Nonequilibrium Phase Transition in a Two-Dimensional Driven Open Quantum System

The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is mediated by the proliferation of topological defects, governs the critical behavior of a wide range of equilibrium two-dimensional systems with a continuous symmetry, ranging from spin systems to superconducting thin films and two-dimensional Bose fluids, such as liquid helium and ultracold atoms. We show here that this phenomenon is not restricted to thermal equilibrium, rather it survives more generally in a dissipative highly nonequilibrium system driven into a steady state. By considering a quantum fluid of polaritons of an experimentally relevant size, in the so-called optical parametric oscillator regime, we demonstrate that it indeed undergoes a phase transition associated with a vortex binding-unbinding mechanism. Yet, the exponent of the power-law decay of the first-order correlation function in the (algebraically) ordered phase can exceed the equilibrium upper limit: this shows that the ordered phase of driven-dissipative systems can sustain a higher level of collective excitations before the order is destroyed by topological defects. Our work suggests that the macroscopic coherence phenomena, observed recently in interacting two-dimensional light-matter systems, result from a nonequilibrium phase transition of the Berezinskii-Kosterlitz-Thouless rather than the Bose-Einstein condensation type.

Popular Summary

There cannot be absolute order in a two-dimensional fluid because disorder-causing mechanisms are exceptionally strong. Nevertheless, there can be an enormous difference between systems of lesser and greater order (e.g., between an ordinary fluid such as water and a superfluid such as liquid helium). The latter can flow without any friction and even escape up and over its container walls. The transition between superfluid and normal behavior in two dimensions, even for closed systems that are allowed to equilibrate, is particularly dramatic: It is caused by the appearance of a large number of topological defects in the form of vortices—tiny tornadoes—that destroy the more ordered state. An open question is what causes the transition for particles that cannot be perfectly trapped and equilibrated in any container, such as photons. Their inevitable escape has to be counterbalanced by an external influx to keep the situation steady. We find that the transition is still caused by proliferating tornadoes.

We focus on a quantum fluid of polaritons, and our analysis, based on a stochastic field formalism, accounts for topological defects and fluctuations. Surprisingly, we find that systems that are externally disturbed can remain a superfluid in an overall less-ordered state than their equilibrium counterparts. Whether these systems are more robust to vortex proliferation or simply more disordered by collective fluctuations remains to be determined. This externally overshaken-but-not-stirred quantum fluid clearly constitutes an interesting new laboratory to explore nonequilibrium phases of matter.

We expect that our results will motivate future studies of nonequilibrium phase transitions in driven-dissipative systems, in particular, optical ones.

Figure 1

Polariton system in the OPO regime. Upper panel: 2D map of the photonic OPO spectrum $|{\psi }_{C,k_x,k_y=0}(\omega ){|}^2$ (logarithmic scale) of energy $\omega$ versus the $k_x$ momentum component (cut at $k_y=0$) for a single noise realization and at a pump power $f_p=1.02436f_p^{\mathrm{th}}$, where $f_p^{\mathrm{th}}$ is the mean-field OPO threshold. The arrows show schematically the parametric process scattering polaritons from the pump state into the signal and idler modes. Dashed (green) lines show the bare upper polariton (UP) and lower polariton (LP) dispersions, while dotted (black) lines are the cavity photon ($C$) and exciton ($X$) dispersions. The solid (black) line underneath the spectrum is the $k_y=0$ cut of the single-shot time-averaged in the steady state momentum distribution $\int dt|{\psi }_{C,k_x,k_y=0}(t){|}^2$, clearly showing the macroscopic occupation of the three OPO pump, signal, and idler states. Lower panels: 2D maps of the filtered space profiles $|{\psi }_{s,p,i}(\mathbf{r},t){|}^2$ at a fixed time $t$ for which a steady state regime is reached—the pump emission intensity is rescaled to 1. Blue (red) dots indicate the vortex (antivortex) core positions.

Figure 2

Binding-unbinding transition and vortex-antivortex proliferation across the OPO threshold. Phase (color map) of the filtered OPO signal ${\psi }_s(\mathbf{r},t)$ and position of vortices (black dots) and antivortices (red dots) for increasing values of the pump power, in a narrow region close to the mean-field OPO threshold $f_p^{\mathrm{th}}$: (a) $f_p=1.00287f_p^{\mathrm{th}}$, (b) $f_p=1.01648f_p^{\mathrm{th}}$, (c) $f_p=1.01719f_p^{\mathrm{th}}$, and (d) $f_p=1.02436f_p^{\mathrm{th}}$. We observe a dramatic decrease of both the number of V’s and AV’s, as well as the typical distance between V and AV within a pair, as a function of the increasing pump power. The filtered profiles are plotted at a late stage of the dynamics, at which a steady state is reached.

Figure 3

The phase diagram and the BKT transition. Inset: Mean-field photonic OPO densities for pump $n_p^{\mathrm{MF}}$ (black line) and signal $n_s^{\mathrm{MF}}$ (orange line) states as a function of increasing pump power $f_p$ rescaled by the threshold value $f_p^{\mathrm{th}}$ (vertical black dashed line). The black square at $f_p\simeq f_p^{\mathrm{th}}$ indicates the tiny pump strength interval close to mean-field threshold analyzed in the main panel. Main panel: We plot with (orange) squares the same mean-field signal density $n_s^{\mathrm{MF}}$ as in the inset. All other data are noise-averaged properties from stochastic simulations as a function of the pump strength. The noise-averaged signal density $n_s$ is plotted with (blue) dots, the average vortex number in the signal rescaled by its average maximum value $N_{\max }=222.8$ with (red) diamonds, and the ratio $b$ of noise-averaged and symmetrized distance between nearest-neighboring vortices of opposite charge and of the same charge with (green) empty squares. The shaded region indicates the pump region for the BKT transition. Note that the left-hand side axis label applies to data marked by orange squares and blue dots, while the right-hand side axis label applies to data marked by red diamonds and green empty squares.

Figure 4

Algebraic and exponential decay of the first-order correlation function across the BKT transition. Main panel: Long-range spatial dependence of $g^{(1)}(\mathbf{r})$ for different pump powers $f_p/f_p^{\mathrm{th}}$ close to the mean-field pump threshold (the symbols are the same ones as in the inset and correspond to the same values of $f_p/f_p^{\mathrm{th}}$). Thick solid (thick dashed) lines are power-law (exponential) fitting, from which values of the exponent $\alpha$ are derived. The $f_p/f_p^{\mathrm{th}}=1.0129$ case (orange squares) is a marginal case where both algebraic and exponential fits apply almost equally well, signaling the BKT transition region. Inset: Power-law algebraic decay exponent $\alpha$ for different pump powers $f_p/f_p^{\mathrm{th}}$; error bars are standard deviations of the time-average.