Abstract
In this work, we experimentally demonstrate for the first time the spontaneous generation of twodimensional excitonpolariton Xwaves. Xwaves belong to the family of localized packets that can sustain their shape without spreading, even in the linear regime. This allows the wavepacket to maintain its shape and size for very low densities and very long times compared to soliton waves, which always necessitate a nonlinearity to compensate the diffusion. Here, we exploit the polariton nonlinearity and uniquely structured dispersion, comprising both positive and negativemass curvatures, to trigger an asymmetric fourwave mixing in momentum space. This ultimately enables the selfformation of a spatial Xwave front. Using ultrafast imaging experiments, we observe the early reshaping of the initial Gaussian packet into the Xpulse and its propagation, even for vanishingly small densities. This allows us to outline the crucial effects and parameters that drive the phenomena and to tune the degree of superluminal propagation, which we found to be in close agreement with numerical simulations.
Introduction
Xwaves (XWs)^{1, 2} are a specific type of nonspreading wave packet that maintain their transverse shape along a large field depth with respect to Gaussian beams or other packets. Another wellknown class of nonspreading waves are solitons. However, the dispersion of solitons is constantly compensated by nonlinearity in the medium. Instead, XWs, as a type of nonspreading wave, are generally formed by Bessel beams and can maintain their shape in the absence of nonlinearity^{3}. XWs are a topic of great interest in multiple fields, spanning from photonics to acoustics, and are relevant in any system that is governed by the wave equation. The first experimental demonstration of such optical waves employed a cw laser light^{4}. These beams are not free from diffraction, but their transverse profile keeps its main peak well confined, whereas the weaker lateral peaks expand upon propagation. The same localization principle holds for pulsed XW packets that, in essence, are a polychromatic superposition of Bessel beams^{5}.
Since the early 1990s, XWs have been extensively studied both theoretically and experimentally using nonspreading acoustic pulses by Lu and Greenleaf^{6, 7}. Later, XWs were obtained with light by injecting subps laser pulses across a dispersive material^{3}, demonstrating the potential for signal transmission and imaging. Indeed, their application spans different systems^{1}: medical ultrasonic scanning; optical coherence tomography; nondestructive evaluation of materials and defect identification, including free space optical and radiobased telecommunication systems; optical tweezers, such as accelerating or guiding beams; plasmonics nearfield manipulation; microscopy; and signal transmission. In nanotechnology, the localized waves allow reliable production of highquality beams, which are required for optical and electronbeam lithography with subdiffraction resolution^{8}. Any of these experimental cases has relevance to, for example, realistic antennas truncated in time and space, and analytical solutions have been found for the finite energy content cases^{9, 10}. Among their fascinating properties, it is worth mentioning that Xwaves in vacuum also correspond to the simplest superluminal waveforms^{9, 10}. This indicates that XWs can have an effective velocity higher than c, which emerges from the superposition of ordinary Bessel beams^{4, 5} and, at the same time, conform to the constraints of special relativity and the causality principle^{1}.
Renewed interest in the XWs is also driven by their potential applicability in the field of atomic BoseEinstein condensates (BEC)^{11} and dissipative polariton condensates^{12}. Both of these systems bear deep similarity to the electromagnetic case because they can be described by nonlinear Schrödinger equations^{13, 14}. Polariton XW solutions were predicted not only for microcavity polaritons^{12}, but also in the case of Bragg polaritons by periodically embedding quantum wells directly into multilayer stacks^{15}. In both cases, the XW solutions rely on the locally hyperbolic dispersion (that is, including both negative and positive curvatures). Several theoretical proposals have been developed on this topic as well as on the possibility to obtain spontaneous Xwaves upon exploitation of the nonlinearities^{16, 17, 18, 19}. Recently, a quantum description of XWs has been developed, highlighting the difference in the entanglement properties between externally imprinted and spontaneously generated states^{20}. However, Xwaves have not yet been imprinted or generated in a BEC.
Our work bridges Xwave concepts to hybrid fluids of light and matter. We report the first experimental selfgeneration of an XW packet in a twodimensional (2D) excitonpolariton superfluid starting from an initial Gaussian photonic pulse. This effect has been achieved upon fine tuning of the polariton nonlinearity and proper balance of the positive/negative effective mass ratios along the transverse/longitudinal directions. Using ultrafast digital holography, the experiments show the initial pulse reshaping and propagating, which demonstrating its longitudinal localization down to vanishing densities of the packet. It is noteworthy that the 2D polariton geometry allows the axial XW density and phase profiles along the propagation direction to be determined. The optical access to the wavefunction phase allows highlighting some peculiar topological defects associated with the specific way we obtain the Xwave. Moreover, upon uniquely changing the initial amount of nonlinearity, we show a tunable superluminal peak speed with respect to the group velocity of the polariton system.
