Quantum fluids of light

Quantum fluids of light

Abstract

This article reviews recent theoretical and experimental advances in the fundamental understanding and active control of quantum fluids of light in nonlinear optical systems. In presence of effective photon-photon interactions induced by the optical nonlinearity of the medium, a many-photon system can behave collectively as a quantum fluid with a number of novel features stemming from its intrinsically non-equilibrium nature. We present a rich variety of photon hydrodynamical effects that have been recently observed, from the superfluid flow around a defect at low speeds, to the appearance of a Mach-Cherenkov cone in a supersonic flow, to the hydrodynamic formation of topological excitations such as quantized vortices and dark solitons at the surface of large impenetrable obstacles. While our review is mostly focused on a class of semiconductor systems that have been extensively studied in recent years (namely planar semiconductor microcavities in the strong light-matter coupling regime having cavity polaritons as elementary excitations), the very concept of quantum fluids of light applies to a broad spectrum of systems, ranging from bulk nonlinear crystals, to atomic clouds embedded in optical fibers and cavities, to photonic crystal cavities, to superconducting quantum circuits based on Josephson junctions. The conclusive part of our article is devoted to a review of the exciting perspectives to achieve strongly correlated photon gases. In particular, we present different mechanisms to obtain efficient photon blockade, we discuss the novel quantum phases that are expected to appear in arrays of strongly nonlinear cavities, and we point out the rich phenomenology offered by the implementation of artificial gauge fields for photons.

Contents

I Historical introduction

In the last decades, the study of the physics of quantum fluids has attracted a tremendous interest in a variety of different many particle systems, ranging from liquid Helium Pines and Nozières (1998); Leggett (2004), electrons in solid-state materials Tinkham (2004); Schrieffer (1999); Mahan (1990), trapped gases of ultracold atoms Dalfovo et al. (1999); Giorgini et al. (2008); Bloch et al. (2008), quark-gluon plasma in colliders Yagi et al. (2005); Satz et al. (2010), nuclei Peter Ring (2004). When the thermal de Broglie wavelength becomes comparable or larger than the average interparticle spacing, the Bose vs. Fermi statistics of the constituent particles starts playing a crucial role in determining the properties of the fluid: in non-interacting Fermi gases, the Pauli principle is responsible for the rigidity of the Fermi sphere and the appearance of a Fermi pressure down to zero temperature, while in Bose gases a macroscopic fraction of the particles accumulate into the lowest energy single-particle state, the so-called Bose-Einstein condensate (BEC). The situation is even richer when quantum degeneracy combines with significant inter-particle interactions to produce a variety of spectacular effects such as superconductivity and superfluidity Leggett (1999); Tilley and Tilley (1994) and the fractional quantum Hall effect Yoshioka (1992); Das Sarma and Pinczuk (1997).

Historically, most of the theoretical and experimental activities in this field of many-body physics have addressed systems of material particles such as atoms, electrons, nucleons, or quarks. However, in the last decades, a growing community of researchers has started wondering whether in suitable circumstances light can be considered as a fluid composed of a large number of corpuscular photons with sizable photon-photon interactions. Even if this point of view is perfectly legitimate within the wave-particle duality in quantum mechanics, it is somehow at odd with our intuitive picture of light: the historical development of our understandings of matter and light have in fact followed very different paths.

The idea of matter being formed by a huge number of elementary corpuscles that combine in different ways to form the variety of existing materials dates back to the ancient age with Demokritos’ atomistic hypothesis, while the wavy nature of particles was put forward only in 1924 by de Broglie and experimentally demonstrated by Davisson and Germer in 1927. On the other hand, the long-standing debate between Newton’s corpuscular and Huygens’ undulatory theories of light appeared to be solved in the early nineteenth century with the observation of fringes in Young’s double slit experiment and of the remarkable Arago’s white spot in the shadow of a circular object. With the microscopic support of Maxwell’s theory of electromagnetism, the undulatory thory was able to explain most experimental observations until the beginning of the twentieth century when the corpuscular concept of a photon as a discrete quantum of light was revived by Einstein’s theory of the photo-electric effect. Within the wave-particle duality, our standard interpretation of light then consists of a dual wave/particle beam that is emitted by the source and then freely propagates through optical devices until it is absorbed.

While this intuitive picture of light is perfectly sufficient to describe most cases of interest, still it is missing a crucial element, namely the possibility of frequent collisions between photons that allow for collective fluid-like behaviors in the many photon system. While photon-photon interactions have been predicted to occur even in vacuum via virtual excitation of electron-positron pairs Heisenberg and Euler (1936), the cross section for such a process is so small that it can hardly be expected to play any role in realistic optical systems. On the other hand, the nonlinear polarization of nonlinear optical media is able to mediate significant interactions between photons Boyd (2008); Butcher and Cotter (2008): upon elimination of the matter degrees of freedom, third-order nonlinearities correspond in the language of Feynman diagrams to four-legged vertices describing, among other, binary collisions between a pair of photons.

Among the many different configurations that have been studied in the last few decades for nonlinear optical applications, systems in the so-called strong light-matter coupling regime have turned out to be particularly promising in order to obtain the relatively strong nonlinear interactions that are necessary for collective behavior. In this regime, the photon is strongly mixed with matter degrees of freedom, which gives rise to a new mixed quasi-particle, the polariton Hopfield (1958). Pictorially, the polariton can be seen as a photon dressed by a matter excitation: a reinforced optical nonlinearity then appears thanks to the relatively strong interactions between matter excitations. This strong coupling regime can be achieved in a number of material systems, from atomic gases Raimond et al. (2001); Berman et al. (1993); Fleischhauer et al. (2005) to semiconducting solid state media both in bulk Yu and Cardona (2005); Klingshirn (2007) and in cavity Weisbuch et al. (1992); Deveaud (2007) geometries, to circuit-QED systems based on superconducting Josephson junctions Schoelkopf and Girvin (2008); You and Nori (2011). In the following of the review, we will consider both photon and polariton excitations, depending on the non-resonant or resonant character of the electronic excitation dressing the photon within the material medium.

To create a stable luminous fluid, it is also crucial to give a finite effective mass to the photon. A simplest strategy to this purpose involves a spatial confinement of the photon by metallic and/or dielectric planar mirrors. In a planar geometry with a dielectric medium of refractive index and thickness enclosed within a pair of metallic mirrors, the photon motion along the perpendicular direction is quantized as , being a positive integer. For each longitudinal mode, the frequency dispersion as a function of the in-plane wavevector has the form

(1)

where the effective mass of the photon and the cut-off frequency are related by the relativistic-like expression

(2)

Using suitable values of the effective mass and the cut-off frequency extracted from microscopic calculations Savona (1999), the generic form (1) of the dispersion can be extended to the case of dielectric mirrors. In the presence of some electronic excitation resonant with the cavity mode, the elementary excitations of the cavity have a polaritonic character with a peculiar dispersion law that reflects their hybrid light-matter nature. An example of such dispersion is shown in the central panel of Fig.1: in spite of the complex light-matter interaction dynamics, the bottom of the lower polariton branch is still well approximated by a parabolic dispersion of the form (1) with an effective mass and the cut-off frequency .

