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We present the theory and experimental demonstration of a topological classification of finite tight binding Hamiltonians with chiral symmetry. Using the graph-theoretic notion of complete matchings, we show that many chiral tight binding structures can be divided into a number of sections, each of which has independent topological phases. Hence the overall classification is $N\mathbb{Z}_2$, corresponding to $2^N$ distinct phases, where $N$ is the number of sections with a non-trivial $\mathbb{Z}_2$ classification. In our classification, distinct topological phases are separated by exact closures in the energy spectrum of the Hamiltonian, with degenerate pairs of zero energy states. We show that that these zero energy states have an unusual localisation across distinct regions of the structure, determined by the manner in which the sections are connected together. We use this localisation to provide an experimental demonstration of the validity of the classification, through radio frequency measurements on a coaxial cable network which maps onto a tight binding system. The structure we investigate is a cable analogue of an ideal graphene ribbon, which divides into four sections and has a $4\mathbb{Z}_2$ topological classification., Comment: 18 pages, 9 figures, 2 tables

We report an experimental study of the disordered Su-Schrieffer-Heeger (SSH) model, implemented in a system of coaxial cables, whose radio frequency properties map on to the SSH Hamiltonian. By measuring multiple chains with random hopping terms, we demonstrate the presence of a topologically protected state, with frequency variation of less than 0.2% over the ensemble. Connecting the ends of the chains to form loops, we observe a topological phase transition, characterised by the closure of the band gap and the appearance of states which are delocalised, despite the strong disorder.

Spin-dependent, directional light-matter interactions form the basis of chiral quantum networks. In the solid state, quantum emitters commonly possess circularly polarised optical transitions with spin-dependent handedness. We demonstrate numerically that spin-dependent chiral coupling can be realised by embedding such an emitter in a waveguide-coupled nanocavity, which supports two near-degenerate, orthogonally-polarised cavity modes. The chiral behaviour arises due to direction-dependent interference between the cavity modes upon coupling to two single-mode output waveguides. Notably, an experimentally realistic cavity design simultaneously supports near-unity chiral contrast, efficient ($\beta > 0.95$) waveguide coupling and enhanced light-matter interaction strength (Purcell factor $F_P > 70$). In combination, these parameters could enable the development of highly coherent spin-photon interfaces, ready for integration into nanophotonic circuits., Comment: 19 pages, 5 figures

The Su-Schrieffer-Heeger (SSH) model is the simplest example of a lattice with non-trivial topology. It supports mid-gap topologically protected states, whose energies are unaffected by disorder. We show that photonic multilayer structures provide an exact implementation of an SSH lattice, provided each layer has the same propagation thickness. From this, it follows that the cavity mode in a conventional semiconductor microcavity is a protected SSH mid-gap state. We demonstrate this experimentally using controlled disorder in a mathematically equivalent system, a radio frequency transmission line made from sections of coaxial cable with high and low impedances. We also show theoretically that transmission lines connected to form networks map onto topologically interesting lattices in in higher dimensions., Comment: 5 pages, 4 figures

We study the effect of photonic spin-orbit coupling (SOC) in micropillar lattices on the topological edge states of a one-dimensional chain with a zigzag geometry, corresponding to the Su-Schrieffer-Heeger model equipped with an additional internal degree of freedom. The system combines the strong hopping anisotropy of the $p$-type pillar modes with the large TE-TM splitting in Bragg microcavities. By resolving the photoluminescence emission in energy and polarization we probe the effects of the resulting SOC on the spatial and spectral properties of the edge modes. We find that the edge modes feature a fine structure of states that penetrate by different amounts into the bulk of the chain, depending on the strength of the SOC terms present, thereby opening a route to manipulation of the topological states in the system., Comment: 5 pages, 3 figures (main text); 8 pages, 5 figures (supplementary material)

We study exciton-polaritons in a two-dimensional Lieb lattice of micropillars. The energy spectrum of the system features two flat bands formed from $S$ and $P_{x,y}$ photonic orbitals, into which we trigger bosonic condensation under high power excitation. The symmetry of the orbital wave functions combined with photonic spin-orbit coupling gives rise to emission patterns with pseudospin texture in the flat band condensates. Our work shows the potential of polariton lattices for emulating flat band Hamiltonians with spin-orbit coupling, orbital degrees of freedom and interactions.

