Photonic Nambu-Goldstone bosons

We study numerically the spatial dynamics of light in periodic square lattices in the presence of a Kerr term, emphasizing the peculiarities stemming from the nonlinearity. We find that, under rather general circumstances, the phase pattern of the stable ground state depends on the character of the nonlinearity: the phase is spatially uniform if it is defocusing whereas in the focusing case, it presents a chessboard pattern, with a difference of $\pi$ between neighboring sites. We show that the lowest-lying perturbative excitations can be described as perturbations of the phase and that finite-sized structures can act as tunable metawaveguides for them. The tuning is made by varying the intensity of the light that, because of the nonlinearity, affects the dynamics of the phase fluctuations. We interpret the results using methods of condensed-matter physics, based on an effective description of the optical system. This interpretation sheds light on the phenomena, facilitating the understanding of individual systems and leading to a framework for relating different problems with the same symmetry. In this context, we show that the perturbative excitations of the phase are Nambu-Goldstone bosons of a spontaneously broken $\text{U}\left(1\right)$ symmetry.

Figure 1

The black circles represent a square photonic lattice, within which a finite-size nonlinear solution for the optical beam propagates (yellow structure). With adequate features, it acts as an effective metawaveguide for phase excitations of the optical field (red path). These phase perturbations behave as photonic Nambu-Goldstone bosons.

Figure 2

Propagation-invariant solutions of the nonlinear equation with $V_0=800, N=16, P/N^2=5$. On the left, $|{\phi }_{\mathbf{Q}}^P(x,y=\frac{1}{2}){|}^2$ within a lattice cell for $\mathbf{Q}=(0,0)$ (solid black line), $\mathbf{Q}=(0,\pi )$ (dashed red line), and $\mathbf{Q}=(\pi ,\pi )$ (dotted blue line). Notice that for the first and third cases, it also corresponds to $|{\phi }_{\mathbf{Q}}^P(x=\frac{1}{2},y){|}^2$ due to the symmetry of the configuration. On the right, a contour plot of $|{\phi }_{\mathbf{Q}}^P{(x,y)|}^2$ with $\mathbf{Q}=(\pi ,\pi )$ for the whole domain. Notice that lattice sites have sides of length 1, which is the period of the refractive index and of $|{\phi }_{\mathbf{Q}}^P{(x,y)|}^2$. Periodic boundary conditions for $\psi$ are imposed at the boundaries of the computational domain of size $N\times N$. All units in figures are dimensionless—see Eq. (1). The dimensionless formalism is connected to photonic propagation in a periodic medium as discussed after Eq. (5).

Figure 3

Evolution, computed by numerical integration of Eq. (1), of three different solutions of Eq. (11) with $g=1, V_0=800, N=16, P/N^2=5$. The unstaggered $\mathbf{Q}=(0,0)$ is represented in panels (a) (phase distribution at $z=0$), (d) (phase distribution at $z=10$), and (g) [the blue solid line represents a one-dimensional cut $|\psi (x,\frac{1}{2}){|}^2(z=10)$ compared to the initial wave function depicted by the red dashed line $|\psi (x,\frac{1}{2}){|}^2(z=0)]$. With the same specifications, $\mathbf{Q}=(0,\pi )$ is represented in panels (b), (e), (h) and $\mathbf{Q}=(\pi ,\pi )$ in panels (c), (f), (i). The figure shows that the staggered configuration $\mathbf{Q}=(\pi ,\pi )$ is the only stable one.

Figure 4

Evolution of the phase for initial conditions with noise, fixing $g=1, V_0=800, N=16, P/N^2=5$. Panels (a)–(c) correspond to the $\mathbf{Q}=(\pi /2,\pi /2)$ case for $z=0, z=0.1$, and $z=4$ respectively. Panels (d)–(f) correspond to $\mathbf{Q}=(\pi ,\pi )$ for the same values of $z$. The noise is introduced by multiplying the initial phase at each point by a random number from a normal distribution of amplitude 0.2. It is visible that the phase structure gets destabilized in the $\mathbf{Q}=(\pi /2,\pi /2)$ case whereas it remains stable in the staggered configuration.

Figure 5

In panels (a)–(d), the values of the integrals of Eqs. (21) are shown for four cases with different nonlinearity and different $\mathbf{Q}$, with fixed $V_0=800, N=24$. Additionally, the low-energy coefficients appearing in Eqs. (23) and (25), for the effective free energy, and in the field potential $\left(26\right)$ are shown in panels (f)–(h) in terms of the normalized power $P/N^2$. In all cases—except in the remarkable case (f)—the blue and red lines are practically overlapping and in the plot are seen as a single solid line corresponding to $g=1$ with $q=0$ and $q=\pi$, respectively. On the contrary, panel (f) unveils that the sign of $b$ is positive for $q=0$ and negative for $q=\pi$.

