## Abstract

The existence of quantized vortices is a key feature of Bose–Einstein condensates. In equilibrium condensates, only quantum vortices of unit topological charge are stable, due to the dynamical instabilities of multiply charged defects, unless supported by strong external rotation. Due to immense interest in the physics of these topological excitations, a great deal of work has been done to understand how to force their stability. Here we show that in photonic Bose–Einstein condensates of exciton–polariton quasiparticles pumped in an annular geometry, not only do the constant particle fluxes intrinsic to the system naturally stabilize multiply charged vortex states, but that such states can indeed form spontaneously during the condensate formation through a dynamic symmetry breaking mechanism. We elucidate the properties of these states, notably finding that they radiate acoustically at topologically quantized frequencies. Finally, we show that the vorticity of these photonic fluids is limited by a quantum Kelvin–Helmholtz instability, and therefore by the condensate radius and pumping intensity. This reported instability in a quantum photonic fluid represents a fundamental result in fluid dynamics.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

From their macroscopic coherence, it follows that Bose–Einstein condensates
(BECs) may support rotational flow only in the form of quantized vortices
[1]. These vortices are thus
topological in nature, and are characterized by a phase rotation of
integer ($\ell$) steps of $2\pi$ around a phase singularity. However,
while in principle quantized vortices may take on any topological charge,
in practice, it is understood that only vortices of charge $\ell = \pm 1$ are dynamically stable: higher-order
vortices quickly shatter into constellations of unit vortices due to the
energetics of the system. This shattering process has been detailed
theoretically and observed experimentally in the context of stationary,
harmonically trapped atomic BECs [2–4]. The case is somewhat different for superharmonically trapped,
rapidly rotated condensates, for which there exists a critical rotation
rate above which the vorticity of the system becomes concentrated within a
single effective core. This state, in which all vorticity is within a
single effective core, has been called the *giant
vortex* state by its first experimental observers [5,6]. Such a giant vortex is, therefore, different from a state in
which there is a single point singularity with topological charge
magnitude greater than one—*a multiply charged
vortex*—however, in practice, it is often impossible to
distinguish between the two. On one hand, the density in the vortex core
is negligible, which hinders the resolution of singularity. On the other
hand, the structure of interest is hydrodynamical, and thus has meaning
only up to the length scales for which the hydrodynamical treatment
applies. The classical field description being a long wavelength
approximation of something that is in reality granular and nonclassical,
the hydrodynamic description applies only down to the healing length.
Singularities of like charge that are bound to within a healing length are
thus, to any probe in the hydrodynamical regime, indiscernible from the
theoretical *multiply charged vortex*. Thus,
from here on, we find it useful to call all such vortex structures *multiply charged*.

In our study, we focus on a BEC away from the thermodynamical equilibrium
supported by continuous gain and dissipation such as polariton [7], photon [8], or magnon [9,10] condensates. To be more specific, we
use the example of polariton condensate; however, the results reported may
be relevant to other nonequilibrium condensates. The exciton–polariton
(polariton) is a bosonic quasiparticle composed of light (photons) and
matter (excitons). Polaritons can be generated in optical semiconductor
microcavities. In a typical experimental system, laser light is
continuously pumped into the cavity to excite excitons (bound
electron–hole pairs) in a semiconductor sample. The photons remain trapped
in the cavity for some time, repeatedly being absorbed by the
semiconductor to excite excitons, and then being re-emitted as the
excitons decay. The excitons form superposition states (polaritons) with
the photons, which behave as neither light nor matter. Due to the finite
confinement times of the cavity photons, the polaritons in the condensate
are themselves short lived. In this way, the polariton condensate is
fundamentally different from other condensates: here, neither energy nor
particle number need be conserved. Thus, while a polariton condensate may
settle into a *steady state* (a state in which
the wavefunction is time invariant up to a global phase shift), such a
state is one in which dissipation is balanced by particle gain. The
corollary is that steady state flows are possible. It is well understood
that the pattern-forming capabilities of nonequilibrium, nonconservative
systems are richer than those of equilibrium, conservative systems [11], making the polariton condensate a
fascinating object with which to explore the possibility of novel quantum
hydrodynamical behaviors [12].