Microcavity exciton polaritons^{21, 22, 23, 24, 25, 26, 27} are bosonic particles that result from the mixing of two quasiparabolic modes, the quantum well (QW) excitons and the microcavity (MC) photons, with dispersions of highly unbalanced curvatures. The anticrossing feature of the bare modes, associated with the strong coupling regime, finally produces a highly nonparabolic shape for one of the new normal modes, namely, the lower polariton branch (LPB)^{28}. In particular, the presence of an inflection point, representing both a maximum of the group velocity (v_{g}=∂ω/∂k) and an inversion of the so called diffusive effective mass [m_{diff}=(∂^{2}ω/∂k^{2})^{−1}] (see Ref. 29), is the fundamental reason for the spontaneous XW formation. Polaritons also exhibit very strong nonlinearities^{30, 31} that are able to achieve superfluid regimes^{32, 33} that support quantized vortices^{34, 35}, or lead to several patterns^{36, 37, 38, 39} and soliton state formations^{40, 41}. However, we note that the XW, a solution that exists in the linear limit, is fundamentally different from the 2D bright solitons discussed in Refs 41, 42. In this case, localization was achieved in the so called bistability regime, in which the soliton wave packet was supported by an additional background pump. While the solitonic wave packets are well suited for polaritonic devices that utilize nonlinearity, such as logic gates or transistors^{43}, they are inherently fragile against particle loss that is unavoidable in any photonic system. On the other hand, linearly localized solutions are fundamentally robust against losses and have potential applications in data transport between distant system components. This approach has been demonstrated to be efficient in overcoming performance bottlenecks in electronic signal processing^{44}.
Materials and methods
Experimental methods
The experiments described here are performed on a GaAs/Al_{x}GaAs MC composed of three QW enclosed by two distributed Bragg reflectors, the details of which can be found in Refs 45, 46. The positions inside the MC are set to have the QWs in the antinodes of the confined photonic field. The strong coupling of the two bare modes, the photonic (ψ_{C}) and excitonic (ψ_{X}) fields, leads to two new hybrid modes: the LPB and the upper polariton branch (UPB). This sample is also grown on a specifically doped GaAs substrate with a transparency window centered at 830 nm, which consequently allows operation in a transmission configuration. The sample is kept at a constant temperature of 10 K using a cryostat to avoid thermal ionization of excitons.
This experiment uses an ultrafast digital holography setup (described in Refs 45, 46), in which the emission signal is allowed to interfere with a homodyne, uniform, reference plane wave. The two beams are sent at slightly different incidence angles onto a chargecoupled device camera to collect the associated interference pattern. The resulting interferograms are analyzed with a digital fast Fourier transformation to obtain the amplitude (ψ) and the phase (φ) of the complex wavefunction in real space. A delay line on the reference optical path allows us to scan the signal with respect to time by changing the time delay of the reference. The temporal resolution of this technique is mainly limited by the 2.5 ps duration of the laser pulse, which allows selective excitation of the lower polariton mode upon proper tuning at ~836 nm. The time step was set to 0.5 ps.
The circular polarization is set in the excitation beam to generate only one spin population and consequently maximize the interactions. The same reshaping effects are obtained upon a double total population density when using a linearly polarized excitation beam. The pump spot is set with a FWHM_{x,y} of 10 μm (FWHM_{kx,ky}=0.6 μm^{−1} in reciprocal space), to facilitate the nonlinear scattering process in real space and have a wide enough spot size to cover the dispersion range of interest in kspace.
Numerical methods
To illustrate the dynamical Xwave formation and localization induced by the nonlinearity, we performed simulations starting from a Gaussian initial state Ψ(x,y)=1/(2πσ^{2}) exp{−(x^{2}+y^{2})/(2σ^{2})}, where σ=FWHM/(2sqrt(2ln2)) and FWHM is the fullwidth at halfmaximum of the Gaussian spot. The GPE described in the text was solved numerically using the fourthorder RungeKutta method. The device parameters were as follows: m_{C}=4.27 × 10^{−5} m_{e}, γ_{C}=0.2 ps^{−1}, γ_{X}=0.2 ps^{−1}, g=2 × 10^{−3} meV μm^{2}, Δ=−0.55 meV and Ω_{R}=5.4 meV. The details of the numerical method are described in Ref. 47. Numerical computations were performed with a Zeus cluster in the ACK ‘Cyfronet’ AGH computer center.