Historically, the first mention of the concept of photon fluid dates back to the work of Brambilla et al. (1991); Staliunas (1993), where the time-evolution of the coherent electromagnetic field in a laser cavity with large Fresnel number was reformulated in terms of hydrodynamic equations for the many photon system analogous to the Gross-Pitaevskii equation for the superfluid order parameter. The local light intensity corresponds indeed to the photon density and the spatial gradient of its phase to the local current; the collective behavior originates from the effective photon-photon interactions stemming from the nonlinear refractive index of the medium as well from saturation of gain. In the following years, the transverse dynamics of the electromagnetic field in cavity devices has attracted a lot of attention, in particular the phenomena related to the spontaneous formation of transverse patterns Staliunas and Morcillo (2003); Denz et al. (2010) and to the generation and control of dissipative cavity solitons Ackemann et al. (2009)

In this pioneering literature on hydrodynamics of the photon fluid, a special attention was paid to the phase singularities of the photon field, that were immediately interpreted as quantized vortices Coullet et al. (1989). After their first experimental observation in bulk nonlinear crystals Swartzlander and Law (1992), most of the following literature addressed the physics of optical vortices in the context of the transverse dynamics of laser or photorefractive oscillators: the drift of vortices under the hydrodynamical effect of the Magnus force due to buoyancy was experimentally studied in Vaupel and Weiss (1995). The hydrodynamical shedding of vortices in a moving photon fluid hitting a large defect was first predicted in the optical context in Staliunas (1993) and a pioneering attempt of experimental investigation of this crucial effect of superfluid hydrodynamics was reported soon after in Vaupel et al. (1996).

From a different perspective, the close analogy between a laser threshold and a second order phase transition was recognized as early as in DeGiorgio and Scully (1970); Graham and Haken (1970): above the laser threshold, the electromagnetic field acquires a well defined phase by spontaneously breaking a symmetry as it happens to the matter Bose field in a Bose-Einstein condensate of material particles Gunton and Buckingham (1968); Huang (1987). This interpretation of lasing as the result of a kind of Bose-Einstein condensation of photons is clearest in spatially extended devices such as vertical cavity surface emitting lasers (VCSELs), where the onset of a coherent laser emission is associated to the appearance of long-range spatial coherence along the cavity plane according to the Penrose-Onsager criterion for off-diagonal long-range order Pitaevskii and Stringari (2004); Huang (1987)

(3)

Of course, the non-equilibrium nature of the laser device introduces crucial differences with respect to standard equilibrium BEC as explained in statistical mechanics textbooks Huang (1987): the steady state of the laser device is in fact not determined by a thermal equilibrium condition, but rather follows from a dynamical balance between the pumping and losses Haken (1975). As we shall see in the following of the review, this feature is responsible for a number of new effects.

From the experimental point of view, the close link between BEC and spontaneous coherence effects in optical systems has been fully recognized only in the last decade, starting with the literature on the so-called BEC of exciton-polaritons in semiconductor microcavities Baumberg et al. (2000); Stevenson et al. (2000); Baas et al. (2006); Kasprzak et al. (2006). In this context, questions related to the analogies and differences between laser operation and photon/polariton BEC have attracted a strong interest from the community, with a special attention to thermalization issues. An experimental observation of microcavity polaritons coherently accumulating in the lowest energy states of a harmonic trap potential according to a Bose distribution was reported in Balili et al. (2007): thermalization of the polariton gas was attributed to polariton-polariton collisions within the gas. A similar, apparently thermalized photon distribution was however observed in Bajoni et al. (2007) also in a weak-coupling regime where photon-photon interactions are very weak, which raises fundamental questions about the nature of fluctuations on top of a photon/polariton condensate. These observations are to be contrasted with the strongly non-equilibrium regimes of laser operation observed in a vertical cavity surface emitting laser (VCSEL) device in Scheuer and Orenstein (1999): thermalization into the lowest state is completely ineffective and the condensate mode displays a complex structure with array of vortices.

In the last years, the quest for condensation effects in photon gases has successfully explored a few other interesting avenues. An early mention of the possibility of a gas of bare photons thermalizing to a Bose condensed state via collisions mediated by the optical nonlinearity is found in Navez (2003). BEC in a thermalized gas of photons was experimentally observed in Klaers et al. (2010a) using a macroscopic optical cavity containing a dye solution: because of the rapid decoherence time of the dye molecules, photons are only weakly coupled to the electronic excitations and their thermalization is believed to occur via repeated absorption-emission cycles, which determine the temperature and the chemical potential of the gas in a grand-canonical picture Klaers et al. (2010b); Klaers et al. (2012). A kinetic condensation of purely classical light waves was observed in the remarkable experiment of Sun et al. (2012): as theoretically discussed in, e.g.,  Connaughton et al. (2005), turbulent wave mixing by the optical nonlinearity leads to a redistribution of energy among the different modes and, eventually, to its accumulation into the lowest, condensate mode. A most remarkable feature of this experiment is the complete absence of quantum features, which emphasizes the fundamentally classical origin of the Bose-Einstein condensation phenomenon. More complex condensation phenomena have been reported also in disordered lasers Conti et al. (2008) and in actively-mode-locked lasers Weill et al. (2010a, b).

Simultaneously to these studies on Bose-Einstein condensation and spontaneous coherence effects, a revived interest has been devoted also to the hydrodynamic properties of the photon gas. The concept of the Bogoliubov dispersion of elementary excitations on top of a photon condensate was first investigated for a planar cavity geometry in the pioneering works Chiao and Boyce (1999); Tanzini and Sorella (1999): thanks to the spatial confinement, photons have a massive dispersion of the form (1) and the photon-photon interactions responsible for the collective behavior of the fluid are provided by the nonlinearity of the cavity medium. None of these works however addressed the crucial role of dissipation in the physics of photon fluids: first steps in this direction appeared in Bolda et al. (2001) where the nucleation of vortices in a moving photon fluid past an impenetrable, cylindrical defect in a planar cavity was theoretically investigated. Differently from previous works on oscillators Staliunas (1993), a coherent pumping was proposed as a way to create the photon superfluid in the cavity: this resulted in a different form of the Gross-Pitaevskii equation as originally studied in Lugiato and Lefever (1987) and, more importantly, in the need to switch off the coherent pump to unlock the condensate phase before vortices can appear.

A completely different approach to the study of superfluidity properties of light involved the paraxial propagation of a light beam through a bulk nonlinear crystal, which can be recast into a superfluid hydrodynamic form under the replacement of the time coordinate with the longitudinal coordinate along the propagation direction. After the pioneering experiment in Swartzlander and Law (1992), many authors have theoretically investigated a number of hydrodynamic features in light propagation, from stable liquid-like solitonic structures Josserand and Rica (1997); Michinel et al. (2006), to vortices Firth and Skryabin (1997); Paz-Alonso and Michinel (2005), to the scattering on a defect potential Khamis et al. (2008), to superfluid motion Leboeuf and Moulieras (2010), to dispersive shock waves El et al. (2007), to the generation of optical analogs of acoustic black holes Fouxon et al. (2010). In spite of all this theoretical activity, not many nonlinear optics experiments have specifically addressed hydrodynamic features of light. Among the few exceptions, the experimental study of vortex and soliton stability and interactions Mamaev et al. (1996b, a); Królikowski et al. (1998), the generation, propagation and interaction of dispersive shock waves in fluids of light Wan et al. (2007); Jia et al. (2007), the propagation of a one-dimensional fluid through a small barrier potential Wan et al. (2010), a study of the generation of vortex pairs in the wake of an obstacle in a two-dimensional geometry Wan et al. (2008), a study of the Rayleigh-Taylor instability in stratified fluids Jia et al. (2012), and the realization of a trans-sonic flow of light in a Laval nozzle configuration Elazar et al. (2012). An alternative interesting perspective on many-body physics of photons was developed in the pioneering literature on quantum solitons in nonlinear optical fiber using a quantum nonlinear Schrödinger equation as well as Bethe ansatz techniques Lai and Haus (1989a, b); Kärtner and Haus (1993); Drummond et al. (1993).

