Abstract
Magnetic monopoles^{1} are pointlike sources of magnetic field, never observed as fundamental particles. This has triggered the search for monopole analogues in the form of emergent particles in the solid state, with recent observations in spinice crystals^{2,3,4} and onedimensional ferromagnetic nanowires^{5}. Alternatively, topological excitations of spinor Bose–Einstein condensates have been predicted to demonstrate monopole textures^{6,7,8}. Here we show the formation of monopole analogues in an exciton–polariton spinor condensate hitting a defect potential in a semiconductor microcavity. Oblique dark solitons are nucleated in the wake of the defect^{9,10} in the presence of an effective magnetic field acting on the polariton pseudospin^{11}. The field splits the integer soliton into a pair of oblique halfsolitons^{12} of opposite magnetic charge, subject to opposite effective magnetic forces. These mixed spinphase excitations thus behave like onedimensional monopoles^{13}. Our results open the way to the generation of stable magnetic currents in photonic quantum fluids.
Main
Magnetic monopoles are the magnetic counterparts of electric charges, characterized by a divergent field. The seminal work of Dirac^{1} showed that monopoles are not forbidden by the laws of quantum mechanics. In particular, he considered particles characterized by a wavefunction with a nodal line and a nonintegrable phase around it. One route to create an object behaving like a monopole is thus to engineer a wavefunction with such characteristics. A model system to do this is a spinor Bose–Einstein condensate^{14,15,16}, demonstrating properties such as superfluidity or persistent currents. In reduced dimensions, not only do these quantum fluids support topological defects^{17}, such as vortices (two dimensional; 2D) or solitons (1D) characterized by a node, but an adequate spin distribution can also provide a vector field with a nonzero divergence, satisfying Maxwell’s equations for a point magnetic charge^{6,7,8}. Monopoles can then be arranged in spinor condensates in the form of mixed spinphase topological excitations, with a magnetic analogue of the Coulomb force acting on them^{13}.
Exciton–polariton (polariton) condensates seem a wellsuited system to evidence and study such original effects in quantum fluids. Polaritons are the quasiparticles arising from the strong coupling between excitons and photons confined in planar semiconductor microcavities (InGaAs/GaAs/AlGaAs in our case)^{18}. Polariton fluids are easy to manipulate with standard optical techniques^{19,20,21,22} and they have recently become a model system for the study of quantum fluid effects such as superfluidity^{23}, vortex formation^{24,25} or oblique solitons^{9,10}. Their spin structure is especially interesting: polaritons are bosons with only two allowed spin projections ±1 on the growth axis of the sample, which couple to circularly polarized (σ_{±}) photons in andout of the cavity. A coherent superposition of different spin populations gives rise to polarization states that can be described by a pseudospin vector S mapped onto a Bloch sphere (Fig. 1a). Another remarkable feature of microcavities is the presence of an effective inplane magnetic field (Fig. 1b) induced by the polarization splitting between the transverse electric (TE)–transverse magnetic (TM) polarization modes^{11}. The effective field interacts with the polariton pseudospin, adding a magnetic energy term to the Hamiltonian (see Supplementary Information) and it provides the analogue of a Coulomb force acting on topological monopoles^{13}. Finally, polariton–polariton interactions are strongly spinanisotropic of the antiferromagnetic type^{26}. This is an absolute requirement for the observation of any stable monopole structure^{7,8}. Indeed, a topological monopole in a twocomponent spinor condensate is stable against its destruction by an inplane effective magnetic field if the difference in the interaction energy between the same and opposite spins exceeds the magnetic energy^{13}.
One kind of spinphase topological defect already reported in polariton quantum fluids are the socalled halfvortices^{27,28}. However, no probing of the monopole behaviour has been possible yet, because of the disorderinduced pinning^{28}. The 1D counterpart of a halfvortex is a dark halfsoliton, characterized by a notch in the polariton density of the fluid, and a simultaneous phase and polarization rotation of up to π/2 in the condensate wavefunction across the soliton^{12}.
