# On Parametric Amplification In Discrete Josephson Transmission Line

Eugene Kogan  Department of Physics Bar-Ilan University Ramat-Gan, Israel
Donostia International Physics Center (DIPC) San Sebastian/Donostia, Spain
<EMAIL_ADDRESS>
###### Abstract

We consider the discrete series-connected lossy Josephson transmission line,
constructed from Josephson junctions, capacitors and resistors (one-
dimensional array of Josephson junctions). We derive equations describing
pump, signal, and idler interaction in the system and calculate the thresholds
for the parametric amplification.

###### Index Terms:

Josephson arrays, Josephson amplifiers, parametric amplifiers

##  I Introduction

Superconducting parametric amplifiers attract a lot of interest, due to their
importance in microwave electronics [1, 2, 3, 4]. Traditional amplifiers
comprise a single Josephson junction (JJ) or an array of junctions in a
resonant cavity which ultimately limits the bandwidth and dynamic range.

Recently, owing to impact of the kinetic-inductance traveling-wave parametric
amplifier, the Josephson traveling-wave parametric amplifiers enabling larger
gain per unit length with lesser pump power have been in the particular focus
of several research groups [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22].

We studied previously [23] the problem of parametric amplification in the
Josephson transmission line (JTL) in the continuum approximation. Now we
consider the problem for the discrete JTL.

##  II RSJ Josephson transmission line

We consider a model of the JTL presented in Fig. 1.

Figure 1: Josephson transmission line composed of superconducting grains

We take as the dynamical variables Josephson phases and the potentials of the
grains $V_{n}$. The circuit equations are [23]

$\displaystyle\frac{\hbar}{2e}\frac{d\varphi_{n}}{dt}=V_{n-1}-V_{n},$ (1a)
$\displaystyle
C\frac{dV_{n}}{dt}+GV_{n}=\partial^{2}_{n}\left(\frac{V_{n}}{R_{J}}+C_{J}\frac{dV_{n}}{dt}\right)$
$\displaystyle+I_{c}\left[\sin\left(\varphi_{n}+\varphi_{0}\right)-\sin\left(\varphi_{n+1}+\varphi_{0}\right)\right].$
(1b)

where $C$ is the ground capacitance, $I_{c}$ is the critical current of the
JJ, $G$ is the conductance of the ohmic resistor shunting the ground
capacitor, $R_{J}$ is the ohmic resistor shunting the JJ; the second-order
discrete differential operator $\partial^{2}_{n}$ is defined as
$\partial^{2}_{n}f_{n}\equiv f_{n-1}-2f_{n}+f_{n+1}$. For the sake of
generality we consider d.c. background Josephson current
$I_{c}\sin\left(\varphi_{0}\right)$ flowing along the JTL, where $\varphi_{0}$
is the d.c. Josephson phase.

Inclusion of the capacitance in parallel to the Josephson elements (inter-
island capacitance) would certainly make the model more physically realistic.
However it complicates the mathematics (there appears the fourth order
derivative in the final equation). In spite of this the parallel capacitance
was included in our previous paper dealing with parametric amplification in
the JTL in the continuum approximation [23]. It turned out that such
generalization of the model doesn’t lead to any qualitative differences. So in
the present paper we decided to give the priority to simplicity over
generality.

We can exclude $V_{n}$ and obtain closed equation for $\varphi_{n}$

$\displaystyle\frac{d^{2}\varphi_{n}}{d\tau^{2}}+Z_{J}G\frac{d\varphi_{n}}{d\tau}$
$\displaystyle=\partial^{2}_{n}\left[\sin\left(\varphi_{n}+\varphi_{0}\right)+\frac{Z_{J}}{R_{J}}\frac{d\varphi_{n}}{d\tau}+\frac{C_{J}}{C}\frac{d^{2}\varphi_{n}}{d\tau^{2}}\right],$
(2)

where we introduced the dimensionless time $\tau=t/\sqrt{L_{J}C}$,
$Z_{J}\equiv\sqrt{L_{J}/C}$ is the characteristic impedance of the JTL, and
$L_{J}=\hbar/(2eI_{c})$. Let us present (II) as

