of the dynamical quantities.

## 4. SQUEEZED HARMONIC POTENTIAL

We now consider a displaced vortex in a squeezed harmonic potential, where the trap is harmonic, so representing another linear system, but with a broken symmetry by the trap that arises from its anisotropic confinement in the two directions. This also gives rise to an interesting and counter intuitive dynamics of the vortex core, with time-varying angular momentum, similar to the nonlinear case above, although now in a fully linear system. The external potential thus now reads as

It is remarkable that such a trivial-looking solution leads to the rich and counterintuitive dynamics that we describe in the following. We provide an example of it in Fig. 5. Initially the vortex core is located at $({x_c} = - 0.5,{y_c} = 0)$ and starts to move counterclockwise with a non-uniform speed. As it accelerates, the core is forced to leave the density cloud and travels at the periphery of the beam after a few rotations. Since this is a topological feature, it cannot disappear altogether even if ejected from the fluid. As was the case for the anharmonic oscillator, there comes a point in time when the core is sent away to infinity, in our case along a diagonal in the $x \gt 0$ part of the plane, coming back from the opposite quadrant with respect to the one from which it left the plane. The comeback is done with a deceleration similar to the acceleration of escape, and as the core comes back to its initial position, the cycle can repeat, now in the opposite sense of rotation and escaping along the diagonal of the previous return. Without dissipation, this cyclical dynamics is sustained forever. The dynamics is also highlighted in Visualization 2. The trajectory of the core is obtained from Eq. (27) as

## 5. RABI-COUPLED FIELDS

We conclude with the case that inspired all the others considered previously, since its experimental observation by Dominici *et al.* [17] triggered our interest into the underlying mechanism, which we have now generalized to several systems. While we will focus on polaritons, other similar systems, such as spin–orbit coupled BEC [31], could also accommodate these results. Rabi-coupled fields [32] also exhibit time-varying angular momentum, this time without any external potential (in free two-dimensional space) but activated for each field by its coupling to the other. The system is described by a coherent coupling between two fields ${\psi _C}$ and ${\psi _X}$, describing a cavity-photon field and a quantum-well excitonic field, respectively, in strong coupling [19], with equations that correspond to two coupled Schrödinger equations:

Here, $\Omega$ is the Rabi frequency that couples the two fields with respective free energies ${E_{C,X}}$ and masses ${m_{C,X}}$. There are typically decay terms to describe such particles, which, however, we do not need to consider for our present discussion, although they also lead to interesting variations of the dynamics. As previously, we assume for the initial condition a displaced core, albeit now with one vortex in each field:

The vortex cores are located at possibly different points in real space, defined by (${r_{c,x}},{\varphi _{c,x}}$) in polar coordinates. One can find a closed-form solution in reciprocal space ($k = \sqrt {k_x^2 + k_y^2}$ and ${\theta _k} = \arg [{k_x} + i{k_y}]$) by turning to the Fourier transform ${\cal F}$ of each field, which yields a general expression for the amplitudes of the two coupled fields:

In the limit of a very large mass imbalance, as is typically the case, this yields $w \gg \sqrt {\hbar /(2{m_C}\Omega)}$, whose right-hand size evaluates to about half a micrometer for standard parameters, showing that the approximations below are actually enforced by the diffraction limit for the entire duration of the observations for typical lifetimes of the particles [17]. Therefore, assuming from now on no effect of the dispersion and taking $k_\Omega ^2 \approx \sqrt {{{(2\hbar \Omega)}^2} + {{(\hbar \delta)}^2}}$, expanding the expression for the fields [Eq. (34)] and considering the initial condition of two displaced vortices [Eq. (33)], we find

