# Quantum simulation in hybrid transmission lines

Alessandro Ferreri  ID Institute for Quantum Computing Analytics (PGI-12),
Forschungszentrum Jülich, 52425 Jülich, Germany  Frank K. Wilhelm  ID
Institute for Quantum Computing Analytics (PGI-12), Forschungszentrum Jülich,
52425 Jülich, Germany

###### Abstract

Platforms based on transmission lines are nowadays employed for the simulation
of standard phenomena in quantum electrodynamics and quantum field theory. In
this work, we propose a hybrid platform, in which a right-handed transmission
line is connected to a left-handed transmission line by means of a
superconducting quantum interference device (SQUID). We examine the
interaction between the two transmission lines, as well as the excitation flow
along the composed platform. We show that, by activating specific resonance
conditions, this platform can be used as a quantum simulator of different
phenomena in quantum optics, multimode quantum systems and quantum
thermodynamics.

##  I Introduction

Quantum simulation occupies a preeminent role in the development of state-of-
the-art quantum technologies for the study of complex quantum phenomena which
could not be investigated via direct observations [1, 2, 3, 4, 5, 6, 7, 8, 9].
Nowadays, we can account for a vast range of quantum systems that can be
employed as quantum simulator. Among these systems, particularly successful
platforms for implementations in quantum simulation are the superconducting
circuits [10, 11, 12]. Depending on the arrangement of the circuit elements,
these devices can simulate e.g. two-level systems (qubits), and quantum
harmonic oscillators, and for this reason, superconducting circuits find
application as quantum simulators of a large number of physical scenarios in
quantum electrodynamics (QED) [13, 14], atomic physics and quantum optics [2,
15].

Superconducting electrical circuits are particularly effective for the
investigation of multimode quantum systems, such as quantum fields or many-
body systems [16]. Typical examples are transmission lines (TLs), which well
describe (1+1)-dimensional quantum fields either in free space or confined in
a resonator [17]. In the last two decades, the solid analogy between TLs and
quantum scalar fields, as well as the possibility to externally drive the
magnetic flux by means of superconducting quantum interference devices
(SQUID), was of great support for the investigation of quantum relativistic
phenomena, with particular focus on those stemming from the stimulation of the
quantum vacuum [18]. For instance, effects of particle creation have been
predicted in diverse waveguidelike TLs [19, 20, 21]; well-known examples
thereof are the dynamical Casimir effect [22, 23, 24] (then observed
experimentally [3, 4]), and the Hawking radiation [25, 26, 27].

Continuous transmission lines are normally right-handed, meaning that the
dielectric constant and the magnetic permeability are both always positive.
Due to the right-handedness, the dispersion relation of such TLs enables the
promotion of the discrete circuit nodes to the continuous limit [13]. However,
almost seventy years ago Vaselago proposed left-handed media [28], which are
characterized by negative dielectric constant and magnetic permeability;
moreover, the wave vector points to the opposite direction with respect to the
Poynting vector [29]. As a consequence of the left-handedness, in a limited
range of frequencies such metamaterial also respond to the electromagnetic
field with a negative refractive index [30, 31]. In circuit QED, circuit
platforms showing left-handed features in the dispersion relation are called
“left-handed transmission lines (LHTLs)” [32, 33].

In this work, we present a hybrid platform realized by joining a left-handed
and a right-handed transmission line (RHTL) [34, 35, 36, 37]. To join the two
TLs, we make use of a SQUID. We place this device on the edge between the two
TLs, such that it can directly couple the magnetic flux stemming from both
TLs. We examine the energy conservation at the SQUID, showing that the
interaction between the two TLs leads to a energy shift term altering the
frequency matching between left-handed and right-handed modes. Importantly, by
tuning the Josephson energy of the SQUID we can externally drive the dynamics
of the composed transmission line (CTL). We observe that, by properly
manipulating the Josephson energy, we can simulate a large spectrum of
phenomena described by a linear dynamics (e.g. quantum optics, including
nonlinear optics phenomena up to the second order susceptibility). Finally, we
propose some applications in quantum thermodynamics.

Figure 1: Schematic representation of the hybrid transmission line. It
consists of a left-handed and a right-handed transmission line, each of N
loops, connected by a SQUID in serie.

The paper is structured as follows: in Sec. II we present the CTL and discuss
the differences between LHTL and RHTL in terms of dispersion relations and
eigenenergies of the Hamiltonian. In Sec. III we address the problem of the
energy conservation at the SQUID, and examine the possibility to propagate
quanta of magnetic flux fields from one TL to the other one. In Sec. IV we
show that the interaction Hamiltonian can display different types of two-mode
couplings depending on the activated resonance condition between the Josephson
energy and the modes of both the LTHL and the RHTL. We also study the
correlation between these modes. In Sec. VI we present different scenarios
wherein our CTL can find application as quantum simulator and quantum thermal
machine. We conclude in Sec. VII. Starting from basic circuit equations, in
appendix A we mathematically achieve the Hamiltonian from the Lagrangian using
the magnetic fluxes as scalar fields, and analyze the single terms of the
Hamiltonian of interaction.

##  II Theoretical background

The hybrid TL is pictorially represented in Fig.1. It is composed of two
different platforms, namely a LHTL and a RHTL, connected together by one SQUID
at the edge in between. We assume that the CTL consists of $2N$ identical unit
cells, where $N$ is the number of discrete unit cells equal in both the LHTL
and RHTL. Each unit cell is characterized by the length unit $\Delta x$,
determining the minimum wave length for the transmission lines,
$\lambda=2\Delta x$, and confining the wave vector within the first Brillouin
zone [35]. The presence of the SQUID with the time-dependent Josephson energy
ensures the interaction between the two TLs; furthermore, we will see that it
allows to impose the energy conservation for selected modes of the CTL.

The Lagrangian of the hybrid transmission line consists of three parts, two
thereof describe the left-handed and the right-handed component individually,
whereas the last term is responsible for their interaction:

$\displaystyle\mathcal{L}=\mathcal{L}_{\textrm{L}}+\mathcal{L}_{\textrm{R}}+\mathcal{L}_{\textrm{I}}$
(1)

where

$\displaystyle\mathcal{L}_{\textrm{L}}=$
$\displaystyle\frac{1}{2}\sum_{n_{\textrm{l}}=1}^{N}\left[C_{\textrm{l}}(\dot{\Phi}_{n_{\textrm{l}}+1}^{\textrm{L}}-\dot{\Phi}_{n_{\textrm{l}}}^{\textrm{L}})^{2}-\frac{(\Phi_{n_{\textrm{l}}}^{\textrm{L}})^{2}}{L_{\textrm{l}}}\right],$
(2a) $\displaystyle\mathcal{L}_{\textrm{R}}=$
$\displaystyle\frac{1}{2}\sum_{n_{\textrm{r}}=1}^{N}\left[C_{\textrm{r}}(\dot{\Phi}_{n_{\textrm{r}}}^{\textrm{R}})^{2}-\frac{(\Phi_{n_{\textrm{r}}+1}^{\textrm{R}}-\Phi_{n_{\textrm{r}}}^{\textrm{R}})^{2}}{L_{\textrm{r}}}\right],$
(2b) $\displaystyle\mathcal{L}_{\textrm{I}}=$ $\displaystyle
E(t)\cos\left[\frac{2\pi}{\phi_{0}}\left(\Phi_{1}^{\textrm{L}}-\Phi_{1}^{\textrm{R}}\right)\right].$
(2c)

In the expressions above, $\Phi_{n_{\textrm{l}}}^{\textrm{L}}$
($\Phi_{n_{\textrm{r}}}^{\textrm{R}}$) represents the value of the magnetic
flux field of the left-handed (right-handed) transmission line at the position
$n_{\textrm{l}}$ ($n_{\textrm{r}}$), $C_{\textrm{l}}$ and $L_{\textrm{l}}$ are
the capacitance and the inductance of the LHTL, whereas $C_{\textrm{r}}$ and
$L_{\textrm{r}}$ are the capacitance and inductance of the RHTL, respectively.
Note that the higher $n_{\textrm{l}}$ and $n_{\textrm{r}}$, the farther the
labelled position with respect to the SQUID. Finally, $E(t)$ is the time-
dependent Josephson energy, and $\phi_{0}=\pi\hbar/e$, is the magnetic flux
quantum.