$\displaystyle\frac{d^{2}\varphi_{n}}{d\tau^{2}}-\partial_{n}^{2}\left(\cos\varphi_{0}\cdot\varphi_{n}+\frac{C_{J}}{C}\frac{d^{2}\varphi_{n}}{d\tau^{2}}\right)$
$\displaystyle=\partial_{n}^{2}\left\\{\frac{Z_{J}}{R_{J}}\frac{d\varphi_{n}}{d\tau}+n.l.[\sin\left(\varphi_{n}+\varphi_{0}\right)]\right\\}-Z_{J}G\frac{d\varphi_{n}}{d\tau},$
(3)

where n.l. stands for the non-linear terms of expansion of the sine function
in Taylor series.

If we ignore the dissipation and the non-linear terms, Eq. (II) has obvious
solution

$\displaystyle\varphi_{n}=\frac{1}{2}\sum_{\alpha}A_{\alpha}e^{i(k_{\alpha}n-\omega_{\alpha}\tau)}+c.c.,$
(4)

where

$\displaystyle\omega_{\alpha}^{2}=\frac{4\cos\varphi_{0}\sin^{2}\left(\frac{k_{\alpha}}{2}\right)}{1+\frac{4C_{J}}{C}\sin^{2}\left(\frac{k_{\alpha}}{2}\right)},$
(5)

and $A_{\alpha}$ are arbitrary constant amplitudes. The nonlinearity and the
dissipation we’ll take into account approximately, by changing (4) to

$\displaystyle\varphi_{n}=\frac{1}{2}\sum_{\alpha}A_{\alpha}(n)e^{i(k_{\alpha}n-\omega_{\alpha}\tau)}+c.c.$
(6)

and assuming that the complex amplitudes $A_{\alpha}(n)$ slowly change with
$n$. Additionally, because we consider the dissipation and nonlinear terms as
being in some sense small, while calculating the discrete second order
derivative of the terms in the parenthesis in (II) we will ignore the
corrections which come from the $n$-dependence of the amplitudes, thus the
discrete second derivative operator acting on a partial wave with the wave
vector $k$ just multiplies this partial wave by
$-4\sin^{2}\left(\frac{k}{2}\right)$. For example,

$\displaystyle\frac{d}{d\tau}\left(\varphi_{n-1}-2\varphi_{n}-\varphi_{n+1}\right)$
$\displaystyle=-2i\sum_{\alpha}\sin^{2}\left(\frac{k_{\alpha}}{2}\right)\omega_{\alpha}A_{\alpha}e^{i(k_{\alpha}n-\omega_{\alpha}\tau)}+c.c.,$
(7)

However, we’ll do better while calculating the discrete second derivative
standing in the l.h.s. of (II). We promote $n$, as argument of the amplitudes,
to the continuous variable $Z$ and approximate

$\displaystyle A_{\alpha}(n\pm 1)=A_{\alpha}(Z)\pm\frac{\partial
A_{\alpha}(Z)}{\partial Z}.$ (8)

As the result, for the discrete second derivative we obtain

$\displaystyle\varphi_{n-1}-2\varphi_{n}+\varphi_{n+1}$
$\displaystyle=-2\sum_{\alpha}\sin^{2}\left(\frac{k_{\alpha}}{2}\right)A_{\alpha}e^{i(\omega_{\alpha}\tau-
k_{\alpha}n)}+c.c.,$
$\displaystyle+i\sum_{\alpha}\sin(k_{\alpha})\frac{\partial
A_{\alpha}}{\partial Z}e^{i(k_{\alpha}n-\omega_{\alpha}\tau)}+c.c.$ (9)

##  III 3 waves mixing

###  III-A Coupled equations for the amplitudes

Now let us take into account the nonlinear terms. Consider the case
$\varphi_{0}\sim 1$ and $\left|\varphi_{n}\right|\ll 1$ Expanding the sine
function in the r.h.s. of (II) in Taylor series and keeping the first two
terms of the expansion we have

