
@article{assmannPolaritonCondensatesHighly2011,
  title = {From Polariton Condensates to Highly Photonic Quantum Degenerate States of Bosonic Matter},
  author = {A{\ss}mann, Marc and Tempel, Jean-Sebastian and Veit, Franziska and Bayer, Manfred and {Rahimi-Iman}, Arash and L{\"o}ffler, Andreas and H{\"o}fling, Sven and Reitzenstein, Stephan and Worschech, Lukas and Forchel, Alfred},
  year = {2011},
  volume = {108},
  pages = {1804--1809},
  abstract = {Bose\textendash{}Einstein condensation (BEC) is a thermodynamic phase transition of an interacting Bose gas. Its key signatures are remarkable quantum effects like superfluidity and a phonon-like Bogoliubov excitation spectrum, which have been verified for atomic BECs. In the solid state, BEC of exciton\textendash{}polaritons has been reported. Polaritons are strongly coupled light-matter quasiparticles in semiconductor microcavities and composite bosons. However, they are subject to dephasing and decay and need external pumping to reach a steady state. Accordingly the polariton BEC is a nonequilibrium process of a degenerate polariton gas in self-equilibrium, but out of equilibrium with the baths it is coupled to and therefore deviates from the thermodynamic phase transition seen in atomic BECs. Here we show that key signatures of BEC can even be observed without fulfilling the self-equilibrium condition in a highly photonic quantum degenerate nonequilibrium system.},
  journal = {Proc. Natl. Acad. Sci.},
  number = {5}
}

@article{byrnesNegativeBogoliubovDispersion2012,
  title = {Negative {{Bogoliubov}} Dispersion in Exciton-Polariton Condensates},
  author = {Byrnes, Tim and Horikiri, Tomoyuki and Ishida, Natsuko and Fraser, Michael and Yamamoto, Yoshihisa},
  year = {2012},
  month = feb,
  volume = {85},
  pages = {075130},
  doi = {10.1103/PhysRevB.85.075130},
  abstract = {Bogoliubov's theory states that self-interaction effects in Bose-Einstein condensates produce a characteristic linear dispersion at low momenta. One of the curious features of Bogoliubov's theory is that the new quasiparticles in the system are linear combinations of creation and destruction operators of the bosons. In exciton-polariton condensates, this gives the possibility of directly observing the negative branch of the Bogoliubov dispersion in the photoluminescence (PL) emission. Here we theoretically examine the PL spectra of exciton-polariton condensates taking into account reservoir effects. At sufficiently high excitation densities, the negative dispersion becomes visible. We also discuss the possibility for relaxation oscillations to occur under conditions of strong reservoir coupling. This is found to give a secondary mechanism for making the negative branch visible.},
  journal = {Phys. Rev. B},
  number = {7}
}

@article{caoPolarizationKineticsSemiconductor2008,
  title = {Polarization Kinetics of Semiconductor Microcavities Investigated with a {{Boltzman}} Approach},
  author = {Cao, Huy Thien and Doan, T. D. and Thoai, D. B. Tran and Haug, H.},
  year = {2008},
  month = feb,
  volume = {77},
  pages = {075320},
  doi = {10.1103/PhysRevB.77.075320},
  abstract = {We investigate with a Boltzmann approach the spin relaxation kinetics of microcavity polaritons after an excitation pulse with near-resonant polarized light and calculate the polarization of the emitted light around and above the threshold for stimulated emission. Considering only the optically active excitons with an angular momentum m={$\pm$}1, we calculate the corresponding 2\texttimes{}2 single-particle density matrix. Our kinetic treatment takes the polariton-acoustic phonon as well as the polariton-polariton scattering as the dominant relaxation processes into account. Both processes are spin conserving. Particularly for excitation with circular light, we find in isotropic crystals above threshold a nearly complete circular polarization degree which lasts (typically 40\textendash{}60ps) much longer than the exciting 3ps pulses due to the dominance of the stimulated spin-conserving scattering processes over the spontaneous spin-flip processes. These and other results also for linearly polarized pump light are in very good agreement with corresponding experiments on GaAs microcavities. In addition, we present time- and wave-number-dependent results which too are in qualitative agreement with the available angle- and polarization-resolved luminescence measurements.},
  journal = {Phys. Rev. B},
  number = {7}
}

@article{dengExcitonpolaritonBoseEinsteinCondensation2010,
  title = {Exciton-Polariton {{Bose}}-{{Einstein}} Condensation},
  author = {Deng, Hui and Haug, Hartmut and Yamamoto, Yoshihisa},
  year = {2010},
  month = may,
  volume = {82},
  pages = {1489--1537},
  doi = {10.1103/RevModPhys.82.1489},
  abstract = {In the past decade, a two-dimensional matter-light system called the microcavity exciton-polariton has emerged as a new promising candidate of Bose-Einstein condensation (BEC) in solids. Many pieces of important evidence of polariton BEC have been established recently in GaAs and CdTe microcavities at the liquid helium temperature, opening a door to rich many-body physics inaccessible in experiments before. Technological progress also made polariton BEC at room temperatures promising. In parallel with experimental progresses, theoretical frameworks and numerical simulations are developed, and our understanding of the system has greatly advanced. In this article, recent experiments and corresponding theoretical pictures based on the Gross-Pitaevskii equations and the Boltzmann kinetic simulations for a finite-size BEC of polaritons are reviewed.},
  journal = {Rev. Mod. Phys.},
  number = {2}
}

