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We show that the multicomponent Kardar-Parisi-Zhang equation describes the low-energy theory for phase fluctuations in a $\mathbb{Z}_{2}$ degenerate non-equilibrium driven-dissipative condensate with global $U(1)\times U(1)$ symmetry. Using dynamical renormalisation group in spatial dimension $d=1$, we demonstrate that coupled stochastic complex Ginsburg-Landau equations exhibit an emergent stationary distribution, enforcing KPZ dynamical exponent $z=3/2$ and static roughness exponent $\chi=1/2$ for both components. By tuning intercomponent interactions, the system can access other regimes, including a fragmented condensate regime from a dynamical instability in the phase fluctuations, as well as a spacetime vortex regime driven by the non-linear terms in the coupled KPZ equations. In stable regimes, we show that in specific submanifolds relevant to polaritons, the RG fixed point offers a transformation to decoupled KPZ equations. Our findings have broad implications for understanding multicomponent KPZ systems in the long-wavelength limit., Comment: 7 pages, 3 figures with supplemental materials attached

Driven-dissipative nonlinear systems exhibit rich critical behavior, related to bifurcation, bistability and switching, which underlie key phenomena in areas ranging from physics, chemistry and biology to social sciences and economics. The importance of rare fluctuations leading to a dramatic jump between two very distinct states, such as survival and extinction in population dynamics, success and bankruptcy in economics and the occurrence of earthquakes or of epileptic seizures, have been already established. In the quantum domain, switching is of importance in both chemical reactions and the devices used in quantum state detection and amplification. In particular, the simplest driven single oscillator model serves as an insightful starting point. Here we describe switching induced by quantum fluctuations and illustrate that an instanton approach within Keldysh field theory can provide a deep insight into such phenomena. We provide a practical recipe to compute the switching rates semi-analytically, which agrees remarkably well with exact solutions across a wide domain of drive amplitudes spanning many orders of magnitude. Being set up in the framework of Keldysh coherent states path integrals, our approach opens the possibility of studying quantum activation in many-body systems where other approaches are inapplicable.

We introduce a novel method of efficiently simulating the non-equilibrium steady state of large many-body open quantum systems with highly non-local interactions, based on a variational Monte Carlo optimization of a matrix product operator ansatz. Our approach outperforms and offers several advantages over comparable algorithms, such as an improved scaling of the computational cost with respect to the bond dimension for periodic systems. We showcase the versatility of our approach by studying the phase diagrams and correlation functions of the dissipative quantum Ising model with collective dephasing and long-ranged power law interactions for spin chains of up to $N=100$ spins.

Topological order offers possibilities for processing quantum information which can be immune to imperfections. However, the question of its stability out of equilibrium is relevant for experiments, where coupling to an environment is unavoidable. In this work we demonstrate the robustness of certain aspects of $Z_2 \times Z_2$ symmetry-protected topological (SPT) order against a wide class of dissipation channels in the Lindblad and quantum trajectory formalisms of an open quantum system. This is illustrated using the one-dimensional $ZXZ$ cluster Hamiltonian along with Pauli-string jump operators. We show that certain choices of dissipation retaining strong symmetries support a steady-state manifold consisting of two non-local logical qubits, and for Hamiltonian perturbations preserving the global symmetry, the manifold remains long-lived. In contrast, this metastability is destroyed upon breaking the above-mentioned symmetry. While the localized edge qubits of the cluster Hamiltonian are not conserved by the Lindbladian evolution, they do correspond to weak symmetries and thus retain a memory of their initial state at all times in the quantum trajectories. We utilize this feature to construct protocols to retrieve the quantum information either by monitoring jumps or error mitigation. Our work thus proposes a novel framework to study the dynamics of dissipative SPT phases and opens the possibility of engineering entangled states relevant to quantum information processing., Comment: 20 pages, 8 figures

Phase fluctuations determine the low-energy properties of quantum condensates. However, at the condensation threshold, both density and phase fluctuations are relevant. While strong emphasis has been given to the investigation of phase fluctuations, which dominate the physics of the quantum system away from the critical point -- number fluctuations have been much less explored, even in thermal equilibrium. In this work, we report experimental observation and theoretical description of fluctuations in a circularly-confined non-equilibrium Bose-Einstein condensate of polaritons near the condensation threshold. We observe critical fluctuations, which combine the number fluctuations of a single-mode condensate state and competition between different states. The latter are analogous to mode hopping in photon lasers. Our theoretical analysis indicates that this phenomenon is of a quantum character, while classical noise of the pump is not sufficient to explain the experiments. The manifestation of a critical quantum state competition unlocks new possibilities for the study of condensate formation while linking to practical realizations in photonic lasers.