$\displaystyle
n.l.\left[\sin\left(\varphi_{n}+\varphi_{0}\right)\right]=-\frac{\sin\varphi_{0}}{2}\varphi_{n}^{2}.$
(10)

After we substitute (4) into (10), the general result for the quadratic term
would be too complicated, so we’ll consider a superposition of only three
waves (pump, signal, and idler), with the frequencies $\omega_{p}$,
$\omega_{s}$ and $\omega_{i}$ respectively, satisfying equation

$\displaystyle\omega_{s}+\omega_{i}=\omega_{p}.$ (11)

Keeping only the terms which have $\omega_{p,s,i}t$ time dependence we obtain

$\displaystyle\varphi_{n}^{2}=\frac{1}{2}A_{s}A_{i}e^{i[(k_{s}+k_{i})n-\omega_{p}t]}+\frac{1}{2}A_{p}A_{i}^{*}e^{i[(k_{p}-k_{i})n-\omega_{s}t]}$
$\displaystyle+\frac{1}{2}A_{p}A_{s}^{*}e^{i[(k_{p}-k_{s})n-\omega_{i}t]}+c.c.$
(12)

Finally, collective all the perturbative terms in (II) we obtain coupled
equations for the amplitudes

$\displaystyle\frac{dA_{p}}{dZ}+\nu_{p}A_{p}$
$\displaystyle=i\mu_{p}A_{s}A_{i}e^{i\Delta k\cdot Z}$ (13a)
$\displaystyle\frac{dA_{s,i}}{dZ}+\nu_{s}A_{s,i}$
$\displaystyle=i\mu_{s,i}A_{p}A_{i,s}^{*}e^{-i\Delta k\cdot Z},$ (13b)

where

$\displaystyle\nu_{\alpha}$
$\displaystyle=\frac{\tan(k_{\alpha}/2)}{\omega_{\alpha}}\left[Z_{J}G+4\sin^{2}\left(\frac{k_{\alpha}}{2}\right)\frac{Z_{J}}{R_{J}}\right]$
(14a) $\displaystyle\mu_{p}$
$\displaystyle=\frac{2\sin\varphi_{0}\cdot\tan(k_{p}/2)}{\omega_{p}^{2}}\sin^{2}\left(\frac{k_{s}+k_{i}}{2}\right)$
(14b) $\displaystyle\mu_{s,i}$
$\displaystyle=\frac{2\sin\varphi_{0}\cdot\tan(k_{s,i}/2)}{\omega_{s,i}^{2}}\sin^{2}\left(\frac{k_{p}-k_{i,s}}{2}\right)$
(14c) $\displaystyle\Delta k$ $\displaystyle=k_{s}+k_{i}-k_{p}.$ (14d)

Comparing Eqs. (13) with the appropriate equations from our previous
publication [23], we see that the basic equations systems in the discrete
consideration and in continuum approximation are the same. Only the parameters
of the systems are different (compare (14) with the appropriate equations from
our previous publication [23]).

###  III-B Weak signal: threshold for the parametric amplification

Let us solve Eq. (13) in the small (relative to the pump) signal and idler
approximation ($|A_{s}|,|A_{i}|\ll|A_{p}|$). In this approximation the
equation for $A_{p}$ becomes decoupled from the other two and takes the form

$\displaystyle\frac{dA_{p}}{dZ}+\nu_{p}A_{p}=0,$ (15)

with the obvious solution

$\displaystyle A_{p}(Z)=e^{-\nu_{p}Z}A_{p}^{0}.$ (16)

Treating the other two equations, we’ll introduce the local approximation, by
treating $A_{p}$ in the r.h.s. of Eqs. (13b) as a constant. In this
approximation the solutions for $A_{s},A_{i}$ are

