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Abstract
We theoretically study an exciton–polariton condensate trapped in a harmonic potential with an annular pump. With a circular pump, predictions were made for a spontaneous rotating vortex lattice packed by singly quantized vortices. If the circular pump is replaced by an annular pump, singly quantized vortices are absorbed into the central hole and form a multiply quantized vortex. For a sufficiently narrow annular width, all vortices are absorbed into the central hole, ultimately forming a giant vortex with supersonic circular supercurrents flowing around it. Vortex–antivortex pairs can be generated if a defect is present in these supersonic circular supercurrents. We further discover that the motion of the vortex–antivortex pairs depends on the position at which they were generated. We suggest that this property can be used to control whether the velocity of the circular supercurrents is above or below the sound velocity.
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1. Introduction
The properties of macroscopic coherent matter waves began to attract interest in the middle of the 20th century and continue to be studied today. Several different systems within the subject have been identified and studied comprehensively, such as superconductors, liquid helium, atomic Bose–Einstein condensates (BECs), and, more recently, exciton–polariton BECs in microcavities [1–3]. Two interesting properties of macroscopic coherent matter waves are the formation of quantized vortices and superfluidity. Abrikosov vortex lattices have been observed in type-II superconductors under magnetic fields [4], rotating liquid helium [5], and rotating ultracold atomic gases [6,7]. In contrast to these equilibrium systems, exciton–polariton quasiparticles have finite lifetimes and require an external laser pump to maintain the condensate density. Vortices can be created spontaneously in such a pump and decay process [8,9].
Rotating Abrikosov-like vortex lattices have been predicted to form spontaneously in harmonic traps [10,11]. For a pumping spot of an appropriate size, the rotationally symmetric steady state of the condensate becomes unstable due to the existence of radial inward supercurrents, and the condensate begins to rotate to maintain the stability of the system, resulting in vortices. The existence of honeycomb lattice vortices has also been proposed [12]. The formation of both vortex lattices and vortex–antivortex pairs have been observed [13–15]. These vorticies can even be arranged in a ring shape, forming an annular vortex chain [16–18].
In nature, singly quantized vortices seem to be more stable; multiquantized vortices are rarely seen. A rotating atomic BEC in a harmonic-plus-quartic potential may generate multiquantized vortices or a giant vortex—a vortex with a high winding number and quasi-one-dimensional circular supercurrents flowing around it [19–21]. If the rotation frequency exceeds the frequency of the harmonic trap, a central hole thus appears, and singly quantized vortices begin to be gradually absorbed into central hole to form a multiquantized vortex. As the rotation frequency further increases, all singly quantized vortices fall into the central hole, ultimately forming a giant vortex with quasi-one-dimensional circular supercurrents flowing around it. Higher-order or multiply quantized optical vortices may also appear in the center of Laguerre-Gauss (LG) laser modes that possess an orbital angular momentum (OAM) of $\ell \hbar$ per photon with $|\ell |>1$ [22–27]. These multiquantized optical vortices are also inherently unstable due to experimental imperfections in the sense that they tend to split up in a series of singly quantized vortices. However, these multiquantized vortices or even giant vortices have not yet been realized in the laboratory due to experimental difficulties in reaching the high rotation frequencies required. By contrast, the realization of an exciton–polariton BEC in a microcavity is more plausible due to the smaller effective mass of polaritons (enabling their formation at elevated temperatures), the maturity of the technologies for fabricating the necessary semiconductor structures, and the convenience of experimental manipulation of these substances. Thus, although qubits have been realized in quantum networks such as superconducting circuits [28], photonic systems [29,30], nuclear magnetic resonance systems [31,32], and trapped ions [33], polariton platforms are also an attractive option for realizing quantum simulators. Single-flux or multiple-flux vortices in circular exciton–polariton currents can be potentially used as flux qubits [34–37].
