Formation and Dynamics of Quantum Hydrodynamical Breathing Ring Solitons

TL;DR

This paper predicts and analyzes a new type of stable, self-localized dark ring breather in exciton-polariton condensates, offering insights into quantum hydrodynamical phenomena and phase transitions.

Contribution

It introduces the concept of quantum hydrodynamical breathing ring solitons, detailing their formation, stability, and potential experimental observation methods.

Findings

Prediction of stable dark ring breather structures in exciton-polariton condensates

Analysis of ring dynamics, interactions, and turbulence

Proposal for experimental schemes to observe and utilize these structures

Abstract

We show that exciton-polariton condensates may exhibit a new fundamental, self-localized nonlinear excitation not seen in other quantum hydrodynamical systems, which takes the form of a dark ring shaped breather. We predict that these structures form spontaneously and remain stable under a combination of uniform resonant and nonresonant forcing. We study single ring dynamics, ring interactions and ring turbulence, and explain how direct experimental observations might be made. We discuss the statistics of ring formation and propose an experimental scheme by which these structures may be exploited to study the smooth cross-over between equilibrium and non-equilibrium critical phase transitions. Finally, we present an alternative mechanism of formation for these topological breathers, in which they circumscribe Gaussian resonant pumps. The observation of a breathing ring soliton would represent the first fundamental breathing soliton within the broad field of quantum hydrodynamics.

Equations

$$i{\partial }_t\psi$$

$${\partial }_t{N}_R$$

$$\tilde{\mu }$$

$$\nabla \cdot (\rho \mathbf{u})$$

Full text

The spontaneous formation of patterns in disordered systems has captivated scientists for generations. Of particular interest are self-localized patterns, which are known as solitons in integrable systems and as solitary waves in nonintegrable systems Whitham (2011); Ablowitz and Segur (1981). These can be understood as the fundamental excitations of nonlinear wave systems, and they are typically found in familiar forms across disparate settings. The first to be discovered were of the bright type, which are stabilized by the counterbalancing of dispersion with nonlinear effects. Other solitons are of the dark type and include vortex and domain wall type phase dislocations Berry and Dennis (2000); Alperin et al. (2019); Vilenkin (1985); Pismen and Pismen (1999); Pismen (2006); Burger et al. (1999); Swartzlander Jr and Law (1992); Staliunas and Sanchez-Morcillo (2003). Nonlinear optical resonators were one of the earliest physical systems in which nonlinear dynamical pattern formation was studied in depth Pismen (2006). The results of that field have thus served to inform the study of many other systems, such as Bose-Einstein condensates (BECs) of ultracold atoms Kevrekidis et al. (2007); Burger et al. (1999) and of magnon Nowik-Boltyk et al. (2012) systems. One fundamental soliton that can be formed in the nonlinear optical resonator is the so called ring dark or phase soliton, formed by domain walls that close on themselves to form loops Staliunas and Sanchez-Morcillo (2003). In equilibrium BECs (such as ultra-cold atomic BECs), it has been shown that ring dark solitons are not stable, decaying either acoustically or into so called vortex necklaces of more stable vortices Carr and Clark (2006); Theocharis et al. (2003). These forms have been observed experimentally, though fleetingly, by trapping them along their transverse axes Shomroni et al. (2009). While with this approach the ring structures are localized in that they repeatedly break down and reform within the trap, they are not self-localized and cannot exist in free space.

In this Letter we show that exciton-polariton condensates – hybrid light-matter quantum fluids with strongly nonlinear properties and inherent nonequilibriation– may support a different type of topological defects: breathing ring solitons. These breathing rings are distinct from other solitary structures found in exciton-polariton condensates as well as from those found in other quantum hydrodynamical systems. Geometrically these excitations are similar to the ring shaped phase solitons found in nonlinear optical systems Taranenko et al. (1998), but are distinct in a critical way: the solitons we present here do not appear to settle into a stationary state, instead continuing to oscillate indefinitely in radius and depth 111We do note that it has been predicted, in the exceptional circumstances of high nonlinearity and nonlocality, breathing optical ring -shaped solitons may exist Lu and Hu (2009). However, these solitons are of the beam type as opposed to the systems of interest here (self-localized structures within an extended system). Thus these are of a distinct class of structure.. We study the mechanisms of formation and stabilization of these structures, and report on their dynamics and interactions, including states of ring solitonic turbulence. We also study the statistics of spontaneous ring formation, showing that breathing ring solitons can be used to study critical phenomena in systems with tunable nonequilibriation. Lastly, we demonstrate that in addition to their spontaneous formation during condensation, breathing ring solitons can be formed explicitly via localized forcing.