Results and discussion
Dispersion and effective masses
The highquality Q factor of our microcavity indicates that the LPB and UPB modes are well separated with respect to their linewidths. The two modes can hence manifest their dispersions, observed upon collecting the offresonance excited fluorescence as shown in Figure 1a. Here, we will focus our attention only on the LPB, which shows a strong nonparabolic behavior at higher k vectors. Further, experimental details can be found in the ‘Methods and Materials’ section. The 3D representation of the LPB dispersion surface E(k_{x},k_{y}) is shown in Figure 1b in a region around the inflection point (k_{x}~1.62 μm^{−1}). In the figure, we highlight the nonparabolic characteristics by reporting two orthogonal crosscuts along the longitudinal direction (// blue curve, centered at k_{y}=0) and along the transverse direction (⊥ red curve, at k_{x}=2.15 μm^{−1}). The noteworthy feature that can be appreciated from the 3D representation is the opposing curvature of the two slices around the inflection point.
Moving along the central longitudinal line, the dispersion geometry always corresponds to a null transverse velocity [∂ω/∂k_{y}(k_{x},k_{y}=0)]. The longitudinal group velocity instead grows to a maximum (1.5 μm ps^{−1}) at the inflection point and decreases for larger, inplane longitudinal momenta (k_{x}), Figure 1c. At the same time, both the longitudinal and transverse curvatures of the dispersion surface change as a function of k_{x}, as clearly illustrated in Figure 1d. In particular, the curvatures have opposite signs inside the investigated region (the explored range is denoted by dots or by vertical ticks in any of the four panels), which corresponds to opposing effective masses.
Polariton Xwave
We resonantly excite the polariton superfluid with 2.5 ps laser pulses tuned at ~836 nm and focused to a ~10 μm diameter spot. In Figure 2, we experimentally show the dynamics of the effect optimized using k=2.35 μm^{−1} and 75 μW of pumping power. Figure 2a displays the modulus, and Figure 2b shows the phase of the polariton Xwave packet. The time zero in the temporal evolution is set when the pump stops injecting polaritons, which are then free to evolve within their lifetime. Initially, the density distribution reveals a Gaussian shape with a rather homogeneous phase (with just a weak radial gradient associated with the beam curvature). However, after 10 ps, the Xwave shape can be clearly distinguished. At the successive t=20 and 30 ps snapshots, we can detect just a small vertical spread of the packet, but without a significant distortion in the shape. Notably, the longitudinal waist size remains essentially constant, despite the polariton lifetime being as short as ~10 ps^{45, 46}.
An interesting feature can be seen in the phase map of Figure 2b: the appearance of four quantized vortices, at the edges of the packet. These vortices are shown in detail in the maps of Figure 2c and 2d, overlapping the streamlines of the phase gradient (red arrows) and the dots of the phase singularities (blue and green arrow circles). The diagonally displaced vortexantivortex pairs are an expression of the hyperbolic topology of the driving inplane momenta. Indeed, as evident in the center of the packet, the flows are pushing the polaritons inwards along the propagation direction, which keeps the signal compact, and outwards in the transverse direction.
The dynamics of the polariton superfluid were successfully modeled within the mean field approximation by a set of coupled equations equivalent to the GrossPitaevskii Equation (GPE):
where m_{C} is the effective mass of the microcavity photons, Ω_{R} is the Rabi frequency coupling the photonic ψ_{C} and excitonic ψ_{X} fields, γ_{C} and γ_{X} are the associated decay rates and g is the nonlinear interaction term in the exciton component. Further details are given in the ‘Methods and Materials’ section. The results shown in Figure 2e and 2f, represent the amplitude and phase maps, respectively, at 27.5 ps, demonstrating strong agreement with the main experimental features. The modulation in the tail of the signal that can be seen in the theoretical predictions shown in Figure 2e and 2f, could be due to the interference with a weak nonlinear scattering to opposite k_{x} states. This modulation may be not visible in the experimental data due to the achievable temporal resolution (2.5 ps limited by the reference pulse).
The opposite transverse and longitudinal effective masses force the GPE, which describes the polariton dynamics, to show a highly hyperbolic character. This behavior is crucial to sustain the Xwave phenomena, which demonstrates that the shape conservation does not rely on the nonlinearity, as in the case with solitons^{37, 41}, but rather on the dispersion morphology. It was previously shown that an Xshaped initial profile can be a stationary solution of the linear GPE model^{12}. We experimentally demonstrate that in a weakly nonlinear regime, an initial Gaussian state can be triggered to spontaneously evolve into a steady Xwave via an early fourwave mixing (FWM) process.
Nonlinear triggering
Although the nonlinearities play no role in the propagation and maintenance of the signal, they are crucial for the initial reshaping of the Gaussian pulse into the Xpacket. Indeed, the choice of the initial spot size in real space (density FWHM ~10 μm) produces a proper extension in the reciprocal (momentum) space (FWHM ~0.6 μm^{−1}), thus exploiting the negative curvature. The nonlinearity allows an asymmetric reshaping in momentum space based on the dispersion shape. Thus, an elongated spot in the reciprocal kspace is created along the direction of propagation, signifying a stronger confinement in real space.
