Theory of small charge solitons in one-dimensional arrays of Josephson junctions

We identify and investigate the parameter regime of small charge solitons in one-dimensional arrays of Josephson junctions. We obtain the dispersion relation of the soliton and show that it unexpectedly flattens in the outer region of the Brillouin zone. We demonstrate Lorentz contraction of the soliton in the middle of the Brillouin zone as well as broadening of the soliton in the flat band regime.

Figure 1

Josephson-junction array.

Figure 2

Effective lattice and effective unit cell for the configurations $|R⟩$ and $|R;1,1⟩$. The dashed box marks a primitive unit cell. The lines denote allowed tunneling between the configurations.

Figure 3

(Color online) Dispersion relation for the one-state approximation, i.e., a single Cooper pair (dashed band), for the two-state approximation (dotted bands), and for the four-state approximation (solid bands) as described in the text. We chose $E_C=20E_J$, $\Lambda =10$.

Figure 4

Band structure for $w_c=7$ with parameters $\Lambda =10$ and (a) $E_C=10E_J$, (b) $E_C=5E_J$, and (c) $E_C=2.5E_J$. For clarity, only the 16 lower bands are shown.

Figure 5

(Color online) Charge-charge correlator $⟨{\psi }_k|n_r{n}_{r^{\prime }}|{\psi }_k⟩$ for $E_C=10E_J$, $5E_J$, and $2.5E_J$. In all plots, $\Lambda =10$.

Figure 6

(Color online) Quadrupole moment of the charge-charge correlator $⟨{\psi }_k|n_r{n}_{r^{\prime }}|{\psi }_k⟩$ for $E_C=10E_J$, $5E_J$, and $2.5E_J$. For $\left|k\right|<\pi /4$, the Lorentz contraction can be observed.