Condensates of exciton-polariton quasiparticles (polariton condensates) have recently been realized in semiconductor microcavities Kasprzak et al. (2006). As inherently nonequilibriated condensates of hybrid light-matter quasiparticles, polariton condensates may be thought of neither as equilibrium condensates or as lasers, but rather as something in between Keeling and Berloff (2008). Polaritonic systems have several advantages with respect to other confined optical systems. One is their extraordinary nonlinear properties, which arise from their excitonic component. Their dispersion curves (the so called lower and upper polariton branches appearing from the hybridization of photons and excitons) allow for the individual control of the photonic and excitonic components via detuning, and their properties and dynamics can be easily accessed by angular-resolved imaging or electroluminescence spectroscopy. For low enough densities, polaritons may be considered as bosonic quasiparticles, and so can form a coherent state (BEC). Polariton condensates are non-equilibrium systems set by balance between pumping and losses due to the short lifetime of polaritons. They can continuously cross from weak coupling at higher temperatures and pumping strengths to strong coupling at lower temperatures and lower pumping intensities. In addition, the lifetime of polaritons in the microcavities systems can be increased by improving the quality and number of dielectric Bragg mirrors. Therefore, depending on the polariton lifetime and pumping intensities polariton condensates continuously cross between strongly non-equilibrium systems – lasers– (short lifetime and large pumping rates) and equilibrium BECs (large lifetimes and small pumping).

Polariton condensates can be excited by two different types of pumping. In the resonant pumping scheme, the energy and angle of incidence of the excitation laser are set to be quasi-resonant with a mode of the lower polariton branch. In the non-resonant pumping scheme, the excitation laser has an energy much higher than that of the lower polariton branch. In this case, polaritons can spontaneously form macroscopic coherent states – polariton lasers or polariton condensates– by the accumulation of particles in the same quantum state Kasprzak et al. (2006). Recent experiments have started to combine resonant and non-resonant pumping Ohadi et al. (2016). In those experiments, chemical etching of a GaAs substrate allowed resonant excitation from the back side of the cavity, preventing backscatter from the non-resonant pumping, while allowing synchronization. This technique makes it possible to independently vary the pumping intensity distributions of resonant and non-resonant excitations.

The mean-field behaviour of polariton condensates is governed by the generalized complex Ginzburg-Landau equation (cGLE), with the condensate wavefunction $\psi({\bf r},t)$ coupled to the hot exciton reservoir density $N_{R}$ Kalinin and Berloff (2018); Keeling and Berloff (2008); Wouters and Carusotto (2007); Carusotto and Ciuti (2013); Keeling and Berloff (2011), so that

[TABLE]

where we set $\hbar=1$ and $m=1/2$. In these coupled equations, $g$ is proportional to the polariton-exciton interaction strength, $\eta$ represents the energy relaxation Wouters (2012); Berloff and Keeling (2013), $\gamma$ and $b_{0}$ are proportional to the inverse lifetime of the polariton and hot excitons, respectively, $b_{1}$ is proportional to the ratio of the interaction strength between the condensate and the hot excitons to that between condensate particles. The incoherent and resonant (at $n:1$ resonance with the condensate frequency) pump sources are described by the pumping intensities $P({\bf r},t)$ and $\bar{P}({\bf r},t)$, respectively 222The fixed parameter values in our simulations are $b_{0}=1$, $b_{1}=1$, $\gamma=0.3$, and $\eta=0.3$. The sample disorder is described by a spatially dependent external potential $V_{\rm ext}({\bf r})$. The steady state of the systems is characterized by the chemical potential $\mu$. Using the Madelung transformation $\psi=\sqrt{\rho({\bf r})}\exp[iS({\bf r})+i\mu t]$, neglecting the energy relaxation and the external potential in the view of their smallness and replacing the steady state of Eq. (2) $N_{R}=P/(b_{0}+b_{1}\rho)$ with its Taylor expansion $N_{R}\approx P/b_{0}-Pb_{1}\rho/b_{0}^{2}$, we transform Eqs. (Formation and Dynamics of Quantum Hydrodynamical Breathing Ring Solitons-2) into the continuity equation and the integrated form of the Bernoulli equation (with ${\bf u}=\nabla S$) as

[TABLE]

with the solution $S=0,\rho=(Pb_{0}-\gamma b_{0}^{2}+\bar{P}b_{0}^{2})/Pb_{1}$, $\tilde{\mu}=(1-gPb_{1}/b_{0}^{2})(Pb_{0}-\gamma b_{0}^{2}+\bar{P}b_{0}^{2})/Pb_{1},$ and $\mu=\tilde{\mu}+gP/b_{0}$. Nontrivial uniform state is formed when $\gamma b_{0}<P+\bar{P}b_{0}$.

