Non-equilibrium Bose-Einstein Condensation Research Articles

Controlling nonequilibrium Bose-Einstein condensation with engineered environments

Controlling nonequilibrium Bose-Einstein condensation with engineered environments

Critical fluctuations in a confined driven-dissipative quantum condensate.

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 nonequilibrium 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 is 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 possibilities for the study of condensate formation while linking to practical realizations in photonic lasers.

Non-equilibrium Bose–Einstein condensation in photonic systems

Non-equilibrium Bose–Einstein condensation in photonic systems

Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate

The theoretical description of non-equilibrium Bose–Einstein condensate (BEC) is one of the main challenges in modern statistical physics and kinetics. The non-equilibrium nature of BEC makes it impossible to employ the well-established formalism of statistical mechanics. We develop a framework for the analytical description of a non-equilibrium phase transition to BEC that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states. We consider the limit of fast thermalization and obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC which also covers the equilibrium case. For the particular cases of 2D and 3D, we investigate the non-equilibrium formation of BEC by finding the temperature dependence of the ground state occupation and second-order coherence function. We show that for a given pumping rate, the macroscopic occupation of the ground state and buildup of coherence may occur at different temperatures. Moreover, the buildup of coherence strongly depends on the pumping scheme. We also investigate the condensate linewidth and show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

Polariton Bose-Einstein condensate from a bound state in the continuum.

Bound states in the continuum (BICs)1-3 are peculiar topological states that, when realized in a planar photonic crystal lattice, are symmetry-protected from radiating in the far field despite lying within the light cone4. These BICs possess an invariant topological charge given by the winding number of the polarization vectors5, similar to vortices in quantum fluids such as superfluid helium and atomic Bose-Einstein condensates. In spite of several reports of optical BICs in patterned dielectric slabs with evidence of lasing, their potential as topologically protected states with theoretically infinite lifetime has not yet been fully exploited. Here we show non-equilibrium Bose-Einstein condensation of polaritons-hybrid light-matter excitations-occurring in a BIC thanks to its peculiar non-radiative nature, which favours polariton accumulation. The combination of the ultralong BIC lifetime and the tight confinement of the waveguide geometry enables the achievement of an extremely low threshold density for condensation, which is reachednot in the dispersion minimum but at a saddle point in reciprocal space. By bridging bosonic condensation and symmetry-protected radiation eigenmodes, we reveal ways of imparting topological properties onto macroscopic quantum states with unexplored dispersion features. Such an observation may open a route towards energy-efficient polariton condensation in cost-effective integrated devices, ultimately suited for the development of hybrid light-matter optical circuits.

Polariton condensation into vortex states in the synthetic magnetic field of a strained honeycomb lattice

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.

Polariton lasing and energy-degenerate parametric scattering in non-resonantly driven coupled planar microcavities

Abstract Multi-level exciton-polariton systems offer an attractive platform for studies of non-linear optical phenomena. However, studies of such consequential non-linear phenomena as polariton condensation and lasing in planar microcavities have so far been limited to two-level systems, where the condensation takes place in the lowest attainable state. Here, we report non-equilibrium Bose–Einstein condensation of exciton-polaritons and low threshold, dual-wavelength polariton lasing in vertically coupled, double planar microcavities. Moreover, we find that the presence of the non-resonantly driven condensate triggers interbranch exciton-polariton transfer in the form of energy-degenerate parametric scattering. Such an effect has so far been observed only under excitation that is strictly resonant in terms of the energy and incidence angle. We describe theoretically our time-integrated and time-resolved photoluminescence investigations by an open-dissipative Gross–Pitaevskii equation-based model. Our platform’s inherent tunability is promising for construction of planar lattices, enabling three-dimensional polariton hopping and realization of photonic devices, such as all-optical polariton-based logic gates.

Quantum Field Theory of Correlated Bose–Einstein Condensates: I. Basic Formalism

Quantum field theory of equilibrium and nonequilibrium Bose–Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz–Pines relation; conservation laws; identities among vertices originating from Goldstone’s theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green’s functions, three- and four-point vertices, and two-particle Green’s functions. It enables us to study correlated Bose–Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green’s functions are found to share poles, i.e., the structure of the two-particle Green’s functions predicted by Gavoret and Nozières for a homogeneous condensate at T = 0 is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.

Dissipative Distillation of Supercritical Quantum Gases.