Figure 6

Calculated effective field potential $V\left(\mathrm{\Phi }\right)$ for a self-focusing ($g=1$) and staggered ($q=\pi$) solution in a lattice with fixed $V_0=800$ and $N=24$ as in Fig. 5. The power per site is $P/N^2=2$ so that $b=-0.3, M=11$, and $\mathcal{G}=2.2$. The same sombrero shape is obtained in all cases (for these values of $V_0$ and $N$) provided $g=1$ and $q=\pi$ or $g=-1$ and $q=0$.

Figure 7

Illustration of the propagation of the excitation of the phase ($g=1, V_0=800, N=24, P/N^2=5$). The initial condition is like that of Fig. 3, multiplying by $e^{i\pi /5}$ the wave function for a column of lattice sites. Panels (a)–(c) depict the phase distribution for three values of $z$. Panels (d)–(f) are $y=0$ cuts of the same quantity. Notice that in (a) the strength of the perturbation ($\pi /5$) is higher than the maximum of the color-map scale, a convention chosen to improve visualization in (b) and (c).

Figure 8

Comparison of the wave velocity obtained from repeated numerical integration (solid line) of Eq. (1) and the one predicted by the effective model (dashed line) in Eq. (31) as a function of the power per lattice cell. The green upper curves correspond to the defocusing unstaggered case and the red lower curves correspond to the focusing staggered case. Notice that the horizontal axis starts at $P/N^2=0.5$ because when this quantity tends to zero the excitation loses its Nambu-Goldstone boson character.

Figure 9

Propagation of a circular wave. Initial conditions are as in Fig. 7, except that the $\pi /5$ initial phase perturbation is performed on the four central cells of the lattice. Notice that in (a) the strength of the perturbation ($\pi /5$) is higher than the maximum of the color-map scale.

Figure 10

One-dimensional cuts of the phase perturbation $\varphi (x,y=\frac{1}{2},z)$ moving leftwards, initiated by initial conditions with a $\pi /5$ phase shift in the right-most column. In this example $g=1$ and $V_0=800$. The different panels correspond to different values of $P/N^2$, namely $P/N^2=2$ in (a), $P/N^2=4$ in (b), and $P/N^2=6$ in (c). The color scale does not cover the full range of the perturbation $\varphi$, but it has been chosen to optimize the visibility of the plots without missing essential information.

Figure 11

Bounce of a Nambu-Goldstone phase wave due to reflection in the edge of the structure, resulting in a shift of the propagation direction. The perturbation of the phase is depicted in the lattice sites where Eq. (3) holds. For the rest, where $V=V_0$, the modulus ${\left|\psi \right|}^2$ is nearly vanishing and we just plot them in a uniform color to help visualization. In this example $g=1, V_0=800$, and $P/N^2=5$. Panels (a)–(f) correspond, respectively to $z=0,4.5,6,7.5,9,10.5$.

Figure 12

Representation of $|u_{\mathbf{Q}}{(x,y)|}^2$ for $V_0=100$ and different values of $\mathbf{Q}$. (a) $Q_x=Q_y=0$. (b) $Q_x=Q_y=\pi /2$. (c) $Q_x=0, Q_y=\pi$. (d) $Q_x=Q_y=\pi$.

Figure 13

Representation of the phase of ${\phi }_{\mathbf{Q}}(x,y)$ for $V_0=100$ and $N=16$. The four panels show the same four values of $\mathbf{Q}$ considered for Fig. 12.

Figure 14

Dispersion relation $\mu (Q_x,Q_y)$ of the lowest band ($V_0=100$).

Figure 15

Square modulus of the Wannier function for $\mathbf{R}=(-\frac{1}{2},-\frac{1}{2})$, with $V_0=100, N=16$. Panel (a) is a color map for $|W_{\mathbf{R}}{\left(\mathbf{x}\right)|}^2$. We also include (in the lower part, in orange) the representation of a one-dimensional section $|W_{\mathbf{R}}(x,-\frac{1}{2}){|}^2$ (arbitrary units). Panel (b) uses a logarithmic scale, $ln|W_{\mathbf{R}}{|}^2$. It shows that $W_{\mathbf{R}}\left(\mathbf{x}\right)$ has support in the whole domain but becomes negligible far from the cell parametrized by $\mathbf{R}$.