Figure 1: Figure from Kasprzak et al., 2006. Upper panel: Sketch of a planar semiconductor microcavity delimited by two Bragg mirrors and embedding a quantum well (QW). The wavevector in the direction perpendicular to the cavity plane is quantized, while the in-plane motion is free. The cavity photon mode is strongly coupled to the excitonic transitions in the QWs. A laser beam with incidence angle and frequency can excite a microcavity mode with in-plane wavevector , while the near-field (far-field) secondary emission from the cavity provides information on the real-space (-space) density of excitations. Central panel: The energy dispersion of the polariton modes versus in-plane wavevector (angle). The exciton dispersion is negligible, due to the heavy mass of the exciton compared to that of the cavity photon. In the experiments, the system is incoherently excited by a laser beam tuned at a very high energy. Relaxation of the excess energy (via phonon emission, exciton-exciton scattering, etc.) leads to a population of the cavity polariton states and, possibly, Bose-Einstein condensation into the lowest polariton state. Lower panel: Experimental observation of polariton Bose-Einstein condensation obtained by increasing the intensity of the incoherent off-resonant optical pump.

The research on exciton-polaritons in semiconductor microcavities approached the physics of luminous quantum fluids following a rather different pathway. For many years, an intense activity has been devoted to the quest for Bose-Einstein condensation phenomena in gases of excitons in solid-state materials Griffin et al. (1996): excitons are neutral electron-hole pairs bound by Coulomb interaction, which behave as (composite) bosons. In spite of the interesting advances in the direction of exciton Bose-Einstein condensation in bulk cuprous oxide and cuprous chloride, bilayer electron systems Eisenstein and MacDonald (2004), and coupled quantum wells Butov (2007); High et al. (2012), so far none of these research axes has led to extensive studies of the quantum fluid properties of the alleged exciton condensate. The situation appears to be similar for what concerns condensates of magnons, i.e. magnetic excitations in solid-state materials: Bose-Einstein condensation has been observed Demokritov et al. (2006); Giamarchi et al. (2008), but no quantum hydrodynamic study has been reported yet.

The situation is very different for exciton-polaritons in semiconductor microcavities, that is bosonic quasi-particles resulting from the hybridization of the exciton with a planar cavity photon mode Weisbuch et al. (1992). Following the pioneering proposal by Imamoğlu et al., 1996, researchers have successfully explored the physics of Bose-Einstein condensation in these gases of exciton-polaritons. Thanks to the much smaller mass of polaritons, several orders of magnitude smaller than the exciton mass, this system can display Bose degeneracy at much higher temperatures and/or lower densities.

Historically, the first configuration where spontaneous coherence was observed in a polariton system was based on a coherent pumping of the cavity at a finite angle, close to the inflection point of the lower polariton dispersion. As experimentally demonstrated in Baumberg et al. (2000); Stevenson et al. (2000), above a threshold value of the pump intensity a sort of parametric oscillationCiuti et al. (2000, 2001); Whittaker (2001) occurs in the planar microcavity and the parametric luminescence on the signal and idler modes acquires a long-range coherence in both time and space Baas et al. (2006). As theoretically discussed in Carusotto and Ciuti (2005), the onset of parametric oscillation in these spatially extended planar cavity devices can be interpreted as an example of non-equilibrium Bose-Einstein condensation: the coherence of the signal and idler is not directly inherited from the pump, but appears via the spontaneous breaking of a phase symmetry.

The quest for Bose-Einstein condensation in a thermalized polariton gas under incoherent pumping required a few more years to be achieved: after some preliminary claims  Dang et al. (1998); Deng et al. (2002); Richard et al. (2005a, b), a conclusive demonstration was reported by  Kasprzak et al., 2006: the onset of Bose-Einstein condensation in a gas of exciton-polaritons was assessed both in -space from the macroscopic accumulation of particles into the low-energy states and in real space from the appearance of long-range coherence. The principle of these experiments is illustrated in Fig.1: the laser pump injects hot electron-hole pairs, whose excess energy can be dissipated via phonon emission and then Coulomb scattering processes. For sufficiently high pump powers, the density of the incoherently injected polaritons exceeds the critical density for Bose-Einstein condensation and a coherent condensate appears. Even if the system is still a driven-dissipative one, a quasi-thermal equilibrium state seems to be achieved once the loss rate is sufficiently slow as compared to the thermalization time of the polariton gas. For a more detailed discussion of the issues related to the relaxation mechanisms and the thermalization of the polariton gases, we refer the reader to the review papers that are already available on the subject, e.g. Deng et al. (2010); Keeling et al. (2007). While most of this research is being carried out in devices fabricated with inorganic GaAs- or CdTe-based alloys, a first observation of spontaneous polariton coherence in a non-resonantly pumped organic single-crystal microcavity was very recently reported in Kéna-Cohen and Forrest (2010).

Figure 2: Figures from Amo et al., 2009a. Polaritons are coherently injected into the microcavity by a nearly resonant laser field: in contrast to the non-resonant and incoherent pumping scheme of Fig. 1, this pumping scheme allows to precisely control the density and the in-plane flow speed of the polariton fluid by changing the parameters (intensity, frequency, incidence angle) of the driving laser. Upper group of panels: experimental images of the real- (panels I-III) and momentum- (panels IV-VI) space polariton density extracted from the near-field and far-field emitted light from the cavity. The different columns to increasing values of the polariton density from left to right refer: for the highest density value, polariton superfluidity is apparent as a suppression of the real-space density modulation (panel III) and the corresponding disappearance of the Rayleigh scattering ring (panel VI). Lower group of panels: corresponding theoretical results obtained by numerically solving the non-equilibrium Gross-Pitaevskii equation of Carusotto and Ciuti (2004).

The possibility of using exciton-polariton gases for studies of many-body physics and, more precisely, superfluid hydrodynamics was first proposed in Carusotto and Ciuti (2004): a coherent pump configuration as in Bolda et al. (2001) was adopted. In contrast to the case of off-resonant and incoherent excitation, in the resonant driving configuration it is possible to perform an ab initio theoretical description in terms of a generalized Gross-Pitaevskii equation without having to cope with a phenomenological description of complex relaxation processes. In spite of the simplicity of the system, the coherently injected condensate shows peculiar superfluid properties when it hits a defect in the cavity. The shape of the resulting density perturbation could be interpreted in terms of the celebrated Landau criterion of superfluidity using the generalized Bogoliubov dispersion of excitations in the non-equilibrium condensate: at low flow speeds, superfluidity manifests itself in the suppression of the real-space modulation around the defect and, correspondingly, in the disappearance of the Rayleigh scattering ring in -space. For larger flow speeds peculiar patterns appear in both the density and momentum distribution of the condensate polaritons. Experimental verification of these predictions was reported in Amo et al. (2009a) and is summarized in Fig.2, where the transition from a dissipative flow (panels I and IV) to a superfluid one (panels III and VI) is apparent in both the real- and the momentum-space images. Remarkably, in the same work it has been directly shown that photon-photon interactions are responsible for the appearance of a sound mode in the polariton fluid, as attested by the appearance of a Cherenkov-Mach cone at “supersonic” flow speeds. Following experimental works have extended the study to strong defects, shown the hydrodynamic nucleation of vortex-antivortex pairs Nardin et al. (2011a); Sanvitto et al. (2011) and of dark solitonsAmo et al. (2011); Grosso et al. (2011) in the flowing superfluid. A remarkable theoretical development of the last few years is to use light superfluids in planar geometries to study quantum hydrodynamics effects, in particular the analog Hawking radiation from acoustic black hole configurations Marino (2008); Solnyshkov et al. (2011); Gerace and Carusotto (2012).

Figure 3: Top panel: figure from Hartmann et al., 2006. In an array of coupled optical cavities, photon hopping occurs thanks to the spatial overlap of the photon modes of adjacent cavities. The strong optical nonlinearity is induced by a coherently driven atomic gas present in the cavities. Middle panel: figure from Greentree et al., 2006, showing a schematic diagram of a two-dimensional array of photonic crystal cavities, each cavity containing a single two-level atom. Bottom panel: a Jaynes-Cummings-Hubbard system obtained with superconducting quantum circuits. Each cavity consists of a superconducting transmission line resonator embedding a superconducting qubit, that is an artificial two-level atom. Figure from Koch et al., 2010.