Here, we report on the experimental observation of oblique halfsolitons and on their separation and acceleration caused by the effective magnetic field present in semiconductor microcavities. Oblique solitons (or halfsolitons in the spinor case) are formed in the wake of a localized potential barrier present in the path of a flowing condensate^{9}. They can be seen as the trajectories of 1D solitons in the direction perpendicular to the flow (x), travelling across a 2D flow. The second spatial coordinate (y, parallel to the flow) represents the time coordinate of the 1D system (t = y/v_{f}, where v_{f} is the flow velocity). This means that the soliton trajectory becomes traceable in a steady state regime^{29}. Studying such trajectories, we demonstrate that an integer oblique soliton separates into a pair of halfsolitons of opposite magnetic charge accelerated in opposite directions.
In our experiments we create a polariton fluid in a semiconductor microcavity (see Methods) at a temperature of 10 K by quasiresonant excitation of the lower polariton branch with a continuous wave Ti:sapphire monomode laser. Polarizationresolved realspace images of the polariton fluid in the transmission geometry are then recorded on a CCD (chargecoupled device) camera owing to the photons escaping out of the cavity (Fig. 1c). The fluid is injected at supersonic speed (inplane momentum of k_{P} = 1.3 μm^{−1}, see Methods), upstream from the potential barrier formed by a structural photonic defect present in our sample. Under these conditions, a circularly polarized excitation beam leads to the formation of pairs of oblique dark solitons in the wake of the barrier^{9}, characterized by a phase jump close to π across each notch (see Supplementary Information). In the present experiments, we create a polariton gas with linear polarization parallel to the flow (TM polarization, along the y direction). This is a key feature needed to explore the nucleation of spinphase topological excitations, which can be evidenced by analysing the circularly polarized components of the emission.
First, we demonstrate the formation of halfinteger solitons. Figure 2aI shows the nucleation of two oblique dark solitons to the right of the barrier wake in the σ_{+} component of the emission. They can be identified as dark straight notches in the polariton density. These solitons are almost absent in the σ_{−} component (Fig. 2aII). In turn, in the σ_{−} emission, a deep soliton (S1) clearly appears to the left of the barrier wake (blue arrow), where only a very shallow one is present in σ_{+} (see the profiles in Fig. 2c). The absence of mirror symmetry between Fig. 2aI and aII arises from the specific and uncontrolled form of the natural potential barrier. The individual dark solitons in each of the S_{z} = ±1 states of the fluid appear as long spatial traces with a high degree of circular polarization (ρ_{c} = (I^{+}−I^{−})/(I^{+}+I^{−}), where I^{±}is the emitted intensity in σ_{±} polarization), as shown in Fig. 3a. Interferometric images obtained by combining the realspace emission with a reference beam of homogeneous phase (Fig. 2bI and bII) give access to the phase jump across each soliton. For instance, for the soliton S1 observed in σ_{−}, 42 μm after the obstacle we measure a phase jump of Δθ_{−} = 0.85π (Fig. 2e; note that it would be π for a strict dark soliton with zero density at its centre^{9}), whereas in the same region the phase in the σ_{+} component does not change (Δθ_{+}≈0).