$\displaystyle\left(\begin{array}[]{c}A_{s}\\\
A_{i}\end{array}\right)=e^{-i\Delta k/2\cdot Z}\left(\begin{array}[]{c}\dots
e^{KZ}\\\ \dots e^{K^{*}Z}\end{array}\right),$ (21)

where dots stand for some constant amplitudes, and $K$ is one of the roots of
the characteristic polynomial

$\displaystyle P(K)\equiv K^{2}+(\nu_{s}+\nu_{i})K$
$\displaystyle+(\nu_{s}-i\Delta k/2)(\nu_{i}+i\Delta
k/2)-\mu_{s}\mu_{i}\left|A_{p}\right|^{2}.$ (22)

Parametric amplification of the signal takes place when one of the roots have
positive real part. The boundary between the parametric amplification and no
parametric amplification we can find by demanding that $K$ for that boundary
is purely imaginary. So, after simple algebra, we obtain the condition for the
parametric amplification:

$\displaystyle\mu_{s}\mu_{i}\left|A_{p}\right|^{2}>\nu_{s}\nu_{p}\left[1+\frac{\left(\Delta
k\right)^{2}}{(\nu_{s}+\nu_{i})^{2}}\right].$ (23)

Looking at Eq. (23) we realise that the momenta mismatch $\Delta k$ acts in
some sense similar to the losses in the system. Both factors together define
the threshold for the parametric amplification.

Though it’s not completely consistent, we bring back the coordinate dependence
of $A_{p}$, given by Eq. (16), and consider (23) as a local condition for the
parametric amplification.

###  III-C Small wave vectors limiting case

Consider the limiting case $k_{p},k_{s},k_{i}\ll 1$. In this case $\Delta
k=0$. We assume additionally that $R_{J}=\infty$, hence $\nu_{\alpha}=\nu$
(doesn’t depend upon $\alpha$). Hence Eqs. (13a), (13b) take the form

$\displaystyle\frac{dA_{p}}{dZ}+\nu A_{p}$ $\displaystyle=i\mu_{p}A_{s}A_{i}$
(24a) $\displaystyle\frac{dA_{s,i}}{dZ}+\nu A_{s,i}$
$\displaystyle=i\mu_{s,i}A_{p}A_{i,s}^{*},$ (24b)

where

$\displaystyle\mu_{p}=\mu_{s}+\mu_{i}.$ (25)

We’ll present the amplitudes as

$\displaystyle A_{p}=ie^{-\nu Z}A_{p}^{(r)},\hskip 14.22636ptA_{s,i}=e^{-\nu
Z}A_{s,i}^{(r)}.$ (26)

In the new variables Eqs. (24a), (24b) take the form (compare with Ref. [24])

$\displaystyle\frac{dA_{p}^{(r)}}{d\tilde{Z}}$
$\displaystyle=\mu_{p}A_{s}^{(r)}A_{i}^{(r)}$ (27a)
$\displaystyle\frac{dA_{s,i}^{(r)}}{d\tilde{Z}}$
$\displaystyle=-\mu_{s,i}A_{p}^{(r)}A_{i,s}^{(r)},$ (27b)

where

$\displaystyle d\tilde{Z}\equiv e^{-\nu Z}dZ,$ (28)

that is

$\displaystyle\tilde{Z}=K-\frac{1}{\nu}e^{-\nu Z},$ (29)

where $K$ is an arbitrary constant.

We’ll consider only the real solutions, postponing the general analysis until
later. From (27a), (27b) follows

$\displaystyle\frac{1}{2}\frac{d}{d\tilde{Z}}\left[{A^{(r)}_{p}}^{2}+{A^{(r)}_{s}}^{2}+{A^{(r)}_{i}}^{2}\right]$
$\displaystyle=\left(\mu_{p}-\mu_{s}-\mu_{i}\right)A_{p}^{(r)}A_{s}^{(r)}A_{i}^{(r)}.$
(30)

Due to (25) the r.h.s. of (III-C) is equal to zero, hence

$\displaystyle{A^{(r)}_{p}}^{2}+{A^{(r)}_{s}}^{2}+{A^{(r)}_{i}}^{2}=\text{const},$
(31)

and we can look for a solution in the form (remember the spherical
coordinates)