For the sake of simplicity, in our description we assume that the capacitance
of the Josephson junctions in the SQUID is small with respect to the
capacitance at both the LHTL and the RHTL, $C_{\textrm{J}}\ll
C_{\textrm{l}},C_{\textrm{r}}$, implying that we can neglect it in our
description. Importantly, we will work in a regime where $\Phi_{0}/\phi_{0}\ll
1$ [23, 24, 15, 20], thereby writing the linearized interaction Lagrangian as

$\displaystyle\mathcal{L}_{\textrm{I}}\simeq$
$\displaystyle-\frac{E(t)}{2}\left(\frac{2\pi}{\phi_{0}}\right)^{2}\left(\Phi_{1}^{\textrm{L}}-\Phi_{1}^{\textrm{R}}\right)^{2}.$
(3)

In Eq.(3) we expanded the cosine to the lowest order in
$2\pi\left(\Phi_{1}^{\textrm{L}}-\Phi_{1}^{\textrm{R}}\right)/\phi_{0}$.

Solving the Euler-Lagrange equations far from the SQUID, namely at
$1<n_{\textrm{l}},n_{\textrm{r}}<N$, we get the discrete mode expansions of
the quantum flux field in both the LHTL and the RHTL in terms of plane wave:

$\displaystyle\hat{\Phi}^{\textrm{L}}(n_{\textrm{l}},t)=$
$\displaystyle\sum_{\lvert
j\rvert=1}^{N/2}\sqrt{\frac{\hbar}{2C_{\textrm{l}}N\omega_{j}}}\left(e^{i(k_{j}n_{\textrm{l}}\Delta
x-\omega_{j}t)}a_{j}+\textrm{h.c.}\right),$ (4)
$\displaystyle\hat{\Phi}^{\textrm{R}}(n_{\textrm{r}},t)=$
$\displaystyle\sum_{\lvert
j\rvert=1}^{N/2}\sqrt{\frac{\hbar}{2C_{\textrm{r}}N\upsilon_{j}}}\left(e^{i(p_{j}n_{\textrm{r}}\Delta
x-\upsilon_{j}t)}b_{j}+\textrm{h.c.}\right),$ (5)

where $\omega_{j}$ and $k_{j}$ are frequency and wave vector of the left-
handed TL, respectively; whereas $\upsilon_{j}$ and $p_{j}$ are frequency and
wave vector of right-handed TL, respectively. The sum over all modes accounts
for plane wave propagating both leftwards (negative sign of $j$) and
rightwards (positive sign of $j$) along the CTL. Here, $a_{j}$ and $b_{j}$ are
the classical amplitudes of the LHTL and RHTL magnetic flux fields
respectively. By means of the Euler-Lagrange equations, we also find that the
two TLs are characterized by the following dispersion relations

$\displaystyle\omega_{j}=$
$\displaystyle\frac{1}{2\sqrt{C_{\textrm{l}}L_{\textrm{l}}}\left\lvert\sin\left(\frac{k_{j}\Delta
x}{2}\right)\right\rvert},$ (6) $\displaystyle\upsilon_{j}=$
$\displaystyle\frac{2\left\lvert\sin\left(\frac{p_{j}\Delta
x}{2}\right)\right\rvert}{\sqrt{C_{\textrm{r}}L_{\textrm{r}}}},$ (7)

for the LHTL and RHTL, respectively. Note that the form of the wave vectors is
identical for both TLs, $w_{j}=\frac{2\pi j}{N\Delta x}$, with $w=k,p$.
Referring to Fig.1, in our notation the positive sign of the wave vectors
always indicates the rightwards propagation of the signal.

The quantization of the two classical magnetic flux fields occurs in accord to
the standard procedure [38, 33]: we first promote the classical fields in
Eq.(4) and Eq.(5), as well as the canonical momenta
$P^{\textrm{L}}(n_{\textrm{l}},t)$ and $P^{\textrm{R}}(n_{\textrm{r}},t)$
defined respectively in Eq.(54) and Eq.(55), to quantum operators: these
fulfill the bosonic commutation relation

$\displaystyle\left[\hat{\Phi}^{\textrm{L}}(n,t),\hat{P}^{\textrm{L}}(m,t)\right]=\left[\hat{\Phi}^{\textrm{R}}(n,t),\hat{P}^{\textrm{R}}(m,t)\right]=i\hbar\delta_{nm},$
(8)

whereas all other commutators are equal to zero. The quantization of the
classical amplitudes is derived by discrete Fourier transforming Eqs.(4), (5),
(54) and (55) and exploiting the commutation rule in Eq.(8) [33]. We get

$\displaystyle\left[\hat{a}_{i},\hat{a}_{j}^{\dagger}\right]=\frac{\delta_{ij}}{4\sin^{2}\left(\frac{k_{j}\Delta
x}{2}\right)},$ (9)

for the magnetic flux field in the LHTL and

$\displaystyle\left[\hat{b}_{i},\hat{b}_{j}^{\dagger}\right]=\delta_{ij},$
(10)

for the magnetic flux field in the RHTL. With little abuse of notation,
sometimes we will refer to the quantum excitations of the magnetic flux fields
as “photons”.

Starting from the Lagrangian in Eq.(1) and the explicit expressions for the
two magnetic flux fields written above, we can derive the Hamiltonian
$\mathcal{\hat{H}}=\mathcal{\hat{H}}_{\textrm{L}}+\mathcal{\hat{H}}_{\textrm{R}}+\mathcal{\hat{H}}_{\textrm{I}}$,
where

$\displaystyle\mathcal{\hat{H}}_{\textrm{L}}=$ $\displaystyle
4\hbar\sum_{\lvert j\rvert=1}^{N/2}\omega_{j}\sin^{2}\left(\frac{k_{j}\Delta
x}{2}\right)\hat{a}_{j}^{\dagger}\hat{a}_{j}$ (11)
$\displaystyle\mathcal{\hat{H}}_{\textrm{R}}=$ $\displaystyle\hbar\sum_{\lvert
j\rvert=1}^{N/2}\upsilon_{j}\hat{b}_{j}^{\dagger}\hat{b}_{j}$ (12)

constitute the two parts of the noninteracting Hamiltonian for the left-handed
and right-handed transmission lines,
$\mathcal{\hat{H}}_{0}=\mathcal{\hat{H}}_{\textrm{L}}+\mathcal{\hat{H}}_{\textrm{R}}$,
and

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=\chi(t)$
$\displaystyle\left\\{\sum_{\lvert
j\rvert=1}^{N/2}\left[\frac{1}{\sqrt{C_{\textrm{l}}\omega_{j}}}\left(e^{-ik_{j}\Delta
x}\hat{a}_{j}^{\dagger}+e^{ik_{j}\Delta x}\hat{a}_{j}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{\sqrt{C_{\textrm{r}}\upsilon_{j}}}\left(e^{-ip_{j}\Delta
x}\hat{b}_{j}^{\dagger}+e^{ip_{j}\Delta
x}\hat{b}_{j}\right)\right]\right\\}^{2}$ (13)

with $\chi(t)=\frac{\hbar E(t)}{2N}\left(\frac{2\pi}{\phi_{0}}\right)^{2}$, is
the interaction Hamiltonian. The mathematical derivation of the Hamiltonian is
reported in Appendix A. We notice that, as long as we do not account for the
interaction at the SQUID, the eigenenergies in Eq.(11) are structurally
identical to the eigenvalues (frequencies) in Eq.(12),

$\displaystyle\epsilon_{j}\equiv
4\hbar\omega_{j}\sin^{2}\left(\frac{k_{j}\Delta
x}{2}\right)=\frac{2\hbar\left\lvert\sin\left(\frac{k_{j}\Delta
x}{2}\right)\right\rvert}{\sqrt{C_{\textrm{l}}L_{\textrm{l}}}}.$ (14)

Therefore, the bare energy grows with the wave vector up to the border of the
first Brillouin zone in both LHTL and RHTL, despite the different structure of
the dispersion relations [33].

