Dynamical development of long-range spatial coherence in non-equilibrium bosonic condensation
Bianca Rae Fabricante, Dąbrówka Biegańska, P. Comaron, Mateusz Król, Matthias Wurdack, Maciej Pieczarka, Mark Steger, David W. Snoke, K. W. West, D. Sanvitto, Dario Ballarini, Dimitrios Trypogeorgos, Loren N. Pfeiffer, Andrew G. Truscott, Marzena Hanna Szymańska, Elena A. Ostrovskaya, Eliezer Estrecho
Development of spontaneous coherence is one of the hallmarks of bosonic condensation in a variety of physical systems, such as cold atoms, confined photons, and hybrid light-matter quasiparticles like exciton polaritons in semiconductors. While spatial coherence is well understood once a steady-state condensate has been established, its temporal evolution as the condensate forms is largely unexplored. Here, we explore the dynamical formation of a non-equilibrium, driven-dissipative exciton-polariton condensate through both time-resolved experiments and numerical modeling. Our study reveals that the spatial coherence is established through two distinct stages. The early-time stage is interaction-driven and features transient oscillations in spatial coherence. This stage is followed by a steady-state regime characterized by a spatially-uniform high degree of coherence that extends over the entire size of the system and persists over time. These stages of spatial coherence development occur in both free flowing and confined exciton-polariton systems that undergo a quench - rapid growth of the condensate starting from two different initial settings. Our study offers a deep insight into the process by which long-range spatial coherence is established in a non-equilibrium bosonic condensate.
Development of spontaneous coherence is one of the hallmarks of bosonic condensation in a variety of physical systems, such as cold atoms, confined photons, and hybrid light-matter quasiparticles like exciton polaritons in semiconductors. While spatial coherence is well understood once a steady-state condensate has been established, its temporal evolution as the condensate forms is largely unexplored. Here, we explore the dynamical formation of a non-equilibrium, driven-dissipative exciton-polariton condensate through both time-resolved experiments and numerical modeling. Our study reveals that the spatial coherence is established through two distinct stages. The early-time stage is interaction-driven and features transient oscillations in spatial coherence. This stage is followed by a steady-state regime characterized by a spatially-uniform high degree of coherence that extends over the entire size of the system and persists over time. These stages of spatial coherence development occur in both free flowing and confined exciton-polariton systems that undergo a quench - rapid growth of the condensate starting from two different initial settings. Our study offers a deep insight into the process by which long-range spatial coherence is established in a non-equilibrium bosonic condensate.
Summary form only given. Bosons can populate quantum states macroscopically, forming various kinds of thermal-equilibrium, and quasi-equilibrium condensates, which exhibit a wide range of collective phenomena including superfluidity and macroscopic coherence in matter. Interferometry has been employed to demonstrate long-range off-diagonal order in cold-atom systems, and, recently, in the rapidly developing field of exciton-polariton condensates. However, all such interferometric measurements have been performed on steady-state condensates, unable to access the temporal off-diagonal order and non-equilibrium dynamics. Moreover, in solid-state systems, interference between two separate condensates has not been observed before. Here we use optical injection to resonantly generate two dynamic exciton-polariton condensates with distinct momenta and at different times. We characterize the interference between condensates, demonstrating their long-range temporal order. We show that the build-up of the temporal long-range order strongly depends on the strength of polariton interactions, as well as the temperature and particle density. Moreover, polariton interactions yield a blue shift of the dynamic condensate energy, acting as an extremely large optical nonlinearity. These results provide direct evidence of temporal long-range order in dynamic condensates as well as demonstrate a new approach for probing the ultrafast coherent dynamics of exciton-polaritons, opening new directions in the fundamental study of coherence in matter as well as in practical applications such as quantum information processing and ultrafast logic.
Polariton condensates provide a versatile platform for exploring non-equilibrium phase transitions and collective phenomena in open quantum systems. Near the condensation threshold, these systems are particularly sensitive to fluctuations and instabilities, yet how these dynamics govern the initial formation of the condensate remains poorly understood. To address this, we directly access the photon statistics and second-order correlation function g (2) (0) of the condensate by using optical trapping and homodyne detection. We show that polariton condensation near the threshold is not a purely static transition, but instead undergoes a dynamical regime characterized by stochastic hopping between condensed and non-condensed states. These intermittent dynamics are accompanied by a gradual reduction of g (2) (0) toward unity, revealing the progressive build-up of coherence even in the presence of strong temporal fluctuations. Numerical simulations, based on a stochastic truncated Wigner description of the driven-dissipative polariton field, reproduce these dynamics and capture the essential role of noise and reservoir interactions. This work demonstrates that the observed temporal hopping is an intrinsic feature of polariton condensation, challenging traditional static descriptions of phase transitions and offering a framework for controlling noise and coherence in driven-dissipative quantum technologies.