$\displaystyle A_{p}^{(r)}$ $\displaystyle=B\cos\theta$ (32a) $\displaystyle
A_{s}^{(r)}$ $\displaystyle=B\sin\theta\cos\phi$ (32b) $\displaystyle
A_{i}^{(r)}$ $\displaystyle=B\sin\theta\sin\phi,$ (32c)

where $B$ is an arbitrary amplitude. For the new variables $\theta$ and $\phi$
we obtain equations

$\displaystyle\frac{d\theta}{d\tilde{Z}}$
$\displaystyle=-\frac{B}{2}\mu_{p}\sin\theta\sin 2\phi$ (33a)
$\displaystyle\frac{d\phi}{d\tilde{Z}}$
$\displaystyle=-\frac{B}{2}\cos\theta\left(\mu_{p}\cos
2\phi-\mu_{s}+\mu_{i}\right).$ (33b)

Dividing one of the equations to the other we obtain

$\displaystyle\frac{d\theta}{\tan\theta}=\frac{\mu_{p}\sin 2\phi
d\phi}{\mu_{p}\cos 2\phi-\mu_{s}+\mu_{i}}.$ (34)

Equation (34) can be easily integrated and we obtain

$\displaystyle\sin^{2}\theta=\frac{D}{\mu_{p}\cos 2\phi-\mu_{s}+\mu_{i}},$
(35)

where $D$ is the integration constant. The solution (35), plotted in the
variables $A_{\alpha}^{(r)}$, is graphically presented in Fig. 2. In the
variables $A_{\alpha}$ the sphere contracts uniformly.

Figure 2: The solution (35) plotted in the variables $A_{\alpha}^{(r)}$. We
have chosen $\mu_{i}/\mu_{s}=1.2$ and $\theta=.1$ for $\phi=0$.

Equation (35) is even simpler than it looks.Taking into account (31), it can
be written down as

$\displaystyle(\mu_{s}-\mu_{i}){A_{p}^{(r)}}^{2}+\mu_{p}\left({A_{s}^{(r)}}^{2}-{A_{i}^{(r)}}^{2}\right)=\text{const}.$
(36)

The latter equation follows immediately from (27a), (27b).

There also exists a singular solution of (33a), (33b), where the constant
$\phi$ is given by the equation

$\displaystyle\mu_{p}\cos 2\phi-\mu_{s}+\mu_{i}=0,$ (37)

and $\theta(\tilde{Z})$, found from (33a), is given by the equation

$\displaystyle\cos\theta=\tanh\left(\frac{B}{2}\mu_{p}\sin
2\phi\cdot\tilde{Z}\right).$ (38)

Equation (38) shows that when $\tilde{Z}\to\infty$, either $\theta\to 0$ or
$\theta\to\pi$, depending upon the sign of $\sin 2\phi$.

However, we are mostly interested not in the dependence of one angle variable
upon the other, but in their dependence upon $\tilde{Z}$. This dependence can
be found by numerical integration of Eqs. (33a), (33b) and is illustrated in
Fig. 3. The left side of the Figure, with $\theta$ increasing from nearly 0,
to $\pi/2$, demonstrates the parametric amplification (provided the losses are
not too big).

Figure 3: Numerical solution of Eqs. (33a), (33b) for $\theta$ (blue solid
line) and $\varphi$ (red dashed line). We have chosen $\mu_{s}=.2$,
$\mu_{i}=.24$, $\theta(0)=.1$ and $\varphi(0)=0$ (no idle signal at
$\tilde{Z}=0$).