##  III Phase matching at the SQUID

In this section we examine the signal propagation along the CTL. In
particular, we want to figure out whether the interaction between two TLs
characterized by different dispersion relations enables energy and momentum
propagation at their intersection. To address this issue properly, we need to
analyze the interaction Hamiltonian in detail.

Once we expand the square in Eq.(13), we observe that the interaction
Hamiltonian can be split conveniently into smaller terms, each representing a
specific type of two-mode coupling:

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=\mathcal{\hat{H}}_{\textrm{ES}}+\mathcal{\hat{H}}_{\textrm{HP}}+\mathcal{\hat{H}}_{\textrm{RM}}+\mathcal{\hat{H}}_{\textrm{1S}}+\mathcal{\hat{H}}_{\textrm{2S}}+\mathcal{\hat{H}}_{\textrm{IS}}$
(15)

More details about the single contributions of the Hamiltonian in Eq.(15) and
their explicit form are discussed in Appendix A. Note that the interaction can
occur both within the same TL, and between modes stemming from the left-handed
and right-handed components.

We now need to figure out whether all terms of the interaction Hamiltonian
contribute to the dynamics with equal weigh, or whether there are resonant
terms whose contribution play a major role. To verify this, we momentarily
move our description to the interaction picture, and observe if there are
static terms in the Hamiltonian. To induce an external time-dependence in the
Hamiltonian at $t>0$ we modulate the Josephson energy of the SQUID around a
fixed value according to $E(t)=E_{0}[1+\epsilon\cos(\Omega t)]$, where
$E_{0}=I_{\textrm{c}}\phi_{0}$ is the average value of the Josephson energy
with critical current $I_{\textrm{c}}$. Here, $\Omega$ and $\epsilon$ are the
oscillation frequency and the dimensionless oscillation amplitude,
respectively.

Once we move to the interaction picture, the only constant term in the
interaction Hamiltonian is

$\displaystyle\mathcal{\hat{H}}_{\textrm{ES}}^{\textrm{I}}=\frac{\hbar
E_{0}}{N}\left(\frac{2\pi}{\phi_{0}}\right)^{2}\sum_{\lvert
j\rvert=1}^{N/2}\left(\frac{1}{C_{\textrm{l}}\omega_{j}}a_{j}^{\dagger}\hat{a}_{j}+\frac{1}{C_{\textrm{r}}\upsilon_{j}}\hat{b}_{j}^{\dagger}\hat{b}_{j}\right),$
(16)

which describes the shift of the bare energies and frequencies in both
transmission lines. We can therefore redefine the noninteracting Hamiltonian
by including this term as
$\mathcal{\hat{H}}_{0}=\mathcal{\hat{H}}_{\textrm{L}}+\mathcal{\hat{H}}_{\textrm{R}}+\mathcal{\hat{H}}_{\textrm{ES}}$.
The energy shift of the bare eigenenergies in the LHTL is simply given by:

$\displaystyle\tilde{\epsilon}_{j}=\epsilon_{j}+\frac{\hbar
E_{0}}{NC_{\textrm{l}}\omega_{j}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}.$ (17)

To see the frequency shift clearly, we now introduce the corrected dispersion
relations of the LHTL and the RHTL as follows:

$\displaystyle\tilde{\omega}_{j}=$
$\displaystyle\omega_{j}+\frac{E_{0}}{4NC_{\textrm{l}}\omega_{j}\sin^{2}\left(\frac{k_{j}\Delta
x}{2}\right)}\left(\frac{2\pi}{\phi_{0}}\right)^{2},$ (18)
$\displaystyle\tilde{\upsilon}_{j}=$
$\displaystyle\upsilon_{j}+\frac{E_{0}}{NC_{\textrm{r}}\upsilon_{j}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}.$
(19)

These are achieved by letting the ladder operators evolve in the interaction
picture, and exploiting the commutation rules in Eqs.(9) and (10):

$\displaystyle\hat{a}_{j}^{\textrm{I}}(t)=\hat{U}_{0}^{\dagger}(t)\hat{a}_{j}\hat{U}_{0}(t)=\hat{a}_{j}e^{-i\tilde{\omega}_{j}t},$
(20)
$\displaystyle\hat{b}_{j}^{\textrm{I}}(t)=\hat{U}_{0}^{\dagger}(t)\hat{b}_{j}\hat{U}_{0}(t)=\hat{b}_{j}e^{-i\tilde{\upsilon}_{j}t},$
(21)

where $\hat{U}_{0}=e^{-i\mathcal{H}_{0}t/\hbar}$. Each corrected frequency
consists of the sum of the bare frequency and a correcting term.
Interestingly, in both LHTL and RHTL such correcting terms decrease
monotonically with the wave vector.

Now we want to focus on the possibility to transmit excitations through the
SQUID. We notice that the only element in the interaction Hamiltonian allowing
the exchange of excitations between the two parts of the CTL is

$\displaystyle\mathcal{\hat{H}}_{\textrm{HP}}^{\textrm{I}}=\sum_{i,j}^{N/2}g_{ij}$
$\displaystyle\left(\hat{a}_{i}\hat{b}_{j}^{\dagger}e^{-i(\tilde{\omega}_{i}-\tilde{\upsilon}_{j})t+i(k_{i}-p_{j})\Delta
x}+\textrm{h.c.}\right).$ (22)

where we defined the coupling constants $g_{ij}=-\frac{\hbar
E_{0}}{N\sqrt{C_{\textrm{l}}C_{\textrm{r}}\omega_{i}\upsilon_{j}}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}$.
This term describes the hopping effect between the two TLs, namely the
annihilation of one excitation on one side of the CTL and the creation of
another excitation on the other side. The hopping effect between two modes of
the two TLs can be activated by imposing the energy and momentum conservation
of the two magnetic flux fields at the SQUID. However, as a consequence of the
discreteness of the mode structure, the propagation of quantum excitations
through the SQUID does not result possible for all mode pairs, but only for
those modes of the magnetic flux fields fulfilling the phase matching
conditions given by $k_{i}=p_{j}$ (momentum conservation) and the
transcendental equation $\tilde{\omega}_{i}=\tilde{\upsilon}_{j}$ (frequency
conservation). Since the momentum conservation of process is fulfilled by
modes with the same mode number, the transcendental equation becomes
$\tilde{\omega}_{j}=\tilde{\upsilon}_{j}$, and it is solved by tuning the set
of parameters
$\\{C_{\textrm{l}},C_{\textrm{r}},L_{\textrm{l}},L_{\textrm{r}}\\}$
opportunely. The discrete mode structure of both TLs therefore allows to
restrict the propagation of excitations between the two TLs to only two
selected modes, thereby limiting undesired hopping effects between other
modes.