We experimentally realize a method to produce nonequilibrium Bose-Einstein condensates with condensed fraction exceeding those of equilibrium samples with the same parameters. To do this, we immerse an ultracold Bose gas of ^{87}Rb in a cloud of ^{39}K with substantially higher temperatures, providing a controlled source of dissipation. By combining the action of the dissipative environment with evaporative cooling, we are able to progressively distil the nonequilibrium Bose-Einstein condensate from the thermal cloud. We show that by increasing the strength of the dissipation it is even possible to produce condensates above the critical temperature. We finally demonstrate that our out-of-equilibrium samples are long lived and do not reach equilibrium in a time that is accessible for our experiment. Due to its high degree of control, our distillation process is a promising tool for the engineering of open quantum systems.

Multivortex states and dynamics in nonequilibrium polariton condensates

In strongly nonequilibrium Bose–Einstein condensates described by the generalized Gross–Pitaevskii equation, vortex motion becomes self-accelerated while the long-range vortex–antivortex interaction appears to be repulsive. We numerically study the impact of these rather unusual vortex properties on the dynamics of multivortex systems. We show that at strong nonequilibrium the repulsion between vortices and antivortices leads to a dramatic slowdown of their annihilation. Moreover, in finite-size samples, relaxation of multivortex systems can lead to the formation of metastable vortex–antivortex clusters, whose shape and size depend, in particular, on the sample geometry, boundary conditions and deviations from equilibrium. We further demonstrate that at strong nonequilibrium the interaction of self-accelerated vortices with inhomogeneous condensate flows can lead to generation of new vortex–antivortex pairs.

Simulating Ising and n-State Planar Potts Models and External Fields with Nonequilibrium Condensates.

Classical spin models with discrete or continuous degrees of freedom arise in many studies of complex physical systems. A wide class of hard real-life optimization problems can be formulated as a minimization of a spin Hamiltonian. Here we show how to simulate the discrete Ising and n-state planar Potts models with or without external fields using the physical gain-dissipative platforms with continuous phases, such as lasers and various nonequilibrium Bose-Einstein condensates. The underlying operational principle originates from a combination of resonant and nonresonant pumping. Our results lay grounds for the physical simulations of a broad range of Hamiltonians with complex interactions that can vary in time and space and with combined symmetries.

Networks of non-equilibrium condensates for global optimization

Recently several gain-dissipative platforms based on the networks of optical parametric oscillators, lasers and various non-equilibrium Bose–Einstein condensates have been proposed and realised as analogue Hamiltonian simulators for solving large-scale hard optimisation problems. However, in these realisations the parameters of the problem depend on the node occupancies that are not known a priori, which limits the applicability of the gain-dissipative simulators to the classes of problems easily solvable by classical computations. We show how to overcome this difficulty and formulate the principles of operation of such simulators for solving the NP-hard large-scale optimisation problems such as constant modulus continuous quadratic optimisation and quadratic binary optimisation for any general matrix. To solve such problems any gain-dissipative simulator has to implement a feedback mechanism for the dynamical adjustment of the gain and coupling strengths.

Driven-dissipative non-equilibrium Bose–Einstein condensation of less than ten photons

Coherence is a defining feature of quantum condensates. These condensates are inherently multimode phenomena and in the macroscopic limit it becomes extremely difficult to resolve populations of individual modes and the coherence between them. In this work we demonstrate non-equilibrium Bose-Einstein condensation (BEC) of photons in a sculpted dye-filled microcavity, where threshold is found for $8\pm 2$ photons. With this nanocondensate we are able to measure occupancies and coherences of individual energy levels of the bosonic field. Coherence of individual modes generally increases with increasing photon number, but at the breakdown of thermal equilibrium we observe multimode-condensation phase transitions wherein coherence unexpectedly decreases with increasing population, suggesting that the photons show strong inter-mode phase or number correlations despite the absence of a direct nonlinearity. Experiments are well-matched to a detailed non-equilibrium model. We find that microlaser and Bose-Einstein statistics each describe complementary parts of our data and are limits of our model in appropriate regimes, which informs the debate on the differences between the two.

Spin polarized semimagnetic exciton-polariton condensate in magnetic field

Owing to their integer spin, exciton-polaritons in microcavities can be used for observation of non-equilibrium Bose-Einstein condensation in solid state. However, spin-related phenomena of such condensates are difficult to explore due to the relatively small Zeeman effect of standard semiconductor microcavity systems and the strong tendency to sustain an equal population of two spin components, which precludes the observation of condensates with a well defined spin projection along the axis of the system. The enhancement of the Zeeman splitting can be achieved by introducing magnetic ions to the quantum wells, and consequently forming semimagnetic polaritons. In this system, increasing magnetic field can induce polariton condensation at constant excitation power. Here we evidence the spin polarization of a semimagnetic polaritons condensate exhibiting a circularly polarized emission over 95% even in a moderate magnetic field of about 3 T. Furthermore, we show that unlike nonmagnetic polaritons, an increase on excitation power results in an increase of the semimagnetic polaritons condensate spin polarization. These properties open new possibilities for testing theoretically predicted phenomena of spin polarized condensate.

