Theory of coherent phase modes in insulating Josephson junction chains

Recent microwave reflection measurements of Josephson junction chains have suggested the presence of nearly coherent collective charge oscillations deep in the insulating phase. Here we develop a qualitative understanding of such coherent charge modes by studying the local dynamical polarizability of the insulating phase of a finite length sine-Gordon model. By considering parameters near the noninteracting fermion limit where the charge operator dominantly couples to soliton-antisoliton pairs of the sine-Gordon model, we find that the local dynamical polarizability shows an array of sharp peaks in frequency representing coherent phase oscillations on top of an incoherent background. The strength of the coherent peaks relative to the incoherent background increases as a power law in frequency as well as exponentially as the Luttinger parameter approaches a critical value. The dynamical polarizability also clearly shows the insulating gap. We then compare the results in the high frequency limit to a perturbative estimate of phase-slip-induced decay of plasmons in the Josephson junction chain.

Figure 1

Absorption of a weakly coupled transmission probe on the left measures the dynamical conductivity ${\sigma }_R\left(\omega \right)=\omega \chi \left(\omega \right)$ at the end of a JJ chain. The JJ chain is composed of an array of islands with ground capacitance $C_g$ coupled by JJs. The JJs have a capacitance $C_J$ in addition to a Josephson coupling with strength $E_J$.

Figure 2

(a) Dispersion ($\varepsilon$ versus $k$) of excitations of the sine-Gordon model with antisolitons at negative energy. The bold arrow shows charge neutral soliton-antisolition pairs with momentum $q$ and energy $\omega$. Large $q$ pairs have a large near degeneracy with $\omega \simeq q$ constituting a coherent excitation peak. (b) Local dynamical polarizability $\chi$ of a sine-Gordon model as a function of $\omega$ shows coherent peaks (see inset) spaced by finite size $1/L$ dominating over the incoherent background as frequency increases.

Figure 3

Luttinger parameter $K$ dependence of resistance oscillations. Change in Luttinger parameter decreases the mass gap $m$ and enhances the oscillation for $K=1.4$ (red dashed line). The solid blue curve shows suppressed oscillations at $K=1$ since the frequency is of order mass.

Figure 4

Local dynamical polarizability $\chi$ of the disordered massive Dirac model as a function of $\omega$ shows coherent peaks at high frequencies and smaller random absorption peaks at low frequencies. Inset shows that the peak shape in the disordered system, which is more symmetric compared to that in the clean system shown in the inset in Fig. 3.

Figure 5

Inverse lifetime of single plasmon states scaled by $g^2/E_J$. The system parameters are $M=1600, D=0.02, E_1/E_J=0.1$ while $E_0$ is changed. Power-law increase of the decay rate at smaller wave vectors (i.e., $q\to 0$) is consistent with weaker peaks at smaller frequencies obtained from the sine-Gordon model. Moreover, the decay rate is suppressed as $E_0$ decreases, which is consistent with the experiment [19].

Figure 6

(a) Phase profile ${\varphi }_n$ for the microscopic JJ chain model. The phase-slip phase profile is discontinuous but local in the sense that it vanishes away from the phase slip. (b) The discontinuity can be removed at the expense of introducing nonlocality to produce an unwound phase slip.