To make a concrete example, in Fig.2 we plot the LHTL eigenenergies in
Fig.(17), as well as the dispersion relations in Eq.(18) and Eq.(19). The two
dispersion relations are obtained by selecting a set of parameters
$\\{C_{\textrm{l}},L_{\textrm{l}},L_{\textrm{r}}\\}$, and fixing a determined
resonant mode pair by solving the trascendental equation
$\tilde{\omega}_{j}=\tilde{\upsilon}_{j}$ with respect to the capacitance of
the RHTL, $C_{\textrm{r}}$. We solved this equation for three set of modes,
namely $\tilde{\omega}_{100}=\tilde{\upsilon}_{100}$,
$\tilde{\omega}_{50}=\tilde{\upsilon}_{50}$ and
$\tilde{\omega}_{30}=\tilde{\upsilon}_{30}$, obtaining three different values
of $C_{\textrm{r}}$, then we plotted the dispersion relations.

Figure 2: Corrected dispersion relations $\tilde{\omega}_{j}$ (black) and bare
eigenenergies $\tilde{\epsilon}_{j}/\hbar$ (gray) of the LHTL. Other colors
indicate the corrected dispersion relations $\tilde{\upsilon}_{j}$ of the
RHTL, achieved by solving the transcendental equation
$\tilde{\omega}_{j}=\tilde{\upsilon}_{j}$, with $j$ labeling the $j$-th mode
of both the LHTL and the RHTL, and using $C_{\textrm{r}}$ as the variable.
Frequencies of the RHTL are plotted ensuring the frequency matching for the
mode pair $\tilde{\omega}_{30}=\tilde{\upsilon}_{30}$ (blue),
$\tilde{\omega}_{50}=\tilde{\upsilon}_{50}$ (red), and
$\tilde{\omega}_{100}=\tilde{\upsilon}_{100}$ (green), thereby obtaining
$C_{\textrm{r}}=0.27$ pF, $C_{\textrm{r}}=1.60$ pF, and $C_{\textrm{r}}=6.39$
pF, respectively. Other parameters are: $C_{\textrm{l}}=0.4$ pF,
$L_{\textrm{r}}=L_{\textrm{l}}=60$ pH, $I_{\textrm{c}}=1.25$ $\mu$A, and
$N=200$.

##  IV Interaction and dynamics

In the last section we observed that we can control the set of parameters
$\\{C_{\textrm{l}},C_{\textrm{r}},L_{\textrm{l}},L_{\textrm{r}}\\}$ to enable
the propagation of quanta of the magnetic flux fields between two modes of the
LHTL and the RHTL. All other terms in Eq.(22) describing the interaction
between two off-resonant modes do not fulfill the phase matching conditions,
and their contribution is in fact negligible. This allows us to drastically
reduce the Hilbert space of the system, thereby focusing on few modes of the
CTL.

The possibility to isolate the contribution of specific mode pairs and ignore
all secularities (i.e. off-resonant terms) can be applied to other interaction
terms accounted in the Hamiltonian in Eq.(15). In particular, since we are
interested in the interaction between the LHTL and the RHTL, we can tailor the
set
$\\{\Omega,C_{\textrm{l}},C_{\textrm{r}},L_{\textrm{l}},L_{\textrm{r}}\\}$,
thereby ensuring the phase matching for only one coupling term in Eq.(15)
containing products of ladder operators stemming from different TLs. Note that
the presence of the modulation frequency $\Omega$ is fundamental for the
activation of Raman coupling or squeezing terms, which otherwise would be
counterrotating and therefore negligible after the rotating wave approximation
(RWA).

The possibility to control the Hamiltonian in Eq.(15) and activate one of the
possible interaction terms according to our needs makes this device an
interesting platform for the simulation of typical scenarios in quantum optics
and quantum information. In what follows, we present some possible application
of our findings. We stress that, although we might consider also the coupling
between modes within the same TL, henceforth we will focus on the interaction
of modes stemming from the LHTL and the RHTL. We will always assume that the
two resonant modes are characterized by the same mode number $j$, thereby
omitting the label $j$ henceforth.

###  IV.1 Photon hopping

As a first scenario, we again consider the interaction

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=\hbar\xi_{\textrm{hp}}\left(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger}\right),$
(23)

with coupling constant

$\displaystyle\xi_{\textrm{hp}}=-\frac{2E_{0}}{N\omega\sqrt{C_{\textrm{r}}C_{\textrm{l}}}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}.$
(24)

This Hamiltonian can be achieved from Eq.(22) by tailoring the parameters such
that $\tilde{\omega}=\tilde{\upsilon}=\omega$ and performing the RWA. Note
that to achieve this result we did not require the modulation of the Josephson
energy at the SQUID. In the Heisenberg picture, the annihilation operators for
the two TLs evolve as follows:

$\displaystyle\hat{a}(t)=$ $\displaystyle e^{-i\omega
t}\left(\hat{a}\cos(\xi_{\textrm{hp}}t)+i\hat{b}\sin(\xi_{\textrm{hp}}t)\right),$
(25a) $\displaystyle\hat{b}(t)=$ $\displaystyle e^{-i\omega
t}\left(\hat{b}\cos(\xi_{\textrm{hp}}t)+i\hat{a}\sin(\xi_{\textrm{hp}}t)\right).$
(25b)

This Hamiltonian describes a collinear beam splitting, in which the
transmission and reflection parameters are modulated in time by the coupling
constant $\xi_{\textrm{hp}}$. This type of interaction therefore expresses the
propagation of excitations along the CTL due to the annihilation of one photon
on one of the two TLs and the creation of one photon with same frequency and
momentum on the other one.

###  IV.2 Raman scattering

A second interesting scenario is delineated by the Hamiltonian

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=\hbar\xi_{\textrm{rm}}\left(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger}\right),$
(26)

which is structurally identical to Eq.(23) but with coupling constant

$\displaystyle\xi_{\textrm{rm}}=-\frac{\epsilon
E_{0}}{N\sqrt{C_{\textrm{r}}C_{\textrm{l}}\omega\upsilon}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}.$
(27)

To obtain this Hamiltonian, we tune the SQUID modulation frequency such that
$\tilde{\omega}=\tilde{\upsilon}+\Omega$, assuming that
$\tilde{\omega}>\tilde{\upsilon}$. In contrast to the previous case, the
frequencies of the two interacting modes differ, and the frequency
conservation is only guaranteed by the modulation of the Josephson energy at
the SQUID. This interaction gives rise to two possible phenomena: the Stokes
Raman scattering, when the SQUID induces the decay of a photon from the LHTL
into a lower-energetic photon of the RHTL; the anti-Stokes Raman scattering,
when the SQUID pumps the mode $\tilde{\upsilon}$ of the RHTL, which decays
into a photon $\tilde{\omega}$ with higher energy. Clearly, we can manipulate
our parameters thereby activating a resonance of the form
$\tilde{\upsilon}=\tilde{\omega}+\Omega$, with
$\tilde{\upsilon}>\tilde{\omega}$.

One of the remarkable facts stemming from combining two TLs with different
dispersion relations is that the Raman effect can also occur between modes of
the two TLs having same bare energy $\tilde{\epsilon}=\hbar\tilde{\upsilon}$.
After the Raman scattering in the SQUID, a signal generated in the RHTL would
therefore propagate to the LHTL with amplified frequency but same wave vector
and bare energy [39].

As before, we report the annihilation operators of the two modes
$\tilde{\omega}$ and $\tilde{\upsilon}$ evolved in the Heisenberg picture:

$\displaystyle\hat{a}(t)=$ $\displaystyle
e^{-i\tilde{\omega}t}\left(\hat{a}\cos(\xi_{\textrm{rm}}t)+i\hat{b}\sin(\xi_{\textrm{rm}}t)\right),$
(28a) $\displaystyle\hat{b}(t)=$ $\displaystyle
e^{-i\tilde{\upsilon}t}\left(\hat{b}\cos(\xi_{\textrm{rm}}t)+i\hat{a}\sin(\xi_{\textrm{rm}}t)\right).$
(28b)

Despite the evident similarities between the annihilation operators written
above and those in Eq.(25), we stress that this interaction mixes two input
modes with different frequencies.

