Spontaneous symmetry breaking of self-trapped and leaky modes in quasi-double-well potentials

We investigate competition between two phase transitions of the second kind induced by the self-attractive nonlinearity, viz., self-trapping of the leaky modes, and spontaneous symmetry breaking (SSB) of both fully trapped and leaky states. We use a one-dimensional mean-field model, which combines the cubic nonlinearity and a double-well-potential (DWP) structure with an elevated floor, which supports leaky modes (quasi-bound states) in the linear limit. The setting can be implemented in nonlinear optics and Bose–Einstein condensates. The order in which the SSB and self-trapping transitions take place with the growth of the nonlinearity strength depends on the height of the central barrier of the DWP: the SSB happens first if the barrier is relatively high, while self-trapping comes first if the barrier is lower. The SSB of the leaky modes is characterized by specific asymmetry of their radiation tails, which, in addition, feature a resonant dependence on the relation between the total size of the system and radiation wavelength. As a result of the SSB, the instability of symmetric modes initiates spontaneous Josephson oscillations. Collisions of freely moving solitons with the DWP structure admit trapping of an incident soliton into a state of persistent shuttle motion, due to emission of radiation. The study is carried out numerically, and basic results are explained by means of analytical considerations.

Figure 1

The symmetric double-well potential for leaky modes (quasi-bound states), defined as per Eq. (6).

Figure 2

(a) Typical examples of trapped symmetric (black) and asymmetric (blue, gray in printed version) states for propagation constant $k=0.1$, the respective norms being $N_{\mathrm{symm}}=1.542$ and $N_{\mathrm{asym}}=1.082$. (b) The same for leaky modes, at $k=-0.085$, with $N_{\mathrm{symm}}=0.642$ and $N_{\mathrm{asym}}=0.630$. The inset in panel (b) zooms in on one spatial period the small oscillatory tail of the symmetric leaky mode. For the sake of comparison, panel (c) displays the symmetric and asymmetric states with the same $k$ as in panel (b), but trapped in DWP (6) with impenetrable outer barriers (i.e., $H=2$ is replaced by $H=\infty$ in them), the respective norms being $N_{\mathrm{symm}}=0.792$ and $N_{\mathrm{asym}}=0.769$. All the modes are produced by potential (6) with the height of the inner barrier $A=0.5$.

Figure 3

(a) A typical example of tails of an asymmetric leaky mode, found at propagation constant $k=-0.075$. (b) The total norms of the right and left tails of the asymmetric leaky mode, along with the tail norm for its symmetric counterpart, vs $k$. Vertical arrows indicate positions of the maxima predicted, through the commensurability condition, by Eq. (12). (c) The asymmetry parameter for the tails' norm, defined as per Eq. (11), vs $k$. These results were obtained for $A=0.5$ in DWP structure (6), and the total size of the system is $L=128$.

Figure 4

Families of symmetric and asymmetric modes, represented by curves for the norm vs the propagation constant in panels (a)–(c), Hamiltonian vs the norm in panels (d)–(f), and the asymmetry [defined per Eq. (15)] vs the norm in panels (g)–(i). The left, middle, and right plots correspond to the height of the central barrier in potential (6) $A=0.05$, $0.2$, and $0.5$, respectively. Continuous (olive) and dotted (red) lines designate stable and unstable states, respectively. The dashed lines in the top and middle panels depict, for the sake of comparison, the $N\left(k\right)$ and $E\left(N\right)$ dependencies for the free-space solitons, given by Eq. (13). The square symbols in panels (a)–(f) designate the location of the bifurcation points.

Figure 5

Values of (a) propagation constant $k$, and (b) total norm $N$ at the SSB bifurcation point vs the height of the inner barrier ($A$). The horizontal line $k=0$ in panel (a) and the square symbol in panel (b) designate the boundary between the trapped and leaky modes.

Figure 6

The evolution of unstable symmetric states is displayed in panels (a)–(c) for potential (6) with $A=0.05$, and in panels (d)–(f) for $A=0.5$; the corresponding SSB bifurcations taking place (a)–(c) at positive $k_{\mathrm{bif}}$, viz., $k_{\mathrm{bif}}\approx 0.313$, and (d)–(f) negative $k_{\mathrm{bif}}$, viz., $k_{\mathrm{bif}}\approx -0.100$, respectively: (a) $k=0.314$; (b) $k=0.40;$ (c) $k=0.90$; (d) $k=-0.08$; (e) $k=-0.01$; (f) $k=0.30[k>0$ in panel (f) is chosen to display unstable evolution which takes place far from the bifurcation point]. The instability gets stronger with the increase of $k$, i.e., moving from the left panels to the right panels.

Figure 7

The chart of outcomes of collisions of free solitons, launched with tilt $c$ [see Eq. (17)], with potential structure (8), in the plane of $(H_0,c)$. The norm of the incident soliton is fixed as per Eq. (18). In this figure, the width of the outer potential barriers is $W_0=4$. In region (1), the soliton bounces back from the left outer barrier. Region (2) refers to trapping the soliton inside the potential structure, where it performs shuttle motion. In region (3), the incident soliton passes the structure. The dashed line is the analytical prediction produced by Eq. (21).

Figure 8

Generic examples of outcomes of collisions of the soliton with the double-well potential structure (8) for $N=4.90\left(k=3\right)$, $W_0=4$, and different heights of the outer initial barriers, $H_0$, and different tilts $c$ of the incident soliton: (a) $[H_0=0.6,c=0.8]$, (b) $[H_0=0.6,c=1.15]$, (c) $[H_0=0.6,c=1.5]$, (d) $[H_0=0.3,c=0.8]$, (e) $[H_0=1.3,c=1.8]$, (f) $[H_0=1.2,c=1.92]$. See further explanations in the text.

Figure 9

The same as in Fig. 7, but varying the width of the outer potential barriers $W_0$ while their height is fixed at $H_0=1$. The dashed curve shows the analytical prediction for the boundary of area (1) given by Eq. (20).

Figure 10

Outcomes of the collisions, for $W_0=1.2$ and $c=1.66$. In this borderline example, the soliton hits the right (second) outer potential barrier and splits into two fragments, one escaping and the other one staying trapped in the potential structure.