To highlight the impact of the nonlinearities, the temporal dynamics at four different pumping powers are shown in Figure 3. At a low density, as shown in Figure 3a, the reshaping is absent and the signal spreads uniformly in both the longitudinal and transverse directions. However, an anisotropy in the intensity distribution between the longitudinal and transverse diffusions starts to appear when increasing the pump power, as shown in Figure 3b and 3c. At 75 μW, the reshaping reaches its optimum, shown in Figure 3d, and the packet shows a very welldefined Xshape, together with a small circular tail. Above this power, the dynamics enter into a strongly nonlinear regime (between 100 and 500 μW), where the redistribution due to high densities involves radial counterflows, which reshape the signal beyond a recognizable Xpacket. Such a regime occurs just before the onset of the dynamical nonlinearity inversion, leading to the real space collapse described in Ref. 37.
The role played by the inplane momentum k_{x} is shown in Figure 3e–3h, where only the injection angle is changed, while keeping the initial density constant. In Figure 3e, despite the large nonlinearities that are as high as in Figure 3d, no redistribution is observed. Upon gradually increasing the injection angle, shown in Figure 3f and 3g, the packet shows again a marked anisotropy in its diffusion along the longitudinal and transverse directions. This is due to the larger difference between the longitudinal and transverse effective masses in the excited region of the dispersion. This difference reaches its maximum at 2.35 μm^{−1}, where the reshaping is optimized, Figure 3h.
Localization and superluminality
We now focus on the propagation of the polariton XWs. Figure 4a shows the time evolution of the normalized polariton population (blue points) together with the pump pulse temporal envelope (solid red curve). The t=0 ps has been chosen to be at the maximum of the polariton population, which is when the pulse has essentially finished its pumping action and the polaritons free evolution starts. The growing longitudinal/transverse anisotropy can be appreciated upon a visual comparison between the associated amplitude time space charts in Figure 4b (longitudinal) and Figure 4d (transverse), as well as from the associated phase charts in Figure 4c and 4e. In the longitudinal charts, the signal propagates for 40 μm with a constant speed of ~1.20 μm ps^{−1}, and a final width very similar to the original shape. However, in the transverse maps, the width reveals the standard wave packet diffusion. This is clearly confirmed in Figure 4g and 4h, where both the longitudinal and transverse profiles are reported to be t=0 and t=30 ps, respectively, together with their associated Gaussian fits.
The power dependence of the differential spreading along the two directions is analyzed in detail in Figure 5a. Here, the temporal evolution of the longitudinal and transverse FWHM densities is shown for different excitation powers, corresponding to the previous Figure 3a–3d. For the lowest power (P=22 μW, red line), the reshaping is completely absent, and the wave packet expands continuously in both directions. The longitudinal (filled dots) and transverse (open dots) spreading have the same spreading rate. At larger injected power, and consequently stronger nonlinearity, the degree of anisotropy between the longitudinal and transverse size gradually increases (P=45 μW and P=53 μW are indicated by the orange and green dots, respectively), leading to the suppression of the longitudinal spread. Strikingly, for P=75 μW, the packet undergoes a longitudinal squeezing during the first 10 ps. This is associated with the nonlinear redistribution into the Xwave packet, whose shape can be neatly distinguished in the previous maps of Figure 3d. Based on these features, we may state that it is possible to qualitatively distinguish between the three main dynamic phases: pulse injection (−5÷0 ps), initial redistribution (0÷10 ps) and propagation (10÷30 ps). The numerical simulations of Figure 5b reproduce the experiments in a perfect agreement with our trends. We again note that this phenomenology is different from the bright solitons that are sustained under a cw background pump beam as in Refs 41, 42. In that case, the pump keeps the background just below the bistability threshold over its width and feeds the nonlinear maintenance of the moving soliton, which can propagate only within the pump spot. Instead, here, while the transverse width is not conserved (in agreement with the lateral positive mass), the longitudinal width is preserved along the propagation length of more than 40 μm, despite the packet arriving to this position with only a very small fraction of the initial population and density (one order of magnitude lower). Furthermore, the bright solitons in Refs 41, 42 exhibit a propagation speed that is set uniquely by the injection k of the cw background pump and, being dissipative solitons, is not affected by the seed pulse. In contrast, the Xwave offers the possibility to tune the group velocity of the packet using the incident inplane momentum, and can furthermore achieve a fine degree of tunability of the peak speed using the power control of the exciting pulse.