@article{doanCondensationKineticsMicrocavity2005,
  title = {Condensation Kinetics of Microcavity Polaritons with Scattering by Phonons and Polaritons},
  author = {Doan, T. D. and Cao, Huy Thien and Tran Thoai, D. B. and Haug, H.},
  year = {2005},
  month = aug,
  volume = {72},
  pages = {085301},
  doi = {10.1103/PhysRevB.72.085301},
  journal = {Phys. Rev. B},
  number = {8}
}

@book{griffinBoseCondensedGasesFinite2009,
  title = {Bose-{{Condensed Gases}} at {{Finite Temperatures}}},
  author = {Griffin, Allan and Nikuni, Tetsuro and Zaremba, Eugene},
  year = {2009},
  month = feb,
  edition = {1},
  publisher = {{Cambridge University Press}},
  address = {{Cambridge ; New York}},
  abstract = {The discovery of Bose-Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases. It uses a simplified microscopic model to give a clear, explicit account of collective modes in both the collisionless and collision-dominated regions. Major topics such as kinetic equations, local equilibrium and two-fluid hydrodynamics are introduced at an elementary level. Explicit predictions are worked out and linked to experiments. Providing a platform for future experimental and theoretical studies on the finite temperature dynamics of trapped Bose gases, this book is ideal for researchers and graduate students in ultracold atom physics, atomic, molecular and optical physics and condensed matter physics.},
  isbn = {978-0-521-83702-6}
}

@article{grossStructureQuantizedVortex1961,
  title = {Structure of a Quantized Vortex in Boson Systems},
  author = {Gross, E. P.},
  year = {1961},
  month = may,
  volume = {20},
  pages = {454--477},
  issn = {1827-6121},
  doi = {10.1007/BF02731494},
  abstract = {For a system of weakly repelling bosons, a theory of the elementary line vortex excitations is developed. The vortex state is characterised by the presence of a finite fraction of the particles in a single particle state of integer angular momentum. The radial dependence of the highly occupied state follows from a self-consistent field equation. The radial function and the associated particle density are essentially constant everywhere except inside a core, where they drop to zero. The core size is the de Broglie wavelength associated with the mean interaction energy per particle. The expectation value of the velocity has the radial dependence of a classical vortex. In this Hartree approximation the vorticity is zero everywhere except on the vortex line. When the description of the state is refined to include the zero point oscillations of the phonon field, the vorticity is spread out over the core. These results confirm in all essentials the intuitive arguments ofOnsager andFeynman. The phonons moving perpendicular to the vortex line are coherent excitations of equal and opposite angular momentum relative to the substratum of moving particles that constitute the vortex. The vortex motion resolves the degeneracy of the Bogoljubov phonons with respect to the azimuthal quantum number.},
  journal = {Il Nuovo Cimento (1955-1965)},
  number = {3}
}

@article{gustViscosityDiluteBose2015,
  title = {The Viscosity of Dilute {{Bose}}\textendash{{Einstein}} Condensates},
  author = {Gust, Erich D. and Reichl, L. E.},
  year = {2015},
  month = oct,
  volume = {T165},
  pages = {014034},
  issn = {1402-4896},
  doi = {10.1088/0031-8949/2015/T165/014034},
  abstract = {The viscosity of a dilute Bose\textendash{}Einstein condensate is obtained for a range of temperatures and densities. The kinetic equation used to derive the viscosity is based on Bogoliubov mean field theory and is a Boltzmann-like equation for the Bogoliubov excitations in the condensate. The viscosity can be assigned a numerical value for any gas for which the mass and scattering length are known. The viscosity decreases slowly as the temperature is decreased, but at low temperature (around 0.01 Tc), begins to show a much more rapid decrease with decreasing temperature. This interesting behavior of the viscosity is related to the behavior of the eigenvalues of the bogolon collision operator at these low temperatures.},
  journal = {Phys. Scr.}
}

@phdthesis{haugBatchLocalizationAlgorithm2019,
  title = {Batch Localization Algorithm for Floating Wireless Sensor Networks},
  author = {Haug, Martin},
  year = {2019},
  month = oct,
  address = {{Berlin, DE}},
  school = {Technische Universit{\"a}t Berlin},
  type = {Bachelor Thesis}
}