We report on the formation of a diverse family of transverse spatial polygon patterns in a microcavity polariton fluid under coherent driving by a blue-detuned pump. Patterns emerge spontaneously as a result of energy-degenerate polariton-polariton scattering from the pump state to interfering high order vortex and antivortex modes, breaking azimuthal symmetry. The interplay between a multimode parametric instability and intrinsic optical bistability leads to a sharp spike in the value of second order coherence $g^{(2)}(0)$ of the emitted light, which we attribute to the strongly superlinear kinetics of the underlying scattering processes driving the formation of patterns. We show numerically by means of a linear stability analysis how the growth of parametric instabilities in our system can lead to spontaneous symmetry breaking, predicting the formation and competition of different pattern states in good agreement with experimental observations., Comment: Main text: 12 pages, 8 figures Supplementary material: 2 pages, 3 figures

We demonstrate that Kerr nonlinearity in optical circuits can lead to both resonant four-wave mixing and photon blockade, which can be used for high-yield generation of high-fidelity individual photon pairs with conjugated frequencies. We propose an optical circuit, which, in the optimal pulsed-drive regime, would produce photon pairs at the rate up to $10^5s^{-1}$ (0.5 pairs per pulse) with $g^{(2)}<10^{-2}$ for one of the conjugated frequencies. We show that such a scheme can be utilised to generate color-entangled photons., Comment: 5+\epsilon pages, 6 figures

We report an extended family of spin textures in coexisting modes of zero-dimensional polariton condensates spatially confined in tunable open microcavity structures. The coupling between photon spin and angular momentum, which is enhanced in the open cavity structures, leads to new eigenstates of the polariton condensates carrying quantised spin vortices. Depending on the strength and anisotropy of the cavity confinement potential and the strength of the spin-orbit coupling, which can be tuned via the excitonic/photonic fractions, the condensate emissions exhibit either spin-vortex-like patterns or linear polarization, in good agreement with theoretical modelling.

We report an ultrafast optical tuning of the reflectivity of AlGaAs/InAlGaAs multiple quantum well photonic crystal waveguides using a reflection geometry, pump-probe technique.

We demonstrate the creation of vortices in a macroscopically occupied polariton state formed in a semiconductor microcavity. A weak external laser beam carrying orbital angular momentum (OAM) is used to imprint a vortex on the condensate arising from the polariton optical parametric oscillator (OPO). The vortex core radius is found to decrease with increasing pump power, and is determined by polariton-polariton interactions. As a result of OAM conservation in the parametric scattering process, the excitation consists of a vortex in the signal and a corresponding anti-vortex in the idler of the OPO. The experimental results are in good agreement with a theoretical model of a vortex in the polariton OPO., Comment: 3 figures

We study the polarisation dependence of the homogeneously broadened nuclear spin resonance in a crystal. We employ a combinatorial method to restrict the nuclear states to a fixed polarisation and show that the centre of the resonance is shifted linearly with the nuclear polarisation by up to the zero polarisation line width. The width shrinks from its maximum value at zero polarisation to zero at full polarisation. This suggests to use the line shape as a direct measure of nuclear polarisation reached under dynamical pumping. In the limit of single quantum of excitation above the fully ferromagnetic state, we provide an explicit solution to the problem of nuclear spin dynamics which links a bound on the fastest decay rate to the observable width of the resonance line., Comment: 8 pages, 2 figures

Several mechanisms are discussed which could determine the spatial coherence of a polariton condensate confined to a one dimensional wire. The mechanisms considered are polariton-polariton interactions, disorder scattering and non-equilibrium occupation of finite momentum modes. For each case, the shape of the resulting spatial coherence function g1(x) is analysed. The results are compared with the experimental data on a polariton condensate in an acoustic lattice from [E. A. Cerda-Mendez et al, Phys. Rev. Lett. 105, 116402 (2010)]. It is concluded that the shape of g1(x) can only be explained by non-equilibrium effects, and that ~10 modes are occupied in the experimental system., Comment: 6 pages, 4 figures

A method is presented for modelling the optical properties of a photonic crystal structure mounted on a substrate which is thick enough that the light reflected from the back is incoherent with reflections from the front. Transmission and reflection spectra are presented for the cases where the structure is a multilayer planar microcavity and an opal photonic crystal. The contributions from the incoherent reflections can be very large, and in the photonic crystal case, change the positions of the spectral peaks.