The aim of this paper is to theoretically describe a trapped polariton condensate with both pumping and decay. The pump geometry is programmed with an annular laser excitation. Numerical modeling based on the open-dissipative Gross–Pitaevskii equation suggests the formation of a giant vortex and spontaneous circular superflow within the annular pump area. By using the numerical procedure in the split-operator method, we demonstrate how the giant vortex is formed in the center of a harmonic trap that gathers a fixed number of phase singularities into a central hole of low polariton density. Surprisingly, although these giant vortices with large winding numbers greater than 20 have a propensity to split into single vortices, they can be dynamically stable in this annular pump configuration. By manipulating the annular excitation width, several different vortex phases can be identified, such as a vortex lattice, a multiply quantized vortex surrounded by vortex rings, and a giant vortex with single or multiple rings of circular supercurrents. All the three phases have the same total winding number (or vortex quanta). Furthermore, the speed of the circular supercurrents can be controlled to be above or below the sound velocity by generating vortex–antivortex pairs in the presence of a defect. These interesting features contrast with those of a typical energy-conserving atomic BEC in a quadratic-plus-quartic potential trap [19–21]. In contrast to these atomic BECs, the spontaneous formation of a giant vortex in the trapped polariton condensate is not subject to any rotation, stirring, or phase imprinting. This property allows for the study of the rotational response and pattern formation of a trapped condensate confined in an annular potential; it also allows for the realization of a controlled supercurrent in solid-state systems.
2. Steady state solutions of annular pump geometry
We consider an exciton–polariton condensate trapped in a harmonic potential with a nonresonant pumping process. This system can be conveniently described with a complex Gross–Pitaevskii equation [10].
Equation (2) is solved numerically using the split-operator method. Figure 1 presents the density profile of the condensate with $R_>$ fixed at 7, $\alpha _0=6$, and $\sigma =0.3$. $\alpha$ and $\sigma$ are chosen based on the estimation from the precursory article by J. Keeling et al.[10]. For $R_<=0$ [Fig. 1(a)], which is equivalent to a uniform circular excitation, rotating triangular vortex lattices may arise, as was predicted in a previous study [40]. Twenty singly quantized vortices are inside the excitation. If $R_<$ is increased to 0.8, the three vortices near the origin merge to form a vortex with a winding number of 3. The remaining vortices are still arranged in two rings [Fig. 1(b)]. For $R_<=2$, the inner vortex ring (with 5 singly quantized vortices) is absorbed into the central hole and forms a vortex with a winding number of 8 [Fig. 1(c)]. For $R_<=3$, all singly quantized vortices are absorbed into the central hole, leaving only a giant vortex with quasi-one-dimensional supercurrents flowing around it [Fig. 1(d)]. The phase maps shown in Fig. 2 can help recognize the generation of giant vortex. The phase singularities are located inside the central hole, and the uniform variation of the phase in the $\theta$ direction manifests as uniform circular supercurrents flowing around the central hole with the phase gradient; that is, $j_{\theta }=\nabla _{\theta } \phi$, where $\phi$ is the phase of the macroscopic wave function of the condensate. As $R_<$ is further increases to 4 and 5, the condensate divides into double and triple rings, respectively [shown in Fig. 1(e) and (f)]. Similar patterns are present in a rotating atomic BEC [41]. These phases occur due to the interference of the direct and indirect inward flows. Indirect inward flow is the flow due to particles that are initially pushed radially outward near the outer edge $(R_>)$ of the annular pumping but then flow back toward the center due to the harmonic potential.
Fig. 1. Steady state density profiles of the annularly pumped condensate. $R_<$ = 0, 0.8, 2, 3, 4, and 5 for (a–f), respectively. The winding number is 20 in all subfigures. Parameters are $R_>=7$, $\alpha _0=6$, $\sigma =0.3$.
Fig. 2. Steady state phase map of the annularly pumped condensate. $R_<$ = 0, 0.8, 2, 3, 4, and 5 for (a–f), respectively. The winding number is 20 in all subfigures. Parameters are $R_>=7$, $\alpha _0=6$, $\sigma =0.3$.
Vortices still exist in singly quantized form in the central hole of the giant vortex. This phenomenon is the same as that predicted for a giant vortex in a rotating atomic BEC in a harmonic-plus-quartic potential. A theoretical analysis of vortices in a rapidly rotating atomic BEC demonstrated that vortex cores never overlap [42]. In an exciton–polariton BEC experiment, doubly quantized vortices tended to split into two singly quantized vortices [43]. Our findings agree with these results.