A dark soliton present in just one spin component of the fluid is the fingerprint of a halfsoliton^{12}. The mixed spinphase character of these topological excitations is further evidenced when analysing them in the linear polarization basis. In the regions where the two circular polarizations are of equal intensity (that is, the fluid surrounding the halfsolitons) we can define a linear polarization angle η = (θ_{+}−θ_{−})/2 and a global phase ϕ = (θ_{+}+θ_{−})/2, where θ_{+} and θ_{−} are the local phases of each circularly polarized component^{12,27}. In our experiments we directly access the phase jump Δϕ and the change of η across the solitons by studying the linearly polarized emission in the diagonal and antidiagonal directions (polarization plane rotated by +45° and −45° with respect to the TM direction). Figure 2d,f shows that the halfsoliton S1 is also present in these polarizations with a phase jump of Δϕ≈0.4π. This confirms that across the halfsolitons, ϕ undergoes a jump Δϕ≈0.85π/2≈(Δθ_{+}+Δθ_{−})/2, that is, onehalf the phase jump observed in the circularly polarized component in which the soliton is present. We also expect a similar jump Δη of the direction of polarization. This is demonstrated in Fig. 3b, where all the halfsolitons present in our fluid (dashed lines extracted from Fig. 2aI and aII) appear as walls between domains of diagonal (magenta) and antidiagonal (green) polarization. Mapping the linear polarization vector in the vicinity of soliton S1 (Fig. 4a), we deduce a jump of the polarization direction of Δη≈0.32π (Fig. 4c), close to Δϕ, the ideal expected value.
Analysing the halfsoliton trajectory from polarizationresolved realspace measurements, we study their acceleration in the field originating from the TE–TM splitting present in the structure^{11}, pointing in the direction of the flow (y, red arrow in Fig. 1b). The acceleration arises from the interaction between this magnetic field and the pseudospin texture of the halfsoliton, shown in Fig. 4b for S1. In the direction perpendicular to the soliton (dotted line), the inplane pseudospin S is divergent, because it points away from S1 on both sides, as expected for a magnetic charge.
We are able to evaluate the force acting on the halfsoliton as the gradient of the magnetic energy with respect to the halfsoliton position x_{0}. The magnetic energy per unit length is , where the integral is performed along the x′ transverse direction, perpendicular to the halfsoliton located at x_{0}. The energy has a positive contribution from the left of the halfsoliton (S and pointing in opposite directions), and a negative one from the right (S and having the same direction). For the magnetic energy to be minimized, a magnetic force appears, pushing the halfsoliton towards the left, increasing the negative contribution. As solitons are density notches, their effective mass is negative^{17} and, therefore, the acceleration is in the direction opposite to the force. Thus, the halfsoliton S1 that appears in the σ_{−} component of the fluid accelerates towards the right, as sketched in Fig. 4b. The direction of the acceleration is opposite for the soliton present in the σ_{+} component (see arrows in Fig. 3b).
The monopole dynamics allows an understanding of the mechanisms of formation of the halfsolitons in our experiments. An integer soliton nucleated right behind the obstacle can be seen as a superposition of two halfsolitons of opposite magnetic charges. The presence of the TE–TM effective magnetic field makes them experience opposite magnetic forces, leading to their separation and to the curved trajectories depicted as dashed lines in Fig. 3, a behaviour similar to the monopole separation in spin ice under a magnetic field^{3}. The halfsolitons pushed towards the centre are slowed down, gaining stability and becoming darker as the trajectory becomes parallel to the field. Those pushed outwards gain velocity and become shallower until they eventually disappear (see Supplementary Information and the black dashed lines in Fig. 3). The trajectories of these expelled secondary halfsolitons are perturbed far from the obstacle axis by the presence of further solitons nucleated by the large barrier, particularly on the right side of the images. The monopole behaviour and soliton separation are well reproduced by a nonlinear Schrödinger equation including spin and the effective magnetic field present in our microcavities (see Fig. 3 and Supplementary Information). The analogy of halfsolitons with magnetic monopoles goes well beyond the behaviour reported here under an applied magnetic field. Our theoretical model predicts repulsive and attractive interactions between halfsolitons depending on their respective charges (see Supplementary Information).
A remarkable feature of halfsoliton monopoles is that they can be thought of as charged partly photonic quasiparticles, propagating with a high velocity in a fluid that supports superflow^{23}. Their generation can be well controlled by the phase and density engineering of the polariton wavefunction, their trajectories can be easily followed using standard optical techniques and their dynamics can be controlled by applying strain or external electric fields, which modify the effective magnetic field^{30}. Furthermore, owing to the developed engineering of the polariton landscape^{19,20}, we open the way to the realization of magnetronic circuits in a polariton chip.