@article{haugBoseEinsteinCondensationNonequilibrium1983,
  title = {Bose-{{Einstein}} Condensation in Nonequilibrium Systems},
  author = {Haug, H. and Kranz, H. H.},
  year = {1983},
  month = jun,
  volume = {53},
  pages = {151--156},
  issn = {1431-584X},
  doi = {10.1007/BF01304202},
  abstract = {The properties of quantum statistically degenerate systems of Bosons which are created by an external pump field and decay within a finite lifetime are investigated by means of a Green's function treatment. These investigations help to understand the physical properties of such condensed Bose-systems as excitons, excitonic molecules and spin aligned hydrogen atoms. As an example, recent experiments by Hulin et al. on degenerate excitons in Cu2O are analyzed and a condensate fraction of about 5\% is obtained.},
  journal = {Z. Physik B - Condensed Matter},
  number = {2}
}

@article{haugElectronTheoryOptical1984,
  title = {Electron Theory of the Optical Properties of Laser-Excited Semiconductors},
  author = {Haug, H. and {Schmitt-Rink}, S.},
  year = {1984},
  month = jan,
  volume = {9},
  pages = {3--100},
  issn = {0079-6727},
  doi = {10.1016/0079-6727(84)90026-0},
  journal = {Progress in Quantum Electronics},
  number = {1}
}

@article{haugQuantumKineticDerivation2014,
  title = {Quantum Kinetic Derivation of the Nonequilibrium {{Gross}}-{{Pitaevskii}} Equation for Nonresonant Excitation of Microcavity Polaritons},
  author = {Haug, H. and Doan, T. D. and Tran Thoai, D. B.},
  year = {2014},
  month = apr,
  volume = {89},
  pages = {155302},
  doi = {10.1103/PhysRevB.89.155302},
  journal = {Phys. Rev. B},
  number = {15}
}

@book{haugQuantumKineticsTransport2008,
  title = {Quantum {{Kinetics}} in {{Transport}} and {{Optics}} of {{Semiconductors}}},
  author = {Haug, Hartmut and Jauho, Antti-Pekka},
  year = {2008},
  edition = {2},
  publisher = {{Springer-Verlag}},
  address = {{Berlin Heidelberg}},
  doi = {10.1007/978-3-540-73564-9},
  abstract = {Nanoscale miniaturization and femtosecond laser-pulse spectroscopy require a quantum mechanical description of the carrier kinetics that goes beyond the conventional Boltzmann theory. On these extremely short length and time scales the electrons behave like partially coherent waves. This monograph deals with quantum kinetics for transport in low-dimensional microstructures and for ultra-short laser pulse spectroscopy. The nonequilibrium Green function theory is described and used for the derivation of the quantum kinetic equations. Numerical methods for the solution of the retarded quantum kinetic equations are discussed and results are presented for high-field transport and for mesoscopic transport phenomena. Quantum beats, polarization decay, and non-Markovian behaviour are treated for femtosecond spectroscopy on a microscopic basis. Since the publishing of the first edition in 1996 the nonequilibrium Green function technique has been applied to a large number of new research topics, and the revised edition introduces the reader to some of these areas, such as molecular electronics, noise calculations, build-up of screening and polaron correlations, and non-Markovian relaxation, among others. Connection to recent experiments is made, and it is emphasized how the quantum kinetic theory is essential in their interpretation.},
  isbn = {978-3-540-73561-8},
  series = {Springer {{Series}} in {{Solid}}-{{State Sciences}}}
}

@article{haugStructureFormationSuperfluid2015,
  title = {Structure Formation, Superfluid Velocity Patterns, and Induced Vortex-Antivortex Oscillations and Rotations in Microcavity Polaritons},
  author = {Haug, H. and Doan, T. D. and Thoai, D. B. Tran},
  year = {2015},
  month = may,
  volume = {91},
  pages = {195311},
  doi = {10.1103/PhysRevB.91.195311},
  abstract = {We use the nonequilibrium Gross-Pitaevskii equation in the form recently derived [H. Haug et al., Phys. Rev. B 89, 155302 (2014)] to study spontaneous pattern formation and the connected superfluid current patterns. Using a nonresonant excitation beam with ring structure we get depending on the details of the structure and of the pump power, different spatial patterns of the condensate density. The corresponding phase profiles allow to identify, e.g., vortex-antivortex pairs, but beyond that, yield an image of the superfluid flow patterns linked with the structured condensate density. The fast superfluid flow driven by the spatially changing phase with velocities of the order of several m{$\mu$}/ps is found to be often supersonic. In order to test dynamically the stability of the spontaneously formed flow patterns under external perturbations, we apply an additional resonant Laguerre-Gauss beam with angular momentum. This beam causes complex response of the phase patterns. This response is shown to be basically an oscillation or rotation of the vortex-antivortex pair depending on the strength of the extra beam. The rotation is induced via a ring of vortices induced by the Laguerre-Gaus beam. The main result of these studies is the extraordinary stability of the vortex-antivortex pair even under strongly perturbing external fields.},
  journal = {Phys. Rev. B},
  number = {19}
}