## 2. SPONTANEOUS FORMATION OF MULTIPLY CHARGED VORTICES

In this paper, we show theoretically that multiply charged vortex states
can appear spontaneously and remain throughout the coherence time in a BEC
of exciton–polariton quasiparticles excited by a ring-shaped laser
profile, without the application of any external rotation, trapping
potentials, or stirring. Previously, the spontaneous formation of multiply
charged vortices of a given charge was theoretically proposed and
experimentally realized in polariton condensates by pumping in an odd
number of spots around a circle [13], or by the engineering of helical pumping geometries [14]. In the first case, the central
vortex in this geometry is created driven by the antiferromagnetic
coupling of the neighboring condensates and the frustration arising from
their odd number. In the second case, the helical patterns are engineered
so that the condensate is pumped explicitly with orbital angular momentum.
Another recent proposal has shown that phase imprinted vorticity might
remain concentrated at a localized mirror defect [15], in contrast to another recent work in which
nonresonant pumping with a higher-order Laguerre–Gaussian beam resulted in
the clear transfer of total vorticity, while failing to form a multiply
charged vortex structure [16]. Yet
other works have exploited the lack of simple connectedness of condensates
confined to annular traps, in both equilibrium [17] and exciton–polariton condensates [18]; as in an annular condensate rotation
does not necessitate a vortex defect. Interestingly, a central
multi-charged vortex was observed in a numerical study of polariton
condensates under the *weak* Mexican-hat-type
pump (see supplemental material in [19]). There, the particle fluxes exist from the center outwards as
the pumping profile peaks at the center. As the pumping intensity
increases, the central vortex of small multiplicity (three in that case)
breaks into the clusters of single-charge vortices trapped by the minima
of the pumping potential. The multiply charged vortices we discuss here
differ from these works in the geometry considered (ring-pumped trapped
condensates, long coherence times), formation mechanism (probabilistic and
spontaneous during condensation, away from the hot reservoir), and the
vortex properties (vortices exist on the maximum density background and so
are truly nonlinear in nature). We describe their formation, stability,
and dynamics. The dynamics of two and more interacting multiply charged
vortices are also studied. We find that our results apply for a wide range
of possible experimental parameters, suggesting that these structures are
general to ring-pumped trapped polariton BECs in the strong coupling
regime.

The dynamics of the polariton BEC in the mean field are described by the complex Ginzburg–Landau equation (cGLE) coupled to a real reservoir equation representing the bath of hot excitons in the sample, nonresonantly excited by the spatially resolved laser pump profile $P(\textbf{r})$ [20–24]:

Polaritons can be confined all-optically by shaping the excitation laser
beam. By using spatial light modulators to shape the optical excitation,
ring-shaped confinements were generated with condensates forming inside
the ring [26,27], and have been predicted to support the spontaneous
formation of unit vortices [28].
Long-lifetime polaritons in ring traps are emerging as a platform for
studies of fundamental properties of polariton condensation largely
decoupled from the excitonic reservoir and therefore having significantly
larger coherence times [29–32]. We represent the profile of the ring
pump by a Gaussian annulus of the form $P(r,\theta ,t) = P{e^{-
\alpha {{(r - {r_0})}^2}}}$ with inverse width $\alpha$ and radius ${r_0}$, which excites local quasiparticles that
then flow outward. The closed-loop pump geometry has two major
implications. The first is that the condensation threshold is first
achieved not where the sample is pumped, but *within* the borders of the pumping ring. This results in the
effective spatial separation of Eqs. (3) and (4),
which makes the parameters related to excitonic reservoirs such as ${b_0}$, ${b_1}$, and $g$ irrelevant to the condensate dynamics up
to a change in pump strength. The second and most critical implication of
the ring pump geometry is the existence of constant fluxes towards the
center of the ring. Such fluxes carry the matter together with
spontaneously formed vortices and force vortices to coalesce.