###  IV.3 Two-mode squeezing

The modulation of the SQUID can give rise to another interesting scenario.
Indeed, by activating the resonance condition
$\Omega=\tilde{\omega}+\tilde{\upsilon}$ we can squeeze the state of two modes
stemming from two different TLs. Once we fix the resonance condition written
above, ensure the momentum conservation $p=-k$, and apply the RWA, the
Hamiltonian describing the two-mode squeezing coupling reads

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=\hbar\xi_{\textrm{sq}}\left(\hat{a}^{\dagger}\hat{b}^{\dagger}+\hat{a}\hat{b}\right),$
(29)

and the annihilation operators in the Heisenberg picture evolve as

$\displaystyle\hat{a}(t)=$ $\displaystyle
e^{-i\tilde{\omega}t}\left(\hat{a}\cosh(\xi_{\textrm{sq}}t)-i\hat{b}^{\dagger}\sinh(\xi_{\textrm{sq}}t)\right),$
(30a) $\displaystyle\hat{b}(t)=$ $\displaystyle
e^{-i\tilde{\upsilon}t}\left(\hat{b}\cosh(\xi_{\textrm{sq}}t)-i\hat{a}^{\dagger}\sinh(\xi_{\textrm{sq}}t)\right),$
(30b)

with squeezing parameter $\xi_{\textrm{sq}}$ equal to $\xi_{\textrm{rm}}$ in
Eq.(27). These operators suggest us that we can squeeze the quantum vacuum of
the two transmission lines at the same time, thereby generating one photon in
the LHTL and one photon in the RHTL. Phenomena of this kind play a crucial
role in nonlinear optics. An example is the parametric down-conversion, in
which a pump laser interacts with a nonlinear material generating two
entangled and spatially distinguishable photons, named signal and idler
photons [40, 41, 42, 43].

##  V Correlations between the TLs

In this section we study the correlations between two modes of the LHTL and
the RHTL interacting resonantly. Our approach is based on the second order
correlation function, which, for our purpose, assumes the following expression
[44]:

$\displaystyle
G^{(2)}(n_{\textrm{l}},t_{1};n_{\textrm{r}},t_{2})=\langle\Phi^{\textrm{L}(-)}(n_{\textrm{l}},t_{1})\Phi^{\textrm{R}(-)}(n_{\textrm{r}},t_{2})\Phi^{\textrm{L}(+)}(n_{\textrm{l}},t_{1})\Phi^{\textrm{R}(+)}(n_{\textrm{r}},t_{2})\rangle.$
(31)

Sometimes we will make use of the normalized second order correlation
function,

$\displaystyle
g^{(2)}(n_{\textrm{l}},t_{1};n_{\textrm{r}},t_{2})=\frac{G^{(2)}(n_{\textrm{l}},t_{1};n_{\textrm{r}},t_{2})}{G^{(1)}(n_{\textrm{l}},t_{1})G^{(1)}(n_{\textrm{r}},t_{2})},$
(32)

where $G^{(1)}(n,t)=\langle\Phi^{(-)}(n,t)\Phi^{(+)}(n,t)\rangle$ is the first
order correlation function.

We initialize the system in the two-mode Fock state
$\lvert\psi(0)\rangle=\lvert s_{\textrm{L}},s_{\textrm{R}}\rangle$, where
$s_{\textrm{L}}$ and $s_{\textrm{R}}$ represent the number of excitations of
the modes with frequencies $\tilde{\omega}$ and $\tilde{\upsilon}$,
respectively. This reduces the correlation function to

$\displaystyle
G_{s_{\textrm{L}},s_{\textrm{R}}}^{(2)}(t_{1},t_{2})=\frac{\hbar^{2}}{N^{2}C_{\textrm{r}}C_{\textrm{l}}\omega\upsilon}\langle\hat{a}^{\dagger}(t_{1})\hat{b}^{\dagger}(t_{2})\hat{a}(t_{1})\hat{b}(t_{2})\rangle.$
(33)

The explicit form of the correlation function depends on the specific
resonance we are interested in. In what follows, we will analyse the three
cases studied before, namely the photon hopping, the Raman scattering and the
two-mode squeezing. Note that, despite the different physical interpretation
of the two scenarios in sections IV.1 and IV.2, the annihilation (and
therefore also the creation) operators in Eq.(25) and Eq.(28) lead to the same
form of the correlation function, therefore these two cases will be presented
together.

###  V.1 Photon hopping and Raman scattering

We now fix the parameters of the CTL such that the energy conservation is
fulfilled either in terms of $\tilde{\omega}=\tilde{\upsilon}$, or
$\tilde{\omega}=\tilde{\upsilon}+\Omega$. As mentioned above, the second order
correlation function is structurally identical for the two scenarios, and
reads

$\displaystyle G_{s_{h},s_{d}}^{(2)}(t_{1},t_{2})=$
$\displaystyle\frac{\hbar^{2}s_{\textrm{L}}s_{\textrm{R}}C_{1}^{2}C_{2}^{2}}{N^{2}C_{\textrm{r}}C_{\textrm{l}}\omega\upsilon}$
$\displaystyle\left[(1-T_{1}T_{2})^{2}+\frac{s_{\textrm{L}}-1}{s_{\textrm{R}}}T_{2}^{2}+\frac{s_{\textrm{R}}-1}{s_{\textrm{L}}}T_{1}^{2}\right],$
(34)

where we defined $C_{j}=\cos(\xi t_{j})$ and $T_{j}=\tan(\xi t_{j})$, with
$\lvert j\rvert=1,2$. Here, $\xi$ corresponds to either $\xi_{\textrm{hp}}$ or
$\xi_{\textrm{rm}}$. Clearly, when the CTL is prepared in the vacuum state, or
when only one photon populates the CTL, the correlation function is
identically zero. However, it is particularly interesting to consider the case
when the CTL is initially populated by two photons, each stemming from a
different TL. In this case, the normalized second order correlation function
becomes

$\displaystyle g_{1_{\textrm{L}},1_{\textrm{R}}}^{(2)}(t_{1},t_{2})=$
$\displaystyle\cos^{2}[\xi(t_{1}+t_{2})].$ (35)

This correlation function tells us that the coincidence probability to detect
two photons at different TLs is modulated, and that periodically this
probability reaches zero. To see what happens to the two photons when
$g_{1_{\textrm{L}},1_{\textrm{R}}}^{(2)}(t_{1},t_{2})=0$, we let the system
evolve in the Schrödinger picture. Since the initial state consists of two
photons in different channels, this is equivalent to applying the creation
operators $\hat{a}^{\dagger}(t)$ and $\hat{b}^{\dagger}(t)$ in Eq.(25) or
Eq.(28) to the vacuum state,
$\lvert\psi(t)\rangle=\hat{a}^{\dagger}(t)\hat{b}^{\dagger}(t)\lvert
0_{\textrm{L}},0_{\textrm{R}}\rangle$, hence obtaining

$\displaystyle\lvert\psi(t)\rangle=$ $\displaystyle
e^{i(\tilde{\omega}+\tilde{\upsilon})t}\cos\left(2\xi\,t\right)\lvert
1_{\textrm{L}},1_{\textrm{R}}\rangle$
$\displaystyle+i\,e^{i(\tilde{\omega}+\tilde{\upsilon})t}\sin\left(2\xi\,t\right)\left(\lvert
0_{\textrm{L}},2_{\textrm{R}}\rangle+\lvert
2_{\textrm{L}},0_{\textrm{R}}\rangle\right).$ (36)

As we can see, the normalized correlation function in Eq.(35) with
$t_{1}=t_{2}=t$ corresponds to the square of the first term of the output
state, which describes the probability to detect two photons in two different
modes. When this probability vanishes at time $t_{\textrm{dip}}=\pi/(4\xi)$,
the probability to find two photons at the same TL is maximized. The state
$\lvert\psi(t_{\textrm{dip}})\rangle$ therefore corresponds to the output
state of a quantum interference scenario, in which two photons interacting in
a beam splitter are only found both in one of the two output channels. This
phenomenon, in which the quantum interference of two photons annuls the
coincidence probability and generates the two-photon NOON state, is referred
to as Hong-Ou-Mandel effect, and it finds applications in many quantum optical
frameworks [45, 46, 47, 48].

