The ground-state energy E 0 for the linear quantum harmonic oscillator as a function of the lattice spacing α, for two values of the trap frequency, Ω = 0.1 and Ω = 0.05. Solid and dotted lines show the energy spectrum as found by solving the QHO eigenvalue problem, while (red) circles and (blue) crosses show the respective solutions obtained from the Mathieu equation (20). The dashed and dashed-dotted lines show the analytical result of Eq. (28). The inset shows the wavefunction profile for α = 10.
Top right panel: Energy of the lowest four excited states as functions of the lattice spacing α, for a trap strength Ω = 0.1. Solid lines corresponding to the 1st-, 2nd-, 3rd, and 4th-excited states indicate the energy obtained by solving the QHO eigenvalue problem, dashed lines show the respective solutions obtained from the Mathieu equation (20), and circles show the respective analytical results of Eq. (25). Top left panels: spatial profiles of the 1st-, 2nd-, 3d-, and 4th-excited states for α = 10 (corresponding to the strongly discrete limit); solid lines depict the 1st- (top) and 3d- (bottom) excited states, while dotted lines depict the 2nd- (top) and 4th- (bottom) excited states, respectively. Bottom panels (from left to right): spatial profiles of the 1st-, 2nd-, 3d-, and 4th-excited states for α = 1 (corresponding to the discrete regime).
The number of atoms N as a function of the chemical potential μ (for α = 0.8 and Ω = 0.1) for the three lowest states: the ground state (solid line), the first excited state (dashed line), and the second excited state (dotted line). Each branch begins from the linear limit (N = 0), where μ equals the energy of the corresponding linear state. The insets show the profiles of these nonlinear states for μ = 1.2.
Top left panel: The normalized number of atoms N/α as a function of the chemical potential μ, for α = 10 (i.e., in the vicinity of the anti-continuum limit) and Ω = 0.1. The black square indicates the region where this panel is magnified, as shown in top right panel. The letters A, B, …, H denote certain points in the diagram for which corresponding wavefunction profiles are shown in the middle and bottom panels. Stable (unstable) branches and respective states are depicted by solid (dashed or dotted) lines.
The pitchfork bifurcation, relevant to the (green) branches D, F (and its parity-symmetric one), and G, as viewed by the difference ψ1 − ψ−1 of the two outer sites as a function of the normalized atom number N/α.
The linear stability analysis for the first-excited state, corresponding to a single discrete dark soliton. The three top panels show the real part of the lowest-order eigenvalues and the two left bottom panels show the maximum of the imaginary part of the eigenvalues, both as functions of the chemical potential μ. The bottom right panel shows the dependence of the critical value of the chemical potential, μ c , for the onset of the instability as a function of α. Branches shown with circles (in red) in the two top left panels denote dynamically unstable modes, which have emerged upon collision of modes with opposite Krein sign. The parameter values are α = 1.2 (left column), α = 0.6 (middle column), and α = 0.1 (right column); the trap strength is in all cases Ω = 0.1.
The top panel shows a spatio-temporal contour plot of the density of the first-excited state (corresponding to a single discrete dark soliton), for parameter values μ = 1, α = 1.2, and Ω = 0.1. The soliton stays at rest, up to t ≈ 1500, and then starts to perform oscillations of growing amplitude. The bottom panel shows the initial density profile.
The BdG analysis for the second-excited state, corresponding to a discrete dark soliton pair. The top (bottom) panels show the real (imaginary) part of the lowest-order eigenfrequencies as functions of the chemical potential μ. Branches shown with circles (in red or green) in the top panels denote dynamically unstable modes, which have emerged upon collision of modes with opposite Krein sign. The parameter values are α = 1.2 (left column), α = 0.6 (middle column), and α = 0.1 (right column); the trap strength is in all cases Ω = 0.1.
Same as Fig. 7, but for the second-excited state, corresponding to a discrete dark soliton pair, for parameter values μ = 0.5 (corresponding to the first instability band – see bottom middle panel of Fig. 8), α = 0.6, and Ω = 0.1.
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