The giant vortex can be spontaneously formed by switching the inner radius when the vortex lattice, as in Fig. 1(a–c), is formed in advance, as displayed in Fig. 3. The vortex ring is first generated by an annular pump beam with inner radius $R_<=3$ (left-top subfigure of Fig. 3), and the pump beam is then varied to $R_<=4$. Subsequently, the giant vortex is generated. However, the pump beam can be directly applied with $R_<=4$ to generate the giant vortex [refer to Fig.1(d)]. Thus, the change of the pump beam shape is not compulsory to generate the giant vortex. If an appropriate annular pump beam is applied in the first time, as in Fig.1(d–f), a giant vortex spontaneously forms. In our previous work [40], we identify an instability boundary in the phase diagram of pump size versus pump power. The significance of this boundary is that when we have a larger pump size with suitable pump power, the condensate supercurrent is flowing inward (towards the center) and there is a supersonic region where the condensate flow velocity is larger than sound velocity. Both these two phenomena will induce spontaneous symmetry breaking and consequently the azimuthal flow of the condensate. Therefore, in theory, as long as we use the pump size with proper pump power and inner annular radius along the instability boundary, it is relatively easy to generate giant vortices.
Fig. 3. Formation of the giant vortex. The state at $T=0$ is the equilibrium state of $R_<=3$; $R_<$ is increased to 4 after $T>0$ (see Visualization 1). Other parameters are identical to those in Fig. 1.
Figure 4 shows the average angular momentum $L_z=\frac {\int {d^2\vec {r} \Psi ^* \hat {L}_z \Psi }}{\int {d^2\vec {r} \Psi ^*\Psi }}$ for the various inner radius under three pump conditions. The average angular momentum increases as the vortices concentrate in the central hole. The average angular momentum is approximately constant for $R_<\geq 4$ because the vortex structures are similar. If $\alpha _0$ is reduced to 3 from the steady state of $\alpha _0=4.4$, the rotation speed increases. Three more vortices are then created at the outer boundary of the condensate and spiral into the condensate to join the vortex arrangement. This property enables the observation of phase slippage in the exciton–polariton condensate [19,44]. If $\alpha _0$ is increased to 10 from the steady state of $\alpha _0=4.4$, no substantial change in the average angular momentum is observed, and no vortices are gained or lost. This result occurs because $\alpha _0=4.4$ is very close to the boundary of the phase diagram of the rotationally symmetric steady states and the rotating states [40]. Increasing $\alpha _0$ to cross the boundary from the rotating states to the rotationally symmetric steady states only increases the density of the condensate and without altering the rotation status. This property enables changing the sound velocity of the system and is discussed further later in this paper.
Fig. 4. Average angular momentum versus inner pumping radius. Parameters are $\alpha _0=3, 4.4, 10$; $\sigma =0.3$; and $R_>=7$.
3. Excitation spectrum and stability of the giant vortex
In the following section, we analyze the stability of the giant vortex. We insert $\Psi =f(r)\psi (\theta,t)$ into Eq. (1) and integrate over r. Assuming that the maximum value of $|f(r)|$ is normalized to 1, we obtain the following:
Fig. 5. Excitation spectra of the giant vortex without a defect. The dotted, dashed and solid lines indicate $\alpha _0=3, 4.4, 10$, respectively. Parameters are $\sigma =0.3$, $R_<=4$, and $R_>=7$.
Fig. 6. Density profiles of the giant vortices in a finite-size defect. (a) $\alpha _0=4.4$. (b) $\alpha _0=10$. (c) Zoom in view of (a). (d) Zoom in view of (b). The parameters: $V_0=10$, $\textbf {r}_0=(5.5,0)$, $L=0.3$, $\sigma =0.3$.
