Universal self-similar evolution of two-dimensional Bose-Einstein condensates in the acoustic regime

Problem Statement
The paper investigates the nonstationary, self-similar evolution of two-dimensional (2D) Bose-Einstein condensates (BECs) driven out of equilibrium in the acoustic regime. Specifically, it addresses how an initially large-scale condensate evolves into a turbulent cascade, transferring energy toward smaller scales (ultraviolet, UV). While self-similar evolution is a common phenomenon in physics (e.g., diffusion, nuclear explosions), the nature of scaling in nonlinear wave systems is often categorized into two classes: "first kind," determined by dimensional analysis and conservation laws, and "second kind," where scaling exponents are determined by nonlinear eigenvalue problems. The authors aim to characterize the self-similar dynamics of 2D BECs, particularly in the presence of acoustic waves, and determine whether the system follows first- or second-kind self-similarity. A specific challenge addressed is the behavior of polariton BECs, which suffer from finite lifetimes and strong dissipation, compared to atomic BECs.

Methodology
The study employs a dual approach combining numerical simulations of the Gross–Pitaevskii equation (GPE) and the Wave Kinetic Equation (WKE):

1. Gross–Pitaevskii Equation (GPE): The authors simulate the 2D GPE, which describes the dynamics of atomic and polariton BECs. The equation includes terms for the speed of sound ($c_s$), healing length ($\xi$), external trapping potential ($V_{trap}$), and a linear damping term ($\gamma$) to model polariton losses.

   • Simulations were performed using the pseudospectral code FROST.
   • Initial conditions involved a ground state perturbed by weak random Bogoliubov waves at large scales.
   • Two setups were used: a homogeneous square trap of size $75\xi$ and a larger periodic box of size $1024\xi$ without a trap to better observe universal features.

2. Wave Kinetic Equation (WKE): To gain theoretical insight, the authors utilized the Weak Wave Turbulence Theory (WWTT) framework. They derived and numerically solved the WKE for large-scale Bogoliubov waves in 2D.

   • The WKE describes the temporal evolution of the energy spectrum ($e_k$) in the weakly nonlinear limit.
   • Simulations were performed using the code WavKinS over a spectral range spanning four decades.
   • The WKE includes a damping term added ad-hoc to model dissipation, acknowledging that rigorous derivation usually assumes no forcing or dissipation.

3. Analysis: The authors tracked the evolution of the energy spectrum and defined a characteristic wavevector $k^*(t)$ (the energy-weighted average) to quantify the propagation of the turbulent front. They sought self-similar solutions of the form $e_k(t) = E_0 f(t) \Phi(k/k^*(t))$.

Key Contributions and Results

• Identification of Self-Similar Regime: Both GPE and WKE simulations reveal a self-similar evolution where a front propagates toward higher wavevectors, leaving behind a power-law spectrum $e_k \sim k^{-1}$. This exponent corresponds to the Kolmogorov-Zakharov (KZ) solution for a direct energy cascade in 2D acoustic waves.
• Second-Kind Self-Similarity: Contrary to the expectation for "infinite capacity" systems (which typically exhibit first-kind self-similarity determined by dimensional analysis), the 2D acoustic system exhibits second-kind self-similarity. The scaling exponents cannot be derived solely from dimensional arguments; instead, they emerge from a nonlinear eigenvalue problem.
• Universal Constant $\beta$: The dynamics of the propagating front are governed by a single dimensionless universal constant, $\beta$. The authors derived an equation for the front position $k^*(t)$:
  $$\frac{1}{k^*} \frac{dk^*}{dt} \ln\left(\frac{k^*}{k_0}\right) = \frac{\beta}{\tau} e^{-t/\tau_D}$$
  where $\tau$ is a characteristic time scale and $\tau_D$ is the dissipation time.
• Numerical Determination of $\beta$: By fitting the numerical evolution of $k^*(t)$ from both GPE and WKE simulations to the theoretical prediction, the authors determined the universal value:
  $$\beta \approx 8.9$$
• Independence from Dissipation: A striking result is that the self-similar form $\Phi(\eta)$ and the constant $\beta$ are independent of the damping coefficient $\gamma$. Despite the presence of dissipation (relevant for polaritons), the system preserves the same universal self-similar form. This is attributed to the fact that the degree of homogeneity of the dissipative term matches that of the collisional integral in the WKE.
• Spectral Structure: The self-similar function $\Phi(\eta)$ exhibits a power-law behavior $\eta^{-1}$ at small $\eta$ (consistent with the KZ solution) and an exponential decay $\exp(-\beta \eta / D)$ at large $\eta$ (small scales), where $D$ is a diffusion coefficient.

Significance and Claims
The paper claims to establish the dynamics of 2D acoustic BECs as a specific example of a degenerate type of second-kind self-similar solution. The significance lies in the following points:

1. Universal Behavior: The system is fully characterized by the pair $(\Phi, \beta)$, where $\beta \approx 8.9$ is a universal constant independent of initial conditions or the presence of dissipation.
2. Borderline System: The authors suggest that 2D acoustic waves represent a "borderline infinite capacity system." While the energy integral for the KZ scaling is only logarithmically divergent (unlike the UV divergence in standard infinite capacity systems), the system inherits properties from both first- and second-kind self-similarity regimes.
3. Theoretical Validation: The work demonstrates that the Weak Wave Turbulence Theory (WWTT) successfully predicts the self-similar evolution of the GPE, even in the presence of damping, bridging the gap between the microscopic GPE dynamics and the macroscopic WKE description.
4. Experimental Relevance: The findings are relevant to both atomic BECs (inviscid case) and polariton BECs (dissipative case), providing a theoretical framework for interpreting non-equilibrium experiments in these systems, particularly regarding the formation of small-scale structures and the propagation of turbulent fronts.

The authors conclude that the scaling exponents in this system are dynamically selected rather than imposed by dimensional constraints, highlighting the complexity of non-equilibrium evolution in 2D quantum fluids.