In the more general case where the two vortices are not aligned, i.e., with a relative displacement, they exhibit some of the displaced-vortex dynamics of the previous sections. The two vortex cores also orbit each other. Angular momentum, alongside density, is coherently transferred and oscillating between the two fields. The net total angular momentum, obtained as the sum of angular momentum of the two fields, is conserved. The total angular momentum in each field, however, is time dependent. Note that the observation of the system is typically performed over one field only (the photon field), so this polaritonic mechanism is intrinsically one that generates periodically time-varying OAM. It originates from the exchange of particles with different positions and momenta [34,35], as is well represented by the cores moving on their off-axis orbits (which are also periodically changing their distances from the center). An example of this polaritonic dynamics is shown in Fig. 6 for ${m_C}/{m_X} \ll 1$, with frames of the density and phase maps for the photon (${\psi _C}$) field (see also Visualization 3). The core moves along a circular orbit, shown in cyan, and the morphology of the vortex, further shown by red isodensity contours, also displays the type of nontrivial oscillatory structure similar to those already encountered in the previous systems. Here, too, the core and the fluid as a whole exhibit peculiar and opposed dynamics, with the core undergoing sequences of accelerations and decelerations, being faster in the outer part of the beam [17,33], while the angular momentum oscillates smoothly and regularly. The essence of this behavior can be reproduced by a simplified version of the coupled-field solutions, namely, in matrix representation (cf. Appendix A):

Again, solutions (38,39) are valid in the regime when $w \gg 1$ only, and would break down when effects from the dispersion play a rule [36]. In this case, further noteworthy dynamics takes over, with vortex–antivortex pair creation and recombination similar to the circular quantum well discussed above, but those are beyond the current scope of this text. The representation Eq. (39) evokes a rotation matrix; however, it differs due to the complex elements. Such a unitary matrix is familiar from the dynamics of binary fields in the linear regime [35], where it bears an interpretation similar to rotation. To show this, we introduce a complex number that defines the quantum state in the basis of the coupled fields ${\cal Z} \equiv {\psi _C}/{\psi _X}$. The matrix representation in Eqs. (38) and (39) reappears as a bilinear transformation ${{\cal Z}_0} \mapsto {\cal Z} = M({{\cal Z}_0})$, where ${{\cal Z}_0}$ is the initial state, and

Assuming $\delta = 0$ hereafter for simplicity, the transformation Eq. (40) has two fixed states ${{\cal Z}_0} = \pm 1$, which correspond to the vortex cores in the dressed states (eigenstates, or normal modes) [by definition, the dressed states provide a basis of states in which Eq. (32) is diagonal with hence no evolution in time]. Rewriting the transformation $M$ in the normal form

The Riemann sphere corresponds to the Bloch sphere of quantum—or polaritonic—states available to a coupled system, and it can be shown [17] that the plane can be identified as the real physical plane where the fields are evolving. This is revealed by the motion of the core—neatly identifiable as the single point of zero density—itself following such an Apollonius circle. Bringing the dynamics on a sphere simplifies it considerably [35], as the Rabi oscillations for any quantum state become simply circles on the sphere (reduced to a point at the poles in the case of polaritonic eigenstates). The main axis of this sphere is here set horizontal with respect to the complex plane. Detuning between the modes has the effect of tilting the plane of the circles on the sphere. It is worth noting that also the complex polar coordinates of the sphere, due to the stereographic projection, are mapped into two families of mutually orthogonal Apollonian circles in the 2D real space as well. An example of this is shown in Visualization 4. The identification of the complex plane from the stereographic projection to the physical plane for the fields allows the interpretation of the two wavefunctions ${\psi _{C,X}}$ as the breakdown of a wider and more natural object, living in a different space, to describe the dynamics. This object, the full wavefunction (as opposed to the cavity wavefunction or the exciton wavefunction), provides the density of any quantum states present in the system at any time. The peculiar dynamics observed in Fig. 6 becomes a simple rigid rotation in time of the full wavefunction on its Bloch sphere, as expected from the linear Rabi dynamics. The role of the displaced vortices (33) in this case is to prepare an initial condition that covers the entire sphere, since the vortex morphology makes it so that any quantum state is realized at any time at one, and only one, point in space. Any such point with a given quantum state undergoes simple Rabi oscillations, corresponding to, on the sphere, its rotation along its own circle. At such, this establishes a homeomorphism between the Riemann sphere (Bloch sphere of quantum states) and the complex plane (real physical space) [17], which accounts very simply for the vortex core dynamics, making it only a particular case, namely, the state perpendicular to that chosen for the observation (in our cases, the pure-excitonic state, since the observation is made with photons). Any other quantum state can then be seen to undergo a similar dynamics. This also explains why the core can, in this case like in previous ones, be sent arbitrarily far, including to infinity, as a result of the stereographic projection. In all cases, the motion is a smooth one at uniform speed on the sphere. If the system is prepared in such a way as to make the corresponding circle on the sphere pass by its projection point, the corresponding trajectory will be distorted in the plane to pass by infinity. Such a case is considered in Appendix B. This considerable simplification of an otherwise intricate dynamics is very appealing and calls for its generalization to the other systems. The phenomenology being so similar, one could expect that there also exist equivalent parametric or phase spaces to be defined in which the intricate dynamics of the anharmonic or squeezed cases becomes trivial and physically transparent. This remains for us, however, an open question.

