Tuning across Universalities with a Driven Open Condensate

Driven-dissipative systems in two dimensions can differ substantially from their equilibrium counterparts. In particular, a dramatic loss of off-diagonal algebraic order and superfluidity has been predicted to occur because of the interplay between coherent dynamics and external drive and dissipation in the thermodynamic limit. We show here that the order adopted by the system can be substantially altered by a simple, experimentally viable tuning of the driving process. More precisely, by considering the long-wavelength phase dynamics of a polariton quantum fluid in the optical parametric oscillator regime, we demonstrate that simply changing the strength of the pumping mechanism in an appropriate parameter range can substantially alter the level of effective spatial anisotropy induced by the driving laser and move the system into distinct scaling regimes. These include (i) the classic algebraically ordered superfluid below the Berezinskii-Kosterlitz-Thouless (BKT) transition, as in equilibrium; (ii) the nonequilibrium, long-wavelength-fluctuation-dominated Kardar-Parisi-Zhang (KPZ) phase; and the two associated topological-defect-dominated disordered phases caused by proliferation of (iii) entropic BKT vortex-antivortex pairs or (iv) repelling vortices in the KPZ phase. Furthermore, by analyzing the renormalization group flow in a finite system, we examine the length scales associated with these phases and assess their observability in current experimental conditions.

Popular Summary

One of the most fascinating observations in the study of complex systems is that they exhibit universality: The same behavior can be found in vastly different physical realizations. A liquid boiling into a gas, for example, behaves exactly like a magnet losing its magnetic properties as it heats up. Similar behaviors among disparate systems can be grouped into what are known as universality classes, with each class describing a different behavior. While this classification is well established for systems that are in thermal equilibrium, it is tricky to classify nonequilibrium systems that are perturbed by some external mechanism. We find a way to experimentally realize one of these classes and show that the same system can be moved between different classes by only slightly changing the strength of the external drive.

Driven systems often exhibit behavior that is very different from their equilibrium counterparts. One example, the Kardar-Parisi-Zhang (KPZ) universality class, comprises a colorful diversity of dynamical processes ranging from the growth of liquid crystals and bacterial colonies to the combustion of paper. Despite an abundance of theoretical predictions, observation of KPZ scaling in two spatial dimensions remains elusive. One candidate system is a fluid of exciton-polaritons—mixed light-matter bosonic particles—in semiconductor microcavities. We show that under parametric driving, not only is KPZ scaling within reach of current experimental setups, but it is also possible to tune the system between fundamentally different universal regimes by making the system more anisotropic.

We provide the missing link between the general theory of the two-dimensional KPZ universality class and an actual physical realization, giving experimentalists a clear guideline in the search for KPZ physics in parametrically driven exciton-polaritons.

Figure 1

Polariton system. Dispersions for the exciton (EX) and cavity photon ($C$) with zero detuning mixing to form upper (UP) and lower polaritons (LP). The external drive introduces pairs of lower polaritons with momentum ${\mathbf{k}}_p$ and energy ${\omega }_p$ ($P$), which scatter into the signal ($S$) and idler ($I$) states while conserving energy and momentum.

Figure 2

Stability diagram. Regions of instability of the pump-only state towards an OPO solution (yellow and green). The simplest three-mode OPO state is stable for parameters marked by green, i.e., at $k_s$ around 0.1 for high $F_p^2$ (regions A and B) and over a more extended higher $k_s$ range at low $F_p^2$ (region C). In the yellow region, both the pump-only and the three-mode ansatz are unstable. The red arrows indicate the direction of increasing pump power shown in Figs. 4, 5, 6. Note that the pump intensity (vertical axis) is normalized to the lower threshold, $F_{pl}^2\equiv (F_p/F_p^{\mathrm{lo}}{)}^2$. Parameters are the same as in the text with zero detuning and $k_p=1.4$.

