Temporal vortex morphology and time--varying angular momentum

It is known that the fundamental mode of rotation in quantum fluids is represented by a quantum vortex. Its quantized value represents the orbital angular momentum (OAM) per particle, which is reduced in case of offset vortices. Here, we study analytically the dynamics of a displaced quantum vortex in a number of 2D systems, including infinite or harmonic confining potentials and a two-component condensate. The vortex core demonstrates peculiar motion in all such systems, however, the OAM content may vary in time or not, depending on the underlying physics. For the off-axis vortex in the radial potentials, the OAM remains invariant, despite the core moves on circular or elliptic orbits and even when secondary vortex-antivortex pairs proliferate due to self-interference of the wavepacket being reshaped. However, including harmonic time perturbations with a gain-dissipation term or adding a spatial anisotropy, confers time-varying OAM to the morphology reshaping. Considering a binary condensate, the Rabi dynamics can imply time-dependent angular momentum and core orbital motion in each of the components, as a natural result of their coherent coupling. The salient result across all the different cases is that a weird motion of the vortex core, and also its dynamical morphology, are distinct from and do not necessarily imply a time-varying OAM per particle, whereas the latter always imply some offset core and its morphology reshaping.


I. INTRODUCTION
Topology is indispensable to different areas of physics, in particular to condensed matter physics [1]. This involves invariant under deformation from ground state to a new configuration for which any path to the ground state is energetically too costly. As such, topological invariant powered by Homotopy groups provide subtle tool to detect holes in a given space; examples are topological defects in system with continuous broken symmetry. Vortex, a typical defect in system with global U(1) broken symmetry, is characterized by a null density region and an integer (topological) charge associated with the singularity in the gradient of the order parameter phase at the core position; indeed, the topological charge (TC) defines the intrinsic orbital angular momentum (OAM) available in the vortex state. Quantum vortices are common in many phases including superconductors [2], superfluids [3], atomic Bose Einstein condensates [4], and polaritons [5], among others.
Vortex beam (VB), confined vortex state in the form of beam with spiral wavefront, is exotic in a sense that it can carry angular momentum [6]; indeed one can distinguish two kinds of VBs, one with non-uniform phase-varying, which is called anisotropic VB, and the other with uniform phase variation, which is referred to as isotropic VB. In both cases, the fields, being associated to a displaced vortex, carry reduced-fixed amount of angular momentum and have modified morphology distributions. For static VB (not varying in time), either isotropic or anisotropic, it is known that the vortex helicity can be described based on the combination of some elementary vortices with integer topological charges which determines the overall morphology distribution and associated angular momentum across the beam [7].
Vortex beams can be dynamic, i.e., the angular content can be time dependent. First report regarding timevarying angular momentum was given in Ref. [8], where two pulses, one delayed with respect to the other, are imprinted on microcavity polariton (coupled photonic and excitonic fields) fluids. This leads to temporally and spatially structured components (photon and exciton), that is, the off-center core in one component (photon) is dynamically matched by an off-axis core in the counterpart field (exciton). The coherent transfer between the two components lets the two cores moving on the same orbit (with a reciprocal delay), resulting in the oscillating linear and angular momentums in the emitted light. Such oscillations happen in the linear regime of the dynamics. Quite recently a similar scheme, with two retarded pulses but now partially overlapping in time, has been used in the nonlinear process of high harmonic generation, to result in the vortex beam with continuously time-varying angular momentum [9].