## 6. SUMMARY AND CONCLUSION

We have shown how displaced vortices can lead to interesting dynamics of both the vortex core itself and the total angular momentum of the field, in a variety of platforms in the linear regime. Our choice of platforms included various types of confining potentials as well as coupled condensates (in the limit of low densities where interactions do not play a role). In all cases, we have highlighted the different behavior and character of the vortex as seen through the dynamics of its core, its morphology (phase map structure), and total angular momentum. While all are intrinsically connected, they can follow entirely different types of dynamics, for instance, with a complex underlying evolution of the vortex morphology, with creation of vortex–antivortex pairs, and cores displaying sequences of accelerations and decelerations with possible transit to infinity, while the net angular momentum can remain unaffected. More specifically, a displaced vortex in a radial potential keeps its OAM constant, as expected, despite the core moving on circular or elliptic orbits with even possible secondary vortex–antivortex pair creations due to self-interferences of the wavepacket. In the presence of an asymmetry of the potential, due to either different types or even simply different magnitudes of the confinement in different directions, the OAM becomes time varying, and the field displays a striking phenomenology, notably for the vortex core that can be sent cyclically to infinity with sequences of extreme accelerations and decelerations. The phenomenology can also hold without confining potentials, as illustrated by Rabi-coupled fields where the results can be further interpreted as a topological homeomorphism linking the Bloch sphere of possible quantum states for the system with the real physical plane. The salient result across all the different cases is that of a striking motion of the vortex core with an intricate dynamical morphology of the beam, being distinct from and not necessarily implying a time-varying OAM per particle, whereas the latter always implies some offset core and reshaping of its morphology. When this is the case, the core cannot be described as a mechanical point-like object with an effective mass that otherwise allows it to account for the dynamical properties of the system, namely, its momenta (linear and angular) and energies (potential and kinetics). The absence of interactions was chosen in the models for simplicity and to provide closed-form solutions, but numerical simulations show that the phenomenology survives in their presence. This provides opportunities for new types of micro-control and manipulations of small objects with exotic wavepackets, similar to Airy beams interacting with light particles [38]. In our case, the time-varying angular momentum can exert a torque on objects immersed in fluids set in motion as shown in the text, or provide other services in precision metrology, such as gyroscopes and even Casimir torque measurements, or to exploit the periodicity of the system, for instance, to realize a hybrid optomechanical torsion pendulum. Much control is available from the beam shape and morphology, which is easily tunable with a sequence of control optical pulses, as already demonstrated experimentally. The topology of the structured light thus emitted could also be useful even in a pure linear context, offering an additional degree of freedom of possible benefit for applications of OAM signal encoding and transmission.