###  V.2 Two-mode squeezing

To conclude this section, we analyze the second order correlation function
when the resonance $\Omega=\tilde{\omega}+\tilde{\upsilon}$ is activated.
Substituting Eq.(30) into Eq.(33) we obtain

$\displaystyle G_{s_{h},s_{d}}^{(2)}(t_{1},t_{2})=$
$\displaystyle\frac{\hbar^{2}s_{\textrm{L}}s_{\textrm{R}}\bar{C}_{1}^{2}\bar{C}_{2}^{2}}{N^{2}C_{\textrm{r}}C_{\textrm{l}}\omega\upsilon}\left[1+\bar{T}_{1}\bar{T}_{2}+\frac{(s_{\textrm{L}}+1)(s_{\textrm{R}}+1)}{s_{\textrm{L}}s_{\textrm{R}}}(\bar{T}_{1}^{2}\bar{T}_{2}^{2}+\bar{T}_{1}\bar{T}_{2})+\frac{s_{\textrm{L}}+1}{s_{\textrm{R}}}\bar{T}_{2}^{2}+\frac{s_{\textrm{R}}+1}{s_{\textrm{L}}}\bar{T}_{1}^{2}\right]$
(37)

where we defined $\bar{C}_{j}=\cosh(\xi_{\textrm{sq}}t_{j})$ and
$\bar{T}_{j}=\tanh(\xi_{\textrm{sq}}t_{j})$, with $\lvert j\rvert=1,2$. This
function describes the correlations between the LHTL and the RHTL when the two
modes of interest are prepared in a Fock state with $s_{\textrm{L}}$ and
$s_{\textrm{R}}$ excitations, respectively. As expected for squeezed states,
the second order correlation function does not vanish when the system is
prepared in the vacuum state. Indeed in this case the normalized second order
correlation function becomes

$\displaystyle g_{0_{\textrm{L}},0_{\textrm{R}}}^{(2)}(t_{1},t_{2})=$
$\displaystyle 1+\coth(\xi_{\textrm{sq}}t_{1})\coth(\xi_{\textrm{sq}}t_{2}),$
(38)

and reduces to

$\displaystyle g_{0_{\textrm{L}},0_{\textrm{R}}}^{(2)}(t)=$ $\displaystyle
1+\coth^{2}(\xi_{\textrm{sq}}t),$ (39)

when $t_{1}=t_{2}$, demonstrating the presence of correlations even when the
system is initially prepared in the vacuum state. Note that in the limit $t\gg
1/\xi_{\textrm{sq}}$ this function asymptotically approaches to
$g_{0_{\textrm{L}},0_{\textrm{R}}}^{(2)}(t)=2$, and the two modes individually
show the statistics of a thermal state.

##  VI Further quantum simulation scenarios

So far we have presented our platform as possible quantum simulator of
standard phenomena in quantum optics. However, the versatility of this device
suggests further possible uses in other branches of physics, such as in
quantum mechanics, multimode quantum systems and quantum thermodynamics. In
what follows, we will briefly discuss these three cases.

###  VI.1 General two-mode coupling for degenerate QHOs

In quantum mechanics, the Hamiltonian of two interacting bosonic modes induces
a linear dynamics when all annihilation and creation operators appear as
combinations of quadratic terms [49]. These types of Hamiltonian are largely
employed in quantum optics, and in this work we already discussed three
scenarios of interest. In section IV.1, for example, we discussed a beam
splitting interaction, in which the Hamiltonian in Eq.(23) stems from the
resonant activation of hopping terms. On the other hand, in section IV.3 the
Hamiltonian in Eq.(29) only contains the two-mode squeezing terms. In both
cases, we made use of the RWA to get rid of highly oscillating terms that do
not fulfil the energy conservation. However, the resonance conditions enabling
the filtering of the hopping terms and the squeezing terms from Eq.(15) do not
exclude themselves reciprocally, but they can be activated together by
properly tuning the modulation of the Josephson energy. Once we assume
$\tilde{\omega}=\tilde{\upsilon}=\omega$ and $\Omega=2\omega$, the interaction
Hamiltonian becomes

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=$
$\displaystyle\hbar\xi_{\textrm{hp}}\left(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger}\right)+\hbar\xi_{\textrm{sq}}\left(\hat{a}^{\dagger}\hat{b}^{\dagger}+\hat{a}\hat{b}\right)$
$\displaystyle+\hbar\xi_{\textrm{sl}}\left[(\hat{a}^{\dagger})^{2}+\hat{a}^{2}\right]+\hbar\xi_{\textrm{sr}}\left[(\hat{b}^{\dagger})^{2}+\hat{b}^{2}\right],$
(40)

where in the second line we introduced the single-mode squeezing coupling
terms, weighed by the squeezing parameters for the LHTL and the RHTL mode,

$\displaystyle\xi_{\textrm{sl}}=-\frac{\epsilon
E_{0}}{NC_{\textrm{l}}\omega}\left(\frac{2\pi}{\phi_{0}}\right)^{2},$ (41)
$\displaystyle\xi_{\textrm{sr}}=-\frac{\epsilon
E_{0}}{NC_{\textrm{r}}\omega}\left(\frac{2\pi}{\phi_{0}}\right)^{2},$ (42)

respectively. This Hamiltonian describes the most general linear interaction
between two distinguishable quantum harmonic oscillators (QHOs) characterized
by the same frequency. The dynamics stemming from Eq.(40) was largely studied
by means of the symplectic formalism in [49].

###  VI.2 Multimode quantum system interaction

In this section we want to discuss the possibility to extend our model to a
larger set of interacting bosonic degrees of freedom. We can realize this
scenario by shaping the modulation of the Josephson energy as the
superposition of $M$ periodic functions:

$\displaystyle
E(t)=E_{0}\left[1+\sum_{m=1}^{M}(\epsilon_{m}\sin(\Omega_{m}t)+\kappa_{m}\cos(\Omega_{m}t))\right]$
(43)

with $\epsilon_{m}$ and $\kappa_{m}$ oscillation amplitudes, and $\Omega_{m}$
being the frequency of the $m$-th periodic function.

By choosing the frequencies $\Omega_{m}$ opportunely, we can therefore
activate further resonances and select further interacting modes. This allows
us on the one hand to extend the scenarios analyzed so far (beam splitting
interactions, Raman scatterings, two-photon squeezing interactions) to a
larger number of interacting mode pairs; on the other hand, we can prepare
exotic Hamiltonian describing multimode interactions.