##  IV 4 waves mixing

###  IV-A Coupled equations for the amplitudes

Now let us consider the case $\varphi_{0}=0$. In this case, expanding the sine
function in the r.h.s. of (II) in Taylor series with respect to phases
difference and keeping the first two nonzero terms of the expansion we have

$\displaystyle
n.l.\left[\sin\left(\varphi_{n}\right)\right]=-\frac{1}{6}\varphi_{n}^{3}.$
(39)

We again consider the superposition of three waves, but the frequencies this
time satisfy the relation

$\displaystyle\omega_{s}+\omega_{i}=2\omega_{p}.$ (40)

Keeping only the terms which have $\omega_{p,s,i}t$ time dependence we obtain

$\displaystyle\varphi_{n}^{3}=\frac{3}{8}\sum_{\alpha,\beta}\left(2-\delta_{\alpha\beta}\right)A_{\alpha}\left|A_{\beta}\right|^{2}e^{i[k_{\alpha}n-\omega_{\alpha}t]}$
$\displaystyle+\frac{3}{4}A_{p}^{*}A_{s}A_{i}e^{i[(k_{s}+k_{i}-k_{p})n-\omega_{p}t]}+\frac{3}{8}A_{p}^{2}A_{i}^{*}e^{i[(2k_{p}-k_{i})n-\omega_{s}t]}$
$\displaystyle+\frac{3}{8}A_{p}^{2}A_{s}^{*}e^{i[(2k_{p}-k_{s})n-\omega_{i}t]}+c.c..$
(41)

Finally, collective all the perturbative terms in (II) we obtain coupled
equations for the amplitudes

$\displaystyle\frac{dA_{p}}{dZ}+\nu_{p}A_{p}$
$\displaystyle=i\gamma_{p}\left(|A_{p}|^{2}+2|A_{s}|^{2}+2|A_{i}|^{2}\right)A_{p}$
$\displaystyle+2i\eta_{p}A_{p}^{*}A_{s}A_{i}e^{i\Delta k\cdot Z}$ (42a)
$\displaystyle\frac{dA_{s,i}}{dZ}+\nu_{s,i}A_{s,i}$
$\displaystyle=i\gamma_{s,i}\left(2|A_{p}|^{2}+|A_{s,i}|^{2}+2|A_{i,s}|^{2}\right)A_{s,i}$
$\displaystyle+i\eta_{s,i}A_{p}^{2}A_{i,s}^{*}e^{-i\Delta k\cdot Z},$ (42b)

where $\nu_{\alpha}$ is given by (14a) with $\cos\varphi_{0}$ being put equal
to 1, and

$\displaystyle\gamma_{\alpha}$
$\displaystyle=\frac{\tan(k_{\alpha}/2)}{2\omega_{\alpha}^{2}}\sin^{2}\left(\frac{k_{\alpha}}{2}\right)$
(43a) $\displaystyle\eta_{p}$
$\displaystyle=\frac{\tan(k_{p}/2)}{2\omega_{p}^{2}}\sin^{2}\left(\frac{k_{s}+k_{i}-k_{p}}{2}\right)$
(43b) $\displaystyle\eta_{s,i}$
$\displaystyle=\frac{\tan(k_{s,i}/2)}{2\omega_{p}^{2}}\sin^{2}\left(\frac{2k_{p}-k_{i,s}}{2}\right)$
(43c) $\displaystyle\Delta k$ $\displaystyle=k_{s}+k_{i}-2k_{p}$ (43d)

Comparing Eqs. (42) with the appropriate equations from our previous
publication [23] we can only repeat what was said in the very end of Section
III-A.

###  IV-B Weak signal

In the small (relative to the pump) signal and idler approximation
($|A_{s}|,|A_{i}|\ll|A_{p}|$), the equation for $A_{p}$ becomes decoupled from
the other two and takes the form

$\displaystyle\frac{dA_{p}}{dZ}+\nu_{p}A_{p}=\frac{i}{2}\gamma_{p}|A_{p}|^{2}A_{p}.$
(44)

It is interesting to compare the result following from equation (44) with the
nonlinear dispersion law obtained in our previous publication [25]. There we
considered equation