In all these works, interactions between single photons are weak and the hydrodynamic behavior of the photon gas originates from the collective interactions of a large number of coherent photons sharing the same orbital wavefunction. Almost in the same period, a series of pioneering theoretical works Hartmann et al. (2006); Greentree et al. (2006); Angelakis et al. (2007) made the first steps in the theoretical investigation of a completely different regime of strongly interacting photon gases, where interactions between single photons are large enough to induce sizable quantum correlations in the photon gas. The starting point of these proposals is the so-called photon blockade phenomenon predicted in Imamoglu et al. (1997) and then experimentally observed in Birnbaum et al. (2005): when the optical nonlinearity of a single-mode cavity is large enough for a single photon to shift the resonance frequency by an amount larger than the cavity linewidth, a resonant laser is able to inject photons in the cavity only one at a time and the stream of transmitted photons are strongly antibunched in time. When this strong optical nonlinearity is inserted in a lattice geometry with many coupled cavities, one can expect that the photon gas will show the rich physics of the Bose-Hubbard model Fisher et al. (1989), including the superfluid to Mott-insulator transition recently observed in atomic gases Greiner et al. (2002). The most promising systems to experimentally address this physics are illustrated in the different panels of Fig.3: macroscopic cavities filled by an optically dressed atomic gas; an array of photonic crystal cavities embedding two-level emitters; superconducting circuits embedding artificial two-level atoms based on Josephson junctions. Soon after, these proposals were generalized to other many-body states, in particular Tonks-Girardeau gases of impenetrable photons in a one-dimensional hollow fiber geometry Chang et al. (2008) and fractional quantum Hall states Cho et al. (2008).

In contrast to the developments in superfluid hydrodynamics of dilute photon fluids, none of these pioneering works on strongly correlated gases specifically addressed the consequences of the losses that are unavoidably present when dealing with light. Dissipation was in fact considered only as a hindrance limiting the available time for manipulation and observation of the quantum state, with a most detrimental effect on the Mott insulator state. A new perspective on strongly correlated photon gases was introduced in Gerace et al. (2009) for the case of a two site photonic Josephson system and, soon later, in Carusotto et al. (2009) for the case of a Tonks-Girardeau gas: a full inclusion of the interplay of driving, dissipation and strong interactions into the model offers the opportunity to observe novel dynamical features in such a driven-dissipative photon gas, and also suggests new tools for its experimental manipulation. In addition to this, the driven-dissipative regime allows for novel mechanisms of photon blockade to be exploited in coupled cavity systems Liew and Savona (2010), Bamba et al. (2011); Bamba and Ciuti (2011). Very recently, the analogy with on-going developments in ultracold atomic gases Dalibard et al. (2011) has opened a new research line in the direction of creating synthethic gauge fields for photons: with a suitable tailoring of the photonic environment, the motion of a photon can be made to experience an effective vector potential Wang et al. (2009); Koch et al. (2010); Hafezi et al. (2011b); Umucalılar and Carusotto (2011) and novel quantum states of the photon fluid analogous to the fractional quantum Hall effect can be realized Cho et al. (2008); Umucalılar and Carusotto (2012).

This review article is organized as follows. In Sec. II, we briefly summarize the main tools of elementary quantum field theory that are used to theoretically describe the non-equilibrium and quantum physics of the photon fluid in nonlinear optical devices. Even though a particular attention is paid to planar microcavity geometries where most of the recent experimental observations were obtained, most of the concepts are fully general and can can be transferred to other systems. The generalized Gross-Pitaevskii equation describing the dynamics of the photon fluid at mean-field level is introduced in Sec.III: particular emphasis will be devoted to the new features stemming from the non-equilibrium nature of the system. The most significant static and dynamic properties of the coherent photon fluid under the different excitation schemes are reviewed in the following three sections: Sec.IV deals with the case of a coherent and quasi-resonant driving, Sec.V deals with the optical parametric oscillation case when the cavity is pumped in the vicinity of the inflection point of the lower polariton branch, and Sec.VI deals with the case of a incoherent pumping. We will review the consequences of the non-equilibrium condition on the condensate shape both in real and momentum space as well as its impact on the elementary excitation spectrum. The superfluidity properties of the photon (polariton) fluid are reviewed in Sec.VII: among the several signatures of superfluidity, most emphasis will be devoted to the density modulation induced in the photon fluid by a weak impurity, for which impressive experimental observations have been recently obtained. Subtle issues related to the generalization of the Landau criterion and the role of the interaction-induced speed of sound will be extensively discussed. More complex hydrodynamic effects involving the nucleation of solitons and vortices in the wake of a large and strong defect are reviewed in Sec.VIII. The review of the emerging field of strongly correlated photons is the subject of Sec. IX: we will discuss in detail the different microscopic mechanisms that can be used to achieve an efficient photon blockade regime and the rich new physics that has been predicted for lattices of strongly nonlinear cavities, in particular in the presence of an synthetic gauge field. Finally, conclusions and future perspectives are drawn in Sec.X.

Ii Quantum field description of nonlinear planar cavities

In this Section, we will review a second quantization formalism approach to describe the physics of 2D nonlinear cavities. Our goal is to present the essential quantum field theoretical tools that will be useful to understand the discussion of quantum fluid effects in the following sections. While we will give particular emphasis to semiconductor microcavity systems where most of the experimental observations have been carried out, much of the theoretical concepts reviewed here can be applied to arbitrary planar optical resonators embedding a nonlinear slab and are easily generalized to other geometries. We will be careful in pointing out when some properties are specific to a given system or not. Comprehensive introductions to semiconductor microcavity systems can be found in the specific literature, for instance Deveaud et al. (2003); Deveaud (2007); Deveaud et al. (2009); Kavokin et al. (2008).

ii.1 Free cavity fields and input-output formalism

The two-dimensional photon field

Since a planar cavity is by definition invariant under in-plane translations, the in-plane wavevector is a good quantum number for the free photon dynamics, which can be described by an Hamiltonian of the form

(4)

where the and operators respectively destroy and create a cavity photon with an in-plane wavevector and a polarization state . The creation and destruction operators satisfy standard Bose commutation rules

(5)
(6)

Two-dimensional real-space cavity photon field operators and are defined as the Fourier transform of ,

(7)

and again satisfy Bose commutation rules

(8)
(9)

While the operator can be interpreted as a sort of two-dimensional photon density on the cavity plane, the physically observable electric field at the three-dimensional position is expressed in terms of the field operators by

(10)

where the -dependence of the photon mode wavefunction keeps track of the metallic or dielectric nature of mirrors. In the simplest case of a planar cavity of refractive index enclosed between metallic mirrors spaced by a distance , the -dependent profile of the lowest mode has a sinusoidal shape extending for ,

(11)

the boundary conditions at the mirrors fix the position of the nodes, and the global amplitude of the field is determined by the energy of a single photon state . The unit vectors are a basis of light polarizations. A detailed discussion of the more complex case of dielectric cavities enclosed between a pair of Distributed Bragg Reflector (DBR) dielectric mirrors can be found in Savona (1999).