@article{horikiriHighlyExcitedExcitonpolariton2017,
  title = {Highly Excited Exciton-Polariton Condensates},
  author = {Horikiri, Tomoyuki and Byrnes, Tim and Kusudo, Kenichiro and Ishida, Natsuko and Matsuo, Yasuhiro and Shikano, Yutaka and L{\"o}ffler, Andreas and H{\"o}fling, Sven and Forchel, Alfred and Yamamoto, Yoshihisa},
  year = {2017},
  month = jun,
  volume = {95},
  pages = {245122},
  doi = {10.1103/PhysRevB.95.245122},
  abstract = {Exciton polaritons are a coherent electron-hole-photon (e-h-p) system where condensation has been observed in semiconductor microcavities. In contrast to equilibrium Bose-Einstein condensation (BECs) for long lifetime systems, polariton condensates have a dynamical nonequilibrium feature owing to the similar physical structure that they have to semiconductor lasers. One of the distinguishing features of a condensate to a laser is the presence of strong coupling between the matter and photon fields. Irrespective of its equilibrium or nonequilibrium nature, exciton polaritons have been observed to maintain strong coupling. We show that by investigating the high-density regime of exciton-polariton condensates, the negative branch is directly observed in photoluminescence. This is evidence that the present e-h-p system is still in the strong-coupling regime, contrary to past results where the system reduced to standard lasing at high density.},
  journal = {Phys. Rev. B},
  number = {24}
}

@article{hugenholtzGroundStateEnergyExcitation1959,
  title = {Ground-{{State Energy}} and {{Excitation Spectrum}} of a {{System}} of {{Interacting Bosons}}},
  author = {Hugenholtz, N. M. and Pines, D.},
  year = {1959},
  month = nov,
  volume = {116},
  pages = {489--506},
  doi = {10.1103/PhysRev.116.489},
  abstract = {In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.},
  journal = {Phys. Rev.},
  number = {3}
}

@book{kadanoffQuantumStatisticalMechanics1962,
  title = {Quantum {{Statistical Mechanics}}},
  author = {Kadanoff, Leo P. and Baym, Gordon},
  year = {1962},
  publisher = {{Westview Press}},
  address = {{Cambridge, Mass}},
  abstract = {This book is a very early systematic treatment of the application of the field-theoretical methods developed after the Second World War to the quantum mechanical many-body problem at finite temperature. It describes various techniques that remain basic tools of modern condensed matter physicists.},
  isbn = {978-0-201-41046-4}
}

@article{kirkpatrickTransportDiluteCondensed1985,
  title = {Transport in a Dilute but Condensed Nonideal {{Bose}} Gas: {{Kinetic}} Equations},
  shorttitle = {Transport in a Dilute but Condensed Nonideal {{Bose}} Gas},
  author = {Kirkpatrick, T. R. and Dorfman, J. R.},
  year = {1985},
  month = feb,
  volume = {58},
  pages = {301--331},
  issn = {1573-7357},
  doi = {10.1007/BF00681309},
  abstract = {A kinetic equation is derived that describes the nonequilibrium quasiparticle distribution function for a dilute, inhomogeneous Bose gas below its {$\lambda$}-point. The derivation is based on a microscopic description of the Bose gas, and is similar to that used to obtain the classical Boltzmann equation and its generalization to higher densities from the Liouville equation. In a subsequent paper the kinetic equation derived here will be used to derive the Landau-Khalatnikov two-fluid equations together with microscopic expressions for the associated transport coefficients.},
  journal = {J Low Temp Phys},
  number = {3}
}

@article{kulczykowskiPhaseOrderingKinetics2017,
  title = {Phase Ordering Kinetics of a Nonequilibrium Exciton-Polariton Condensate},
  author = {Kulczykowski, Micha{\l} and Matuszewski, Micha{\l}},
  year = {2017},
  month = feb,
  volume = {95},
  pages = {075306},
  doi = {10.1103/PhysRevB.95.075306},
  abstract = {We investigate the process of coarsening via annihilation of vortex-antivortex pairs, following the quench to the condensate phase in a nonresonantly pumped polariton system. We find that the late-time dynamics is an example of universal phase-ordering kinetics, characterized by scaling of correlation functions in time. Depending on the parameters of the system, the evolution of the characteristic length scale L(t) can be the same as for the two-dimensional XY model, described by a power law with the dynamical exponent z{$\approx$}2 and a logarithmic correction, or z{$\approx$}1 which agrees with previous studies of conservative superfluids.},
  journal = {Phys. Rev. B},
  number = {7}
}

