Fermionic Light in Common Optical Media

Recent experiments have proved that the response to short laser pulses of common optical media, such as air or oxygen, can be described by focusing Kerr and higher order nonlinearities of alternating signs. Such media support the propagation of steady solitary waves. We argue by both numerical and analytical computations that the low-power fundamental bright solitons satisfy an equation of state which is similar to that of a degenerate gas of fermions at zero temperature. Considering, in particular, the propagation in both ${\mathrm{O}}_2$ and air, we also find that the high-power solutions behave like droplets of ordinary liquids. We then show how a grid of the fermionic light bubbles can be generated and forced to merge in a liquid droplet. This leads us to propose a set of experiments aimed at the production of both the fermionic and liquid phases of light, and at the demonstration of the transition from the former to the latter.

Figure 1

Plot of the central amplitude $\Phi (0)$ vs $\mu /{\mu }_{\infty }$ for the family of fundamental eigenstates of Eq. (1). Note that $\Phi (0)$ has a maximum for $\mu /{\mu }_{\infty }\approx 0.7$, due to the defocusing nonlinearities. Left-hand inset: Radial profiles of three eigenfunctions of Eq. (1) with $\mu /{\mu }_{\infty }=0.004$ (black solid line), $\mu /{\mu }_{\infty }=0.3$ (red dashed line), and $\mu /{\mu }_{\infty }=0.97$ (blue dash-dotted line), respectively. Right-hand inset: Radial pressure profiles corresponding to these solutions. The low-power profile is magnified by a factor of ${10}^3$ for clarity. Labeled points on the main curve refer to the eigenstates displayed within the insets.

Figure 2

Plot of the logarithms of $p_c$ vs $R$ for the fundamental solitons (black line). The fermionic behavior of Eq. (4) (red dotted line) and the liquid YL equation (green dashed line) are compared with the numerics. The labeled points correspond to the same eigenstates displayed in Fig. 1. For each value of $R$, two different outcomes are possible depending on the power, corresponding to two different phases of light. Inset: Plot of $\ln (p_c)$ vs $\ln ({\rho }_c)$ (black solid line) overlapped with Eq. (4) (red dashed line) for comparison.

Figure 3

Isosurface intensity plot of the dynamical phase transition from a square grid of fermionic light solitons to an individual liquid light soliton in ${\mathrm{O}}_2$. All solitons in the grid are eigenstates with $\mu /{\mu }_{\infty }=0.3$ displayed in Fig. 1. They are forced to merge in the center of the computational window by means of an external harmonic potential. We can see three pseudocolor plots displaying the initial condition (left, $\eta =0$), the collapse of the light bubbles (middle, $\eta =200$), and the final state (right, $\eta =500$). In the latter we observe a flattop soliton with radius $R\approx 10$ and ${\rho }_c\approx 0.76$ arising after the massive coalescence of the fermionic solitons. The width of the square spatial domain displayed is $w_{\xi ,\chi }=200$.

Figure 4

Proposal for the generation of the initial condition of Fig. 3 in experimental setups. The (a) pseudocolor plot shows the bidimensional cosine squaredlike phase distribution to be imprinted onto the ingoing beam. We assume that the beam is much wider than the filaments, and restrict our simulation to a square domain $\xi ,\chi \in [-75,75]$ where the wave front is assumed to be homogeneous. In (b),(c) ($\eta =100,200$) we observe how the beam profile starts to destabilize, leading to the formation of a grid of spatial solitons at $\eta =300$ [see (d)].