We study non-equilibrium polariton superfluids in the optical parametric oscillator (OPO) regime using a two-component Gross-Pitaevskii equation with pumping and decay. We identify a regime above OPO threshold, where the system undergoes spontaneous symmetry breaking and is unstable towards vortex formation without any driving rotation. Stable vortex solutions differ from metastable ones; the latter can persist in OPO superfluids but can only be triggered externally. Both spontaneous and triggered vortices are characterised by a generalised healing length, specified by the OPO parameters only., Comment: 5 pages, 4 figures

Real and momentum space spectrally resolved images of microcavity polariton emission in the regime of condensation are investigated under non resonant excitation using a laser source with reduced intensity fluctuations on the timescale of the exciton lifetime. We observe that the polariton emission consists of many macroscopically occupied modes. Lower energy modes are strongly localized by the photonic potential disorder on a scale of few microns. Higher energy modes have finite k-vectors and are delocalized over 10-15 microns. All the modes exhibit long range spatial coherence comparable to their size. We provide a theoretical model describing the behavior of the system with the results of the simulations in good agreement with the experimental observations. We show that the multimode emission of the polariton condensate is a result of its nonequilibrium character, the interaction with the local photonic potential and the reduced intensity fluctuations of the excitation laser., Comment: 9 pages, 12 figures

A theoretical model is presented which explains the dominant decoherence process in a microcavity polariton condensate. The mechanism which is invoked is the effect of self-phase modulation, whereby interactions transform polariton number fluctuations into random energy variations. The model shows that the phase coherence decay, g1(t), has a Kubo form, which can be Gaussian or exponential, depending on whether the number fluctuations are slow or fast. This fluctuation rate also determines the decay time of the intensity correlation function, g2(t), so it can be directly determined experimentally. The model explains recent experimental measurements of a relatively fast Gaussian decay for g1(t), but also predicts a regime, further above threshold, where the decay is much slower., Comment: 5 pages, 1 figure

Our model comprehensively simulates modern nanoscale semiconductor microcavities incorporating cavity quantum electrodynamics within both the weak and strong coupling regimes, using on-resonant laser excitation and nonresonant excitation due to a wetting layer. For weak coupling, the most significant effect is photon antibunching with nonresonant emission. We investigate how the antibunching characteristics change as the cavity finesse is increased towards the strong coupling regime. Antibunching can also be observed in a strongly coupled system with resonant excitation, using the photon blockade mechanism which has been demonstrated in atom systems. We calculate what cavity parameters are required to observe this effect. Experimentally these studies are equivalent to nonlinear pump probe measurements, where a strong pump, either resonant or nonresonant, is used to excite the coupled system, and the resulting state is characterized using a weak, resonant probe beam., Comment: 5 pages, 5 figures, OECS conference 2007, Sicily, Italy the paper is withdrawn due to a crucial correction on one of the graphs which has been successfully adjusted before acceptance in Phys. Rev. B

A single atom in a cavity is the model system of cavity quantum electrodynamics (CQED). The strong coupling regime between the atom and cavity-confined photon corresponds to the reversible exchange of energy between the two modes, and underpins a wide range of CQED phenomena with applications in quantum information science, including for example as quantum logic gates and as sources of entangled states. An important advance was achieved recently when strong coupling between excitons and cavity photons was reported for the first time for localized quantum dots (QDs) in micron-size solid state cavities. This has significance in terms of scalability and integration with other optical devices, and could lead to the emergence of "quantum optics on a chip" technology. However the results presented so far for quantum dots are in the linear regime, corresponding to coupling to the vacuum field (vacuum Rabi splitting); they are not a true QED effect and can equally well be described by classical physics as the coupling between two oscillators. In this paper, we present evidence for a purely quantum phenomenon for the QD/cavity photon system, namely the increase in splitting of the levels when the mean number of photons in the cavity is increased. This corresponds to non-linearities on the single-photon scale: the presence of a single excitation in the cavity changes the level structure, affecting the emission energies for a second photon. Such results are a first step in demonstrating the promise of quantum dots for CQED applications., Comment: 11 Pages, 3 Figures