Here, we study analytically some simple physical systems that can sustain a dynamical vortex, in the sense that the vortex morphology is temporally being stretched or folded up, while the angular momentum content may vary with time. Indeed, one may expect a time-varying morphology also results in a time-varying angular momentum. We rather show that the time-varying morphology can be associated to but does not necessarily imply time-varying angular momentum. To this end, we consider an off-axis vortex in some elementary physical examples: infinite (circular) quantum well potential, harmonic potential perturbed by spatially anharmonic component and time-harmonic term, and two coupled condensates. We consider in Sec. II the case of displaced arXiv:2006.07173v1 [cond-mat.quant-gas] 12 Jun 2020 vortex in quantum well with hard boundary, for which we introduce the mechanism of pair excitement induced exclusively by quantum effects associated with interference. While the vortex morphology is peculiarly varied in time, the angular momentum content remains steady. Section III accounts for the off-center vortex in the subtle example of harmonic potential, where we extend the study to consider the effect of the spatial anisotropic anharmonicity (deviation from harmonic potential) and time-harmonic perturbation (oscillatory in time) with a gain/dissipative term. We show that while the harmonic potential alone sustains an angular momentum which is constant, on the other hand the time harmonic perturbation induces periodically varying angular momentum. Interestingly, anharmonicity modifies greatly the morphology of the vortex alongside with the vortex core motion, as removing the degeneracy of the excited states of harmonic potential mediates oscillatory time-varying angular momentum. The other possibility to observe the time-varying angular momentum is through binary condensates, namely, Rabi coupled fields, which is studied in Sec. IV. Such behavior was shown recently for the case of large imbalance between masses of coupled fields, corresponding to the microcavity polaritons. Finally, Sec. V provides conclusion remarks. It is worth noting that all examples considered here are in the linear regime, where the self-interaction between particles are negligible.

II. QUANTUM WELL
It might be one of the simplest case to consider an offcenter vortex in an infinite quantum well. We consider the Hamiltonian of a circular quantum well of radius R: where where r = x 2 + y 2 and ϕ = arg(x + iy). Solving equation H qw χ n,l = E n,l χ n,l , one can find states and energies through: where J |n| is the Bessel function of integer order, N n,l is the normalization constant, l = 0, ±1, ±2, · · · , and β n,l = k n,l R is the nth zero of the |l|th Bessel function for which J |l| (β n,l ) = 0; this is a condition imposed by wavefunction on the boundary, namely χ n.l (R, ϕ) = 0. In addition to stationary solutions χ n,l , one can solve Schroedinger equation i ∂ t ψ = Hψ for a given arbitrary initial state. To end this, we note that: ψ(r, ϕ, t) = n,l α n,l e −iE n,l t/ χ n,l (r, ϕ) , (4a) α n,l = dϕ rdrψ 0 χ * n,l (r, ϕ) , where ψ 0 (r, ϕ) ≡ ψ(r, ϕ, 0). Here, initial condition is a superposition of two Gaussian wavepackets one with topological charge TC = 1 and another with TC = 0; it reads: where w is the spot size of the wavepacket and x c ≡ r c cos ϕ c , y c ≡ r c sin ϕ c gives the position of the core. A similar initial state could be implemented in an experiment with two delayed resonant pulses [8] in a polariton fluid. The wavepacket is given by: ψ(r, ϕ, t) = n α n,1 e −iEn,1t/ χ n,1 + α n,0 e −iEn,0t/ χ n,0 .
The dynamics is intriguingly highlighted. Examples are shown in Fig. 1. We can detect different stages in the dynamics; at first we see an increment of the wavepacket size inside the well, which brings the density null and crest in larger and larger separation; however, the wavepacket should be zero on the boundary and there is a time instant when diffusion stops and a second crest starts to form , in half-moon shape, which is thinner than the first one but larger in radius. At the same time the vortex core rotates and the distortion of the phase excites further vortices, and precisely a vortex-antivortex pair, that move until they again recombine. Very shortly after the outer density reforms to an inner density, followed by rotation of the core and once again by creation and then annihilation of the vortex pairs. Such processes are repeated during the time evolution of the dynamics. Despite the complex motion of the core, the angular momentum content remains steady, one can show that: The desired quantity is the angular momentum per particle, which in the current physics is less than one (as expected for an off-center vortex [10]), namely, l ≡ L z /N < 1, where N is the total number of the particles and is given by: Dynamics of an off-center vortex in quantum well with hard boundary. Upper panels show the density profile of the wavpacket: |ψ|. The lower panels illustrate the corresponding phase map: arg[ψ]. In (h) phase gets so distorted that a pair is excited, which is shown by V (vortex) and AV (antivortex). The pair moves until V and AV recombine. It can be excited later, as is shown in (j). Here, we used: R = 2 µm, and w = 0.5 µm.