To form ring solitons on the top of this nontrivial uniform state, we must enter the regime in which Ising-type phase domains are supported. This regime is that of phase bistability, which can be achieved in the complex Ginzburg-Landau equation via parametric forcing induced by a relatively large second order resonant pump Coullet and Emilsson (1992). We thus set the resonant forcing term to $n=2$ in Eq. (Formation and Dynamics of Quantum Hydrodynamical Breathing Ring Solitons). This regime surrounds a fork-type Hopf bifurcation near the generation threshold as opposed to a sub or supercritical one Staliunas and Sanchez-Morcillo (2003). Assuming that the condensation process arises from a noisy initial state, the condensate forms with independently coherent regions separated by domain walls of either Bloch or Ising type. The healing length of the domain wall $L_{h}$ – the characteristic distance of the density recovery when locally perturbed – is defined by the length scale at which the potential and kinetic energies are equal, which by Eq. (3-4) is $L_{h}\approx|\mu|^{-1/2}$ defined above. In typical GaAs experiments the reservoir relaxation times are large so increasing $P(\bar{P})$ or/and $g$ is the most natural way to decrease the healing length of the domain wall.

Ising walls are required to either end at the boundary of the condensate, or to form closed loops. In the latter case, the dark ring grows or shrinks until it reaches a characteristic radius Staliunas and Sanchez-Morcillo (2003). In the optical case the ring of characteristic radius finds itself in a stable, local energy minima, and thus remains stationary. However, in polariton condensates, after reaching the critical radius the behavior of the rings diverges from the behavior of geometrically similar solitons observed in other systems. We determined that the ring solitons self-annihilates (i) in the fast reservoir regime $b_{0}\gg\gamma$, (ii) in small reservoir detuning regime $g\ll 1$, (iii) in long-lived polaritonic systems. All these regimes are physically relevant to some experiments Nelsen et al. (2013); Kalinin and Berloff (2019); Berloff et al. (2017). However, a slow reservoir evolution ($b_{0}\lesssim\gamma$), for short-lived polaritons and a sufficiently large reservoir detuning (all of which correspond to values of typical GaAs microcavity experiments Kasprzak et al. (2006); Deng et al. (2007); Manni et al. (2012)) prevent the ring soliton from disappearing and lead to appearance of a ring breather: the dissipative decrease in the radius of the ring soliton is accompanied by the increase in the reservoir profile density in the ring core, which imposes a repulsive force in the outward direction to make the ring expand. This process repeats itself as shown in Supplemental Video 1. This nonlinear excitation is self-localized by an explicitly dynamical interaction.

As all of the destabilizing mechanisms (i)-(iii) have similar effects on the existence and dynamics of the ring solitons, we concentrate on the effect of varying the polariton-exciton interaction strength (parameterized by $g$). The detuning between the cavity photon energy and the exciton resonance determines the relative photonic/excitonic character of the polariton and, therefore, its effective mass and the strength of the polariton-exciton interactions Carusotto and Ciuti (2013). The detuning $g$ can be further changed by the pumping geometry by considering trapped condensates separated from the pumps Cristofolini et al. (2013). Finally, implanting protons into the quantum wells or into the top of distributed Bragg reflectors allows for an independent spatial control of both the exciton and the cavity photon energies, and, therefore, affects $g$ Schneider et al. (2016) as well. By these mechanisms, the experimental ranges of our dimensionless parameter $g$ can vary between $0.1-2$.

Our simulations show that in the case of low $g$, rings form ad infinitum. This makes for a sustained state of ring turbulence. A condensate density for a particular moment of time in this regime is shown in Fig. 1(a). While the dynamics of the low $g$ case are not uninteresting from a theoretical perspective, experiments on exciton-polariton condensates are currently limited to time-averaged imaging, and thus this case is not directly observable. However we find that in the high $g$ case, rings are formed only during the condensation process. They appear to interact attractively, and upon contact a pair of rings will either merge into one or annihilate each other. Eventually the decay of rings ends, and a quasistationary state is reached with rings being pinned by the system disorder represented in our simulations by $V_{\rm ext}$. This is demonstrated in Supplemental Video 2, time snapshots from which are shown in Fig. 1(d,e) illustrating the condensate density contour plots. Figure 1 shows a time snapshot of a spontaneously formed breathing ring soliton after the system has reached its final, quasistationary state (b), as well as a time integrated image of that state (c). Thus, we predict that long-lived breathing ring solitons are directly observable, and that their ring shaped character, radii, locations and numbers are directly measurable as well.