Polarization effects and effective spin-orbit interaction

Figure 4: Top panel: snapshots of the propagation of cavity excitations created by a tightly focused and linearly polarized laser pulse in a semiconductor planar microcavity. Left and right panels show the optical intensity in the same (left) and in the orthogonal (right) polarization. The polarization-dependent patterns are due to the TE/TM splitting of the planar cavity modes (12). Top panel taken from Langbein et al., 2005. Middle panels labeled as (a-c): the effective optical spin-orbit interaction (12) is equivalent to a -dependent magnetic field (orange arrows) that rotates the pseudo spin associated to the photon polarization state on the Poincaré sphere. Lower panel: as anticipated in Kavokin et al. (2005a), the combination of the effective spin-orbit interaction with disorder-induced resonant Rayleigh scattering of a linearly polarized pump leads to the optical analogue of the spin Hall effect that is visible in the -space pattern of the -polarization shown in the bottom panel. Middle and lower panels taken from Leyder et al., 2007

In writing the Hamiltonian (4), we have implicitly assumed that the polarization states are degenerate. While this assumption is exact at , the reflection amplitudes off a dielectric mirror for the TE (Transverse Electric) and TM (Transverse Magnetic) linear polarization states are generally different at , introducing a frequency splitting of the TE/TM modes proportional to in the small limit Panzarini et al. (1999).

Physically, the resulting TE/TM splitting can be interpreted as a kind of spin-orbit interaction term coupling the orbital (the wavevector ) and pseudospin (the polarization) degrees of freedom of the cavity photon and can be described by a Hamiltonian term of the form Kavokin et al. (2004); Shelykh et al. (2010)

(12)

where are the Pauli matrices, the polarization indices run here over the circular polarizations basis and quantifies the -dependent magnitude of the TE/TM splitting. In the alternative form shown in the last line, is the angle between the wavevector and the axis on the plane 1

If one represents the two polarization states of the cavity photon as a -pseudospin, then the spin-orbit interaction (12) is equivalent to an effective momentum-dependent magnetic field, which induces a precession of the pseudospins. As shown in Fig. 4, this kind of optical spin-orbit interactions can give rise to spectacular effects such as a strongly anisotropic polarization-dependent propagation Langbein et al. (2005). When combined with disorder-induced scattering, this effect is responsible for an optical analogue of the spin Hall effect Kavokin et al. (2005a); Leyder et al. (2007). More recently, this same physics was investigated in a purely photonic cavity in Maragkou et al. (2011).

It is worth noting that the term spin Hall effect of light is sometimes used in the photonic literature to denote a different family of effects stemming from the spin-orbit coupling experienced by photons propagating in bulk optical media with weak spatial inhomogeneities of the refractive index Liberman and Zel’dovich (1992); Bliokh and Bliokh (2004a); Onoda et al. (2004); Bliokh and Bliokh (2004b); Bliokh and Bliokh (2006). Recently, this coupling was shown to have remarkable consequences such as a sizable lateral shift of the trajectory of a light beam beyond the geometrical optics prediction Hosten and Kwiat (2008); Bliokh et al. (2008).

Pumping and losses: input-output theory and master equation

A quantum description of the driving of the cavity photon mode by an incident coherent laser beam can be obtained using the input-output theory of optical cavities. A complete discussion of this theory for single mode cavities can be found in quantum optics textbooks Gardiner and Zoller (2004); Walls and Milburn (2006). Its extension to the case of planar microcavities with a continuum of in-plane modes can be found in Ciuti and Carusotto (2006): the in-plane translational symmetry of the device guarantees that the in-plane vector is conserved, so that an external radiation of frequency and incident angle only couples to the cavity field component of in-plane wavevector . Correspondingly, the cavity field component of in-plane can only decay into external radiation emitted at an angle satisfying the analogous condition : this latter condition is schematically illustrated in the right-most part of the upper panel of Fig.1.

The Hamiltonian term describing the external driving of the cavity by a coherent incident field of amplitude can be written in -space as

(13)

where is the Fourier transform of and the coefficient is proportional to the transmission amplitude of the front mirror for light with in-plane wavevector .

The finite transmittivity of the front and the back mirrors of the cavity is responsible for the re-emission of light from the cavity with an amplitude proportional to the in-cavity field operator. Combining this secondary emission with the direct reflection of the coherent laser light off the front mirror, input-output theory leads to the form

(14)
(15)

for the quantum operators describing the transmitted and reflected fields, respectively. Analogously to the input coefficients in (13), the output coefficients are proportional to the transmission amplitude of the mirrors and can depend on the in-plane wavevector and the polarization . An implicit assumption of this formalism is that transmission through the mirrors is almost instantaneous, so that the relations between the external and in-cavity fields (13) are local in time. In frequency space, this means that the and coefficients do not depend on the frequency . The validity of this Markovian assumption is intuitively understood for thin metallic mirrors; a discussion for dielectric DBR mirrors was recently reported Sarchi and Carusotto (2010).

As usual, the emission of light by the cavity is accompanied by a radiative damping of the cavity field at a wavevector- and polarization-dependent rate , proportional to the sum of the mirror transmittivities . Tracing out the radiative modes of the field outside the cavity, dissipation results in additional terms to be included in the master equation for the evolution of the density matrix of the cavity field Walls and Milburn (2006); Gardiner and Zoller (2004); Breuer and Petruccione (2002),

(16)

Under the assumptions that the temperature of the radiative modes outside the cavity is much lower than the frequency of the cavity mode and that the radiative coupling is Markovian Breuer and Petruccione (2002), the super-operator accounting for the dissipative effects has the zero temperature Lindblad form

(17)

with a frequency-independent decay rate . Additional, non-radiative decay channels due, for instance, to absorption in the cavity material can be included in the model by including into the master equation (16) additional terms of the same form (17) and proportional to the non-radiative loss rate . In the following we shall indicate with the total decay rate of a cavity photon.

A complete calculation of the value of the , and parameters for specific configurations goes beyond the scope of this review and generally requires a microscopic solution of Maxwell equations for the field propagation across the device; in a planar geometry, a tool commonly used for this kind of calculations is the so-called transfer matrix method, reviewed e.g. in Burstein and Weisbuch (1995); Savona (1999). Useful general relations can be mentioned for the case when the front and back mirrors of the cavity are identical: under this assumption, , and the radiative decay rate . Furthermore, explicit expressions for the and coefficients can be given for the case of a planar cavity enclosed by loss-less metallic mirrors of transmittivity separated by a distance , namely and .

ii.2 Optical nonlinearities and effective photon-photon interactions

So far, we have discussed the dynamics of the cavity field at the level of linear optics, where the quantum dynamics of the non-interacting cavity field reduces to the classical wave equation stemming from Maxwell’s electrodynamics in material media. The situation changes when the cavity layer (or the mirrors) embeds a material with a sizable optical nonlinearity: the non-linear dependence of the matter polarization on the applied electric field is responsible for a number of wave-mixing processes coupling different cavity modes and generating new frequency components Butcher and Cotter (2008); Boyd (2008). Moreover, it can lead to strong modifications of the quantum fluctuation properties of the cavity field, such as a reduced noise on some field quadrature or even the generation of entangled states for the field Walls and Milburn (2006). The present review being devoted to the quantum fluid aspects of the photon gas, we will concentrate our attention on third-order nonlinearities proportional to the nonlinear polarizability that can be described in terms of binary interactions between pairs of photons.

Under the standard rotating-wave approximation (that is valid here provided the photon mass is larger than all other energy scales, e.g. losses, kinetic energy, interactions), the total number of photons is conserved and the nonlinear process can be described by a four-operator Hamiltonian term of the form,

(18)

where the matter degrees of freedom have been traced out and summarized into the effective photon-photon interaction potential . The polarization index runs over the circular polarization states and spin angular momentum is implicitly assumed to be conserved in the collision process. Total momentum is also conserved in the process of two photons of initial in-plane wavevector and scattering into the new wavevector states of in-plane momentum and .

As the typical length scale of the electron dynamics in typical bulk material media and in semiconductor heterostructures Bastard (1991) used for quantum fluid effects is much shorter than the optical wavelength along the plane, the interaction potential can be approximated with its zero-momentum value for : in real-space, this corresponds to assuming that photon-photon interaction occur via a local potential,

(19)

In two and three dimensions, the use of a strictly local interaction potential beyond the Born approximation often leads to UV divergences: standard techniques to make the theory regular involve renormalization of the interaction potential on a discrete lattice Mora and Castin (2003) or the use of suitably defined pseudo-potentials Huang (1987).

