Topologically driven Rabi-oscillating interference dislocation

3.1 Dislocation propagation

Based on the pumping scheme previously introduced, we first pump the system with the pulse p₁, where a topological charge is brought first to the photon field and then, due to the Rabi oscillations, the field is transferred along with the topological charge to the exciton field. As the transfer proceeds, the topological charge gets distributed in both fields simultaneously. Equivalently, there are two cores in the polariton fields, but they are now, like the fields themselves, stationary. The density map of any of these fields includes a central null point (vortex core), where the density is zero. After performing some Rabi oscillations, the second pulse is introduced, to occupy a small region in real space, therefore with a fast diffusion. Also, the action of the second pulse is to immediately displace the vortex cores because of the incoming pulse interference with the previously created polaritons. As long as the wavepacket expands radially in space without interferences, its contribution to OAM can be well identified. The interference of the radial and azimuthal flows sculpts out spiral waveforms whose contributions to OAM can no longer be tracked. The Rabi oscillations and the SIP effect result in an interesting new phenomenology.

We present in Figure 2 the amplitude map of the photon field. Among different fields inside the cavity, only the photonic part, that escapes the cavity, can be detected via external devices. By using a spectral resolution or filtering tool, the upper and lower modes could be also detected separately in their photonic component, at the price of loosing some time resolution though. As we expect from the physics of SIP, rings of high and low densities are created; but there are new features due to the topology of the initial vortex. First, the rings of high (or low) densities are not symmetric (high density is, in this case, shifted to the left) whereas they remain symmetric in the absence of the vortex (see panel (c) of Figure 1). This is similar to the physics of the displaced vortices [9], but here this happens for the rings. The displacement can therefore occur in any direction, depending on the relative phase. This could have applications for clocking, e.g., by transforming a time delay into an angle. A symmetric wavepacket, such as a Gaussian, has a mean linear momentum ⟨−iℏ∇⟩ equal to zero. There is a diffusion of the wavepacket but this never leads to an overall or average motion, that is, the centroid of the wavepacket does not move. This changes for an asymmetric wavepacket, such as a displaced vortex state in the SIP regime, which can acquire an NTLM from the asymmetry in the system without any linear momentum being brought from the outside. To describe this, we compute the linear momentum [36]

Here, ⟨p _i ⟩ is the NTLM of the i = C,X,L,U fields. Typical variations of ⟨p _i ⟩ for the lower polariton wavepacket are shown in Figure 3(a). Since the upper packet diffuses fast, its associated NTLM is almost negligible and hence we are focused on the portion in the lower packet. The arrows show the evolution of the NTLM’s direction in time (with a time step between each arrows of 0.01 ps, the magnitudes are shown in panel (c)). Upon sending the second pulse, the transverse linear momentum (TLM) is formed and its net value eventually points in a fixed and stationary direction in the dressed (upper and lower polariton) fields, while it oscillates in the bare (photon and exciton) fields. Interestingly, the fixed direction of the NTLM can be manipulated by the time delay between the two pulses. In panel (a), we assume a time delay of Δt = 1.5 ps and in panel (b), different values of Δt result in a different final direction. This allows to trigger and orientate a net motion of the centroid of the wavepacket ( ⟨ r ⟩ = I ̂ ⟨ x ⟩ + J ̂ ⟨ y ⟩ ) in the SIP regime. Panel (d) shows the trajectories of ⟨r⟩ for the bare and the lower polariton wavepackets, contrasting the spiral trajectories of the former as opposed to the fixed direction maintained by the NTLM for the polariton. Indeed, by tuning the time delay, a propagating wavepacket can be oriented in any specific direction. This may have applications in atoms/molecules segregation [37] or phase transformation [38].

Figure 2:

The regime of Rabi interference dislocation, shown in the amplitude of the photon component of polaritons.

After sending the second pulse, the mixed dynamics of SIP and the moving Rabi vortices produce a moving dislocation in the pattern. The dislocation stems from the misalignment of the rings, induced by the motion of the vortex core. Here, different frames along one Rabi period have been chosen. In each frame, the position of the vortex core is shown by a blue point. It follows an expanding orbit that is shown (in one period) by the dark-dashed green line in the first panel. The starting point is marked as a green dot and the end point as a red one. For the numerical simulations, we used: ℏΩ = 4 meV, ω = −Ω/3, W₁ = 5 μm, W₂ = 1.2 μm, t₁ = 0, t₂ = 1.5 ps, δ _t = 0.3 ps, R₁ = 1 ps⁻¹, and R₂ = 4 ps⁻¹.