As a first example, we can prepare two distinguishable and nondegenerate
interacting QHOs [50]. To do this, we tailor the Josephson energy of the SQUID
in Eq.(43) (with $M=2$) by imposing
$\Omega_{1}=\tilde{\omega}_{h}-\tilde{\upsilon}$,
$\Omega_{2}=\tilde{\omega}_{h}+\tilde{\upsilon}$,
$\epsilon_{1}=\epsilon_{2}=0$ and $\kappa_{1}=\kappa_{2}=2$. Once we set the
parameters as above, the interaction Hamiltonian reduces to

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=\hbar\xi\left(\hat{a}+\hat{a}^{\dagger}\right)\left(\hat{b}+\hat{b}^{\dagger}\right),$
(44)

where

$\displaystyle\xi_{\textrm{hp}}=\xi_{\textrm{sq}}=\xi=-\frac{\epsilon
E_{0}}{N\sqrt{C_{\textrm{r}}C_{\textrm{l}}\omega\upsilon}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}.$
(45)

The eigenmodes deriving from the diagonalization of this Hamiltonian are
sometimes referred to as polaritons [51], and their dynamics is well known in
literature [49].

Interestingly, the possibility to activate more resonances by means of the
modulation of the Josephson energy in Eq.(43) permits a drastic extension of
the number of interacting modes, and consequently of the Hilbert space. As an
example, we can prepare the general Hamiltonian of a three-mode system:

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}=$
$\displaystyle\hbar\left[\xi_{1}\left(\hat{c}_{i}+\hat{c}_{i}^{\dagger}\right)\left(\hat{c}_{j}+\hat{c}_{j}^{\dagger}\right)+\xi_{2}\left(\hat{c}_{i}+\hat{c}_{i}^{\dagger}\right)\left(\hat{c}_{k}+\hat{c}_{k}^{\dagger}\right)\right.$
$\displaystyle\left.+\xi_{3}\left(\hat{c}_{j}+\hat{c}_{j}^{\dagger}\right)\left(\hat{c}_{k}+\hat{c}_{k}^{\dagger}\right)\right],$
(46)

with $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$ coupling constants, and with $c=a,b$
representing one mode of either the LHTL or the RHTL. This Hamiltonian
describes three selected modes of the CTL, each thereof is interacting with
the other one.

Figure 3: Schematic representation of the cascade quantum network. The
modulation frequencies $\Omega_{m}$ of the Josephson energy are set thereby
activating the Raman resonances between modes of the LHTL and the RHTL. Every
resonant mode of the CTL is coupled to a bath at temperature
$T_{n_{\textrm{x}}}$, with $\textrm{x}=\textrm{l},\textrm{r}$. Higher
frequency modes of the CTL are coupled to baths with higher temperatures.
Therefore, the SQUID extracts work at each step of the Raman cascade. The
subscripts on the frequencies simply differentiate the mode frequencies and do
not refer to any specific modes.

###  VI.3 CTL as quantum thermal machine

Finally, we want to discuss some interesting applications of our system in
quantum thermodynamics. The idea is to employ the CTL as a quantum thermal
machine [52, 53], which can either convert heat into work (quantum heat
engine) [54, 55] or cool down a cold bath (quantum refrigerator) [54, 56]. A
model of quantum heat engine is the quantum amplifier, in which two bosonic
modes, coupled to two baths at different temperatures, interact by means of an
external drive to generate output power [54]. To let our device work as a
quantum amplifier, we need to thermalize two modes of the CTL, say
$\tilde{\omega}$ and $\tilde{\upsilon}$, by coupling them to two baths at
different temperature: assuming $\tilde{\omega}>\tilde{\upsilon}$, the LHTL
mode is coupled to a hot bath at temperature $T_{\textrm{h}}$, whereas the
RHTL is coupled to a cold bath at temperature $T_{\textrm{c}}$. The required
resonance condition we need to exploit is the Raman scattering
$\Omega=\tilde{\omega}-\tilde{\upsilon}$. The interaction Hamiltonian written
in the Schrödinger picture is:

$\displaystyle\mathcal{\hat{H}}_{\textrm{I}}(t)=\hbar\xi\left(e^{i\Omega
t}\hat{a}\hat{b}^{\dagger}+e^{-i\Omega t}\hat{a}^{\dagger}\hat{b}\right),$
(47)

where we again assumed a modulation of the Josephson energy of the form
$E(t)=E_{0}(1+\epsilon\cos(\Omega t))$. By carrying out the time derivative of
this Hamiltonian and averaging it with respect to the state, we get the
amplifier output power [54]:

$\displaystyle\mathcal{P}(t)=i\hbar\Omega\xi\left(e^{i\Omega
t}\langle\hat{a}\hat{b}^{\dagger}\rangle-e^{-i\Omega
t}\langle\hat{a}^{\dagger}\hat{b}\rangle\right).$ (48)

Another powerful application of our CTL is the realization of a quantum
network [57, 58]. In the context of quantum thermodynamics, a quantum network
consists of an ensemble of wires and junctions which allows the heat
propagation and the extraction of net work. The multimode structure of the
CTL, along with the possibility to activate multiple resonances (as shown in
the previous section), suggest that the CTL itself can act as a quantum
network. A model of quantum network can be realized by coupling the modes of
the LHTL and the RHTL to different baths and activating chains of Raman
resonances by means of a tailored modulation of the Josephson energy. The
cascade quantum amplifier is pictorially represented in Fig.3. The higher the
mode frequency, the higher the temperature of the bath the mode is coupled
with. The chain of Raman resonances activates the heat flow from the higher
temperature baths towards the cooler temperature ones, whereas the SQUID
extracts power at each Raman decay. More details about the implementation of
this quantum heat engine on our CTL platform, as well as the analysis of the
heat flows and the extracted power, are left for a future publication.

##  VII Summary and Conclusion

We have presented a hybrid platform realized by connecting a left-handed and a
right-handed transmission line by means of a superconducting quantum
interference device. After quantizing the magnetic flux fields in both TLs, we
have investigated the phase matching at the SQUID, showing that the
propagation of quantum excitations from one TL to the other one can occur only
for two degenerate modes of the CTL.

We observed that the interaction between the two TLs occurs only between two
modes fulfilling specific resonance conditions. We have shown that we can
select the interacting modes, as well as the type of interaction, by
controlling the set of circuit parameters. Among all possible interactions we
could have activated, we have presented the hopping effect, the Raman
scattering, and the two-mode squeezing. Considered more relevant for our
discussion, we have focused our attention only on interactions between LHTL
and RHTL modes.

As a result of our analysis, we have shown that our hybrid transmission line
can simulate standard processes in quantum mechanics and quantum optics, such
as the parametric down-conversion and the HOM interference at the microwave
scales. Among other technical uses, our platform can Raman amplify an incoming
signal preserving the wave vector, and it can be exploited in the framework of
quantum thermodynamics as a quantum heat engine.

As extensions of this work we envisage the realization of chains of hybrid
metamaterial TLs in which we can control the spectral feature of the
propagating quanta with higher precision. In the framework of quantum
metrology, this platform can find application in the realization of circuit
versions of well-known interference platforms, such us collinear Mach-Zehnder
and SU(1,1) interferometers. This work paves the way for future theoretical
and experimental implementations of hybrid metamaterial transmission lines
working at the quantum scales.

##  VIII Acknowledgments

The authors thank David Edward Bruschi for the helpful comments. The authors
acknowledge support from the joint project No. 13N15685 “German Quantum
Computer based on Superconducting Qubits (GeQCoS)” sponsored by the German
Federal Ministry of Education and Research (BMBF) under the framework “Quantum
technologies–from basic research to the market”.