Indeed, our wave packet exhibits one of the most interesting signatures of Bessel Xpulses, superluminality. This effect is driven by the Bessel cone angle θ associated with the Xpulses^{48, 49, 50, 51}, whose peak moves (in vacuum) at v=c/cos(θ). In any system, the role of c is played by the group velocity v_{g} as obtained from the specific dispersion slope (illustrated for polariton waves in Figure 1). Here, we experimentally observe an increase in the speed of the density peak with respect to the centerofmass speed, up to a value of 6% in the case of the largest power, as shown in Figure 5c. We can evaluate that a maximum angle θ~18° is reached for P=75 μW. In terms of transverse inplane momentum, this angle corresponds to a δk_{y}~±0.6 μm^{−1}. These lateral kstates are induced in the initial FWM along the (nearly flat) transverse direction of the dispersion. Numerical simulations performed at different initial populations confirm the trend of the increase in the peak velocity with respect to the centerofmass, as shown in Figure 5d. We stress that different degrees of superluminal speed could be achieved without changing any other parameter (for example, spot width in real/kspace, central momentum, central energy, pulse width) but only the pulse power, consequently tuning the strength of the nonlinearity.
A complementary nonlinear effect is obtained along the longitudinal direction. As introduced in Figure 2, a specific feature of our structured polariton XW is represented by the leading and trailing islands that are developed around the main packet during the initial reshaping. These features can be neatly resolved due to the strong coherence properties of polaritons. Indeed, such coherence is maintained during the interference phenomena between the nonlinearly induced counterpropagating flows. In particular, the faster v_{g}(k_{x}) leading and the slower v_{g}(k_{x}) trailing subpackets represent the FWM states that are initially created at smaller (k_{x}−δk_{x}) and larger (k_{x}+δk_{x}) longitudinal momentum, respectively. The counter intuitive association between the group velocity and momentum differentials are due to the negative curvature of the dispersion. Instead, the circular shape of the interference between the three packets suggests that the two excited FWM states have a larger transverse extension in momentum space (k_{y}) with respect to the primary transverse profile. In Figure 6a, we report both the amplitude and phase longitudinal profiles (corresponding to P=75 μW and t=7.5 ps) to highlight the presence of two sharp πjumps in front and behind of the main packet, which is in a perfect spatial correspondence to dark dips in the density profile. The ignition time of such πjumps is also visible in the time space charts of Figure 4c at t≈7.5–8.0 ps. This is also the time of the vortex–antivortex pair generation in real space (see the main sequence in Figure 2). It is interesting to note that the appearance of such dark lines themselves is like a couple of phase singularities (quantum vortex) in the time space domain of Figure 4c. In general, the nonlinear selfdevelopment of a πjump may be a signature of a dark soliton^{37, 52}, which can be sustained in 2D condensates by repulsive interactions^{53}. In Figure 6b, we report the evolution of the phase profiles at equidistant time frames (every 2.5 ps). The profiles indicate how the sharp πjump is only present in a given frame at early times, before being smoothed as expected, due to a loss in the intensity. Hence, we may conclude that the dark soliton is a transient structure, a result of the nonlinear way we ignite the XW in the polariton fluid. The dark soliton is then washed out, without representing an intrinsic feature of the XW itself, as opposed to the longitudinal localization, which is instead preserved in time.
Conclusions
We have experimentally demonstrated the possibility to excite a peculiar class of traveling localized wave packets, called Xwaves, in 2D excitonpolariton fluids. Selfgeneration of an Xwave out of a Gaussian excitation spot is obtained via a weakly nonlinear asymmetric process with respect to two directions of the nonparabolic polariton dispersion. The dynamics of the packet are observed using ultrafast imaging, revealing a propagation over tens of micrometers, only limited by the polariton dissipation. We have tuned the nonlinearity and the injected inplane momentum to achieve both the optimal effect and preserve the longitudinal localization, even when the density fades away. Different degrees of superluminality have been achieved and associated with the variable transverse angular aperture induced by the nonlinear process in its early stage. Polaritonbased alloptical platforms are devised as robust candidates to study the fundamental science connected to 2D Xwave packets and possible future applications exploiting them in signal propagation.
Alternative 2D platforms are represented by, for example, multilayer stacks supporting Bloch surface waves (BSWs) at the external interface. These surface modes naturally exhibit very large inplane speed, which converts into longrange propagation, with negative mass dispersion and have exhibited exploitable nonlinearity upon coupling with an organic layer that is stable up to room temperature^{54}. The BSWs are enabling competitive systems compared to surface plasmon resonance for labelfree highsensitivity biosensing^{55}. Such systems also offer the possibility to easily pattern the external open surface to realize planar guiding or focusing elements^{56}, or even tilted, topgrating, launching, diffractionfree surface waves^{57}, analogous to what has been previously realized with plasmonic systems^{58}. Hence, BSW polaritons are a natural evolution for the study of Xwave pulse propagation over hundreds of μm and their exploitation for novel 2D optical tweezers and sensing combined functionalities.
