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Vortices in Polariton OPO Superfluids

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Exciton Polaritons in Microcavities

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 172))

Abstract

This chapter reviews the occurrence of quantised vortices in polariton fluids, primarily when polaritons are driven in the optical parametric oscillator (OPO) regime. We first review the OPO physics, together with both its analytical and numerical modelling, the latter being necessary for the description of finite size systems. Pattern formation is typical in systems driven away from equilibrium. Similarly, we find that uniform OPO solutions can be unstable to the spontaneous formation of quantised vortices. However, metastable vortices can only be injected externally into an otherwise stable symmetric state, and their persistence is due to the OPO superfluid properties. We discuss how the currents characterising an OPO play a crucial role in the occurrence and dynamics of both metastable and spontaneous vortices.

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Notes

  1. 1.

     In rotating condensates, a vortex can be created if the angular velocity is higher than a critical value [75, 76]. When the rotation is halted, then the vortex will spiral out of the condensate [77].

  2. 2.

      Given a complex field or wavefunction, \(\vert \psi (\mathbf{r},t)\vert {\mathrm{e}}^{\mathrm{i}\phi (\mathbf{r},t)}\), describing either a quantum particle of mass m or a macroscopic number of particles condensed in the same quantum state, the current is defined as [76]:

    $$\begin{array}{rcl} \mathbf{j}(\mathbf{r},t) = \frac{\hslash } {m}\vert \psi (\mathbf{r},t){\vert }^{2}\nabla \phi (\mathbf{r},t) = \vert \psi (\mathbf{r},t){\vert }^{2}{\mathbf{v}}_{\mathrm{s}}(\mathbf{r},t),& & \end{array}$$
    (6.18)

    where \({\mathbf{v}}_{s}(\mathbf{r},t)\) is the flow velocity. In the following, with a slight abuse of notation, we will refer to the current as the gradient of the phase only, \(\nabla \phi (\mathbf{r},t)\).

  3. 3.

     Note also that we find that the value of the pump threshold for OPO is not altered by the presence of a weak photonic disorder.

  4. 4.

     D. Sarkar (University of Sheffield), private communication.

  5. 5.

     When generated by a noise pulse, both stable and metastable vortices have equal probability to have either charge ± 1. Similarly, when vortices are triggered via a Laguerre-Gauss probe, their vorticity can flip during the transient period. In particular, flipping can follow the appearance of two antivortices at the edge of the signal, one recombining with the triggered vortex. Note that the vorticity flipping conserves the total orbital angular momentum, in the sense that when for the signal m flips, say, from + 1 to − 1, for the idler the opposite happens, i.e. m flips from − 1 to + 1.

  6. 6.

    This is in reality an oversymplified version of the full equation satisfied by ψs, i({ r}), because one can show that there are small terms breaking the rotational symmetry, implying the vortex is not a pure angular momentum state, though it still has a definite winding number [D. Whittaker, private communication]. This, however, does not affect the expression of the healing length.

  7. 7.

     Note that, differently from the cw laser beam, the energy distribution spectrum of which is essentially δ-like, a pulsed beam has an intrinsic width in energy, proportional to the inverse pulse duration, σ t  − 1.

  8. 8.

     If the cw pump drives the system into the OPO regime, then the parametric scattering triggered by the pulsed probe will emerge in addition to the one related to OPO. However, as discussed later, the TOPO regime can be reached also in absence (below threshold) of the OPO.

  9. 9.

     See, for example, Fig. 3 of [82], where the intensity maximum of the extra population is reached within 4 ps after the maximum of the pulsed probe, is followed by a slow decay.

  10. 10.

     In the regime where the probe generates only a weak parametric scattering, aside the strong emission from the pump state, the dispersion is simply that of the LP and thus is not surprising that the signal propagates with a group velocity given by \({v}_{{k}_{\mathrm{pb}}}^{\mathrm{LP}}\). Remember that here, the cw pump is below the threshold for OPO.

  11. 11.

     Note, however, that even if in the atomic and polaritonic cases the same Laguerre-Gauss laser field is used, the mechanism of spinning the BEC atoms is different from the one which rotates polaritons.

  12. 12.

     We checked that m =  ± 1 (m =  ∓ 1) vortex solutions can appear only into the OPO signal (idler). A vortex probe pulse of any charge m injected resonantly to the pump momentum, and energy gets immediately transferred to an m =  ± 1 (m =  ∓ 1) vortex in the signal (idler), leaving the pump vortex-less.

  13. 13.

     Later, in Sect. 6.6.1, in connection to the stability of multiply quantised vortices, we also describe vortices in the TOPO regime, where we follow the vortex dynamics not of the OPO like here, but of the extra population only.

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Acknowledgements

We would like to acknowledge fruitful collaborations with A. Berceanu, E. Cancellieri, D. Sanvitto, C. Tejedor, and D. M. Whittaker on some of the topics discussed in this review, as well as the collaboration with the experimental group at UAM in Madrid (C. Antón, M. Baudisch, G. Tosi, L. Viña). We are particularly grateful to I. Carusotto and J. Keeling for stimulating discussions and for the critical reading of this manuscript. We also would like to thank C. Creatore, B. Deveaud-Plédran, M. Maragkou, and Y. Yamamoto for useful suggestions and points of discussion. F.M.M. acknowledges the financial support from the programs Ramón y Cajal and POLATOM (ESF). This work has been also supported by the Spanish MEC (MAT2008-01555, QOIT-CSD2006-00019), CAM (S-2009/ESP-1503), and FP7 ITN “Clermont4” (235114).

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Authors and Affiliations

  1. Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, Madrid, 28049, Spain

    Francesca M. Marchetti

  2. Department of Physics, University of Warwick, Coventry, CV4 7AL, UK

    Marzena H. Szymańska

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  1. Francesca M. Marchetti
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  2. Marzena H. Szymańska
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Correspondence to Francesca M. Marchetti .

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  1. Institute of Solid State Physics, Institutskaya 2, Chernogolovka, 142432, Russia

    Vladislav Timofeev

  2. NNL, Istituto Nanoscienze - CNR, Via Arnesano, Lecce, 73100, Italy

    Daniele Sanvitto

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Marchetti, F.M., Szymańska, M.H. (2012). Vortices in Polariton OPO Superfluids. In: Timofeev, V., Sanvitto, D. (eds) Exciton Polaritons in Microcavities. Springer Series in Solid-State Sciences, vol 172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24186-4_6

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