In the simplest case when the photon frequencies that are involved in the photon fluid dynamics are very far away from electronic resonances in the nonlinear optical material, the optical transitions are virtual and the population of the excited electronic states remains negligible. In this regime, the photon-photon potential can be expressed as in terms of the nonlinear refractive index normalized in a way such that the effective refractive index is . A more sophisticated theory of photon-photon interactions in a planar cavity device embedding a collection of anharmonic atoms appeared in Chiao et al. (2004). In the case of dielectric cavities, the cavity layer thickness has to be replaced by the effective thickness of the cavity mode  Savona (1999).

While sitting far from resonance generally allows to minimize undesired absorption losses in the material, non-resonant optical nonlinearities have the serious drawback of being generally very small and requiring a large number of photons to be observable. While this is not too much a concern for quantum fluid experiments, strong nonlinearities are required to observe photon blockade effects and generate strongly correlated photon fluids. In the next Subsection we shall extend our discussion to systems where the cavity mode is strongly coupled to narrow transitions in the optical medium: in this case, the photon inherits the strong nonlinearity of the matter excitations.

ii.3 Strong light-matter interaction and cavity polaritons

In the last two decades, the physics of strong light-matter interaction has flourished in many interesting domains, including the fields of atomic cavity QED Raimond et al. (2001); Berman et al. (1993); Fleischhauer et al. (2005), semiconductor microcavities and superconducting circuit QED Schoelkopf and Girvin (2008); You and Nori (2011). In this section of our review we shall concentrate our attention on the case of planar semiconductor microcavities embedding one or many quantum wells Deveaud et al. (2003); Deveaud (2007); Deveaud et al. (2009); Kavokin et al. (2008). A brief account of other systems will be given later in Sec.IX.

Quantum well excitons coupled to the cavity mode

As detailed in more specialized reviews such as Bastard, 1991 and Deveaud et al., 2003, a quantum well (QW) consists of a thin semiconductor layer (a few nm thick) embedded in a different semiconductor compound acting as ’barrier’ material. The chemical composition of the well is chosen to have the bottom of the conduction (the top of the valence) band at a lower (higher) energy than the surrounding material, thus producing quantum confinement of both electrons and holes. The lowest energy optical transition corresponds to the excitation a two-dimensional (hydrogen-like) electron-hole pair confined in the QW layer, the so-called exciton. In typical microcavity samples, one or more QWs are embedded in the cavity layer, with their plane parallel to the cavity plane. To have a strong and quasi-resonant coupling of their electronic degrees of freedom with the cavity mode, the QWs are placed at the antinodes of the cavity field and the cavity mode frequency is tuned in the vicinity of the lowest QW exciton with frequency .

At the level of linear optics Klingshirn (2007); Bastard (1991), the contribution of the quantum well exciton to the optical properties of the cavity can be described in terms of a resonant contribution to the dielectric polarizability of the structure of the form

(20)

where the quantum well is approximated as a very thin layer located at and the excitonic transition has an oscillator strength surface density . The exciton resonant frequency is weakly dependent on the in-plane wavevector , with an exciton mass of the order of the electron mass, i.e. orders of magnitude larger than the effective photon mass . The linewidth accounts for all non-radiative decay channels of the exciton.

In a quantum picture, the quantum well exciton can be described in terms of destruction (creation) operators () that destroy (create) an exciton with total momentum and their real-space counterparts and defined as their Fourier transform as in (7). In standard QWs Bastard (1991), electrons can have spin projection along the growth axis , while holes can have . Among the four exciton states with spin projection that exist in a QW, here we shall restrict our attention to the that are coupled to the cavity mode; the ones with are optically inactive. Provided the interparticle distance remains much larger than their Bohr radius, excitons behave as bosonic particles, whose creation and destruction operators satisfy Bose commutation rules of the same form as equations (5-6) for cavity photons. The corrections to bosonic behavior due to the composite nature of the exciton have been theoretically addressed in Combescot et al. (2008). Discussions of exciton physics with a special eye to planar microcavities can be found in the recent review papers by Deveaud et al., 2003, Deveaud et al., 2009 and Deng et al., 2010.

In a second quantized formalism, the Hamiltonian describing in space the exciton dynamics and its coupling to the cavity field can be written as

(21)

The last term describes the coherent conversion of an exciton into a cavity photon at a Rabi frequency proportional to the product of the electric dipole of the quantum well exciton transition times the local amplitude of the cavity photon electric field. In terms of the classical dielectric properties (20) of the quantum well, the Rabi frequency in a planar metallic cavity can be related to the exciton oscillator strength by

(22)

where the final fraction accounts for the displacement of the quantum well position from an antinode of the cavity mode. In the general case of DBR cavities, the distance between the mirrors has to be replaced by the effective thickness of the cavity mode Savona (1999). If quantum wells are present in the cavity, linear combinations of the exciton states are dark, while the single bright one is coupled to the cavity mode with an enhanced coupling . When the QWs are located in equivalent positions, the collective coupling enhancement factor is . Note that, in contrast to the case of a quantum well in free space Andreani et al. (1991); Tassone et al. (1992), the presence of the microcavity eliminates the direct coupling of the quantum well exciton to the external radiative modes and radiative decay of the exciton can only take place via the lossy cavity mode.

In addition to these effects stemming from its coupling to radiation, the exciton is also subject to non-radiative recombination processes. In a master equation formalism, these processes can be described by a Lindblad term of the form (17) with a (generally weak) decay rate , as well as dephasing processes at a rate  Liew and Savona (2011) due to, e.g., interactions with carriers and spatial inhomogeneities in the quantum well thickness. The corresponding term in the master equation reads:

(23)

In view of the on-going theoretical and experimental developments, it is important to remind that the form (21) of the light-matter coupling term is based on the so-called rotating-wave approximation where anti-resonant terms proportional to are neglected. This approximation is very accurate as long as , which is generally the case for microcavity systems under examination here where is of the order of  meV for typical III-V based samples and of the order of  meV for II-VI based samples. Significant deviations appear in the opposite ultra-strong coupling regime where is comparable or larger than . Such a regime has been recently theoretically predicted and experimentally observed in systems where the cavity mode is strongly coupled to intersubband electronic transitions in doped quantum wells Ciuti et al. (2005), Anappara et al. (2009); Guenter et al. (2009); Todorov et al. (2010), cyclotron transitions of two-dimensional electron gases Hagenmüller and De Liberato (2010); Scalari et al. (2012) and in circuit-QED systems Devoret et al. (2007)Bourassa et al. (2009); Nataf and Ciuti (2011)Niemczyk et al. (2010); Fedorov et al. (2010). In addition to going beyond the rotating-wave approximation in the light-matter coupling term in (21), a consistent theoretical description of systems in the ultra-strong coupling regime also requires a more sophisticated modeling of the frequency dependence of the dissipative baths Ciuti and Carusotto (2006); Carusotto et al. (2012); Ridolfo et al. (2012).

Exciton-polaritons

The coupled dynamics of photons and excitons in a microcavity is described by the Hamiltonian terms (4) and (21) that involve the product of two field operators: as a result, the quantum dynamics exactly recovers the classical Maxwell wave equations inserting a thin layer of material with the susceptibility (20). Furthermore, simple linear combinations of the and operators Hopfield (1958) can be used to transform the Hamiltonian into a diagonal form

(24)

where the and operators correspond to hybridized excitations resulting from the linear superposition of exciton and cavity photon modes, the so-called exciton-polaritons. Thanks to the translational symmetry along the cavity plane, the wavevector and the spin remain good quantum numbers for the new quasi-particles.