The fact that l < 1 comes from the initial field (5), where we introduce two kinds of particles with a fraction only of them carrying (integer and equal to one) angular momentum. Here the infinite external potential is a purely confining one, its shape is radially symmetric and azimuthally homogeneous. Considering the potential gradient and its symmetry, its boundaries act both as a local and net force on the fluid at any moment, however there is no net torque acting on the fluid. Basically, while the center of mass of the fluid and its net linear momentum keep changing due to the continuous Phase distortion resulting from the interference between two waves, one is oblique with respect to the other and carries topological charge TC ≤ 1. Frames are for different k, wave number, where its increasing distorts the resulting phase. If the total field e ikx + re iϕ − rce iϕc (described in the text) is written in its imaginary and real form, the condition for zero density is given as: − cos(kx) = x − xc and − sin(kx) = y − yc. Here we take xc = 1 µm and yc = 0, and then the roots of − cos(kx) = x − xc give the possible numbers of vortices. In (a) and (b) there is only one root, corresponding to the intersection of two curves x − xc (shown in cyan) and − cos(kx) (shown red). In (c) there are three roots, which represent two vortices (denoted by V) and an antivortex (denoted by AV).
bouncing of the fluid against the well boundaries, its net angular momentum remains constant. Yet, some peculiar morphology reshaping, involving also the topological charges, happens during the evolution.
Once the expanding cloud reaches the boundary and is partially reflected, a circular ripple of lower density with larger phase gradient (visible starting from Fig. 1b,g on) is formed. It can be understood in terms of interference between the outward diffusing and the inward reflected waves. The secondary vortex-antivortex pair is nucleated starting from this loop of local low density. The pair proliferation is understandable in the realm of wave interference as well. Consider the interference of two waves, both with a Laguerre Gaussian type envelope of same amplitude: one propagating in the k 1 direction, A 1 = e −ik1·r e −r 2 /(2w 2 ) , and the other propagating in the different direction k 2 but with a topological charge TC ≤ 1, A 2 = e −ik2·r e −r 2 /(2w 2 ) re iϕ − r c e iϕc . If, for simplicity, A 1 travels along the x direction, for which k 1 = kî, and for A 2 we assume k 2 = 0, then the two interfering waves can yield a total density I ∝ |e ikx + re iϕ − r c e iϕc |. Depending on wavenumber k, such a density can have one root, corresponding to the initial topological charge TC ≤ 1, or can sustain three roots corresponding to a vortex and a vortex-antivortex pair. Example of this behavior of the phase is shown in Fig. 2, where we take x c = 1 µm and y c = 0 (that is obtained by polar parameters r c = 1 µm and ϕ c = 0). Upon increasing k, the phase gets distorted, resulting in vortex pair nucleation. This condition also depends on the relative amplitude of the two waves, which we didn't considered in order to highlight the importance of the momentum. Coming back to the displaced vortex in the circular quantum well, the excitement of the pair comes to play when a reflected wave from the hard boundary interferes with the field inside the well, as described partially in the above example, where a vortex-antivortex pair is created and annihilated repeatedly. There could be higher energetic initial conditions for which a larger number of vortex-antivortex pairs are cyclically observed. All these dynamics and morphology reshaping happen keeping a constant angular momentum.

III. HARMONIC POTENTIAL
Next, we consider the displaced vortex in a harmonic potential. The main equation is i ∂ t ψ = H so ψ with: where ω so stands for the natural frequency of the oscillations. Ring potentials in general have been shown to sustain rotating patterns [11,12], based on combinations of rotational modes of different eigenenergies. The harmonic potential provides the staple model for many vibrating systems. Here, we show interesting features of the vortex fluid inside such potential and add a further degree, that is, temporally periodic angular momentum. To this end, we start our analysis with a spatially equipotential initial condition, namely, where the width parameter w = β ≡ /(mω so ) is assumed in the initial configuration expressed by Eq. (5).