Symmetry considerations are key towards our understanding of the fundamental laws of Nature. The presence of a symmetry implies that a physical system is invariant under specific transformations and this invariance may have deep consequences. For instance, symmetry arguments state that a system will remain in its initial state if incentives to actions are equally balanced. Here, we apply this principle to a chain of qubits and show that it is possible to engineer the symmetries of its Hamiltonian in order to keep quantum information intrinsically protected from both relaxation and decoherence. We show that the coherence properties of this system are strongly enhanced relative to those of its individual components. Such a qubit chain can be realized using a simple architecture consisting of a relatively small number of superconducting Josephson junctions.

We investigate the phase diagram of a two-dimensional driven-dissipative system of polaritons coupled to the excitonic reservoir. We find that two critical points exists. The first corresponds to the quasi-condensation and the second to a first-order phase transition from the non-uniform state with spatially modulated density to a uniform state. The latter is related to the modulational instability of a homogeneous state due to the repulsive interactions with the non-condensed reservoir. The first-order character of the transition is evidenced by a discontinuity in the density and the correlation length as well as the phase coexistence and metastability. Moreover, we show that a signature of a Berezinskii-Kosterlitz-Thouless-like transition can be observed in the non-uniform phase.

Photonic materials are a rapidly growing platform for studying condensed matter physics with light, where the exquisite control capability is allowing us to learn about the relation between microscopic dynamics and macroscopic properties. One of the most interesting aspects of condensed matter is the interplay between interactions and the effect of an external magnetic field or rotation, responsible for a plethora of rich phenomena -- Hall physics and quantized vortex arrays. At first sight, however, these effects for photons seem vetoed: they do not interact with each other and they are immune to magnetic fields and rotations. Yet in specially devised structures these effects can be engineered. Here, we propose the use of a synthetic magnetic field induced by strain in a honeycomb lattice of resonators to create a non-equilibrium Bose-Einstein condensate of light-matter particles (polaritons) in a rotating state, without the actual need for external rotation nor reciprocity-breaking elements. We show that thanks to the competition between interactions, dissipation and a suitably designed incoherent pump, the condensate spontaneously becomes chiral by selecting a single Dirac valley of the honeycomb lattice, occupying the lowest Landau level and forming a vortex array. Our results offer a new platform where to study the exciting physics of arrays of quantized vortices with light and pave the way to explore the transition from a vortex-dominated phase to the photonic analogue of the fractional quantum Hall regime., Comment: Final version submitted to SciPost

Being able to describe accurately the dynamics and steady-states of driven and/or dissipative but quantum correlated lattice models is of fundamental importance in many areas of science: from quantum information to biology. An efficient numerical simulation of large open systems in two spatial dimensions is a challenge. In this work, we develop a tensor network method, based on an infinite Projected Entangled Pair Operator (iPEPO) ansatz, applicable directly in the thermodynamic limit. We incorporate techniques of finding optimal truncations of enlarged network bonds by optimising an objective function appropriate for open systems. Comparisons with numerically exact calculations, both for the dynamics and the steady-state, demonstrate the power of the method. In particular, we consider dissipative transverse quantum Ising and driven-dissipative hard core boson models in non-mean field limits, proving able to capture substantial entanglement in the presence of dissipation. Our method enables to study regimes which are accessible to current experiments but lie well beyond the applicability of existing techniques.

Methods for modeling large driven dissipative quantum systems are becoming increasingly urgent due to recent experimental progress in a number of photonic platforms. We demonstrate the positive-P method to be ideal for this purpose across a wide range of parameters, focusing on the archetypal driven dissipative Bose-Hubbard model. Notably, these parameters include intermediate regimes where interactions and dissipation are comparable, and especially cases with low occupations for which common semiclassical approximations can break down. The presence of dissipation can alleviate instabilities in the method that are known to occur for closed systems, allowing the simulation of dynamics up to and including the steady state. Throughout the parameter space of the model, we determine the magnitude of dissipation that is sufficient to make the method useful and stable, finding its region of applicability to be complementary to that of truncated Wigner. We then demonstrate its use in a number of examples with nontrivial quantum correlations, including a demonstration of solving the urgent open problem of large and highly non-uniform systems with even tens of thousands of sites., Comment: 17 pages, 14 figures

