Spectral properties of the ladder-like Josephson junction array

Josephson plasmons are elecromagnetic waves that propagate along the array in the $X$ direction. At this point we remind the Josephson plasmon dispersion law for the $N$-row JJA obtained in [20]. This dispesion law consists is derived from the discrete system of the sine-Gordon equations that are linearized around the groundstate (4). It consists of $2N-1$ branches:

In Eq. (6) the sign of the superscript in $\omega_{\pm}$ is defined only by the sign near the radical and not by the superscripts in $W^{\pm}$. The function $\omega_{i}(q)$ is the plasmon dispersion law of the simple dc biased parallel JJA with only one row of vertical junctions [27]:

The coefficients $\alpha_{n}$ always lie in the interval $]-1,3[$. For $N=2$ rows there is only one coefficient $\alpha_{2}=1$, for $N=3$ there are two coefficients, $\alpha_{2}=0$, $\alpha_{3}=2$, for $N=4$ $\alpha_{n}=1-2\cos\frac{\pi(n-1)}{4}\Rightarrow$ $\alpha_{2}=1-\sqrt{2}$, $\alpha_{3}=1$, $\alpha_{4}=1+\sqrt{2}$ and so on.

The Josephson plasmon spectrum consists of $2N-1$ branches. Within the enumeration used in Eqs. (5)-(18) these branches are positioned as follows: (i) highly dispersive branches $1<\omega_{1}<\omega_{2}<\cdots\omega_{N-1}$; (ii) flat band with the Jopsehson plasma frequency $\omega_{0}=1$; (iii) almost flat branches $\omega_{-N+1}<\cdots<\omega_{-2}<\omega_{-1}<1$. Thus, the highly dispersive branches are numbered by the positive superscripts, lie above the flat band and look similar to the dispersion law of the simple parallel array $\omega_{i}(q)$ but are shifted from each other. In the unbiased case $\gamma=0$ all the $\omega_{-|n|}$ branches become flat: $\omega_{-N+1}(q)=\omega_{-N+2}(q)=\cdots=\omega_{-2}(q)=\omega_{-1}(q)=\omega_{0}=1$. Otherwise they form a very narrow subband that is confined in the following frequency intherval: $(1-\gamma^{2})^{1/4}\leq\omega_{-n}<\omega_{-n}(\pi)<1$. Even for rather high values of $\gamma$ this interval will be much smaller than $1$.

Group velocity characterizes the speed of the plasmon wavepackage propagation. By looking at it one can have an idea how different Josephson plasmon modes transport energy. The group velocity for each of the $\omega_{n}(q)$ modes of (5)-(6) has the following form:

It is obviously zero for the flat branch $\omega_{0}=1$, similarly $v_{gr,-n}=0$ for the almost flat branches if the array is not biased, $\gamma=0$. If $\gamma\neq 0$, the respective velocities for the $n\neq 0$ modes are shown in Fig. 2. There are three curves for each of the subbands, for the strongly dispersive branches $\omega_{1,2,3}$ and for the almost flat branches $\omega_{-1,-2,-3}$.

Since $\omega_{1}<\omega_{2}<\omega_{3}$, the same is true for the group velocities: $v_{gr,1}<v_{gr,2}<v_{gr,3}$. The group velocity for the strongly dispersive modes, $v_{gr,|n|}$, does not deviate strongly from the $\sin q$ dependence. The maximal values of $v_{gr,|n|}(q)$ are positioned near $q=\pi/2$. The situation is more complex for the almost flat modes. The dispersion law for these modes is parabolic near the origin $q=0$ but then reaches a plateau. The inflexion point of the $\omega_{-|n|}(q)$ dispersion laws which is the maximum of the group velocity shifts closer to the origin as $\eta$ increases. It should be noted that the following is true for the almost flat modes only close to the origin: $v_{gr,-1}>v_{gr,-2}>v_{gr,-3}$. Away from the maximum these velocities begin to cross. The group velocity for the highest mode $\omega_{-1}$ is the sharpest and its maximum is the closest to the origin. Thus, we conclude that the group velocity for the almost flat modes is at least one order of magnitude smaller than for the strongly dispersive modes. However, the almost flat modes, especially the mode $\omega_{-1}$, exhibit a sharp peak not far from the $q=0$ point.

The density of states (DoS) in various condensed matter systems contains useful information that can be accessed experimentally. We compute the Josephson plasmon density of states (DoS) per unit cell as a function of the frequency from the dispersion laws (5)-(6). The general expression for the plasmon DoS reads

The sum runs over the all non-flat bands $n\neq 0$, and the flat branch contribution is represented by the Diracs $\delta$-function term $\delta(\omega-1)$. The Heaviside $\theta$-function is used to account for the overlaping branches explicitly. This point will be discussed below.

Before discussing the plasmon DoS for the general array with an arbitrary number of rows let us mention the simplest case of the JJ array of parallelly shunted junctions. The dispersion law for the plasmons, $\omega_{i}(q)$ is given by equation (9). The respective density of states reads

It it not hard to notice that this function has two singularities at the edges of the plasmonic band: $\omega_{i}(0)=(1-\gamma)^{1/4}$ and $\omega_{i}(\pi)=\sqrt{\sqrt{1-\gamma^{2}}+4/\beta_{L}}$. There is a minimum somewhere between these two singularities.