In the Gross–Pitaevskii theory, the size of a vortex can be characterized with the healing length which is the length scale over which the condensate spatial extent changes significantly, and can be calculated through $\xi =\hbar /\sqrt {2mgn}$. The healing length of Fig. 6(a) and Fig. 6(b) are 0.25 and 0.18, respectively. If the defect is larger than the healing length, vortex–antivortex pairs can be generated behind a defect [47]. Vortices tend to rotate in the same direction as the giant vortex, whereas antivortices rotate in the opposite direction. Furthermore, the motion of the vortex–antivortex pairs depends on the position at which they were generated. Placing a defect at a dense area, such as $\textbf {r}_0=(5.5,0)$ with $V_0=30$ and $L=0.3$, a vortex–antivortex pair is generated behind a defect [Fig. 7(a)]. Then, the antivortex is attracted into the central giant vortex hole first and annihilates a winding number [Fig. 7(b)] such that the system cannot afford to sustain the original circular supercurrents and begins to slightly flow radially inward. Therefore, the other vortex of the pair moves with the radial flow into the central hole and supplies a winding number [Fig. 7(c)]. This process repeats numerous times, and the circular supercurrents with constant velocity can be sustained. However, if the defect is placed further outside, such as at $\textbf {r}_0=(6.3,0)$, the vortex is immediately pushed out due to the radially outward flow near the outer boundary of the annular pump after the generation of the vortex–antivortex pair [Fig. 7(d)]. Only the antivortex goes with circular supercurrents. It is eventually attracted into the central hole and annihilates a winding number [Fig. 7(e)]. This process continues until the velocity of the supercurrents is insufficient to support the generation of any vortex–antivortex pairs [Fig. 7(f)]. In this final equilibrium state, only nine winding numbers remain, and the velocity of the supercurrents $v_s=2.16$ is below the sound velocity $c=3.42$ (numerical). Additional concentric rings also appear in the central hole due to interference from the radial component of the supercurrents. We suggest that this phenomenon can be used to control whether the velocity of the supercurrents is above or below the sound velocity through the placement of a defect in a dense or sparse area.
Fig. 7. Time evolution of the circular superflow against defects placed at two different radial positions. For (a–c), $\textbf {r}_0=(5.5,0)$ (see Visualization 2) and for (d–f), $\textbf {r}_0=(6.3,0)$ (see Visualization 3). $V_0=30$, $L=0.3$, $\alpha _0=10$, and $\sigma =0.3$. "v" denotes a vortex, and "a" denotes an antivortex.
4. Theoretical drag force and supercurrent velocity
The superfluidity of the annularly pumped exciton-polariton condensate can be theoretically investigated by studying the drag force exerted by the flowing condensate on an obstacle. Here, a point defect is placed at the position $\textbf {r}_0=(x_{0},0)$ with the maximum condensate density, and the theoretical perturbed wavefunction can be obtained for further evaluation of the drag force (see the Methods section for details). We are interested in the states with a giant vortex in the center and a single ring of supersonic circular supercurrents. Figure 8 presents the azimuthal drag force in terms of the supercurrent velocities. For the same excitation strengths of $\alpha _0=3, 4.4$, increasing the inner radius from $R_<=4$ to $R_<=5$ (decreasing the annular width) only slightly changes the steady-state condensate distribution, and the state with the giant vortex with two rings of supersonic circular supercurrents has not yet been achieved. However, the drag force decreases because the effective condensate density, represented by ($\chi +E_{i})/(\beta \sigma )$, is decreased. Instead of placing the defect at the maximum density, we shift the defect to a sparse area at $\textbf {r}_0=(6.3,0)$ as displayed in Fig. 8. The antivortex for each generated vortex pairs is attracted into the central hole and annihilates a winding number. Consequently, the total winding number decreases and the condensate slows down; the generation of vortex pairs then stops. In this situation, the drag force is significantly decreased. In the annular pump geometry, the best way to control the drag force and flow velocity is by placing a defect at the proper position.
Fig. 8. Numerical (a) and theoretical (b) drag force in terms of the supercurrent flow velocity. The triangles, circles, and squares indicate $\alpha _0=3, 4.4, 10$, respectively. Parameters are $\sigma =0.3$ and $R_>=7$. Default defect position without extra note is $x_{0}=5.5$.
If the numerical wavefunctions are used for the evaluation of the drag force and the point defect is replaced by a Gaussian defect $V(r)$—where $V(r)=V_0e^{-(\textbf {r}-\textbf {r}_0)^2/L^2}$, $V_0=30$ is the defect strength, $\textbf {r}_0=(x_0,0)$ is the defect position, and $L=0.3$ is the defect size—the azimuthal drag force in terms of the velocity of supercurrents can be calculated, as presented in Figure 9. The numerical drag force is similar to the theoretical drag force for the high pump strength of $\alpha _{0}=10$. However, at low pump strengths, the higher numerical density variation around the monitored defect position causes the numerically calculated drag force to deviate from the theoretical drag force.