We study an emergent semiclassical time crystal composed of two interacting driven-dissipative bosonic modes. The system has a discrete $\mathbb Z_2$ spatial symmetry which, depending on the strength of the drive, can be broken in the time-crystalline phase or it cannot. An exact semiclassical mean-field analysis, numerical simulations in the quantum regime, and the spectral analysis of the Liouvillian are combined to show the emergence of the time crystal and to prove the robustness of the oscillation period against quantum fluctuations., Comment: Final version accepted in NJP

Optical parametric oscillations (OPOs) - a non-linear process involving the coherent coupling of an optically excited two particle pump state to a signal and an idler states with different energies - is a relevant mechanism for optical amplification as well as for the generation of correlated photons. OPOs require states with well-defined symmetries and energies: the fine-tuning of material properties and structural dimensions to create these states remains a challenge for the realization of scalable OPO-based functionalities in semiconductor nanostructures. Here, we demonstrate a pathway towards this goal based on the control of confined microcavity exciton-polaritons modulated by the spatially and time varying dynamical potentials produced by a surface acoustic waves (SAW). The exciton-polariton are confined in um-sized intra-cavity traps fabricated by structuring a planar semiconductor microcavity during the epitaxial growth process. OPOs in these structures benefit from the enhanced non-linearities of confined systems. We show that SAW fields induce state-dependent and time-varying energy shifts, which enable the energy alignment of the confined levels with the appropriate symmetry for OPO triggering. Furthermore, the dynamic acoustic tuning, which is fully described by a theoretical model for the modulation of the confined polaritons by the acoustic field, compensates for fluctuations in symmetry and dimensions of the confinement potential thus enabling a variety of dynamic OPO regimes. The robustness of the acoustic tuning is demonstrated by the synchronous excitation of an array of confined OPOs using a single acoustic beam, thus opening the way for the realization of scalable non-linear on-chip systems., Comment: Main text: 10 pages, 7 figures Supplementary: 6 pages, 3 figures

Quantum fluids of light are realized in semiconductor microcavities by exciton-polaritons, solid-state quasi-particles with a light mass and sizeable interactions. Here, we use the microscopic analogue of oceanographic techniques to measure the excitation spectrum of a thermalised polariton condensate. Increasing the fluid density, we demonstrate the transition from a free-particle parabolic dispersion to a linear, sound-like Goldstone mode characteristic of superfluids at equilibrium. Notably, we show that excitations are created with a definite direction with respect to the condensate, analogous to how a sea breeze develops surface waves aligned with the wind. These results reveal the effect of asymmetric pumping on the collective excitations of a condensate. Furthermore, we measure the critical sound speed for polariton superfluids close to equilibrium.

We investigate the critical behavior of the open coherently-driven Bose-Hubbard dimer under nonlocal dissipation. A conserved quantity arises from the nonlocal nature of the dissipation, rendering the dimer multistable. In the weak-coupling semiclassical limit, the displayed criticality takes the form of amplitude bistability and breaking of spatial and temporal symmetries. A period-bistable time crystal is formed, consisting of Josephson-like oscillations. Mean-field dynamics and quantum trajectories complement the spectral analysis of the Liouvillian in the approach to the semiclassical limit., Comment: Accepted in PRB. 6 pages, 2 figures. Supplemental material included. Comments are welcome

We study the spin vortices and skyrmions coherently imprinted into an exciton-polariton condensate on a planar semiconductor microcavity. We demonstrate that the presence of a polarization anisotropy can induce a complex dynamics of these structured topologies, leading to the twist of their circuitation on the Poincar\'e sphere of polarizations. The theoretical description of the results carries the concept of generalized quantum vortices in two-component superfluids, which are conformal with polarization loops around an arbitrary axis in the pseudospin space., Comment: 8 pages, 4 figures, SI 3 pages, SI 6 movies at http://www.pnas.org/content/113/52/14926?tab=ds and https://drive.google.com/open?id=0B0QCllnLqdyBUkVaSFpqSHB0cHM

We report the observation of the Berezinskii-Kosterlitz-Thouless transition for a 2D gas of exciton-polaritons, and through the joint measurement of the first-order coherence both in space and time we bring compelling evidence of a thermodynamic equilibrium phase transition in an otherwise open driven/dissipative system. This is made possible thanks to long polariton lifetimes in high-quality samples with small disorder and in a reservoir-free region far away from the excitation spot, that allow topological ordering to prevail. The observed quasi-ordered phase, characteristic for an equilibrium 2D bosonic gas, with a decay of coherence in both spatial and temporal domains with the same algebraic exponent, is reproduced with numerical solutions of stochastic dynamics, proving that the mechanism of pairing of the topological defects (vortices) is responsible for the transition to the algebraic order. Finally, measurements in the weak-coupling regime confirm that polariton condensates are fundamentally different from photon lasers and constitute genuine quantum degenerate macroscopic states.