Both QWMC and BSW polariton platforms represent nanophotonic technologies that are characterized by a strong and tunable χ^{(3)} nonlinearity resulting from polaritonpolariton interactions. The thirdorder nonlinearity governs not only the FWM process but also other useful phenomena, such as self and crossKerr modulation. Thus, we expect that they will be highly pronounced in our polariton superfluid. Tunable, efficient, nonlinear interactions are a ‘holy grail’ in photonic and optical systems^{59, 60} and quantum computing^{61} for building the optical gates necessary to construct a quantum computer.
Author contributions
LD and MM proposed the experiments. AG, LD, DB, MDG, GG and DS set up the laboratory configurations. AG and LD performed the experiments and analyzed the data. OV, MM and MS developed the theory, performed numerical simulations and provided the theoretical interpretations. All the authors discussed the results. AG, LD, OV, MM, MS and DS wrote the manuscript. DS supervised the research.
References
 1
Recami E, ZamboniRached M, HernandezFigueroa HE . Localized waves: a scientific and historical introduction. In: HernándezFigueroa HE, ZamboniRached M, Recami E, editors. Localized Waves, Wiley Series in Microwave and Optical Engineering. Hoboken, N.J.: WileyInterscience; 2008.
 2
Recami E, ZamboniRached M . Localized waves: a review. In: Hawkes P, editor. Advances in Imaging and Electron Physics, Vol. 156. Amsterdam: Elsevier; 2009. p235–353.
 3
Sõnajalg H, Rätsep M, Saari P . Demonstration of the BesselX pulse propagating with strong lateral and longitudinal localization in a dispersive medium. Opt Lett 1997; 22: 310–312.
 4
Durnin J, Miceli Jr JJ, Eberly JH . Diffractionfree beams. Phys Rev Lett 1987; 58: 1499–1501.
 5
Salo J, Fagerholm J, Friberg AT, Salomaa MM . Unified description of nondiffracting X and Y waves. Phys Rev E 2000; 62: 4261–4275.
 6
Lu JY, Greenleaf JF . Ultrasonic nondiffracting transducer for medical imaging. IEEE Trans Ultrason Ferroelect Freq Control 1990; 37: 438–447.
 7
Lu JY, Greenleaf JF . Nondiffracting X wavesexact solutions to freespace scalar wave equation and their finite aperture realizations. IEEE Trans Ultrason Ferroelect Freq Control 1992; 39: 19–31.
 8
Yalizay B, Ersoy T, Soylu B, Akturk S . Fabrication of nanometersize structures in metal thin films using femtosecond laser Bessel beams. Appl Phys Lett 2012; 100: 031104.
 9
ZamboniRached M, Recami E, HernándezFigueroa H . New localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies. Eur Phys J D 2002; 21: 217–228.
 10
ZamboniRached M, Fontana F, Recami E . Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finiteenergy case. Phys Rev E 2003; 67: 036620.
 11
Conti C, Trillo S . Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice. Phys Rev Lett 2004; 92: 120404.
 12
Voronych O, Buraczewski A, Matuszewski M, Stobiñska M . Excitonpolariton localized wave packets in a microcavity. Phys Rev B 2016; 93: 245310.
 13
Efremidis NK, Siviloglou GA, Christodoulides DN . Exact Xwave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential. Phys Lett A 2009; 373: 4073–4076.
 14
Conti C . Generation and nonlinear dynamics of X waves of the Schrödinger equation. Phys Rev E 2004; 70: 046613.
 15
Sedov ES, Iorsh IV, Arakelian SM, Alodjants AP, Kavokin A . Hyperbolic metamaterials with bragg polaritons. Phys Rev Lett 2015; 114: 237402.
 16
Couairon A, Gaižauskas E, Faccio D, Dubietis A, Di Trapani P . Nonlinear Xwave formation by femtosecond filamentation in Kerrmedia. Phys Rev E 2006; 73: 016608.
 17
Kolesik M, Wright EM, Moloney JV . Dynamic nonlinear X waves for femtosecond pulse propagation in water. Phys Rev Lett 2004; 92: 253901.
 18
Conti C, Trillo S, Di Trapani P, Valiulis G, Piskarskas A et al. Nonlinear electromagnetic X waves. Phys Rev Lett 2003; 90: 170406.
 19
Di Trapani P, Valiulis G, Piskarskas A, Jedrkiewicz O, Trull J et al. Spontaneously generated Xshaped light bullets. Phys Rev Lett 2003; 91: 093904.
 20
Ciattoni A, Conti C . Quantum electromagnetic X waves. J Opt Soc Am B 2007; 24: 2195–2198.
 21
Sanvitto D, KénaCohen S . The road towards polaritonic devices. Nat Mater 2016; 15: 1061–1073.
 22
Byrnes T, Kim NY, Yamamoto Y . Excitonpolariton condensates. Nat Phys 2014; 10: 803–813.