A theoretical investigation is presented of the statistics of excitons in quantum dots (QDs) of different sizes. A formalism is developed to build the exciton creation operator in a dot from the single exciton wavefunction and it is shown how this operator evolves from purely fermionic, in case of a small QD, to purely bosonic, in case of large QDs. Nonlinear optical emission spectra of semiconductor microcavities containing single QDs are found to exhibit a peculiar multiplet structure which reduces to Mollow triplet and Rabi doublet in fermionic and bosonic limits, respectively., Comment: Extensively expanded revision, 14 pages, 12 figures, submitted to Phys. Rev. B

A detailed description is provided of a numerical model of the microcavity optical parametric oscillator (OPO). The model is solved on a two dimensional grid, in the time domain, using an alternating-direction implicit method. Results are presented for a system pumped with a gaussian spatial profile beam. It is shown that at high pump powers the OPO signal forms a ring, with low emission at the centre where the pumping is strongest., Comment: 4 pages, 1 figure

The CW microcavity optical parametric oscillator (OPO) state is investigated using a theoretical treatment which includes the contributions of the pump, signal and idler populations to the renormalisation of the polariton energies (the `blue-shift'). The theory predicts the pumping conditions under which the OPO switches on, showing that a pump angle > 10 degrees is required, but there is no particular significance to the `magic-angle' where pump, signal and idler are all on resonance. The signal and idler renormalisation contributions also causes the signal emission to be driven towards the normal direction as the pump power increases above threshold., Comment: 9 pages, 6 figures, submitted to PRB

Polarisation effects in the microcavity parametric oscillator are studied using a simple model in which two chi-three optical parametric oscillators are coupled together. It is found that there are, in general, a number of steady states of the model under continuous pumping. There are both continuous and discontinuous thresholds, at which new steady-states appear as the driving intensity is increased: at the continuous thresholds, the new state has zero output intensity, whereas at the discontinuous threshold it has a finite output intensity. The discontinuous thresholds have no analog in the uncoupled device. The coupling also generates rotations of the linear polarisation of the output compared with the pump, and shifts in the output frequencies as the driving polarisation or intensity is varied. For large ratios of the interaction between polarisations to the interaction within polarisations, of the order of 5, one of the thresholds has its lowest value when the pump is elliptically polarised. This is consistent with recent experiments in which the maximum output was achieved with an elliptically polarised pump., Comment: 7 pages, 4 figures

I present a theoretical treatment of parametric scattering in strong coupling semiconductor microcavities to model experiments in which parametric oscillator behaviour has been observed. The model consists of a non-linear excitonic oscillator coupled to a cavity mode which is driven by the external fields, and predicts the output power, below threshold gain and spectral blue shifts of the parametric oscillator. The predictions are found to be in excellent agreement with the experimental data., Comment: 4 pages, LaTex, Revtex

Microcavity inhomogeneous line widths are calculated numerically using a microscopic two dimensional model of the polariton interaction with quantum well disorder. The calculations show show that in most structures the line widths are determined by disorder scattering between polariton and exciton states. This is because motional narrowing effectively removes the contribution due to multiple scattering between polariton states., Comment: 4 pages, Revtex, postscript figures input with epsf

A new, exact method for calculating excitonic absorption in superlattices is described. It is used to obtain high resolution spectra showing the saddle point exciton feature near the top of the miniband. The evolution of this feature is followed through a series of structures with increasing miniband width. The Stark ladder of peaks produced by an axial electric field is investigated, and it is shown that for weak fields the line shapes are strongly modified by coupling to continuum states, taking the form of Fano resonances. The calculated spectra, when suitably broadened, are found to be in good agreement with experimental results., Comment: 9 pages Revtex v3.0, followed by 4 uuencoded postscript figures, SISSA-CM-94-001