The wavepacket can be obtained in a close form: This gives the orbit where the core moves on, which is a circle of radius r c (0); such a motion happens with a constant speed v c = r c (0)ω so . Although the core oscillates in time, the angular momentum content remains steady: This is consistent with a fractional value due to the offset core, which remains constant due to the fixed radius of its orbit. Then, the rotational content depends on the initial radial distance of the core from the origin, ranging from L z = 1 for r c (0) = 0 to some very tiny value for large r c (0). Interesting dynamics happens when in the initial condition we set w = β, corresponding to a nonequipotential isotropic case. Depending on the ratio of w/β , approximate solution can be provided through: where H so φ n,m = E n,m φ n,m , and α n,m (t) = e −itEn.m/ ψ 0 φ n,m dxdy with n, m = 0, 1, 2, · · · . The evolution of the density is shown in Fig. 3 for w = 1.2β and for the core located initially in (−1, 0). First, the wavepacket shrinks in size and at the same time the core also moves, but this time it moves on an ellipse rather than a circle: (βx c (t)/w) 2 +(wy c (t)/β) 2 = (β/w) 2 . Then, the wavepacket shapes to its initial size, and 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, that is being temporally sped up or down, and also the timevarying morphology, the quantum average of angular momentum is yet constant. In general and for any initial condition, one can show that for a simple harmonic potential the expectation value of angular momentum reads as: which is clearly time-independent. This is once again (as in the case of the circular quantum well) in agreement with the symmetry of the radial force field (energy gradient) of such potentials, which do have a zero net torque. In their interaction with fluid, they cannot exert any change of its overall angular momentum, which keeps constant. Although the angular momentum is steady in the above cases, the time-varying angular momentum could be induced by means of some perturbations; here we consider two examples. The first is an external time harmonic (and spatially homogeneous) potential V (t) = V 0 cos(ω p t) + iV 1 sin(ω p t) which could be induced by an external oscillating field such as an incident laser. Here, we also add a complex term to take into account decay and/or gain. The coefficient α n,m in Eq. 12 is given by: Corresponding average of the angular momentum reads as: which is periodically time dependent. The amplitude of oscillations depend strongly on ratio of V 1 / ω p , and may go beyond , for example, L z > 2 . The morphology of the vortex now attains an extra prefactor weight of e −V1 cos(ωpt)/ ωp which results in oscillatory angular momentum.
The second example accounts for the effect of spatial anharmonicity, namely, a quartic contribution which is added to the harmonic oscillator potential. Indeed, a fluid in power law trap demonstrated some rich vortex states including crossover from vortex lattice to giant vortex [13]. We do not consider such transition at large rotational velocity here, as we rather focus on the possibility of time-varying angular momentum with anharmonicity, which could be naturally the part of every vibrating system. Here for simplicity we consider anistropic anharmonic term H an ≡ λx 4 , where λ > 0. For this case the coefficient α n,m is given as: where γ n ,n ≡ φ n ,k |x 4 |φ n,k . Due to parity of the eigenstates φ n,m , nonzero values of γ n ,n are given for n + n equal to an even nonzero integer number. It leads to some simplification in computing α n,m (t). We showed that the angular momentum is steady, not varying in time, for a spatial harmonic potential (which is associated to a symmetric central force field). However, even a small deviation from this configuration, such as anisotropic anharmonicity (which makes the force field a not-central one), can result in time-varying angular momentum. Considering initial condition with w = β in Eq. (5), the inosotropic anharmonicity removes the degeneracy of the excited states of the harmonic potential, and for small λ, the wavepacket in Eq. (10) modifies to where we introduce γ n ≡ λγ n,n / . Here, the core will move on the following orbit: which is given in the parametric representation. Examples of the dynamics are shown in Fig 4. The vortex core (white point) moves on a boomerang shape trajectory (shown in cyan), while the morphology of the vortex, which is shown with the red contours, is being modified. As the example of the anisotropic vortex, the vortex shape is initially stretched and then oriented along one branch of the core trajectory, while the core itself moves to a position too far from the vision, at infinite distance (Fig 1a-c). As the core comes back along its orbit, the vortex changes its orientation and then is stretched again along the other branch of the core trajectory; the core also repeats its previous motion. (Fig 1d-h). Moreover, there is a possibility to observe the edge dislocation, that is, when t cr ≡ N π 2(γ1−γ0) , with N as the natural number. Indeed one of the overall movements of the fluid can be thought as partially oscillating right and left due to the anisotropic term while also moving up and down due to the harmonic trap. This makes dark interference lines along the diagonal directions, which also recall the crossing lines between the two terms of the potential. Another interesting feature related to the anisotropic anharmonicity is the induced time-varying angular momentum which for the wavepacket (17) reads as: where the angular momentum oscillates in time. In contrast to the time harmonic perturbation, the amplitude of such oscillations is bounded to (− , ) interval. Interesting feature is that one can find some moments (t cr ), when the angular content is zero (as seen above, when the diagonal edge dislocation is maximally manifested and the vortex core is at infinite distance).