@article{leeNonequilibriumAtomicCondensates2016,
  title = {Non-Equilibrium Atomic Condensates and Mixtures: Collective Modes, Condensate Growth and Thermalisation},
  shorttitle = {Non-Equilibrium Atomic Condensates and Mixtures},
  author = {Lee, Kean Loon and Proukakis, Nick P},
  year = {2016},
  month = nov,
  volume = {49},
  pages = {214003},
  issn = {0953-4075, 1361-6455},
  doi = {10.1088/0953-4075/49/21/214003},
  abstract = {The non-equilibrium dynamics of trapped ultracold atomic gases, or mixtures thereof, is an extremely rich subject. Despite 20 years of studies, and remarkable progress mainly on the experimental front, numerous open question remain, related to the growth, relaxation and thermalisation of such systems, and there is still no universally accepted theory for their theoretical description. In this paper we discuss one of the state-of-the-art kinetic approaches, which gives an intuitive picture of the physical processes happening at the microscopic scale, being broadly applicable both below and above the critical region (but not within the critical region itself, where fluctuations become dominant and symmetry breaking takes place). Specifically, the `Zaremba\textendash{}Nikuni\textendash{}Griffin' (ZNG) scheme provides a self-consistent description of the coupling between the condensate and the thermal atoms, including the collisions between these two subsystems. It has been successfully tested against experiments in various settings, including investigation of collective modes (e.g. monopole, dipole and quadrupole modes), dissipation of topological excitations (solitons and vortices) as well as surface evaporative cooling. Here, we show that the ZNG model can capture two important aspects of nonequilibrium dynamics for both single-component and two-component BECs: the Kohn mode (the undamped dipole oscillation independent of interactions and temperature) and (re) thermalisation leading to condensate growth following sudden evaporation. Our simulations, performed in a spherically symmetric trap reveal (i) an interesting two-stage dynamics and the emergence of a prominent monopole mode in the evaporative cooling of a single-component Bose gas, and (ii) the long thermalisation time associated with the sympathetic cooling of a realistic two-component mixture. Related open questions arise about the mechanisms and the nature of thermalisation in such systems, where further controlled experiments are needed for benchmarking.},
  journal = {J. Phys. B: At. Mol. Opt. Phys.},
  number = {21}
}

@article{manniSpontaneousPatternFormation2011,
  title = {Spontaneous {{Pattern Formation}} in a {{Polariton Condensate}}},
  author = {Manni, F. and Lagoudakis, K. G. and Liew, T. C. H. and Andr{\'e}, R. and {Deveaud-Pl{\'e}dran}, B.},
  year = {2011},
  month = sep,
  volume = {107},
  pages = {106401},
  doi = {10.1103/PhysRevLett.107.106401},
  abstract = {Exciton-polariton condensation can be regarded as a self-organization phenomenon, where phase ordering is established among particles in the system. In such condensed systems, further ordering can occur in the particle density distribution, under particular experimental conditions. In this work we report on spontaneous pattern formation in a polariton condensate under nonresonant optical pumping. The slightly elliptical ring-shaped excitation laser that we employ forces condensation to occur into a single-energy state with periodic boundary conditions, giving rise to a multilobe standing-wave patterned state.},
  journal = {Phys. Rev. Lett.},
  number = {10}
}

@article{margolinAntidiffusiveVelocitiesMultipass1989,
  title = {Antidiffusive Velocities for Multipass Donor Cell Advection},
  author = {Margolin, G. and Smolarkiewicz, K.},
  year = {1989},
  volume = {20},
  pages = {907--929},
  journal = {Siam J Sci Comput},
  number = {3}
}

@article{pieczarkaGhostBranchPhotoluminescence2015,
  title = {Ghost {{Branch Photoluminescence From}} a {{Polariton Fluid Under Nonresonant Excitation}}},
  author = {Pieczarka, Maciej and Syperek, Marcin and Dusanowski, {\L}ukasz and Misiewicz, Jan and Langer, Fabian and Forchel, Alfred and Kamp, Martin and Schneider, Christian and H{\"o}fling, Sven and Kavokin, Alexey and S{\k{e}}k, Grzegorz},
  year = {2015},
  month = oct,
  volume = {115},
  pages = {186401},
  doi = {10.1103/PhysRevLett.115.186401},
  journal = {Phys. Rev. Lett.},
  number = {18}
}

@article{pieczarkaObservationQuantumDepletion2020,
  title = {Observation of Quantum Depletion in a Non-Equilibrium Exciton\textendash{}Polariton Condensate},
  author = {Pieczarka, Maciej and Estrecho, Eliezer and Boozarjmehr, Maryam and Bleu, Olivier and Steger, Mark and West, Kenneth and Pfeiffer, Loren N. and Snoke, David W. and Levinsen, Jesper and Parish, Meera M. and Truscott, Andrew G. and Ostrovskaya, Elena A.},
  year = {2020},
  month = jan,
  volume = {11},
  pages = {1--7},
  issn = {2041-1723},
  doi = {10.1038/s41467-019-14243-6},
  abstract = {Many aspects of polariton condensate behaviour can be captured by mean-field theories but interactions introduce additional quantum effects. Here the authors observe quantum depletion in a driven-dissipative condensate and find that deviations from equilibrium predictions depend on the excitonic fraction.},
  copyright = {2020 The Author(s)},
  journal = {Nat Commun},
  number = {1}
}