It is well known that vortices can form during the rapid condensation of a
Bose gas via the Kibble–Zurek mechanism [33–42]. However, in our system, there exists
a different mechanism of spontaneous defect generation that requires a
relatively *slow* condensate formation. Due to
the inward flow of particles in our system, the condensation threshold is
reached first in the center of the system. Assuming a large enough ratio
of new particle flow to dissipation, this young condensate will grow into
a relatively uniform disk within the boundary of the pump. However, in
between these two stages, radial matter wave interference is to be
expected, with higher frequency during the early stages of condensation.
The zeros of the radial interference pattern are well studied under a
different name: the dark ring soliton [11,43,44]. As has been shown previously, these dark solitons
are unstable to transverse (“snake”) perturbations, and break apart into
pairs of unit vortices of opposite charge [45,46]. Thus, for a slowly
condensing system, it is reasonable to expect that these solitons have
enough time to break down to produce a chaotic array of vortex
singularities. This process resembles a two-dimensional case of the
collapsing bubble mechanism of vortex nucleation [47]. As the condensation process completes and the vortex
turbulence decays, there is some finite chance of the condensate being
left with a net topological charge, as vortex pairs may unbind near the
boundary, and one or the other may leave to annihilate with its image.
These like-charged vortices would then coalesce in the center of the
condensate.

Direct numerical integration of Eqs. (3) and (4) not only confirms that this process can take place, but that for low pump power, the condensate takes on a net topological charge more often than not. (Fourth order Runga–Kutta integration is used. The initial wavefunction is set to a profile of low amplitude random noise. All simulations are repeated for many of these profiles.) We reiterate that this coalescence of vortices exists despite the lack of external rotation or sample nonuniformity. Repeating the numerical experiment with many iterations of random initial wavefunction noise, we find multiply charged vortex states of stochastic sign and magnitude. The average topological charge magnitude is found to depend significantly on the radius of the pump ring, increasing for larger radii. An example of these dynamics is presented in Fig. 1, which shows the main steps in the process by which the condensate spontaneously adopts a topological charge of two: the formation of a central condensate surrounded by annular discontinuities in Fig. 1(a), the breakdown of an annular discontinuity into vortex pairs in Fig. 1(b), vortex turbulence in Fig. 1(c), and the final bound vortex state in Fig. 1(d). For Fig. 1, we use the system parameters $\eta = 0.3$, $\gamma = 0.05$, $g = 1$, and ${b_0} = 1$, ${b_1} = 6$, but the result was found not to depend sensitively on these choices; up to a rescaling of pump strength, this behavior was reconfirmed for a large range of sample parameters: $g \in [0.1 - 2]$, ${b_0} \in [0.01 - 10]$ for $\gamma \in [0.05 - 0.1]$, and for all reasonably physical values of $\eta$ (including $\eta = 0$.)

An advantage of the spatial separation of the condensate from the reservoir
in ring-pumped geometry is in the enhanced coherence time that exceeds the
individual particle lifetime by three orders of magnitude [50]. Therefore, spontaneously created
multiply charged vortices might soon be observable in single-shot
experiments within one condensate realization. However, for now, only the
average wavefunctions of many iterations of the stochastic condensate
formation are observable in experiment. In a perfectly uniform sample, one
would expect an equal chance of the stochastic formation of vortex charges
of either handedness, which would cancel in the experimentally observable
mean wavefunction. However, it has been established in experiments that
the slight inhomogeneities inherently present in all physical samples act
to favor one handedness over the other [13]. Thus, the only experimental observable is the mean *magnitude* of the vorticity distribution. This
magnitude is well above zero for a wide range of parameters, making the
experimental observation of this effect highly feasible within the current
state of the art. (For example: mean vorticity amplitudes determined from
direct simulations of Eqs. (3) and (4) for pump
radius $12\,\,\unicode{x00B5}{\rm
m}$ tend linearly from 2.5 to 4.9 as the pump
strength is decreased from $P = 9$ to $P = 5$.)