The dispersion of the upper and lower polariton branches and has the typical anticrossing form

(25)

that is shown in the middle panel of Fig.1: the bare cavity photon and exciton branches (dashed lines) are mixed by the light-matter coupling term into the polariton branches; the minimum splitting of the two polariton branches is obtained for . The so-called strong light-matter coupling regime is defined when , i.e., light-matter coupling exceeding losses.

The photonic and excitonic content of the polariton modes is quantified by the real-valued Hopfield coefficients, which relate the cavity mode and exciton field amplitude to the ones in the lower and upper polariton modes,

(26)
(27)

Close to the crossing point, both polariton modes have approximately equal photon and exciton content , while farther away the two polariton acquire a purely excitonic or photonic character. Differently from the general case of Hopfield (1958), the absence of the creation operators in the Bogoliubov transormation (26-27) is a consequence of the rotating-wave approximation.

The effective masses of the lower and upper polariton branches are calculated from the curvature of the dispersion (25). In the most remarkable case where , the condition implies that and ; furthermore, the two polaritons have equal photon and exciton content as long as . Most of the experiments that will be discussed in the next sections only involve the lowest states of the lower polariton branch for small wavevectors and a description in terms of a single polariton branch with parabolic dispersion of mass is accurate. More details on the basic properties of microcavity polaritons can be found in Deveaud et al. (2003); Deveaud (2007); Deveaud et al. (2009); Kavokin et al. (2008)

Polariton-polariton interactions

As it is typically done in the many-body physics, a widespread procedure is to describe the system in terms of an effective model Hamiltonian that is able to reproduce the exciton-exciton interactions without invoking its elementary constituents: the idea is to replace the complex Coulomb interactions between the electron and holes by a simple two-body interaction potential involving the exciton as a whole. In the simplest version, low-energy scattering at a relative wave vector much smaller than the exciton Bohr radius can be accurately described by a contact two-body interaction potential term,

(28)

where the spin indices run over the circular polarization basis . Rotational invariance for a contact interaction potential imposes that total exciton spin is conserved and that and .

A first estimation of in terms of a microscopic quantum well electron-hole model of the exciton was provided in Ciuti et al. (1998); Tassone and Yamamoto (1999) within the Born approximation, giving where is the dielectric constant of the material and is the exciton Bohr radius. This positive and featureless value of the triplet interaction constant appears to be in reasonable agreement with experimental measurements Amo et al. (2009a); Ferrier et al. (2011) and provides the most relevant contribution when the microcavity is excited by circularly polarized light.

For other polarization configurations, the much richer physics of the singlet channel needs to be considered: different experiments Kavokin et al. (2005b); Amo et al. (2010b) have reported values of the singlet interaction constant with opposite signs; first systematic studies of the singlet interaction constant as a function of the microcavity parameters were recently performed in Vladimirova et al. (2010); Paraïso (2010). In analogy to the Feshbach resonance phenomenon in atom-atom scattering when the energy of the incident atomic pair is in the vicinity of a molecular intermediate state Chin et al. (2010), several authors Savasta et al. (1999, 2003); Wouters (2007); Carusotto et al. (2010) have suggested the importance of the biexciton intermediate state (i.e. a two-electron, two-hole bound complex in the singlet scattering channel: in the vicinity of the scattering resonance, the effective interaction constant in the singlet channel is expected to be strongly enhanced and to change sign. More details on the theory of biexciton Feshbach effects will be presented in Sec.IX.2.3.

In principle, exciton-exciton interactions may also transform a pair of bright excitons into a pair of spin dark excitons Ciuti et al. (1998). While these processes are important in isolated quantum wells, they no longer conserve energy in microcavities: because of the Rabi coupling between photons and bright excitons, the dark spin excitons are at much higher energy than the lower polariton branch and can only play a role as non-resonant intermediate states of high-order processes.

An additional interaction channel originates from saturation of the exciton oscillator strengthTassone and Yamamoto (1999); Rochat et al. (2000); Glazov et al. (2009): Pauli exclusion principle for electrons and holes forbids that another exciton be created at a distance shorter than the Bohr radius from an existing exciton. At the lowest order in the exciton density and retaining only the bright exciton states, this can be modeled as an effective quartic Hamiltonian term of the form

(29)

with a saturation potential . As it is discussed in detail in the recent paper by Glazov et al., 2009, the exact value of the saturation density depends on the specific shape of the wavefunction of the relative motion. For typical values of semiconductor microcavities, the saturation term is generally significantly smaller than the exciton-exciton contribution.

As previously mentioned, in many relevant experimental conditions, one can restrict to the bottom of the lower polariton branch and approximate the dispersion as parabolic. In this regime, we can rewrite the interaction Hamiltonians (28) and (29) in terms of polariton operators and keep only the terms involving the lower polariton branch. This finally leads to an effective polariton-polariton contact interaction

(30)

with

(31)

Recent experiments Cristofolini et al. (2012) have demonstrated strong coupling of a cavity photon with a hybrid exciton in a double quantum well geometry, where the direct and indirect exciton states are mixed by coherent electron tunneling events across the barrier. As a result, the polariton acquires a finite electric dipole moment, that is expected to strongly reinforce polariton-polariton interactions and, possibly, induce new effects due to long-range dipole form of the interaction potential Astrakharchik et al. (2007); Böning et al. (2011).

ii.4 External potentials affecting the in-plane motion of cavity excitations

In analogy to magnetic and optical traps for atomic gases Pitaevskii and Stringari (2004), several strategies have been explored to implement spatially- and spin-dependent external potentials for cavity photons and excitons, creating Hamiltonian terms like

(32)

Provided the amplitudes of the potentials and are much smaller than the Rabi energy and for a sufficiently smooth spatial variation, the resulting potential acting on lower polaritons can be written as

(33)

We will start by considering several experimental protocols to generate scalar potentials of the form .

The polariton trap used in the Bose-Einstein condensation experiment of Balili et al., 2007 was created by applying a mechanical stress, resulting in an energy red-shift of the exciton state Negoita et al. (1999) and thus an attractive potential for excitons () with an approximately harmonic profile at the bottom.

Figure 5: Top panel: figure from El Daïf et al., 2006. An effective lateral confinement potential for cavity photons (and hence for cavity polaritons) can be obtained with a position-dependent cavity thickness (achievable in semiconductor microcavities via growth, etching and regrowth). Bottom panel: figure from Sanvitto et al., 2011. An effective lateral confinement potential for cavity photons can be obtained by using control laser spots with different polarizations as first achieved in Amo et al. (2010b).

Another concept of polariton trap exploits the dependence of the cavity mode frequency on the thickness of the cavity layer. This feature is usually exploited to obtain a full scan of the exciton-photon detuning across the resonance on a single microcavity sample by growing a wedge-shaped wafer Weisbuch et al. (1992). In the experiment of Sermage et al. (2001), a polariton acceleration effect was observed as a consequence of the cavity wedge: within a parabolic band approximation, the polariton motion can be described by a Newton-like equation .

A more sophisticated development of this same idea was implemented in Lu et al. (2005); El Daïf et al. (2006); Kaitouni et al. (2006) to create polariton boxes with an overgrowth technique able to create a position-dependent thickness of the cavity spacer (see top panel of Fig. 5). This technique provides a quite flexible tool to design external potentials with arbitrary shapes on characteristic spatial lengths in the few m range.

A more brutal way of confining polaritons is to design a micropillar structure by etching away all the layers forming the top mirror and the cavity layer (and possibly also the lower mirror down to the substrate): in this case, light is confined in the in-plane directions by the large refractive index mismatch at the air-semiconductor interface Burstein and Weisbuch (1995). As with the regrowth technique, arbitrary geometries can be realized with full three-dimensional confinement Gérard et al. (1996); Reithmaier et al. (1997); Ohnesorge et al. (1997) or with free motion along photonic wires Zhang et al. (1995); Kuther et al. (1998); Wertz et al. (2010). The direct processing of micropillar devices may come at the price of larger lateral losses at the cavity/air interface. The recent realization of double pillar configurations showing a sizable tunnel coupling of the photonic wavefunctions de Vasconcellos et al. (2011) is very promising in view of creating more complex structures such as cavity arrays.