Figure 3:

(a) Creation, evolution and stabilization of the NTLM vector, here for the lower polariton field. The final direction is shown by the red vector. (b) Different values of Δt (time delay between the pulses) yield different final NTLM directions. (c) Amplitude of the total linear momentum ⟨p⟩ = ⟨p_C⟩ + ⟨p_X⟩ = ⟨p_L⟩ + ⟨p_U⟩ for different Δt. Linear momentum is created with the second pulse and remains constant when its direction stabilizes. (d) Wavepacket centroids for various fields, showing with curly curly oscillations of the bare fields and straight linear motion for the dressed ones (the small deviations in the L direction at earlier times are not visible on this scale).

Other interesting features pertain to the time dynamics. For instance, rather than a mere diffusion of the ripples, they oscillate with the same rhythm as the vortex core motion, which is itself induced by the Rabi oscillations. Another example follows from the azimuthal misalignment of the rings, mostly visible in the low density region. This produces a dislocation (fork-like signature) in the pattern of fringes. This is particularly visible in panels (b) and (c) of Figure 2. In panels (a) and (d), the dislocation is also there but repelled at the frontier of the packet, where it is less visible. This is due to the motion of the vortex core positioned in the outermost region, adding an extra low density ring to the inner ripples. The dislocation oscillates at the Rabi frequency, since the vortex core does too. Dislocations or forks generally appear in interference patterns and are associated to vortices. For example, interference of a vortex with a plane wave or a vortex with itself can result in the emergence of a dislocation [39]. However, the appearance of a dislocation in the SIP regime is of a different origin, namely, the displaced vortex changes the alignment of the ripples, inducing a dislocation at distances far from the vortex itself. We provide movie animations of each density profile in the Supplementary Movies SM1-SM4 (see Supplementary Material for descriptions). An additional movie SM5 is also attached that displays the dynamics of a slice of the photon density near the maximum point of the density (contour of Max|ψ_C|(1 − ϵ), where ϵ is small). This shows clearly a see-saw motion for the NTLM, rhythmed by the Rabi oscillations. This could have application for a more forceful, sweeping particle clearing [40].

3.2 Polarimetric analysis: topology imaging

The concept of transverse light polarization naturally arises in many optical situations and its spatial distribution or texture can be non-homogeneous. The phase singularity in a plain quantum vortex can become a polarization singularity in so-called spin vortices. For a scalar field, topological defects of two kinds can be found by looking at the phase isolines orientation [41, 42] and their textures, with a negative integer winding (−1) for saddle points and positive (+1) for dislocations. Dislocations, also known as phase singularities or phase vortices, can be further distinguished by their other quantum number (−1 or +1), associated to the phase winding around the singularity. In the spin vortices of a vector field, the textures can be associated to the polarization of the fields, e.g., also encoded in the orientation and eccentricity of the polarization ellipses [43–47]. In such a case, and similar to polarization states in optics, we can extend the representation and concepts of the ellipse geometry to our binary field (see Supplementary Material for further details). Namely, we use a Poincaré or Bloch sphere representation, associated to the complex polariton state, and choose the dressed modes (upper and lower) basis as the vertical axis of such a sphere. One can define the quantity

where the nodes of σ give the circular state (or C-points [31, 32]) of the elliptical field, and also the vortex core positions. Indeed, here the C-points are representing a pure lower (upper) state; and as such they require a zero density in the opposite upper (lower) field. For such a reason, a vortex in one field is also a C-point in this representation (the vortex core bears a zero density so it is a pure state of the opposite field). Also, the phase of the quantity in Eq. (6) is the relative phase between the upper and lower fields, arg[σ] = arg[ψ_L] − arg[ψ_U].