IV. RABI-COUPLED FIELDS
Rabi-coupled fields [14] can also show time-varying angular momentum in each one of the two (or more) components. To study such system, let consider a coherent coupling between the two fields ψ C and ψ X , where the main equations are given as: where Here, Ω is the Rabi frequency, and m C (m X ) shows the mass in the C(X) field. We consider no decay term in any of the two fields. For the initial condition we assume: where the vortex cores are located in different points in real space, defined by (r c,x , ϕ c,x ) in polar coordinates. We are not considering here the degree represented by a different amplitude factor between the two fields. One can find a close form solution in reciprocal space (k = k 2 x + k 2 y and θ k = arg[k x + ik y ]) by applying the Fourier transform F on each field, and obtain a general expression for the amplitudes of two coupled fields [15]: whereψ C,X = F[ψ C,X ] and we introduce M ± = Ω 2 (1 ± (m C /m X )) and k 2 Ω = (2 Ω) 2 + ( δ + k 2 M − ) 2 with δ ≡ E C − E X as the energy detuning. In general it is not possible to find a close form solution in real space, due to polaritonic factor [16], however, with respect to the initial conditions (22), we can find some approximate solutions in the regime of w 1 which is the case for the experimental observations [8]. In this regime, one notes that the dynamics is limited to a very small in-terval in reciprocal space, say of the order k max ∼ 0.1. It leads to some simplifications in Eqs. (23), namely, k 2 Ω ≈ (2 Ω) 2 + ( δ) 2 . If we expand the expression for the fields and apply to the initial condition of the two vortices (22), then we find where ω R ≡ (2Ω) 2 + δ 2 /2 is introduced as the frequency of oscillations, and Ω ≡ Ω(1 + m r ) with m r ≡ m C /m X . It is noted that if initially vortex cores are positioned in the same point of the real space (r c e iϕc = r x e iϕx ), they do not move at all; however, detuning (δ = 0) can induce dephased density oscillations [16][17][18], while the morphology remains constant, otherwise morphology undergoes a complex orbiting reshaping. The mean angular momentum content of each field is then given as: which is clearly independent of m r , however, oscillatory in time. Again, for zero detuning it can be observed that for co-aligned vortices, the momentum stays constant. If however δ = 0, even though the cores do not move, there are some spatial density waves and angular momentum oscillations induced by detuning. Most interestingly, in the general case, the two vortex cores keep orbiting each other. Then the angular momentum, alongside density, is coherently transferred and oscillating between the coupled fields, while the total angular momentum in the sum of the two fields is conserved. In other terms, due to the exchange of particles with different position and momentum, also the average angular momentum per particle in each of the two fields is periodically time-varying. This is well represented by the cores moving in the off-axis orbit (which is, periodically changing their distance from the center). In fact, average quantities, such as angular momentum, depend eventually on the nature of the dynamics, which is here Rabi oscillations between coupled fields [17,18]. Example of the dynamics is shown in Fig. 5 for m r 1, corresponding to the strong coupling regime of cavity photons and quantum well excitons [19].