Methods
Sample description.
The experimental observations have been performed at 10 K in a 2λ GaAs microcavity containing three In_{0.05}Ga_{0.95} As quantum wells. The top and bottom Bragg mirrors embedding the cavity have, respectively, 21 and 24 pairs of GaAs/AlGaAs alternating layers with an optical thickness of λ/4, with λ being the wavelength of the confined cavity mode. The resulting Rabi splitting is 5.1 meV, and the polariton lifetime is about 10 ps. During the molecular beam epitaxy growth of the distributed Bragg reflectors, the slight mismatch between the lattice constants of each layer results in an accumulated stress that relaxes in the form of structural defects. These photonic defects create high potential barriers in the polariton energy landscape.
Excitation scheme.
To create a polariton fluid we excite the microcavity with a continuouswave singlemode Ti:sapphire laser resonant with the lower polariton branch. We use a confocal excitation scheme in which the laser is focused in an intermediate plane where a mask is placed to hide the upper part of the Gaussian spot. Then, an image of this intermediate plane is created on the sample, producing a spot with the shape of a halfGaussian. Polaritons are resonantly injected into the microcavity with a welldefined wave vector, in the region above the defect. In these conditions, polaritons move out of the excitation spot with a free phase, no longer imposed by the pump beam. This is essential for the observation of quantum hydrodynamic effects involving topological excitations with phase discontinuities^{9}.
The momentum of the injected polaritons is set by the angle of incidence α of the excitation laser on the microcavity. This allows us to control the inplane wave vector of the polariton fluid through the relation k = k_{0} sin(α), where k_{0} is the wave vector of the laser field. At the injected polariton momentum k = 1.3 μm^{−1}, the polariton velocity is v_{f} = ℏk/m_{pol} = 1.5 μm ps^{−1} (m_{pol} = 10^{−4}m_{electron}). This velocity is higher than the speed of sound of the fluid for the polariton densities of our experiments, as evidenced by the presence of ship waves upstream of the obstacle in Fig. 2a (see ref. 23). For this value of the momentum, we measure a TE–TM splitting of 20 μeV, resulting in the effective magnetic field sketched in Fig. 1b.
Polaritons are photocreated in our microcavity with TM linear polarization. This corresponds to a pseudospin pointing in the direction of the flow as marked by the arrow in Fig. 1a.
Detection scheme.
The observations reported in this work require the complete knowledge of the polariton spin. To gain this, a complete polarization tomography of the emission is performed through the measurement of the three Stokes parameters S_{1} = (I_{TE}−I_{TM})/I_{tot}, S_{2} = (I_{D}−I_{A})/I_{tot} and S_{3} = (I_{σ+}−I_{σ−})/I_{tot}, where I_{j} is the light intensity emitted with polarization j, and I_{tot} is the total emitted intensity. This requires measuring six different polarizations: I_{H}, I_{V}, I_{+45}, I_{−45}, I_{σ+} and I_{σ−}, which, respectively, represent linear horizontal, linear vertical, linear diagonal, linear antidiagonal, left circularly and right circularly polarized emitted intensity. A combination of wave plates and polarizing beam splitters is used to image each of the polarization components of the emitted light on a CCD camera.
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Acknowledgements
We thank R. Houdré for the microcavity sample and P. Voisin for fruitful discussions. This work was supported by the Agence Nationale de la Recherche (contract ANR11BS10001), the RTRA (contract Boseflow1D), IFRAF, the FP7 ITNs Clermont4 (235114) and SpinOptronics (237252), and the FP7 IRSES ‘Polaphen’ (246912). A.B. is a member of the Institut Universitaire de France.
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Hivet, R., Flayac, H., Solnyshkov, D. et al. Halfsolitons in a polariton quantum fluid behave like magnetic monopoles. Nature Phys 8, 724–728 (2012). https://doi.org/10.1038/nphys2406
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DOI: https://doi.org/10.1038/nphys2406
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