## APPENDIX A: SIMPLIFIED COUPLED-FIELD EXPRESSIONS

It is possible to find a simpler version of the solutions given in Eqs. (38) and (39). Starting from the exact expression for the two coupled fields in the reciprocal space [Eq. (34)], and ignoring all terms with a $k$ dependency, the dynamics remains in the real-space domain within the extent fixed by the size of the initial packet. We can then ignore diffusive effects or dispersive ones in the dynamics of the solutions. Note that real-space solutions to Eq. (36) include diffusion, which comes from the exponential term ${e^{- i{k^2}{M_ +}t/2\hbar}}$ in Eq. (34). Without such effects, one gets the desired solutions:

## APPENDIX B: VORTEX CORE MOTION ON THE RIEMANN SPHERE

We provide one example of the dynamics of the ${\cal Z}$ point on the Riemann sphere, which involves infinity. We consider vortex cores in the photon and exciton fields located at, respectively, (${-}w,0$) and (0,0) in real space. We assume no energy detuning. The corresponding ${\cal Z}$ points are then given as ${{\cal Z}_C}(t) = i\tan \Omega t$ and ${{\cal Z}_X}(t) = - i\cot \Omega t$, which move along the imaginary axis of the complex plane (distinct form 2D real space). At $t = 0$, ${{\cal Z}_X}$ is a point at infinity; at $t = \pi /(2\Omega)$, it reaches the origin of the complex plane; and then at $t = \pi /\Omega$, it is again sent to infinity. In contrast, ${{\cal Z}_C}$ is initially positioned at the origin, then goes to infinity, and finally at $t = \pi /\Omega$, it reappears at the origin. In both cases, $\frac{d}{{dt}}{{\cal Z}_{C,X}}$ is not a constant, and such trajectories involve infinite accelerations and decelerations. However, points on the Riemann sphere have a smooth, uniform-speed dynamics. Indeed, the corresponding points on the sphere are ($0, - \sin 2\Omega t,\cos 2\Omega t$) for ${{\cal Z}_C}$ and ($0,\sin 2\Omega t, - \cos 2\Omega t$) for ${{\cal Z}_X}$, as is obtained directly from Eq. (42). This is the equation for a circle on the sphere, on which the points move with a constant angular speed of $2\Omega$. The described situation can be realized starting from a specific initial condition of the form Eq. (33). The same picture can be extended, with different parameters but in the same form, to points in the real space, where the relevant ${\cal Z}$ point moves on an associated Apollonius circle [17].

## APPENDIX C: CAPTIONS TO SUPPLEMENTARY MOVIES

**Visualization 1**. Dynamics of an off-centered vortex in an infinite circular quantum well. Some frames of the movie are shown in Fig. 1.

**Visualization 2**. Squeezing the harmonic potential in two directions leads to the weird dynamics of the vortex. This is for one period of the motion. Some frames of the movie are shown in Fig. 5.

**Visualization 3**. Rotating vortex in Rabi-coupled fields in the polariton case. Here, we show the photon field dynamics in one period of the motion. Some frames of the photon field (${\psi _C}$) are shown in Fig. 6.

**Visualization 4**. Time dynamics of the photon field (intensity in black, red, yellow, white color scales) undergoing Rabi oscillations. Here, a decay term has been added to illustrate how the phenomenology evolves in its presence (it produces a tilting of the sphere and its circles with respect to the plane of projection), while retaining the main features. The black circle lines represent the loci of isophase in the bare modes basis, $\arg (X + iY) = \arg ({\cal Z})$ = constant, while the white circles represent the loci of photon–exciton isocontent, $Z={\rm constant}$. The nodal points to the isophase circles represent the position of the vortices in the two bare modes.

## Funding

Tecnopolo per la medicina di precisione (CUP: B84I18000540002); Tecnopolo di Nanotecnologia e Fotonica per la medicina di precisione (CUP: B83B17000010001); Iran National Science Foundation.

## Acknowledgment

Dr. Lorenzo Dominici is grateful to the “Tecnopolo per la medicina di precisione” (TecnoMed Puglia) - Regione Puglia: DGR n.2117 del 21/11/2018, CUP: B84I18000540002 and “Tecnopolo di Nanotecnologia e Fotonica per la medicina di precisione” (TECNOMED) - FISR/MIUR-CNR: delibera CIPE n.3449 del 7-08-2017, 4 CUP: B83B17000010001. Dr. Amir Rahmani acknowledges the Iran National Science Foundation (INSF).

## Disclosures

The authors declare no conflicts of interest.

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