@article{pitaevskiiVortexLinesImperfect1961,
  title = {Vortex Lines in an Imperfect {{Bose}} Gas},
  author = {Pitaevskii, LP},
  year = {1961},
  volume = {13},
  pages = {451--454},
  journal = {Sov Phys JETP},
  number = {2}
}

@book{popovFunctionalIntegralsQuantum1983,
  title = {Functional {{Integrals}} in {{Quantum Field Theory}} and {{Statistical Physics}}},
  author = {Popov, V. N.},
  year = {1983},
  publisher = {{Springer Netherlands}},
  abstract = {Functional integration is one of the most powerful methods of contempo\- rary theoretical physics, enabling us to simplify, accelerate, and make clearer the process of the theoretician's analytical work. Interest in this method and the endeavour to master it creatively grows incessantly. This book presents a study of the application of functional integration methods to a wide range of contemporary theoretical physics problems. The concept of a functional integral is introduced as a method of quantizing finite-dimensional mechanical systems, as an alternative to ordinary quantum mechanics. The problems of systems quantization with constraints and the manifolds quantization are presented here for the first time in a monograph. The application of the functional integration methods to systems with an infinite number of degrees of freedom allows one to uniquely introduce and formulate the diagram perturbation theory in quantum field theory and statistical physics. This approach is significantly simpler than the widely accepted method using an operator approach.},
  isbn = {978-90-277-1471-8},
  series = {Mathematical {{Physics}} and {{Applied Mathematics}}}
}

@article{proukakisFinitetemperatureModelsBose2008,
  title = {Finite-Temperature Models of {{Bose}}\textendash{{Einstein}} Condensation},
  author = {Proukakis, Nick P. and Jackson, Brian},
  year = {2008},
  month = oct,
  volume = {41},
  pages = {203002},
  issn = {0953-4075},
  doi = {10.1088/0953-4075/41/20/203002},
  abstract = {The theoretical description of trapped weakly interacting Bose\textendash{}Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the `maze' of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain `classes of approaches' identified by their physical content, rather than their particular mathematical implementation. This tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions, such as the occupation number representation of second quantization, are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross\textendash{}Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean field, number conserving and stochastic) according to their particular needs. In addition to its `training element', we hope that this tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.},
  journal = {J. Phys. B: At. Mol. Opt. Phys.},
  number = {20}
}

@book{proukakisQuantumGasesFinite2013,
  title = {Quantum Gases, {{Finite Temperature}} and {{Non}}-{{Equilibrium Dynamics}}},
  author = {Proukakis, Nick and Gardiner, Simon and Davis, Matthew and Szyma{\'n}ska, Marzena},
  year = {2013},
  publisher = {{IMPERIAL COLLEGE PRESS}},
  doi = {10.1142/p817},
  eprint = {https://www.worldscientific.com/doi/pdf/10.1142/p817}
}

@article{roumposSingleVortexantivortexPair2011,
  title = {Single Vortex-Antivortex Pair in an Exciton-Polariton Condensate},
  author = {Roumpos, Georgios and Fraser, Michael D. and Loeffler, Andreas and H{\"o}fling, Sven and Forchel, Alfred and Yamamoto, Yoshihisa},
  year = {2011},
  month = feb,
  volume = {7},
  pages = {129--133},
  issn = {1745-2473},
  doi = {10.1038/NPHYS1841},
  abstract = {In a homogeneous two-dimensional system at non-zero temperature there can be no ordering of infinite range(1,2). However, for a Bose liquid under such conditions, a superfluid phase is predicted(3-5). Bound vortex-antivortex pairs dominate the thermodynamics and phase coherence properties in this superfluid regime. It is believed that several systems share this behaviour when the parameter describing their ordered state has two degrees of freedom(6). This theory has been tested for some of them(7-12), but there has been no direct experimental observation of a quasi-condensate that includes a bound vortex-antivortex pair. Here we present an experimental technique that can identify a single vortex-antivortex pair in a two-dimensional exciton-polariton condensate. The pair is generated through the inhomogeneous spot profile of the pumping laser, and is revealed in the time-integrated phase maps acquired using Michelson interferometry. Numerical modelling based on the open-dissipative Gross-Pitaevskii equation suggests that the pair evolution is distinctly different in this non-equilibrium system compared with atomic condensates(13).},
  journal = {Nature Physics},
  keywords = {ARRAYS,BOSE-EINSTEIN CONDENSATION,GASES,MICROCAVITY,QUANTIZED VORTICES,SUPERFLUID,TRANSITION},
  number = {2}
}

@article{shelykhSpinDynamicsExciton2005,
  title = {Spin Dynamics of Exciton Polaritons in Microcavities},
  author = {Shelykh, I. A. and Kavokin, A. V. and Malpuech, G.},
  year = {2005},
  month = sep,
  volume = {242},
  pages = {2271--2289},
  issn = {1521-3951},
  doi = {10.1002/pssb.200560965},
  abstract = {In this chapter we address a complex set of optical phenomena linked to the spin dynamics of exciton polaritons in semiconductor microcavities. When optically created, polaritons inherit the spin and...},
  journal = {Phys. Status Solidi B},
  number = {11}
}