$\displaystyle\frac{d^{2}\varphi_{n}}{d\tau^{2}}=\sin\varphi_{n-1}-2\sin\varphi_{n}+\sin\varphi_{n+1}$
(45)

and obtained in the up to the second order with respect to the amplitude
approximation the solution

$\displaystyle\varphi_{n}(\tau)=a\cos[kn-\omega(k)\tau]$ (46)

with the dispersion law

$\displaystyle\omega(k)=2\left(1-\frac{a^{2}}{16}\right)\sin\left(\frac{k}{2}\right).$
(47)

In the absence of dissipation, the solution of (44) can be taken as

$\displaystyle A_{p}(Z)=ae^{i\gamma_{p}a^{2}/2Z},$ (48)

where $a$ is a real constant. Substituting this solution into (6) and taking
into account (43a) we obtain

$\displaystyle\varphi_{n}(\tau)=a\cos[(k+a^{2}\tan(k/2)/8)n-\omega\tau],$ (49)

where $\omega=2\sin(k/2)$. let us introduce

$\displaystyle k^{\prime}=k+a^{2}\tan(k/2)/8.$ (50)

Then up to the lowest order with respect to $a^{2}$ corrections

$\displaystyle k=k^{\prime}-a^{2}\tan(k^{\prime}/2)/8,$ (51)

and the dispersion law can be presented as

$\displaystyle\omega=2\sin[k^{\prime}/2-a^{2}\tan(k^{\prime}/2)/16].$ (52)

Expanding (52) in series with respect to $a^{2}$ and keeping the terms up to
the linear order with respect to $a^{2}$ we obtain

$\displaystyle\omega(k^{\prime})=2\left(1-\frac{a^{2}}{16}\right)\sin\left(\frac{k^{\prime}}{2}\right).$
(53)

which coincides with Eq. (47).

Note that in our previous publications [25] the pump wave hire harmonics,
which exist simultaneously with the main one, due to nonlinearity of the
system, are also considered. However there influence on the signal amplitude
would be an effect of the higher order with respect to the parameter
$A_{p}^{2}$, than the influence of the main harmonic.

###  IV-C Threshold for the parametric amplification

Equations (42b) in the weak signal approximation can be written down as

$\displaystyle\frac{dA_{s,i}}{dZ}+\left(\nu_{s,i}-i\gamma_{s,i}|A_{p}|^{2}\right)A_{s,i}=i\eta_{s,i}A_{p}^{2}A_{i,s}^{*}e^{-i\Delta
k\cdot Z}.$ (54)

In the framework of the local approximation, we treat $A_{p}$ in the r.h.s. of
Eqs. (54) as a constant, so we recover (21), where $K$ this time is one of the
roots of the characteristic polynomial

$\displaystyle P(K)\equiv
K^{2}+(\nu_{s}+\nu_{i})K+(\nu_{s}-2i\gamma_{s}|A_{p}|^{2}-i\Delta k/2)$
$\displaystyle\cdot(\nu_{i}+2i\gamma_{i}|A_{p}|^{2}+i\Delta
k/2)-\eta_{s}\eta_{i}\left|A_{p}\right|^{4}.$ (55)

We again find the boundary between the parametric amplification and no
amplification by demanding that $K$ for that boundary is purely imaginary. So,
after simple algebra, we obtain the condition for the parametric
amplification:

$\displaystyle\eta_{s}\eta_{i}\left|A_{p}\right|^{4}>\nu_{s}\nu_{i}\left\\{1+\frac{\left[\Delta
k+2(\gamma_{s}+\gamma_{i})\left|A_{p}\right|^{2}\right]^{2}}{(\nu_{s}+\nu_{i})^{2}}\right\\}.$
(56)

Here we can only repeat what was said after Eq. (23). Again, we bring back the
coordinate dependence of $A_{p}$, given by Eq. (44) and consider (56) as a
local condition for the parametric amplification.

##  V Discussion

The present paper generalized the previously obtained results on parametric
amplification in JTL [5, 11, 25] to the discrete model of the line. The
discrete character of the line leads to the renormalization of the parameters
of the coupled amplitude equations.