5. Discussion
Spatial pattern formations in polariton condensates have been demonstrated to arise spontaneously in quasi-one-dimensional systems [20,40]. Manni et al. reported the spontaneous formation of an optically induced quasi-one-dimensional untrapped polariton condensate comprised of a number of lobes under a ring-shaped (slightly elliptical) laser excitation [38]. The corresponding standing-wave pattern was interpreted as a consequence of condensation into a single-energy state with periodic boundary conditions featuring a multilobe density pattern. Another study by Dreismann et al. demonstrated the spontaneous formation of an untrapped petal-shaped polariton condensate generated by two concentric circles of light [39]. The imposed double-ring excitation geometry supports the “petal states" arising due to the interaction of two superfluids counter-propagating in the circular waveguide defined by the optical potential. At sufficiently high pump powers, the polaritons traveling away from the outer ring toward the center of the pump geometry were slowed by the potential associated with the inner ring and accumulated in the region between both rings to form condensation and a petal state. The polariton luminescence then lies between the outer and the inner pump rings, and the excitation pattern (i.e., the number of lobes) can be adjusted by modifying the pump geometry. However, the effect of disorder was not included in these two theoretical models; disorder might influence the condensate flow and, by consequence, the kinetic energy of the polaritons to further break the symmetry of the system. The resultant pattern formation and induced condensate currents differ substantially, enabling the exploration of novel patterns such as giant vortices, which have not been investigated in polariton condensate systems.
In this article, we demonstrated the possible existence of a multiquantized vortex and in particularly giant vortex in an exciton–polariton condensate trapped in a harmonic potential by using an annular pump. Previously, the multiquantized vortices with topological charges ($-3$ to $+3$) can be theoretically formed by nonresonantly exciting an odd number of polariton condensates at the vertices of a regular polygon [48]. The spontaneous formation of multiquantized vortices can be also done by the engineering of helical pumping geometries, in which the helical patterns are engineered so that the condensate is pumped explicitly with orbital angular momentum [49]. The multiply charged vortices we discuss here differ from these works in the geometry considered (we use annularly pumped trapped condensates). A more recent comprehensive article also uses a ring-shaped laser profile to study spontaneous multiquantized vortices [50]. However, some difference still exist, we use a harmonic trap and we skip the reservoir dynamics representing the bath of hot excitons. The most distinct result is that, the articles aforementioned proposed nicely the ways to generate multiply quantized vortices, and our work here is to propose a way to generate "giant" vortices.
For different annular pump shapes, different vortex phases can be observed, including a vortex lattice with a central multiquantized vortex, a multiquantized vortex surrounded by vortex rings, and a giant vortex with single or multiple rings of supersonic circular supercurrents flowing around it. The stability of the giant vortex was also been analyzed. The sound velocity can be controlled by varying the pumping strength. However, the velocity of the circular supercurrents is always higher than the sound velocity. In this high-velocity regime, vortex–antivortex pairs can be generated in the presence of a defect. We further discover that the motion of the vortex–antivortex pairs depends on the position at which they are generated (a position near the defects). If the vortex–antivortex pairs are generated sufficiently close the outer boundary of the annular pump, the vortices are pushed out and only antivortices remain in the system. These antivortices are eventually attracted into the central hole and annihilate the winding numbers of the giant vortex. The velocity of the circular supercurrents thus gradually decreases until they can no longer support the generation of vortex–antivortex pairs. We suggest that this mechanism can be used to control the velocity of the circular supercurrents above or below the sound velocity. We believe that the controllable properties of the system provides an avenue for studying supercurrents in exciton–polariton BECs.
6. Methods
Perturbed wavefunction in a point defect. Here, we consider the case in which a static point defect $V(r,\theta )=V_{0}\delta (r-r_{0})\delta (\theta )$ exists in the condensate. A small perturbation can then be applied around the stationary state $\psi _{def}(\theta,t)=e^{-i\mu t}\left [Ae^{in\theta }+\delta \psi \right ]$. In the steady state condition, Eq. (3) becomes a matrix problem, as follows:
Funding
Ministry of Science and Technology, Taiwan (MOST110-2112-M-153-007, MOST111-2112-M-034-001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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