The mechanism by which breathing ring solitons have been shown to form for high $g$ resembles Kibble-Zurek (KZ) mechanism of defect formation in equilibrium systems Damski and Zurek (2010); Zurek (1996). The KZ mechanism was first understood in the context of the phase transitions in the early Universe Kibble (1976, 1980); Zurek (1985), and later in liquid 4He and 3He, liquid crystals, superconductors Bäuerle et al. (1996); Ruutu et al. (1996); Bowick et al. (1994); Maniv et al. (2003), equilibrium Bose-Einstein condensates Berloff and Svistunov (2002); Weiler et al. (2008). The similarities and differences between the KZ transition and pattern formation in nonequilibrium systems are the subject of intense exploration, with an emphasis on the common mechanism of the defect formation: locally uniform symmetry breaking in separate parts of the system which cannot communicate in a finite time, and which thus form to be globally nonuniform to a degree set by the speed of the phase transition (the quench rate). The main difference between the KZ transition and pattern forming in nonequilibrium systems is that in the former, it is assumed that the system is driven out of equilibrium only in the vicinity of the phase transition Zurek (1985). In spite of extensive research on both the KZ transition and on pattern formation in a wide variety of nonequilibrium systems, questions remain regarding the nature of the cross-over between the two mechanisms, and regarding the types of the defects that they can result in. We investigate this relationship by counting the number of quasistationary rings formed spontaneously from random initial noise in the presence of a small sample disorder. This disorder does not hamper the formation of rings, but rather acts like sandpaper, resisting their movement across its surface. Fig. 2 shows the resulting linear, positive correlation between pump power and ring soliton density. To elucidate the effect of the quenching time on the defect formation, we repeated our simulations linearly increasing the pumps from zero to $P=\bar{P}$ over different timescales. The results, shown in the inset of Fig. 2, reveals a $k=-2$ power law. We note that recent theoretical work on nonequilibrium holographic superfluids have shown qualitatively similar results: a linear dependence of excitation strength (temperature in that context) on defect (vortex) density, and a power law dependence of quench time on defect density Chesler et al. (2015).

In addition to the their spontaneous nucleation during the condensate formation, breathing ring solitons can be formed explicitly, by either first or second order resonant forcing with a non-uniform (e.g. Gaussian) profile. With direct numerical integration we simulate the effect of an exciton-polariton condensate excited by a uniform nonresonant pump of uniform amplitude $P$ combined with a small resonant pump with the Gaussian profile $\bar{P}=A_{\rm res}\exp^{-\beta r^{2}}$ where $\beta$ is an inverse width parameter. Either at first ($n=1$) or second ($n=2$) order resonant frequency, the resonant pump locally breaks gauge symmetry and fixes the phase of the condensate Burke et al. (2008). The surrounding condensate remains gauge symmetric, however, and the phase along the ring shaped interface between these regions cycles regularly; when the phase difference between the interfacial ring and the resonant spot is $\pi$, the ring is momentarily singular. Unlike spontaneously formed rings, these rings may be also formed under the fast-reservoir approximation and across wider range of parameters. Figure 3 shows cross sections of rings formed by localized resonant excitation for two different resonant pumping profiles (black and gray). We plot the time-averaged profiles (solid lines) as well as two different time-snapshots of the intensity profiles. The ring radius over time is plotted for one of these simulations (right inset) over two periods. The discontinuities of the ring radius illustrate that the ring does not survive continuously, but instead regularly explodes and reforms from outside of the interfacial region. The null density of the ring minimum corresponds to phase discontinuity, indicative of a topological defect.

In conclusion, we have theoretically predicted the spontaneous formation of stable breathing ring solitons in exciton-polariton condensates. The proposed experimental realisation for such novel topological defects is well within the current experimental conditions and properties of existing microcavities. These structures would represent the first fundamental breather and the first stable ring soliton in any quantum hydrodynamical system. We have discussed how the stability of breathing ring solitons can be exploited to study nonequilibrium defect formation statistics, and thus to probe the fundamentals of nonequilibrium phase transitions. Further, we have proposed an experimental scheme by which these statistics could be probed over the continuous crossover between equilibrium and nonequilibrium phase transitions. Finally, we have elucidated how ring solitons can be formed explicitly, via localized resonant excitation of either second or first order resonant pumping frequency.

Aknowledgements. NGB acknowledges the support from Skoltech-MIT NGP. Both authors are grateful to Kirill Kalinin for discussions.