Figure 3

Crossover between nonequilibrium and equilibrium-like universal regimes. We show stable three-mode OPO configurations in the ${\mathrm{\Gamma }}_0-{\kappa }_0$ space for three different detunings, ${\delta }_{CX}=-1.07$ (green line), $-1.075$ (blue line), $-1.08$ (red line), and for $k_p=1.4$ and $k_s=0.1$ (region A in Fig. 2). The arrows indicate the direction of the increasing external drive strength. The dashed line shows the BKT phase boundary [see Eq. (25)] between algebraically ordered and disordered phases. By increasing the external pump power, for ${\delta }_{CX}=-1.07$, we cross from the nonequilibrium (WA) to the equilibrium-like (SA) disordered regimes. For ${\delta }_{CX}=-1.075$, the systems shows reentrance; it starts in the WA, enters the SA algebraically ordered regime, and finally goes back to the disordered but now SA regime. For ${\delta }_{CX}=-1.08$, we are in the SA equilibrium-like regime for all pump powers, and the system undergoes a BKT transition from an algebraically ordered to a disordered phase by increasing $F_p$.

Figure 4

Changing the effective anisotropy by tuning the driving strength. Left panel: Anisotropy parameter $\mathrm{\Gamma }$ as a function of pump power $F_p^2$ (in the same range as region A in Fig. 2) for different detunings ${\delta }_{CX}=-1.06$ (blue line), $-1.07$ (red line), $-1.08$ (green line) in the stable three-mode OPO regime. Parameters are as in Fig. 3. For ${\delta }_{CX}=-1.075$, the anisotropy parameter crosses zero (dotted red line). Note that the extent of region A, i.e., the range of pump powers for which the three-mode OPO ansatz is stable, depends on the value of the detuning. Right panel: Anisotropy parameter $\mathrm{\Gamma }$ as a function of normalized pump power, where $F_{pl}\equiv F_p/F_p^{\mathrm{lo}}$, at zero detuning but close to the lower OPO threshold, indicated by the green vertical line. The pump and signal have large momenta $k_p\approx 1.84$ and $k_s=1.0$. The dotted red line marks the $\mathrm{\Gamma }=0$ line. This case corresponds to region C in Fig. 2.

Figure 5

Characteristic length scales in the WA regime. Left and right panels: Healing length ${\xi }_0$ (blue line), length scale for the vortex-dominated disordered phase $L_v$ (brown line), and the KPZ length scale $L_{\mathrm{KPZ}}$ (red line) as a function of the pump strength (normalized to the upper threshold, $F_{pu}\equiv F_p/F_p^{\mathrm{up}}$). The vertical dotted (green) line indicates the BKT transition where V-AV pairs proliferate (taken from Ref. [33]), and the vertical solid (green) line shows the mean-field OPO threshold. We use the same parameters as those discussed in the text for the bad microcavity at zero detuning ${\delta }_{CX}=0$, $k_p=1.4$, and $k_s=0.1$. The two different panels correspond to two different ranges in pump strength. Central panel: Logarithmic RG scale for the vortex-dominated phase ($\mathrm{\Delta }{l}_v$) and for the KPZ phase ($\mathrm{\Delta }{l}_{\mathrm{KPZ}}$) as a function of pump strength normalized to the pump at the upper threshold (vertical solid green line). The vertical dotted green line indicates the BKT transition. Note that $L_v$ is expected to be strongly renormalized close to the threshold as explained in Appendix pp2.

Figure 6

Characteristic length scales in the WA regime at intermediate pump powers. Left panel: Nonlinear parameter $g$ as a function of the pump strength normalized to the pump at the lower threshold for a stable three-mode OPO solution with zero detuning. The vertical green lines indicate the lower and upper mean-field thresholds, and the red dotted line marks the $g=1$ border. These are the stable solutions at intermediate values of $F_p$ (region B in Fig. 2). Right panel: Characteristic length scales at the intermediate $F_p$ shown in the left panel. Healing length ${\xi }_0$ (blue line), length scale for vortex-dominated disordered phase $L_v$ (green line), and the KPZ length scale $L_{\mathrm{KPZ}}$ (red line) as a function of pump strength normalized to the pump at the upper threshold. Note that to the left of the nearly vertical increase of $L_{\mathrm{KPZ}}$, its value is indistinguishable from the healing length.