Turbulence of Weak Gravitational Waves in the Early Universe.

We study the statistical properties of an ensemble of weak gravitational waves interacting nonlinearly in a flat space-time. We show that the resonant three-wave interactions are absent and develop a theory for four-wave interactions in the reduced case of a 2.5+1 diagonal metric tensor. In this limit, where only plus-polarized gravitational waves are present, we derive the interaction Hamiltonian and consider the asymptotic regime of weak gravitational wave turbulence. Both direct and inverse cascades are found for the energy and the wave action, respectively, and the corresponding wave spectra are derived. The inverse cascade is characterized by a finite-time propagation of the metric excitations-a process similar to an explosive nonequilibrium Bose-Einstein condensation, which provides an efficient mechanism to ironing out small-scale inhomogeneities. The direct cascade leads to an accumulation of the radiation energy in the system. These processes might be important for understanding the early Universe where a background of weak nonlinear gravitational waves is expected.

Interaction and motion of vortices in nonequilibrium quantum fluids

We study numerically the motion of vortices in nonequilibrium Bose–Einstein condensates, that are described by a generalized Gross–Pitaevskii equation. We analyze how the vortex properties are modified when moving away under deviation from equilibrium. We find that far from equilibrium, the radial component dominates over the azimuthal one in the distribution of vortex currents at large distances from the vortex core. The modification of the current pattern has a strong effect on the vortex–antivortex interaction energy, that can become entirely repulsive. The vortex trajectories are also strongly affected by the driving and dissipation. Self acceleration of vortices is observed in the strong nonequilibrium case.

Bright solitons in non-equilibrium coherent quantum matter.

We theoretically demonstrate a mechanism for bright soliton generation in spinor non-equilibrium Bose-Einstein condensates made of atoms or quasi-particles such as polaritons in semiconductor microcavities. We give analytical expressions for bright (half) solitons as minimizing functions of a generalized non-conservative Lagrangian elucidating the unique features of inter and intra-competition in non-equilibrium systems. The analytical results are supported by a detailed numerical analysis that further shows the rich soliton dynamics inferred by their instability and mutual cross-interactions.

Spontaneous formation and nonequilibrium dynamics of a soliton-shaped Bose-Einstein condensate in a trap.

The Bose-stimulated self-organization of a quasi-two-dimensional nonequilibrium Bose-Einstein condensate in an in-plane potential is proposed. We obtained the solution of the nonlinear, driven-dissipative Gross-Pitaevskii equation for a Bose-Einstein condensate trapped in an external asymmetric parabolic potential within the method of the spectral expansion. We found that, in sharp contrast to previous observations, the condensate can spontaneously acquire a solitonlike shape for spatially homogeneous pumping. This condensate soliton performs oscillatory motion in a parabolic trap and, also, can spontaneously rotate. Stability of the condensate soliton in the spatially asymmetric trap is analyzed. In addition to the nonlinear dynamics of nonequilibrium Bose-Einstein condensates of ultracold atoms, our findings can be applied to the condensates of quantum well excitons and cavity polaritons in semiconductor heterostructure, and to the condensates of photons.

Fluctuations of the order parameter of a mesoscopic Floquet condensate

We suggest that nonequilibrium Bose-Einstein condensates may occur in time-periodically driven interacting Bose gases. Employing the model of a periodically forced bosonic Josephson junction, we demonstrate that resonance-induced ground-state-like many-particle Floquet states possess an almost perfect degree of coherence, as corresponding to a mesoscopically occupied, explicitly time-dependent single-particle orbital. In marked contrast to the customary time-independent Bose-Einstein condensates, the order parameter of such systems is destroyed by violent fluctuations when the particle number becomes too large, signaling the nonexistence of a proper mean-field limit.

Quantum Langevin model for nonequilibrium condensation

We develop a quantum model for nonequilibrium Bose-Einstein condensation of photons and polaritons in planar microcavity devices. The model builds on laser theory and includes the spatial dynamics of the cavity field, a saturation mechanism, and some frequency dependence of the gain: quantum Langevin equations are written for a cavity field coupled to a continuous distribution of externally pumped two-level emitters with a well-defined frequency. As an example of application, the method is used to study the linearized quantum fluctuations around a steady-state condensed state. In the good-cavity regime, an effective equation for the cavity field only is proposed in terms of a stochastic Gross-Pitaevskii equation. Perspectives in view of a full quantum simulation of the nonequilibrium condensation process are finally sketched.