Here we show some frames of the phase map for the photon (ψ C ) field. Initial vortex states can be introduced through resonant pulse pumping and their offset tuned by means of two-pulses coherent control [8]. The core moves along a circle orbit, shown in cyan, and similarly the morphology of the vortex, shown by red isodensity contours, illustrates nontrivial oscillatory shapes. While the core and the vortex itself evolve oddly, in a sense that the core is faster in the outer part of the beam [8,16], the angular momentum oscillates regularly.
Such a peculiar dynamics can be perceived readily by the following simplified version of the coupled field solutions in matrix representation (refer to appendix A for more details): with (27) Once more, it is noted that the solutions (26-27) are valid in the regime of w 1 and not correct in general cases, for example, when the dispersion effect plays a central rule [20].
One may have déjà vu that the aforementioned representation is reminiscent of rotation matrix in linear algebra, however, there is stark contrast due to the complex elements. Indeed, such a unitary matrix is common in the dynamics of binary fields in linear regime [18], and essentially is different than merely a rotation matrix as happens in real space. Yet, one can be hopeful for an interpretation similar to rotation. To this end, let introduce the complex number defining the polariton state (in the basis of the coupled fields) Z ≡ ψ C /ψ X so that the matrix representation in Eqs. (26-27) reappears as a bilinear transformation Z 0 → Z = M (Z 0 ), where Z 0 is the initial state and we introduce: Assuming δ = 0 hereafter for simplicity, the transformation (28) holds two fixed states Z 0 = ±1, which also represent the vortex cores in the dressed states (normal modes) [21]. Rewriting the transformation M in the nor- Time is in ps. The core is initially located at (−w, 0). Here, the dynamics is shown for the limit of mr = 0.001 1 for ψC in Eq. 24. The position of the core is shown by a white point, and trajectory of its motion is illustrated in cyan. The red contours show the shape (morphology) of the vortex. mal form: one finds that any Z moves in the complex plane mapping the polariton state around Apollonius circles that are symmetric with respect to fixed points, and then M maps each circle to itself. In advanced complex analysis, the specific transform M is an example of elliptic transform [22]. Furthermore, one can find an explicit representation of the Z point in terms of its stereographic projection on the Riemann sphere (here also coinciding with the Bloch sphere of polariton states). Denoting (X, Y, Z) as a point from the Riemann sphere, corresponding stere-ographic expressions read as: The representation on the sphere helps to visualize the smooth motion of the dynamics. On the sphere indeed the evolution of every point is a rotation around the axis passing along the two poles which represent the normal modes. Such axis is here set horizontal with respect to the complex plane. The previous considerations only concern the polariton states evolution, independently from any mapping or distribution they could assume in real space. Very interestingly, it is the specific and special initial condition with the two offset vortices of the said form (22) that creates a homeomorphism which links the Riemann sphere to the real space too [8] and mirror the complex plane into it. The vortex core moving along a finite size circle orbit in Fig. 5 is such an example, obtained upon a given set of parameters. A further example of straight line orbit (circle at infinite) is considered in appendix B.

V. CONCLUSIONS
Understanding the quantum fluid systems with timevarying angular content matures a further degree of freedom in the capacity of structured wavepacket to exert torque to different objects, ranging from some small size of nanometer to the size as large as milliliter. We studied analytically some possible systems, either based on confining potentials or on coupled condensates, for which the output vortex beams can carry dynamical orbital angular momentum. One can find a peculiar motion of the vortex core that is nested into a nontrivial morphology of the packet, which could be considered as a kind of temporal-anisotropic vortex structure, namely, its shape is being modified in the time course. In general, one can find the vortex core in motion with some local angular momentum distributed in space, however the expectation value of the angular content of the beam, which is done over the whole of wavepacket as a global quantity, may keep constant or be time-varying, depending on the physical system and the specific settings. The torque applied onto an object immersed into the beam would hence depend on both the beam and object morphologies, and can be made, for example periodic such as in a hybrid optomechanical torsion pendulum. This further degree of time-varying vortex and orbital control could benefit applications of OAM signal encoding and transmission, or precision metrology, such as gyroscopes and even Casimir torque measurements.