## 3. VORTEX IMPRINTING

Another way to study multiply charged vortices is to imprint them explicitly upon a fully formed, uniform condensate [51]. This allows for the study of the structure and dynamics of carefully controlled systems of vortices. To model the result of experimental pulsed phase imprinting, we first model the formation of fully developed non-singular condensate disks. To prevent the spontaneous formation of vortices by the process described above, a relatively strong pump amplitude is used, so that the condensate forms too quickly for the decay of ring singularities into vortices. After the background condensate is formed, phase singularities are imprinted instantaneously, and their dynamics are observed. To first understand the structure of isolated multiply charged vortices, we imprint a series of condensates with different topological charges, and allow these structures to form steady states. When imprinted in equilibrium BEC, multiply charged vortices quickly break into vortices of a single unit of quantization [52].

From the spatial separation of the condensate and the reservoir, the reservoir density is negligible near the central core of the multiply charged vortex, so that Eq. (3) takes the familiar form of the damped nonlinear Schrödinger equation (dNLSE): $i{\partial _t}\psi = - {\nabla ^2}\psi + |\psi {|^2}\psi - i\gamma \psi$.

Under the Madelung transformation $\psi = {\cal A}\exp [iS - i\mu t]$, where $\mu$ is the chemical potential, the velocity is the gradient of the phase $S$: $\textbf{u} = \nabla S$, the density is $\rho (r) = {{\cal A}^2},$ and the imaginary part of the dNLSE yields $\nabla \cdot (\rho \textbf{u}) = - \gamma \rho$. Except for a narrow spatial region where the density heals itself from zero to the density of the vortex-free state, the density is almost a constant, so the radial component of the velocity becomes ${u_r} = - \gamma r$. The real part of the dNLSE reads

We have shown above that the inward fluxes necessitated by the closed pumping geometry result in an effective trapping potential—independent of any effective trapping from the reservoir near the edge of the condensate—which drives the vortices closer together. Our analysis, which shows that the forces from the inward fluid fluxes overcome the topological repulsion of like-signed vortices, applies when the condensate is nearly uniform, which is the case until the vortices begin to overlap. At this stage, there is further interaction between vortices: it has long been understood that in nonequilibrium systems, the topological repulsion of like-signed vortices can be balanced due to the nontopological force emerging from the effective variations in the supercriticality stemming from the density decrease surrounding the defects [53]. The variable-supercriticality force is negligible until the vortices are close enough for significant overlap between their associated density structures. In our system, the radial flux forces bring the vortices of like sign to within the regime at which they may bind to form a multiply charged vortex.

## 4. VORTEX MERGER

Next we consider the arrangements of multiple multiply charged vortices imprinted away from the trap center and brought together by the radial fluxes. Figure 3 shows two examples of the coalescence dynamics of imprinted phase defects. In the first case, three unit vortices coalesce while moving in inward spirals towards the center of the condensate, where there is no net lateral flow. In the second case, which shows the coalescence of two doubly charged vortices, it is observed that both doubly charged vortices hold together for a while before merging in the center to form a single vortex of multiplicity four. These results are found to be repeatable for a wide range of system parameters, suggesting that this behavior is to be expected for any system parameter that allows the formation of a trapped condensate within a ring pump. We note that in this system, the center of the condensate corresponds to the location of maximum background fluid density, in stark contrast to systems designed to collect virtual vorticity into a low-density area [17].

As two (or more) vortices merge while spiraling around the center, they excite density waves in the otherwise uniform background fluid. These acoustic excitations are long lived, and take on a frequency set by the angular frequency of the vortex spiral. For well-separated vortices, this frequency increases consistently as time progresses and their separation shrinks. However, as the vortices begin to share a core, these dynamics become even more complicated, and the new physics dominated by the processes in the vortex core emerges [54].

Figure 4 shows the relative amplitudes of the density waves radiated during the motion of two vortices of unit charge imprinted with a large initial separation. The average frequency of acoustic radiation is found to increase with time at a fixed rate until the vortex cores begin to overlap (left vertical line). During this phase, the frequency distribution narrows, and the average radiation frequency increases linearly at a much lower rate than in the well-separated vortex regime. This continues until the singularities within the core overlap within a healing length (right vertical line), after which a fixed narrow band of acoustic radiation is emitted.