An example of the structuring of the complex function σ is given in Figure 4, where the panels show the morphology based on the relative phase and amplitude map—phase and amplitude isolines—and ellipses representation, respectively, together with the C-points. Panel (a) shows the positions of the singularities, or vortex cores, overlapped to the phase map (arg[σ]). The vortex core in the upper field (green point) is connected to the vortex core in the lower field (yellow point) via relative phase isolines of all possible values (whereas the 2π isoline is the one clearly visible in the map). A space chart of s = | ψ U | 2 − | ψ L | 2 | ψ U | 2 + | ψ L | 2 is shown in panel (b). This effectively defines the latitude of a quantum state on the Bloch sphere (see the Supplementary Material for further details). Interestingly, s is not homogeneous in space. While the relative phase maps also the longitude of the Bloch sphere, the value of s that varies in the interval [−1, 1], maps the latitude on the sphere, with s = cos(θ) with θ the polar angle. The parameter s is +1(−1) at the vortex core of the lower (upper) polariton, while it is 0 at the positions of equal upper/lower content. The C, X vortex cores are moving in an orbit described by the s = 0 line. Correspondingly, the isolines of relative phase along with the isocontent s, are shown in panel (c) (black and solid lines, respectively). The panel (d) shows the corresponding elliptical representation. The ellipse is a very useful tool to describe the polarization state of the electric fields in optics, as well as any other vector field, both in 3D as well as in 2D space. In particular, its eccentricity and orientation can be associated to the (latitude and longitude of the) points on the surface of a unique topological structure, namely the Poincaré sphere. Similarly here, the spatial distribution of normalized ellipses, describing now the binary polariton state on a Bloch pseudospin sphere, is shown in panel (d). Here we find two kinds of C-points; one indicated by a yellow circle, that corresponds to the lower state, and another depicted by a green circle, associated to the upper vortex core. They are associated to opposite winding of the relative phase around each singularity. At the C-points, the ellipses become circles, and one can find the opposite winding of the ellipses axis around two different C-points. Indeed, the orientation of the ellipse axis is directly mapping (half of) the longitude angle on the sphere, which is given by the relative phase. The typical texture of a so-called star spin vortex is visible around the lower mode vortex core (yellow C-point), while the opposite so-called lemon texture is more spread in space around the upper mode core (green C-point), and hence less recognizable. The two X, C vortex cores are associated to a diagonal, antidiagonal linear state (red and blue segments in the panel). They move, thanks to the Rabi oscillations, along the inner closed orbit, around the lower mode vortex core (Figure 4(e) and (f)). Such an orbit is also the s = 0 isocontent line, and represents a so called L-line (the one where the star pattern is visible). The external L-line, the big circle in panel (c), does not represent an orbit, in the sense that it has the same linear elliptical state at all times. The orientation of the ellipses are varying in space, describing some 2D domain on the sphere [33]. The texture of the sphere coordinates (longitude and latitude), represented by arg[σ] and s isolines, is not conformally mapped to the real space (the two isoline families are not mutually orthogonal everywhere in space, as they are on the sphere) differently from the conformal full Bloch beams [8, 44]. We ascribe this to the achieved configuration being a superposition of LG₀₀ and LG₀₁, both with the same center but not the same width. However, the integral of the Bloch sphere surface density in real space is equal to 4π, hence suggesting that the emitted beam is analogous to a distorted full Poincaré beam, where sphere texture is not conformal to the real space.

Figure 4:

Various representations of the fields morphology, namely, based on their relative phase σ = arg ψ U * ψ L and amplitude s = { | ψ U | 2 − | ψ L | 2 } { | ψ U | 2 + | ψ L | 2 } (in upper–lower polariton basis), isolines of σ and s, ellipses (in upper–lower polariton basis), and amplitude (the square root is used to enhance the contrast) of the photon and exciton fields together with vortex cores and saddle points, at t = 3.15 ps.

(a) The phase map of σ, also shows the positions of vortices in the upper–lower basis. Indeed, the unitary vortex charges in each of the two fields compose a vortex dipole in the relative phase map. The displaced vortices (located at yellow and green rings) also induce two saddle points (indicated by two cross symbols) in the relative phase. (b) The relative amplitude s (or equivalently the amplitude of quantum states) varies between −1 and 1 and has an inhomogeneous profile. Panels (a) and (b) imply that all quantum states of the Poincaré sphere are simultaneously present in the 2D space. (c) The isolines of relative phase (shown as black curves, at π/3 intervals) are superimposed with isolines of the relative amplitudes: s (white lines, at 0.4 intervals). (d) Elliptical representation of the polariton state and its spatial variations in 2D space. The green (yellow) point, where the state of the ellipse becomes a circle, corresponds to the upper (lower) vortex core. The state of the ellipse is linear at the exciton and photon vortex cores (blue antidiagonal and red diagonal, respectively). Indeed, at the exciton and photon vortex cores, the polariton state is purely photonic and excitonic, respectively. (e) Squared amplitude of the photon field, with the position of the vortex core at intersection of s = 0 and σ = 0. (f) Squared amplitude map of the exciton field, with the position of the vortex core at the intersection of s = 0 and σ = π. The vortex cores in real space move along the inner white curve during a Rabi cycle.

3.3 Orbital angular momentum oscillations

In a standard symmetric vortex, the amount of rotation of the density is not directly visible, since for symmetry reason, the intensity ring remains despite of the rotation in the phase, that is nevertheless able to exert a physical torque. Indeed, the symmetric vortex has a total OAM ⟨L _z ⟩ and an OAM per particle ⟨ L ∼ z ⟩ ≡ ⟨ L z ⟩ / N , where we define [36]

In the standard symmetric vortex of unitary charge, the OAM per particle is unitary as well, while if the core is off-axis, also leading to asymmetric shapes, the OAM per particle can be fractional [7, 48, 49]. It is therefore interesting to find what is the OAM per particle in our structured case, that not only displays offset vortex cores but also spiral patterns, that are furthermore reshaping in time.