Another strategy to apply external potentials to exciton-polaritons is to deposit metal films on the surface of the cavity: depending on the geometry, this can either increase the photon energy by squeezing the tails of electric field inside the DBR mirror Kim et al. (2008) or even strongly mix the Tamm plasmon with the exciton and the cavity photon modes as predicted in Kaliteevski et al. (2009). Experimental evidence of the strong coupling of a Tamm plasmon with quantum well excitons in a single DBR mirror-metal/air gap microcavity was reported in Symonds et al. (2009); Grossmann et al. (2011). Using metallic layers also allows for a dynamic tuning of the external potential by applying DC electric fields to the structure: an electric field indeed acts on polaritons by reducing the quantum well oscillator strength by spatially separating the electron/hole pairs and by lowering its energy via the quantum confined Stark effect Kim et al. (2008) and thermal expansion Grossmann et al. (2011).

A technique to generate a periodic potential using surface acoustic waves (SAWs) propagating in the semiconductor structure was developed in de Lima et al. (2006) and then applied to fragment a polariton condensate into an array of strongly elongated gases with reduced coherence Cerda-Méndez et al. (2010). Microscopically, the SAW strain field simultaneously changes the thickness and the refractive index of the cavity layer, modulating the exciton energy via a deformation potential. Each of these effects is proportional to the real part of the local amplitude of the sound wave. As a result the effective potential moves in space at a speed equal to the SAW phase velocity, on the order of the sound speed in the material ( m/s in Cerda-Méndez et al., 2010), i.e. much smaller than all other characteristic speeds of the polariton system.

An even more flexible all-optical technique was developed in Amo et al. (2010b): a strong laser field is used to inject polaritons into the cavity with a given spatial profile. Assuming that one can neglect spin-flip processes and that a large number of polaritons is injected, polariton-polariton interactions between counter-polarized polaritons create an effective potential for polaritons, whose geometry and time-dependence can be easily controlled via the laser field. As it is shown in the bottom panel of Fig. 5, quite localized potentials can be generated by this technique. For instance, they were used in Sanvitto et al. (2011) to study vortex dynamics in polariton superfluids. With respect to SAWs, optical potentials have the advantage that they can be modulated on extremely short time scales and designed with almost any desired spatial shape down to the m scale.

Even though the external potential generated by all these methods is mostly a spin-independent one, in some cases it is essential to take into account in the model its spin-dependent component: the different reflection amplitude of TE/TM polarization states at the cavity/air interface in a micropillar device is responsible for a splitting of the two linear polarized states Kuther et al. (1998); an analogous effect may also arise from the coupling of the exciton with mechanical stress in a laterally patterned device Dasbach et al. (2002). In some proposed applications such as Umucalılar and Carusotto (2011), it is essential to be able to impose a specific form of spin-dependent potential to the polaritons. The Zeeman shift under a static magnetic field can be used to split the components of the exciton. A (possibly spatially modulated) strain field van Doorn et al. (1996) or a lateral patterning with submicrometer periodicity Flanders (1983) can be used to generate a linear birefringence in the cavity material, which would then result in a spin-dependent photonic potential of the form

(34)

where the angle fixes the orientation of the birefringence axis and the basis of circularly polarized states is used.

Iii The driven-dissipative Gross-Pitaevskii equation

iii.1 The mean-field approximation

In the previous section we have introduced a quantum mechanical model for the coupled quantum field dynamics of cavity photons and quantum well excitons: the interaction terms describing for exciton-exciton collisions make the dynamics non trivial and are responsible for a number of nonlinear and quantum phenomena. A standard approximation to attack this kind of problems is the so-called mean field approximation, based on classical evolution equations for the expectation values of the quantum field operators and : the former is (approximately) proportional to the expectation value of the in-cavity electric field, the latter is proportional to the matter polarization due to the exciton transition. The mean-field equations of motion for and are obtained from the Heisenberg equations of motion for and by replacing every instance of an operator with the corresponding expectation value.

Of course, this approach is exact as long as we are restricting ourselves to terms in the Hamiltonian that involve at most two operators, e.g. the free dynamics of the coupled cavity photons and excitons, their mutual interconversion, as well as the one-body pumping and loss terms discussed in the previous section. An approximation is instead made on the interactions terms, for which one is assuming,

(35)

In nonlinear optics, a mean-field approximation of this kind is implicitly made whenever one writes the polarization of a medium as the product of a nonlinear susceptibility times the -th power of the classical electric field Boyd (2008); Butcher and Cotter (2008): this approach provides accurate results in most optical media as nonlinear effects require the presence of a large number of photons. If one is interested in the interplay of the optical nonlinearity with quantum fluctuations, a linearized treatment is generally enough to account for the relatively weak quantum fluctuations Walls and Milburn (2006). An alternative, semiclassical approach is based on the truncated-Wigner representation of the quantum field, as discussed in the next subsection. The quantitative validity of these approaches has been tested by comparing with exact calculations for the simplest model of single-mode nonlinear cavity Carusotto (2001) as well as with multimode Wigner simulations Verger et al. (2007). Of course, this approximation breaks down completely as soon as one enters the photon blockade regime Imamoglu et al. (1997) where a single photon is able to substantially modify the response of a device and the discrete, quantum nature of photons starts playing a crucial role.

In the context of quantum gases of material particles, a classical partial differential equation for the superfluid order parameter was first written by Gross, 1961 and Pitaevskii, 1961 to describe quantum vortices in liquid Helium. Starting from modern formulations of the Bogoliubov theory of the dilute Bose gas, a Gross-Pitaevskii equation (GPE) of the form

(36)

can be derived from first principles to describe the dynamics of the condensate wavefunction : the macroscopic fraction of particles that populate the condensate mode behave in a collective way and the quantum atomic matter field behaves as a classical field  Pitaevskii and Stringari (2004); Leggett (2001). Generally speaking, the GPE is quantitatively accurate as long as the occupation of modes other than the condensate one is small: in three dimensions this requires that the temperature is much lower than the transition temperature for Bose-Einstein condensation and that the gas is dilute, i.e. the atom-atom scattering length is much shorter than the mean interparticle spacing .

From a physical standpoint, the GPE is then for the matter field what Maxwell equations are for quantum electrodynamics in nonlinear media. An important difference is however worth emphasizing: while the global phase of the electromagnetic field has a direct and observable physical meaning, conservation of the total number of particles implies that the expectation value of the matter field of massive particles exactly vanishes. Even if treatments based on a spontaneous breaking of the phase symmetry are popular in the literature and have a deep physical foundation in terms of the BEC phase transition Gunton and Buckingham (1968), some doubts may remain on the consistence of descriptions based on a classical atomic field, especially for finite systems. The picture has been reconciled by recent theoretical works that have developed particle-number conserving versions of the Bogoliubov theory Castin and Dum (1998); Gardiner (1997) and have investigated in full detail the meaning of the condensate phase Leggett (1995); Javanainen and Yoo (1996); Cirac et al. (1996); Castin and Dalibard (1997).

iii.2 Generalized GPE under coherent driving

In many experimental circumstances, it is not necessary to work with the pair of equations of motions for the photonic and excitonic fields and one can restrict to a single classical field describing the lower polariton field in a single spin state. This simplified description is generally legitimate provided the Rabi frequency is much larger than all other energy scales of the problem, namely the kinetic and interaction energies, the pump detuning from the bottom of the lower polariton, and the loss rates. The resulting Gross-Pitaevskii equation for polaritons has the form