Of course, multiply charged vortices may also collide: we find that from merger of two equal multiply charged vortices of increasing topological charge, the characteristic acoustic resonances have decreasing frequency, in the near-terahertz regime. This is because the effective mass of the vortex increases with topological charge, so that vortices of larger multiplicity orbit more slowly. As expected, we see that in contrast, when multiple singly charged vortices placed evenly about a common radius merge, they emit higher frequency radiation as the number of unit vortices is increased. Once the multiply charged vortex has formed and is allowed to settle, low-energy density perturbations can be applied to the condensate. To model this, we simulate the effect of a small Gaussian laser pump pulse centered on the vortex. The observed effect is the emission of an acoustic energy pulse at the characteristic frequency of the vortex, as is seen from the merger of the equivalent number of unit vortices. As in any physical system, there exist many small perturbations due to intrinsic disorder, and it is likely that multiply charged vortices in a real system are regularly being excited and emitting acoustic radiation.

*Kelvin–Helmholtz instability.* Next we will
establish the limit on the vortex multiplicity that the trapped condensate
can support. This limit is set by the maximum counterflow velocity that
can be supported between the condensate and the reservoir, therefore
determined by the onset of Kelvin–Helmholtz instability (KHI). KHI is the
dynamical instability at the interface of two fluids when the counterflow
velocity exceeds a criticality. It appears in a variety of disparate
systems, both classical and quantum, but has never been discussed in the
context of polaritonic systems. In quantum fluids, KHI manifests itself
via nucleation of vortices at a counterflow velocity exceeding the local
speed of sound: ${v_c} = \sqrt
{\frac{{{U_0}\rho}}{m}}$. It has been extensively studied for the
interface between different phases of $^3{\rm He}$ [55], two components in atomic BECs [56], or for the relative motion of superfluid and normal
components of $^4{\rm He}$ [57,58]. In the trapped
condensates considered here, the counterflow is that between the
condensate of radius $R$ (which rotates with velocity $\frac{\hbar}{m}\frac{{|\ell
|}}{R}$ at the boundary) and the reservoir
particles along the ring, which are stationary. Thus, it is expected that
KHI should be initiated when the topological charge of the multiply
charged vortex state is high enough that the velocity of condensate
particles at the ring pump radius reaches ${v_c}$. Thus, the maximum topological charge ${\ell _c}$ allowed is set by

## 5. CONCLUSION

In conclusion, we have shown that exciton–polariton condensates excited by an annular pump can spontaneously rotate despite a uniform sample and no angular momentum applied, forming multiply charged vortices. The formation, dynamics, and structure of these vortices were studied. We emphasize that the KHI mechanism is quite specific to the ring-like pumping configuration we considered. In gain-dissipative condensate systems, particle fluxes exist even in the steady state, connecting the regions where they are predominantly created to the regions where they are predominantly dissipated. With the ring-like pumping, the fluxes are oriented strictly towards the center, stabilizing the multiply charged vortex while preventing the formation of localized vortex clusters elsewhere. When other pumping profiles are considered, a more complicated flux distribution emerges. In some localized parts of the sample, radial fluxes may exist similar to our ring-like pumping, but much weaker and less controllable. This may create conditions for the formation of multiply charged vortices, but of rather small multiplicity and that are quickly destroyed by the small changes in parameters (e.g., pumping intensity or the pumping spot size). For instance, the Mexican hat pumping profile in [19] gives rise to outward particle fluxes from the center in addition to inward fluxes from the annular pump, so a vortex of small multiplicity (two or three) is destroyed as the pumping intensity is increased, instead bringing about clusters of vortices stabilized where the fluxes from the center meet the radially inward fluxes. This destruction of the multiply charged vortices in this and similar cases cannot be attributed to KHI.

## Disclosures

The authors declare no conflicts of interest.

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