Figure 7

Crossover between SA (negative ${\mathrm{\Gamma }}_0$) and WA (positive ${\mathrm{\Gamma }}_0$) regimes at finite length scales. Length scale for the vortex-dominated phase in the WA regime $L_v$ (green line) and the disordered phase in the SA $L_{\mathrm{BKT}}$ (blue line) as a function of the anisotropy parameter, tunable by the drive, for the polariton system with ${\delta }_{CX}=-1.07$. The double red arrow A indicates a transition from the disordered WA to an algebraically ordered SA phase by increasing the external pump power in a system of size $L$ such that $L_v<L<L_{\mathrm{BKT}}$. The double red arrow B indicates a transition from the ordered WA regime to the ordered SA regime by increasing the external pump power in a system of size $L$, where $L<L_v<L_{\mathrm{BKT}}$.

Figure 8

Crossover between the KPZ scaling and the SA (negative ${\mathrm{\Gamma }}_0$) regime with algebraic order. Left panel: Regions of detuning between the cavity photons and the excitons ${\delta }_{CX}$ and signal momentum $k_s$ (dark blue) for which the system can exhibit a crossover between the KPZ phase and algebraically ordered phase by tuning the pump strength $F_p$. The yellow region marks stable KPZ solutions. Right panel: $g$ and $\mathrm{\Gamma }$ as a function of the normalized pump strength $F_{\mathrm{pl}}^2$ for $k_s=-0.15$ and ${\delta }_{CX}=-0.25$ (small red circle in the left panel). The crossover between the KPZ and the SA regimes occurs at $F_{\mathrm{pl}}^2\simeq 3.909$: $\mathrm{\Gamma }$ changes sign (vertical red dotted line) while $g>1$ (dotted blue horizontal line). Here, $k_p=1.4$, and we consider all values of $k_s$ for which there is a nonzero mean-field solution and a stable aKPZ equation. Note that the stability analysis indicates that in the blue regions, the three-mode ansatz is unstable. This is confirmed by numerical studies [77], which show that the steady-state polariton field can develop several secondary modes in addition to the three main ones. However, the secondary modes are typically orders of magnitude weaker, and we expect them to give only minor quantitative corrections to the values of $g$ and $\mathrm{\Gamma }$ we find here.

Figure 9

Regions potentially exhibiting a KPZ phase for different experimental microcavities. The KPZ nonlinearity parameter $g$ (in a region where $g>1$) is shown as a function of the normalized pump strength $F_{pl}^2$ for four different microcavities: bad (blue line), typical (red dots), good (black dots), and organic (green dots) cavities. We present the regions that could potentially show KPZ scaling at all length scales beyond the healing length, i.e., $g>1$ and $\mathrm{\Gamma }>1$ (see Sec. 4a). The dashed horizontal line marks the $g=1$ border, whereas the dashed vertical line indicates the normalized lower mean-field threshold, i.e., $F_{pl}^2=1$. Note that only for the bad cavity is the three-mode ansatz stable in the region shown. In this plot, we consider $k_p=1.4$ and $k_s=0.1$ for the bad and good cavities and $k_p=1.6$ and $k_s=0.06$ for the typical and organic cavities.

Figure 10

The aKPZ coefficients for the bad cavity configuration with zero detuning. Left panel: Diffusion coefficients ($D_x$, $D_y$), nonlinearities (${\lambda }_x$, ${\lambda }_y$), anisotropy parameter ($\mathrm{\Gamma }$), and drift term $B_x$ as a function of the pump power normalized to the upper threshold. These parameters vary little with the external pump power. Right panel: Dimensionless nonlinearity $g$ and noise parameter $\mathrm{\Delta }$ as a function of the normalized pump power. We observe the asymptotic growth of these parameters close to the upper OPO threshold.

Figure 11

Dependence of the nonlinear parameter $g$ on exciton-exciton interaction strengths. We show $g$ for three different values of $g_X$ as a function of the normalized external pump power $F_{pl}$ with the lower threshold for a cavity with zero detuning. We observe that the system shows larger values of $g$ when increasing the exciton-exciton interaction.