By extracting from the dispersion law (6) the dependence of the wavenumber $q$ on the frequency $\omega$ it is possible to write down an exact expression for the plasmon density of states:

where the auxilliary functions are given by

and the $\mbox{sign}(n)$ is used in order to take into account the $n>0$ or $n<0$ branches. The quantity $\omega_{|n|}^{2}(0)=[1+(1+\alpha_{|n|+1})/(\eta\beta_{L})]$ is the minimum frequecy for the $n$th strongly dispersive branch.

Before analyzing the plasmon DoS given by Eqs. (14) it is useful to discuss the behaviour of the edge frequency values $\omega_{n}(0)$ and $\omega_{n}(\pi)$ of the strongly dispersive modes ($n>0$). They are shown in Fig. 3 as a function of the inverse anisotropy parameter $1/\eta$.

In the limit $1/\eta\to\infty$ ($\eta\to 0$) the branches are well separated and there are gaps between them. We remind that this limit corresponds to the situation when most of the plasma oscillations takes place in the horizontal subsystem. As $\eta$ increases the gaps start to close and the dispersion curve begin to overlap.

Being equipped with this knowledge we can analyze the plasmon DoS plots presented in Figs. 4-5. In those figures the DoS is plotted as a function of the plasmon frequency for different values of $\eta$. The situation if Fig. 4(a) corresponds to the non-overlaping branches. For $\omega>1$ one can observe three well separated smooth curves with two van Hove singularities each that correspond to the edge frequency values. As the anisotropy is increased from $\eta=0.25$ to $\eta=0.5$ the separate branches begin to overlap [see Fig. 4(b)]. The gaps that were visible in the previous figure now are closed. The singularities remain but now $\omega_{2}(0)<\omega_{1}(\pi)$ and $\omega_{3}(0)<\omega_{2}(\pi)$.

Further increasing of $\eta$ causes the singularities at $\omega=\omega_{1,2,3}(0)$ to move further to the left, while the singularities at $\omega=\omega_{1,2,3}(\pi)$ move to the right. The limit $\eta\gg 1$ corresponds to the situation when the amplitude of the oscillations in the vertical subsystem is much larger then in the horizontal. In this limit the dispersion curves are nested inside each other rather deeply and the DoS singularities concentrate at the respective edges of the band [compare Figs. 4(c) and (d)].

We have not discussed yet the properties of the plasmon DoS in the frequency region $\omega<1$. At $\omega=1$ we have a perfect flat band and there is a solid vertical line in Fig. 4 to demonstrate it. The almost flat branches $\omega_{-|n|}(q)$ are densely squeezed in the interval $\omega_{-|n|}(0)\equiv(1-\gamma^{2})^{1/4}\leq\omega_{-|n|}(q)\leq\omega_{-|n|}(\pi)<1$. Even for rather large bias values this interval is very narrow. For example, $(1-\gamma^{2})^{1/4}\approx 0.931$ for $\gamma=0.5$. The point $q=0$ is $(N-1)$-fold degenerate. For the Brillouin zone edge we have $\omega_{-3}<\omega_{-2}<\omega_{-1}$. The full expressions for $\omega_{-|n|}(\pi)$ is rather complicated and, therefore, we omit it. So, the dispersion curves are completely nested inside each other. As a result, we always have $N$ van Hove singularities, one at $\omega_{-|n|}(0)$ and $N-1$ singularities at $\omega_{-|n|}(\pi)$. When $\eta$ is increased the singularities at $\omega_{-|n|}(\pi)$ concentrate on the right side of the DoS dependence.

In figure 5 the dependence of the plasmon DoS on the discreteness parameter $\beta_{L}$ is demonstrated.

There are $N=6$ rows, therefore we observe $3N-2=16$ van Hove singularities. We observe that the change of $\beta_{L}$ does not bring any qualitative difference. The discreteness parameter influences the total width of the upper ($\omega>1$) plasmon band. If $\beta_{L}$ increases, the discreteness of the array increases and band narrows; it widens otherwise. Since the width of the lower ($\omega<1$, almost flat) subband depends mainly on the bias current, no visible changes are seen there when $\beta_{L}$ is varied. As a final remark we note that the minimal value of DoS is several orders of magnitude larger for the almost flat branches as compared to the strongly dispersive branches.

The spatial structure of the plasmon modes that correspond to each of the branches $\omega_{n}(q)$ is given by the matrix equation

that appears directly from the equations of motion [20]. Here the vectors $\mathbf{A}^{(v,h)}$ are amplitudes of the plasmon waves, $\mathbb{I}_{N}$ is $N\times N$ unit matrix, $\hat{D}_{h}$ is a tridiagonal $N\times N$ matrix

and $\hat{S}$ is a $(N-1)\times N$ bidiagonal matrix with $S_{11}=S_{22}=\cdots=S_{N-1,N-1}=1$, $S_{12}=S_{23}=\cdots=S_{N-1,N}=-1$ and $S_{ij}=0$ elsewhere. We can introduce a normalized real $(2N-1)$-component eigenvector

that consists of $N-1$ components for the vertical junctions and $N$ components for the horizontal junctions.

In the long-wave limit ($q=0$) the horizontal and vertical subsystems decouple and the respective eigenvectors are easy to find. Previously only qualitative description of the eigenvectors [20] has been given. In this paper the complete set of solutions will be presented.