 23
Dagvadorj G, Fellows JM, Matyjaśkiewicz S, Marchetti FM, Carusotto I et al. Nonequilibrium phase transition in a twodimensional driven open quantum system. Phys Rev X 2015; 5: 041028.
 24
Deng H, Haug H, Yamamoto Y . Excitonpolariton BoseEinstein condensation. Rev Mod Phys 2010; 82: 1489–1537.
 25
Amo A, Sanvitto D, Laussy FP, Ballarini D, del Valle E et al. Collective fluid dynamics of a polariton condensate in a semiconductor microcavity. Nature 2009; 457: 291–295.
 26
Kasprzak J, Richard M, Kundermann S, Baas A, Jeambrun P et al. BoseEinstein condensation of exciton polaritons. Nature 2006; 443: 409–414.
 27
Balili R, Hartwell V, Snoke D, Pfeiffer L, West K . BoseEinstein condensation of microcavity polaritons in a trap. Science 2007; 316: 1007–1010.
 28
Kavokin AV, Baumberg JJ, Malpuech G, Laussy FP . Microcavities, 2 edn. Oxford, New York: Oxford University Press; 2017.
 29
Colas D, Laussy FP . Selfinterfering wave packets. Phys Rev Lett 2016; 116: 026401.
 30
Walker PM, Tinkler L, Skryabin DV, Yulin A, Royall B et al. Ultralowpower hybrid lightmatter solitons. Nat Commun 2015; 6: 8317.
 31
Vladimirova M, Cronenberger S, Scalbert D, Kavokin KV, Miard A et al. Polaritonpolariton interaction constants in microcavities. Phys Rev B 2010; 82: 075301.
 32
Amo A, Lefrère J, Pigeon S, Adrados C, Ciuti C et al. Superfluidity of polaritons in semiconductor microcavities. Nat Phys 2009; 5: 805–810.
 33
Berceanu AC, Dominici L, Carusotto I, Ballarini D, Cancellieri E et al. Multicomponent polariton superfluidity in the optical parametric oscillator regime. Phys Rev B 2015; 92: 035307.
 34
Amo A, Pigeon S, Sanvitto D, Sala VG, Hivet R et al. Polariton superfluids reveal quantum hydrodynamic solitons. Science 2011; 332: 1167–1170.
 35
Sanvitto D, Marchetti FM, Szymańsk MH, Tosi G, Baudisch M et al. Persistent currents and quantized vortices in a polariton superfluid. Nat Phys 2010; 6: 527–533.
 36
Whittaker CE, Dzurnak B, Egorov OA, Buonaiuto G, Walker PM et al Polariton pattern formation and its statistical properties in a semiconductor microcavity. Preprint at: arXiv:161203048, 2016.
 37
Dominici L, Petrov M, Matuszewski M, Ballarini D, De Giorgi M et al. Realspace collapse of a polariton condensate. Nat Commun 2015; 6: 8993.
 38
Manni F, Lagoudakis KG, Liew TCH, André R, DeveaudPlédran B . Spontaneous pattern formation in a polariton condensate. Phys Rev Lett 2011; 107: 106401.
 39
Wertz E, Ferrier L, Solnyshkov DD, Johne R, Sanvitto D et al. Spontaneous formation and optical manipulation of extended polariton condensates. Nat Phys 2010; 6: 860–864.
 40
Ostrovskaya EA, Abdullaev J, Desyatnikov AS, Fraser MD, Kivshar YS . Dissipative solitons and vortices in polariton BoseEinstein condensates. Phys Rev A 2012; 86: 013636.
 41
Sich M, Krizhanovskii DN, Skolnick MS, Gorbach AV, Hartley R et al. Observation of bright polariton solitons in a semiconductor microcavity. Nat Photonics 2012; 6: 50–55.
 42
Egorov OA, Gorbach AV, Lederer F, Skryabin DV . Twodimensional localization of exciton polaritons in microcavities. Phys Rev Lett 2010; 105: 073903.
 43
Ballarini D, De Giorgi M, Cancellieri E, Houdré R, Giacobino E et al. Alloptical polariton transistor. Nat Commun 2013; 4: 1778.
 44
Sun C, Wade MT, Lee Y, Orcutt JS, Alloatti L et al. Singlechip microprocessor that communicates directly using light. Nature 2015; 528: 534–538.
 45
Colas D, Dominici L, Donati S, Pervishko AA, Liew TC et al. Polarization shaping of Poincaré beams by polariton oscillations. Light Sci Appl 2015; 4: e350, doi:10.1038/lsa.2015.123.
 46
Dominici L, Colas D, Donati S, Cuartas JPR, De Giorgi M et al. Ultrafast control and Rabi oscillations of polaritons. Phys Rev Lett 2014; 113: 226401.