We formulate the local approximation which allows explicitly find the
threshold for parametric amplification in the case of weak signal. In addition
we present the exact particular solution for the coupled equations for the
pump, signal and idler amplitudes in the particular case of small wave vectors
and present the solution graphically. Finally, we would like to mention that
in real case of the line of finite length, the frequency dependence of the
line impedance leads to matching problems and hence to the wave reflection
(e.g., [26]). However we leave the study of the problem until later.

## Acknowledgment

I am grateful to T. H. A. van der Reep for the discussion. I am also very
grateful to Donostia International Physics Center (DIPC) for the hospitality
during my visit, when the paper was finalised.

## References

  * [1] M. A. Castellanos-Beltran and K. W. Lehnert, Appl. Phys. Lett. 91, 083509 (2007).
  * [2] P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, Rev. Mod. Phys. 84, 1 (2012).
  * [3] B. H. Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas, Nat. Phys. 8, 623 (2012).
  * [4] J. Aumentado, IEEE Microwave magazine, 21, 45 (2020).
  * [5] O. Yaakobi, L. Friedland, C. Macklin, and I. Siddiqi, Phys. Rev. B 87, 144301 (2013).
  * [6] K. O’Brien, C. Macklin, I. Siddiqi, and X. Zhang, Phys. Rev. Lett. 113, 157001 (2014).
  * [7] C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, Science 350, 307 (2015).
  * [8] B. A. Kochetov, and A. Fedorov, Phys. Rev. B. 92, 224304 (2015).
  * [9] M.T. Bell and A. Samolov, Phys. Rev. Applied 4, 024014 (2015).
  * [10] T. C. White et al., Appl. Phys. Lett. 106, 242601 (2015).
  * [11] A. B. Zorin, Phys. Rev. Applied 6, 034006 (2016).
  * [12] L. Fasolo, A. Greco, E. Enrico, J. Thirumalai, and S. I. Pokutnyi, Adv. Condensed - Matter Materials Physics - Rudimentary Res. Topical Technol. IntechOpen, 2019.
  * [13] D. M. Basko, F. Pfeiffer, P. Adamus, M. Holzmann, and F. W. J. Hekking, Phys. Rev. B 101, 024518 (2020).
  * [14] T. Dixon, J. W. Dunstan, G. B. Long, J. M. Williams, Ph. J. Meeson, C. D. Shelly, Phys. Rev. Applied 14, 034058 (2020)
  * [15] A. B. Zorin, Phys. Rev. Applied 12, 044051 (2019).
  * [16] A. Miano and O. A. Mukhanov, IEEE Trans. Appl. Supercond. 29, 1501706 (2019).
  * [17] T. H. A. van der Reep, Phys. Rev. A 99, 063838 (2019).
  * [18] Ch. Liu, Tzu-Chiao Chien, M. Hatridge, D. Pekker, Phys. Rev. A 101, 042323 (2020).
  * [19] A. Greco, L. Fasolo, A. Meda, L. Callegaro,and E. Enrico, Phys. Rev. B, 104, 184517 (2021).
  * [20] Y. Yuan, M. Haider, J. A. Russer, P. Russer, and Ch. Jirauschek. In European Quantum Electronics Conference, 22, Optical Society of America (2021).
  * [21] C. Kow, V. Podolskiy, and A. Kamal, arXiv:2201.04660.
  * [22] H. Katayama, N. Hatakenaka, T. Fujii, and M. P. Blencowe, Phys. Rev. Res. 5, L022055 (2023).
  * [23] E. Kogan, Phys. Stat. Sol. (b), Vol. 260, p. 2300005, 2023.
  * [24] A. L. Cullen, Proceedings of the IEE-Part B: Electronic and Communication Engineering 107, 101 (1960).
  * [25] E. Kogan, Phys. Stat. Sol. (b), Vol. 259, p. 2200160, 2022.
  * [26] S. Kern, P. Neilinger, E. Il’ichev, et al., arXiv: 2203.02448v2 (2023).