Spontaneous symmetry breaking in a split potential box

We report results of an analysis of the spontaneous symmetry breaking (SSB) in a basic (actually, simplest) model that is capable of producing the SSB phenomenology in a one-dimensional setting. It is based on the Gross-Pitaevskii–nonlinear Schrödinger equation with the cubic self-attractive term and a double-well potential built as an infinitely deep potential box split by a narrow ($\delta$ functional) barrier. The barrier's strength $\varepsilon$ is the single free parameter of the scaled form of the model. It may be implemented in atomic Bose-Einstein condensates and nonlinear optics. The SSB bifurcation of the symmetric ground state (g.s.) is predicted analytically in two limit cases, viz., for deep or weak splitting of the potential box by the barrier ($\varepsilon \gg 1$ or $\varepsilon \ll 1$, respectively). For the generic case, a variational approximation (VA) is elaborated. The analytical findings are presented along with systematic numerical results. The stability of stationary states is studied through the calculation of eigenvalues for small perturbations and by means of direct simulations. The g.s. always undergoes the SSB bifurcation of the supercritical type, as predicted by the VA at moderate values of $\varepsilon$, although the VA fails at small $\varepsilon$, due to inapplicability of the underlying ansatz in that case. However, the latter case is correctly treated by the approximation based on a soliton ansatz. On top of the g.s., the first and second excited states are studied too. The antisymmetric mode (the first excited state) is destabilized at a critical value of its norm. The second excited state undergoes SSB bifurcation, like the g.s., but, unlike it, the bifurcation produces an unstable asymmetric mode. All unstable modes tend to spontaneously reshape into the asymmetric g.s.

Figure 1

Setting under consideration. An infinitely deep potential box of width 1 is split in the middle by a narrow tall barrier, approximated by $U\left(x\right)=\varepsilon \delta \left(x\right)$ [see Eq. (3)]. Wave functions of symmetric (even), antisymmetric (odd), and asymmetric stationary states are schematically shown by red, blue, and green curves, respectively.

Figure 2

Norm at the symmetry-breaking bifurcation point versus the strength of the splitting barrier $\varepsilon$. Along with the numerically found and VA-predicted dependences, shown are the analytical approximations based on (a) Eq. (21) and (b) Eq. (27), which are relevant, respectively, for large and small $\varepsilon$ [note that in (a) the range starts from $\varepsilon \approx 2.7]$.

Figure 3

Eigenvalue for the energy of the g.s. in the linear version of Eq. (7) ($g=0$) versus the strength of the splitting barrier $\varepsilon$ as obtained from the numerical solution of Eq. (12) and as predicted by the VA in the form of Eq. (39). The horizontal line shows the constraint imposed by Eq. (13).

Figure 4

(a) Asymmetry of the stationary solutions, defined as per Eqs. (9) and (10), as a function of the total norm, at $\varepsilon =3$. For the VA solution, the asymmetry is calculated through Eq. (31). (b) Dependence between the total norm and energy eigenvalue for the same solutions.

Figure 5

Typical examples of (a) unstable symmetric and (b) stable asymmetric modes, produced by both the VA and numerical solution of Eq. (7) for $\varepsilon =3$ and $N=10$. The respective energy eigenvalues are (a) ${\mu }_{\mathrm{VA}}=-3.53$ and ${\mu }_{\mathrm{num}}=-4.39$ and (b) ${\mu }_{\mathrm{VA}}=-10.04$ and ${\mu }_{\mathrm{num}}=-12.90$.

Figure 6

Instability growth rate ${\lambda }_{\mathrm{i}}\equiv \mathrm{Im}\left(\lambda \right)$ of small perturbations added to the unstable symmetric mode at $\varepsilon =3$ and $N>N_{\mathrm{bif}}$ [see Eq. (41)]. The growth rate was found as a numerical solution of Eq. (42). The instability sets in at exactly the same bifurcation point at which the SSB starts in Fig. 4.

Figure 7

Typical example of the spontaneous conversion of a perturbed unstable symmetric mode into a stable asymmetric one, at $\varepsilon =3, \mu =3$, and $N=10.39$.

Figure 8

(a) Bifurcation diagram shown by means of the $\mathrm{\Theta }\left(N\right)$ dependence, as predicted by the VA at $\varepsilon =1$. Red circles denote two asymmetric solutions for $N=12$, shown in Fig. 9. (b) Counterpart of the bifurcation diagram from (a), as produced by the numerical solution of Eq. (7). (c) Numerically generated $N\left(\mu \right)$ dependences corresponding to (b).

Figure 9

Dashed and dash-dotted profiles represent the VA-predicted asymmetric solutions corresponding, respectively, to the top and bottom red circles in Fig. 8, obtained at $\varepsilon =1$ and $N=12$. The solid profile depicts the solution obtained from the numerical solution of Eq. (7) with the same parameters. The values of the energy eigenvalue pertaining to the dashed, dash-dotted, and solid profiles are $\mu =-24.7, -20.7$, and $-32.0$, respectively.

Figure 10

(a) Numerically generated $N\left(\mu \right)$ dependence for the family of odd (antisymmetric) modes at $\varepsilon =3$. Stable and unstable subfamilies are represented by the continuous and dotted segments, respectively. (b) Instability growth rate [see Eq. (41)] for the unstable portion of the family versus $N$.

Figure 11

Examples of stable and unstable (shown by the solid and dotted lines, respectively) odd modes, for $\varepsilon =3$. The respective values of the norm and energy eigenvalue are $N=9.63$ and $\mu =+5$, and $N=15.63$ and $\mu =-5$.

Figure 12

Example of the spontaneous evolution of an unstable odd mode, shown by the dotted line in Fig. 11 (for $\varepsilon =3, N=15.63$, and $\mu =-5$), towards a stable asymmetric state.

Figure 13

Antisymmetric (odd) modes, which are the lowest excited states, are stable in the region below the boundary shown here in the plane of $(\varepsilon ,N)$.

Figure 14

(a) Juxtaposed $N\left(\mu \right)$ dependences for the symmetric (even), antisymmetric (odd), and asymmetric modes at $\varepsilon =3$. (b) Hamiltonian versus the norm for the same modes, calculated as per Eqs. (5) and (43). In both panels, solid and dotted segments represent stable and unstable solutions, respectively.

Figure 15

Numerically found second excited states, as solutions of Eq. (7), for $\varepsilon =3$ and $\mu =-10$: (a) a symmetric one, with norm $N=38.13$ and (b) an asymmetric mode, with $N=37.39$. (c) Numerically simulated evolution of the unstable asymmetric mode from (b).