 47
Voronych O, Buraczewski A, Matuszewski M, Stobińska M . Numerical modeling of excitonpolariton BoseEinstein condensate in a microcavity. Comput Phys Commun 2017; 215: 246–258.
 48
Bonaretti F, Faccio D, Clerici M, Biegert J, Di Trapani P . Spatiotemporal amplitude and phase retrieval of BesselX pulses using a HartmannShack Sensor. Opt Express 2009; 17: 9804–9809.
 49
Mugnai D, Ranfagni A, Ruggeri R . Observation of superluminal behaviors in wave propagation. Phys Rev Lett 2000; 84: 4830–4833.
 50
Bowlan P, ValtnaLukner H, Lôhmus M, Piksarv P, Saari P et al. Measuring the spatiotemporal field of ultrashort BesselX pulses. Opt Lett 2009; 34: 2276–2278.
 51
ValtnaLukner H, Bowlan P, Lôhmus M, Piksarv P, Trebino R et al. Direct spatiotemporal measurements of accelerating ultrashort Besseltype light bullets. Opt Express 2009; 17: 14948–14955.
 52
Dominici L, Dagvadorj G, Fellows JM, Ballarini D, De Giorgi M et al. Vortex and halfvortex dynamics in a nonlinear spinor quantum fluid. Sci Adv 2015; 1: e1500807.
 53
Rodrigues AS, Kevrekidis PG, CarreteroGonzález R, CuevasMaraver J, Frantzeskakis DJ et al. From nodeless clouds and vortices to gray ring solitons and symmetrybroken states in twodimensional polariton condensates. J Phys Condens Matter 2014; 26: 155801.
 54
Lerario G, Ballarini D, Fieramosca A, Cannavale A, Genco A et al. Highspeed flow of interacting organic polaritons. Light Sci Appl 2017; 6: e16212, doi:10.1038/lsa.2016.212.
 55
Sinibaldi A, Danz N, Descrovi E, Munzert P, Schulz U et al. Direct comparison of the performance of Bloch surface wave and surface plasmon polariton sensors. Sens Actuators B Chem 2012; 174: 292–298.
 56
Yu LB, Barakat E, Sfez T, Hvozdara L, Di Francesco J et al. Manipulating Bloch surface waves in 2D: a platform conceptbased flat lens. Light Sci Appl 2014; 3: e124, doi:10.1038/lsa.2014.5.
 57
Wang RX, Wang Y, Zhang DG, Si GY, Zhu LF et al. Diffractionfree Bloch surface waves. ACS Nano 2017; 11: 5383–5390.
 58
Lin J, Dellinger J, Genevet P, Cluzel B, de Fornel F et al. Cosinegauss plasmon beam: a localized longrange nondiffracting surface wave. Phys Rev Lett 2012; 109: 093904.
 59
Kishida H, Matsuzaki H, Okamoto H, Manabe T, Yamashita M et al. Gigantic optical nonlinearity in onedimensional MottHubbard insulators. Nature 2000; 405: 929–932.
 60
Deng L, Hagley EW, Wen J, Trippenbach M, Band Y et al. Fourwave mixing with matter waves. Nature 1999; 398: 218–220.
 61
Gottesman D . The Heisenberg representation of quantum computers. Preprint at: arXiv:quantph/9807006, 1998.
Acknowledgements
We thank R Houdré and A Bramati for the microcavity device. AG, LD, DB, MDG, GG and DS are supported by the European Research Council POLAFLOW Grant 308136 and the Italian MIUR project Beyond Nano. MS and OV are supported by the NCN Grant no. 2012/04/M/ST2/00789 and MNiSW Iuventus Plus project no. IP 2014 044873. MS acknowledges support from the FNP project FIRST TEAM/20162/17. MM acknowledges support from NCN Grant 2015/17/B/ST3/02273.
Author information
Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no conflict of interest.
Additional information
Note: Supplementary Information for this article can be found on the Light: Science & Applications’website .
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialNoDerivs 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/byncnd/4.0/
About this article
Cite this article
Gianfrate, A., Dominici, L., Voronych, O. et al. Superluminal Xwaves in a polariton quantum fluid. Light Sci Appl 7, 17119 (2018). https://doi.org/10.1038/lsa.2017.119
Received:
Revised:
Accepted:
Published:
Issue Date:
Keywords
 negative mass
 nonlinearity
 polaritons
 superluminal
 Xwaves
Further reading

Nonlocal nonlinear optical X waves and their active control in a Rydberg atomic gas
Physical Review A (2020)

Universal SelfCorrecting Computing with Disordered ExcitonPolariton Neural Networks
Physical Review Applied (2020)

Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices
Physical Review A (2020)

Formation of nonlinear X waves in condensed matter systems
Physical Review B (2019)

Defect production by pure phase twist injection as AharonovBohm effect
Physical Review E (2019)