@article{shiTheoryDecayLuminescence1994,
  title = {Theory of the Decay Luminescence Spectrum of a {{Bose}}-Condensed Interacting Exciton Gas},
  author = {Shi, H. and Verechaka, G. and Griffin, A.},
  year = {1994},
  month = jul,
  volume = {50},
  pages = {1119--1125},
  doi = {10.1103/PhysRevB.50.1119},
  abstract = {Recent experimental studies by Wolfe, Snoke, and Lin have given strong evidence that paraexcitons in optically excited Cu2O crystals form a Bose condensate. In these studies, the observed recombination decay luminescence line shape I({$\omega$}) has been analyzed in terms of an ideal Bose-condensed gas. We derive a general expression for I({$\omega$}) in terms of the exact single-particle spectral density A(k,{$\omega$}) of a Bose fluid. This expression is then used to calculate I({$\omega$}) for a dilute, weakly interacting Bose gas. Our analytic results, based on the approximate theory due to Popov, are valid in the region just below the Bose-Einstein transition temperature. When formally extended down to T=0, they reproduce the well-known Bogoliubov approximation.},
  journal = {Phys. Rev. B},
  number = {2}
}

@article{smolarkiewiczFullyMultidimensionalPositive1984,
  title = {A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small Implicit Diffusion},
  author = {Smolarkiewicz, Piotr K},
  year = {1984},
  month = may,
  volume = {54},
  pages = {325--362},
  issn = {0021-9991},
  doi = {10.1016/0021-9991(84)90121-9},
  abstract = {The idea of the simple positive definite advection scheme presented previously in Monthly Weather Review (III (1983), 479) is improved for an optional multidimensional case and is presented in a generalized format. The accuracy of the scheme is discussed and a review of existing options is presented and illustrated through numerical tests.},
  journal = {Journal of Computational Physics},
  number = {2}
}

@article{solnyshkovHybridBoltzmannGrossPitaevskii2014,
  title = {Hybrid {{Boltzmann}}--{{Gross}}-{{Pitaevskii}} Theory of {{Bose}}-{{Einstein}} Condensation and Superfluidity in Open Driven-Dissipative Systems},
  author = {Solnyshkov, D. D. and Ter{\c c}as, H. and Dini, K. and Malpuech, G.},
  year = {2014},
  month = mar,
  volume = {89},
  pages = {033626},
  doi = {10.1103/PhysRevA.89.033626},
  abstract = {We derive a theoretical model which describes Bose-Einstein condensation in an open driven-dissipative system. It includes external pumping of a thermal reservoir, finite lifetime of the condensed particles, and energy relaxation. The coupling between the reservoir and the condensate is described with semiclassical Boltzmann rates. This results in a dissipative term in the Gross-Pitaevskii equation for the condensate, which is proportional to the energy of the elementary excitations of the system. We analyze the main properties of a condensate described by this hybrid Boltzmann\textendash{}Gross-Pitaevskii model, namely, dispersion of the elementary excitations, bogolon distribution function, first-order coherence, dynamic and energetic stability, and drag force created by a disorder potential. We find that the dispersion of the elementary excitations of a condensed state fulfills the Landau criterion of superfluidity. The condensate is dynamically and energetically stable as longs as it moves at a velocity smaller than the speed of excitations. First-order spatial coherence of the condensate is found to decay exponentially in one dimension and with a power law in two dimensions, similarly with the case of conservative systems. The coherence lengths are found to be longer due to the finite lifetime of the condensate excitations. We compare these properties with those of a condensate described by the popular ``diffusive'' models in which the dissipative term is proportional to the local condensate density. In the latter, the dispersion of excitations is diffusive which as soon as the condensate is put into motion implies finite mechanical friction and can lead to an energetic instability.},
  journal = {Phys. Rev. A},
  number = {3}
}

@article{sunBoseEinsteinCondensationLongLifetime2017,
  title = {Bose-{{Einstein Condensation}} of {{Long}}-{{Lifetime Polaritons}} in {{Thermal Equilibrium}}},
  author = {Sun, Yongbao and Wen, Patrick and Yoon, Yoseob and Liu, Gangqiang and Steger, Mark and Pfeiffer, Loren N. and West, Ken and Snoke, David W. and Nelson, Keith A.},
  year = {2017},
  month = jan,
  volume = {118},
  pages = {016602},
  doi = {10.1103/PhysRevLett.118.016602},
  abstract = {The experimental realization of Bose-Einstein condensation (BEC) with atoms and quasiparticles has triggered wide exploration of macroscopic quantum effects. Microcavity polaritons are of particular interest because quantum phenomena such as BEC and superfluidity can be observed at elevated temperatures. However, polariton lifetimes are typically too short to permit thermal equilibration. This has led to debate about whether polariton condensation is intrinsically a nonequilibrium effect. Here we report the first unambiguous observation of BEC of optically trapped polaritons in thermal equilibrium in a high-Q microcavity, evidenced by equilibrium Bose-Einstein distributions over broad ranges of polariton densities and bath temperatures. With thermal equilibrium established, we verify that polariton condensation is a phase transition with a well-defined density-temperature phase diagram. The measured phase boundary agrees well with the predictions of basic quantum gas theory.},
  journal = {Phys. Rev. Lett.},
  number = {1}
}