We report a record-size, two-dimensional polariton condensate of a fraction of a millimeter radius free from the presence of an exciton reservoir. This macroscopically occupied state is formed by the ballistically expanding polariton flow that relaxes and condenses over a large area outside of the excitation spot. The density of this trap-free condensate is < 1 polariton/{\mu}m^2, reducing the phase noise induced by the interaction energy. Moreover, the backflow effect, recently predicted for the nonparabolic polariton dispersion, is observed here for the first time in the fast-expanding wave packet.

Quantised vortices are remarkable manifestations on a macroscopic scale of the coherent nature of quantum fluids, and the study of their properties is of fundamental importance for the understanding of this peculiar state of matter. Cavity-polaritons, due to their double light-matter nature, offer a unique controllable environment to investigate these properties. In this work we theoretically investigate the possibility to deterministically achieve the annihilation of a vortex with an antivortex through the increase of the polariton density in the region surrounding the vortices. Moreover we demonstrate that by means of this mechanism an array of vortex-antivortex pairs can be completely washed out.

Topological defects such as quantized vortices are one of the most striking manifestations of the superfluid nature of Bose-Einstein condensates and typical examples of quantum mechanical phenomena on a macroscopic scale. Here we demonstrate the formation of a lattice of vortex-antivortex pairs and study, for the first time, its properties in the non-linear regime at high polarion-density where polariton-polariton interactions dominate the behaviour of the system. In this work first we demonstrate that the array of vortex-antivortex pairs can be generated in a controllable way in terms of size of the array and in terms of size and shape of it fundamental unit cell. Then we demonstrate that polariton-polariton repulsion can strongly deform the lattice unit cell and determine the pattern distribution of the vortex-antivortex pairs, reaching a completely new behaviour with respect to geometrically generated vortex lattices whose shape is determined only by the geometry of the system.

Solid state quantum condensates often differ from previous examples of condensates (such as Helium, ultra-cold atomic gases, and superconductors) in that the quasiparticles condensing have relatively short lifetimes, and so as for lasers, external pumping is required to maintain a steady state. On the other hand, compared to lasers, the quasiparticles are generally more strongly interacting, and therefore better able to thermalise. This leads to questions of how to describe such non-equilibrium condensates, and their relation to equilibrium condensates and lasers. This chapter discusses in detail how the non-equilibrium Green's function approach can be applied to the description of such a non-equilibrium condensate, in particular, a system of microcavity polaritons, driven out of equilibrium by coupling to multiple baths. By considering the steady states, and fluctuations about them, it is possible to provide a description that relates both to equilibrium condensation and to lasing, while at the same time, making clear the differences from simple lasers.

We use the standard fermionic and boson-fermion Hamiltonians to study the BCS-BEC cross-over near the 202 G resonance in a two-component mixture of fermionic 40K atoms employed in the experiment of C.A. Regal et al., Phys. Rev. Lett. 92, 040403 (2004). Our mean-field analysis of many-body equilibrium quantities shows virtually no differences between the predictions of the two approaches, provided they are both implemented in a manner that properly includes the effect of the highest excited bound state of the background scattering potential, rather than just the magnetic-field dependence of the scattering length. Consequently, we rule out the macroscopic occupation of the molecular field as a mechanism behind the fermionic pair condensation and show that the BCS-BEC cross-over in ultra-cold 40K gases can be analysed and understood on the same basis as in the conventional systems of solid state physics., Comment: 16 pages, 10 eps figures; final version

We measured polarization-dependent photoluminescence-excitation spectra of highly uniform T-shaped quantum wires at 5 K. We attribute one peak to the 1D-exciton ground state and the continuous absorption band to 1D continuum states. These had similar polarization-dependences. We also observed some other peaks above the 1D-exciton ground state and attribute them to exciton states consisting of excited hole subbands. These results show good agreement with a model calculation of a single electron-hole pair in T-shaped geometry with exact diagonalizations of the Coulomb interaction.