To answer this, we consider the phases involved in the dynamics at two given times, close to the arrival time of the second pulse and after a few picoseconds, as shown in Figure 5. Each row (Figure 5(a)–(d)) corresponds to a time, while each column corresponds to the relative phase between the various fields involved (Figure 5(a)–(d) for the photon–exciton and lower–upper relative phase, respectively). Shortly after sending the second pulse, the vortex cores in each fields are displaced. The simplest dynamics happens for the core in the upper field. Due to a fast diffusion of the upper polaritons injected by the second pulse (which is very tight), the vortex core is less affected by interferences and is displaced only slightly from the origin, where it remains almost at all times (green circle or label in Figure 5). Such a feature of the dynamics can be partially controlled by the energy of the pulse (ω), namely, for more negative energy, the core in the upper field is less displaced.

Figure 5:

Relative phases and vortex cores positions at different times.

(a) and (c) Show the phase map of σ = ψ U * ψ L , which is the relative phase between the upper and lower fields, at t = 3.15 ps and t = 8.14 ps, respectively. The corresponding phase map of ψ C * ψ X , between the exciton and photon fields, is shown in panels (b) and (d), at the same times as (a–c). The position of the vortex cores in all the fields are shown by C, X, L, U labels (for photon, exciton, lower polariton and upper polariton). The upper mode, excited by the second and tight pulse, undergoes a rapid diffusion, and the initial vortex core remains little affected by interferences, remaining almost at the center of the packet, close to the origin of the 2D map (green circle). The core in the lower mode experiences a larger interference with the tight excitation diffusing more slowly, for such a reason it is displaced at the boundary of the packet (yellow circle). The core in the lower mode is then set aside due to the delayed arrival of different radial momenta (changing the interfering phase and hence azimuthal position of the vortex core). However, the cores in the photon and exciton fields are displaced at the boundary as well, and due to Rabi oscillations, they orbit, e.g., around the lower mode core. Differential diffusion between the two normal modes then leads to a loss of their overlap, damping the Rabi oscillations and reducing the C, X vortex cores orbits.

The dynamics for the other core is different. Indeed, the fraction injected by the second pulse, despite being a tight beam, is subject to the group velocity limitations of the lower polariton mode, and undergoes the SIP effect. Only a portion of the momenta composing the packet travels fast outwards. The remaining part, filling the center, moves the vortex core of zero density at a larger distance, as a result of interferences (yellow circle or label in Figure 5). The combination of structured diffusion and azimuthal phase winding from the vortex, results then in the greater lateral movement of the core in the lower field (panels a, c). Hence, it can be seen that the OAM per particle has been altered in both of the normal modes, due to the superposition of the two pulses, and this is reflected in their vortex cores displacements. However, the photon and exciton fields oscillate due to the Rabi oscillations, and their vortex cores orbit along some path (blue and red circles or labels in Figure 5), which is reflected in their OAM oscillations. As far as the differential diffusion between the normal modes reduces their overlap, both the OAM oscillations decrease in time as the orbits shrink.

Such a structured packet can be engineered by tuning the time delay between the two pulses and/or the size of the wavepackets. The key point is that for smaller wavepackets, the Rabi oscillations go off faster. After the second pulse is received and the vortex core is displaced, the wavepacket undergoes density rotation. As long as the Rabi oscillations continue in time, the wavepacket carries extrinsic OAM and rotates. We show in Figure 6 the time-varying total OAM and OAM per particle for different cases. A common feature of these panels is that the oscillations attenuate in time. The rate at which the oscillations decrease depends on the wavepacket size, laser energy detuning, and the energy detuning between the bare fields. Their combination eventually accounts for the reduction of the overlap of the injected l = 0 fraction with the previous l = 1 part, faster in time when using smaller packets (which is, for tighter beams) and for more negative laser energy. Here, the oscillations in ⟨ L ∼ z ⟩ implies a varying distance of the vortex core in a given field. Compared to the previously discussed time-varying OAM [8], [9], [10], here the oscillations stop in a few picoseconds, Figure 6(a) and (b), despite the modes having no decay and no central linear momentum (i.e., no NTLM) being imparted. This comes fully from the differential diffusion of the packets, which separates the dressed fields in a few picoseconds. We also consider the effect of energy detuning, defined as δ = E_C − E_X. A positive δ can trigger more oscillations, and also larger OAM per particle ⟨ L ∼ z ⟩ as shown in Figure 6(c) and (d). For a negative detuning, the oscillations are fading away very fast, as shown in Figure 6(e) and (f).

Figure 6:

Time-varying orbital angular momentum.

Although there is no decay of the fields, oscillations dampen in time. This is due to the fast diffusion of the packet in the upper mode and the consequent loss of overlap with the lower mode. This happens in a small fraction of the polariton lifetime and can be engineered either by energy or packet size manipulations. In (a–b) the energy detuning is δ = 0. We assume δ = 1.5 in (c–d) and δ = −1.5 in (e–f). We used W = 2 μm for the first pulse and w = 0.5 μm for the second pulse.