@article{szymanskaNonequilibriumQuantumCondensation2006,
  title = {Non-Equilibrium Quantum Condensation in an Incoherently Pumped Dissipative System},
  author = {Szymanska, M. H. and Keeling, J. and Littlewood, P. B.},
  year = {2006},
  month = jun,
  volume = {96},
  pages = {230602},
  issn = {0031-9007, 1079-7114},
  doi = {10.1103/PhysRevLett.96.230602},
  abstract = {We study spontaneous quantum coherence in an out of equilibrium system, coupled to multiple baths describing pumping and decay. For a range of parameters describing coupling to, and occupation of the baths, a stable steady-state condensed solution exists. The presence of pumping and decay significantly modifies the spectra of phase fluctuations, leading to correlation functions that differ both from an isolated condensate and from a laser.},
  archivePrefix = {arXiv},
  eprint = {cond-mat/0603447},
  eprinttype = {arxiv},
  journal = {Phys. Rev. Lett.},
  keywords = {Condensed Matter - Other Condensed Matter,Condensed Matter - Superconductivity},
  number = {23}
}

@article{utsunomiyaObservationBogoliubovExcitations2008,
  title = {Observation of {{Bogoliubov}} Excitations in Exciton-Polariton Condensates},
  author = {Utsunomiya, S. and Tian, L. and Roumpos, G. and Lai, C. W. and Kumada, N. and Fujisawa, T. and {Kuwata-Gonokami}, M. and L{\"o}ffler, A. and H{\"o}fling, S. and Forchel, A. and Yamamoto, Y.},
  year = {2008},
  month = sep,
  volume = {4},
  pages = {700--705},
  issn = {1745-2481},
  doi = {10.1038/nphys1034},
  abstract = {The observation of so-called Bogoliubov excitations provides the first sign of possible superfluid behaviour in an exciton-polariton condensate.},
  copyright = {2008 Nature Publishing Group},
  journal = {Nature Phys},
  number = {9}
}

@article{woutersExcitationsNonequilibriumBoseEinstein2007,
  title = {Excitations in a {{Nonequilibrium Bose}}-{{Einstein Condensate}} of {{Exciton Polaritons}}},
  author = {Wouters, Michiel and Carusotto, Iacopo},
  year = {2007},
  month = oct,
  volume = {99},
  pages = {140402},
  doi = {10.1103/PhysRevLett.99.140402},
  journal = {Phys. Rev. Lett.},
  number = {14}
}

@article{zarembaDynamicsTrappedBose1999,
  title = {Dynamics of {{Trapped Bose Gases}} at {{Finite Temperatures}}},
  author = {Zaremba, E. and Nikuni, T. and Griffin, A.},
  year = {1999},
  month = aug,
  volume = {116},
  pages = {277--345},
  issn = {1573-7357},
  doi = {10.1023/A:1021846002995},
  abstract = {Starting from an approximate microscopic model of a trapped Bose-condensed gas at finite temperatures, we derive an equation of motion for the condensate wavefunction and a quantum kinetic equation for the distribution function for the excited atoms. The kinetic equation is a generalization of our earlier work in that collisions between the condensate and non-condensate (C12) are now included, in addition to collisions between the excited atoms as described by the Uehling\textendash{}Uhlenbeck (C22) collision integral. The continuity equation for the local condensate density contains a source term {$\Gamma$}12which is related to the C12collision term. If we assume that the C22collision rate is sufficiently rapid to ensure that the non-condensate distribution function can be approximated by a local equilibrium Bose distribution, the kinetic equation can be used to derive hydrodynamic equations for the non-condensate. The {$\Gamma$}12source terms appearing in these equations play a key role in describing the equilibration of the local chemical potentials associated with the condensate and non-condensate components. We give a detailed study of these hydrodynamic equations and show how the Landau two-fluid equations emerge in the frequency domain {$\omega\tau\mu$} {$\ll$} {$\tau\mu$}is a characteristic relaxation time associated with C12collisions. More generally, the lack of complete local equilibrium between the condensate and non-condensate is shown to give rise to a new relaxational mode which is associated with the exchange of atoms between the two components. This new mode provides an additional source of damping in the hydrodynamic regime. Our equations are consistent with the generalized Kohn theorem for the center of mass motion of the trapped gas even in the presence of collisions. Finally, we formulate a variational solution of the equations which provides a very convenient and physical way of estimating normal mode frequencies. In particular, we use relatively simple trial functions within this approach to work out some of the monopole, dipole and quadrupole oscillations for an isotropic trap.},
  journal = {Journal of Low Temperature Physics},
  number = {3}
}