Abstract The emergence of spatial coherence in a confined two-dimensional Bose gas of exciton polaritons with tuneable interactions offers a unique opportunity to explore the role of interactions in a phase transition in a driven-dissipative quantum system, where both the phase transition and thermalisation are mediated by interactions. We investigate, experimentally and numerically, the spatial coherence and steady-state properties of the gas over a wide range of interaction strengths by varying the photonic/excitonic fraction of the polaritons and their density. We find that the first order spatial correlation function exhibits algebraic decay consistent with the Berezinskii-Kosterlitz-Thouless (BKT) phase transition. The exponent of the algebraic decay is inversely proportional to the coherent fraction of polaritons, in analogy to superfluid fraction of equilibrium quantum gases above the BKT transition, but with a different proportionality constant. Our work paves the way for future investigations of the phenomenon of phase transitions and superfluidity in a driven-dissipative setting.
We review recent results on the coherence and superfluidity of driven dissipative condensates, i.e., systems of weakly interacting nonconserved bosons, such as polariton condensates. The presence of driving and dissipation has dramatically different effects depending on dimensionality and anisotropy. In three dimensions, equilibrium behaviour is recovered at large scales for static correlations, while the dynamical behaviour is altered by the microscopic driving. In two dimensions, for an isotropic system, drive and dissipation destroy the algebraic order that would otherwise exist; however, a sufficiently anisotropic system can still show algebraic phase correlations. We discuss the consequences of this behaviour for recent experiments measuring phase coherence and outline potential measurements that might directly probe superfluidity. Introduction This chapter is dedicated to superfluidity and its relation to Bose-Einstein condensation (BEC), a topic with a long history. Many reviews of the concepts of condensation and superfluidity in thermal equilibrium can be found; see for example Refs. [1, 2, 3, 4]. The focus of this chapter is on how these concepts apply (or fail to apply) to driven dissipative condensates – systems of bosons with a finite lifetime, in which loss is balanced by continuous pumping. We focus entirely on the steady state of such systems, neglecting transient, time-dependent behaviour. Experimentally, the most studied example of a driven dissipative condensate has been microcavity polaritons, an overview of which is given in Ref. [5] and Chapter 4. However, similar issues can arise in many other systems, most obviously photon condensates [6] (see also Chapter 19), magnon condensates [7] (see also Chapters 25–26), and potentially exciton condensates (although typical exciton lifetimes are much longer than for polaritons). Even experiments on cold atoms (see, e.g., Chapter 3) could be driven into a regime in which such physics occurs, when considering continuous loading of atoms balancing threebody losses [8] or atom laser setups [9, 10, 11]. Experiments on polaritons are two-dimensional, and in two dimensions it is particularly important to clearly distinguish three concepts often erroneously treated as equivalent: superfluidity, condensation, and phase coherence. This is because no true Bose-Einstein condensate exists in a homogeneous two-dimensional system.
Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity is a macroscopically populated coherent quantum state subject to concurrent pumping and decay. Debates about the fundamental nature of the condensed phase in this open quantum system still persist. Here, we gain a new insight into the spontaneous condensation process by imaging long-lifetime exciton polaritons in a high-quality inorganic microcavity in the single-shot optical excitation regime, without averaging over multiple condensate realisations. In this highly non-stationary regime, a condensate is strongly influenced by the `hot' incoherent reservoir, and reservoir depletion is critical for the transition to the ground energy and momentum state. Condensates formed by more photonic exciton polaritons exhibit dramatic reservoir-induced density filamentation and shot-to-shot fluctuations. In contrast, condensates of more excitonic quasiparticles display smooth density and are second-order coherent. Our observations show that the single-shot measurements offer a unique opportunity to study formation of macroscopic phase coherence during a quantum phase transition in a solid state system.
We discuss the dynamics of classical Dicke-type models, aiming to clarify the mechanisms by which coherent states could develop in potentially non-equilibrium systems such as semiconductor microcavities. We present simulations of an undamped model which show spontaneous coherent states with persistent oscillations in the magnitude of the order parameter. These states are generalizations of superradiant ringing to the case of inhomogeneous broadening. They correspond to the persistent gap oscillations proposed in fermionic atomic condensates, and arise from a variety of initial conditions. We show that introducing randomness into the couplings can suppress the oscillations, leading to a limiting dynamics with a time-independent order parameter. This demonstrates that non-equilibrium generalizations of polariton condensates can be created even without dissipation. We explain the dynamical origins of the coherence in terms of instabilities of the normal state, and consider how it can additionally develop through scattering and dissipation.