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##  Appendix A Hamiltonian of hybrid transmission line

In this section we want to discuss the derivation of our Hamiltonian, as well
as the single contribution to the interaction Hamiltonian, in more details. We
start from the linearized Lagrangian in Eq.(2a), (2b) and Eq.(3). By means of
the Legendre transformation,

$\displaystyle\mathcal{H}=\sum_{n}\left(\frac{\partial\mathcal{L}}{\partial\dot{\Phi}_{n}^{\textrm{L}}}\dot{\Phi}_{n}^{\textrm{L}}+\frac{\partial\mathcal{L}}{\partial\dot{\Phi}_{n}^{\textrm{R}}}\dot{\Phi}_{n}^{\textrm{R}}\right)-\mathcal{L},$
(49)

we obtain the total Hamiltonian

$\displaystyle\mathcal{\hat{H}}=\mathcal{\hat{H}}_{\textrm{L}}+\mathcal{\hat{H}}_{\textrm{R}}+\mathcal{\hat{H}}_{\textrm{I}},$
(50)

where

$\displaystyle\mathcal{H}_{\textrm{L}}=$
$\displaystyle\frac{1}{2}\sum_{n_{\textrm{l}}=1}^{N}\left[\sum_{i=1}^{n_{\textrm{l}}}\frac{(P_{n_{\textrm{l}}}^{\textrm{L}})^{2}}{C_{\textrm{l}}}+\frac{(\Phi_{n_{\textrm{l}}}^{\textrm{L}})^{2}}{L_{\textrm{l}}}\right]$
(51) $\displaystyle\mathcal{H}_{\textrm{R}}=$
$\displaystyle\frac{1}{2}\sum_{n_{\textrm{r}}=1}^{N}\left[C_{\textrm{r}}(P_{n_{\textrm{r}}}^{\textrm{R}})^{2}+\frac{(\Phi_{n_{\textrm{r}}+1}^{\textrm{R}}-\Phi_{n_{\textrm{r}}}^{\textrm{R}})^{2}}{L_{\textrm{r}}}\right]$
(52) $\displaystyle\mathcal{H}_{\textrm{I}}=$
$\displaystyle\frac{E(t)}{2}\left(\frac{2\pi}{\phi_{0}}\right)^{2}\left(\Phi_{0}^{\textrm{L}}-\Phi_{0}^{\textrm{R}}\right)^{2},$
(53)

with conjugated momenta

$\displaystyle P_{n_{\textrm{l}}}^{\textrm{L}}=$
$\displaystyle\frac{\partial\mathcal{L}^{\textrm{L}}}{\partial\dot{\Phi}_{n_{\textrm{l}}}}=C_{\textrm{l}}\left(2\dot{\Phi}_{n_{\textrm{l}}}^{\textrm{L}}-\dot{\Phi}_{{n_{\textrm{l}}}+1}^{\textrm{L}}-\dot{\Phi}_{{n_{\textrm{l}}}-1}^{\textrm{L}}\right)$
$\displaystyle=$ $\displaystyle-i\sum_{\lvert
j\rvert=1}^{N/2}W_{j}\left(e^{i(k_{j}n_{\textrm{l}}\Delta
x-\omega_{j}t)}a_{j}-\textrm{h.c.}\right),$ (54) $\displaystyle
P_{n_{\textrm{r}}}^{\textrm{R}}=$
$\displaystyle\frac{\partial\mathcal{L}^{\textrm{R}}}{\partial\dot{\Phi}_{n_{\textrm{r}}}}=C_{\textrm{r}}\dot{\Phi}_{n_{\textrm{r}}}^{\textrm{R}}$
$\displaystyle=$ $\displaystyle-i\sum_{\lvert
j\rvert=1}^{N/2}\sqrt{\frac{\hbar
C_{\textrm{r}}\upsilon_{j}}{2N}}\left(e^{i(p_{j}n_{\textrm{r}}\Delta
x-\upsilon_{j}t)}b_{j}-\textrm{h.c.}\right).$ (55)

with $W_{j}=\sqrt{\frac{8\hbar
C_{\textrm{l}}\omega_{j}}{N}}\sin^{2}\left(\frac{k_{j}\Delta x}{2}\right)$,
where in the last lines we made use of the mode decomposition of the magnetic
flux fields in Eq.(4) and (5). Substituting the fields and the canonical
momenta in Eqs.(4), (5), (54) and (55) into the Hamiltonian in Eq.(50),
summing with respect to the spatial indices $n_{\textrm{l}}$ and
$n_{\textrm{r}}$, and exploiting the commutation relations in Eqs.(9) and
(10), we obtain the Hamiltonian contributions in Eqs.(11), (12) and (13).

Once we expand the square of the interaction Hamiltonian in Eq.(13), we
distinguish the single contributions to the interaction, as done in Eq.(15).
In what follows, we will present the explicit form of the single terms in
Eq.(15) and describe their role in the dynamics.

The first term we want to present is

$\displaystyle\mathcal{\hat{H}}_{\textrm{ES}}=\frac{\hbar
E(t)}{N}\left(\frac{2\pi}{\phi_{0}}\right)^{2}\sum_{\lvert
j\rvert=1}^{N/2}\left(\frac{1}{C_{\textrm{l}}\omega_{j}}a_{j}^{\dagger}\hat{a}_{j}+\frac{1}{C_{\textrm{r}}\upsilon_{j}}\hat{b}_{j}^{\dagger}\hat{b}_{j}\right).$
(56)

It describes the shift of the bare frequencies of the CTL, as discussed in
section III.

The interaction Hamiltonian describes both internal couplings, namely the
interactions within the same TL, and external couplings, in which modes from
the LHTL and the RHTL interact with each other. We now focus on the internal
coupling terms, starting from

$\displaystyle\mathcal{\hat{H}}_{\textrm{RM}}=2\chi(t)\sum_{\begin{subarray}{c}i,j\\\
i\neq j\end{subarray}}^{N/2}$
$\displaystyle\left[\frac{1}{C_{\textrm{l}}\sqrt{\omega_{i}\omega_{j}}}\hat{a}_{i}\hat{a}_{j}^{\dagger}e^{i(k_{i}-k_{j})\Delta
x}\right.$
$\displaystyle\left.+\frac{1}{C_{\textrm{r}}\sqrt{\upsilon_{i}\upsilon_{j}}}\hat{b}_{i}\hat{b}_{j}^{\dagger}e^{i(p_{i}-p_{j})\Delta
x}\right].$ (57)

This contribution to the Hamiltonian is related to the internal Raman effect.
Indeed, it describes the possibility to annihilate one particle and generate
another particle with a different energy, depending on whether the TL absorbs
from or releases energy to the SQUID. Clearly, the activation of this effect
strongly depends on the time-dependence of the Josephson energy at the SQUID.

The next two terms include the possibility to squeeze the state of a single TL
by tuning the time-dependent Josephson energy. In particular

$\displaystyle\mathcal{\hat{H}}_{\textrm{1S}}=\chi(t)\sum_{\lvert
j\rvert=1}^{N/2}$
$\displaystyle\left[\frac{1}{C_{\textrm{l}}\omega_{j}}\hat{a}_{j}^{2}e^{2ik_{j}\Delta
x}+\frac{1}{C_{\textrm{r}}\upsilon_{j}}\hat{b}_{j}^{2}e^{2ip_{j}\Delta
x}+\textrm{h.c.}\right]$ (58)

describes the squeezing of a single mode of the CTL, generating therefore
photon pairs within the same mode, whereas

$\displaystyle\mathcal{\hat{H}}_{\textrm{2S}}=\chi(t)\sum_{i,j}^{N/2}$
$\displaystyle\left[\frac{1}{C_{\textrm{l}}\sqrt{\omega_{i}\omega_{j}}}\hat{a}_{i}\hat{a}_{j}e^{i(k_{i}+k_{j})\Delta
x}\right.$
$\displaystyle\left.+\frac{1}{C_{\textrm{r}}\sqrt{\upsilon_{i}\upsilon_{j}}}\hat{b}_{i}\hat{b}_{j}e^{i(p_{i}+p_{j})\Delta
x}+\textrm{h.c.}\right],$ (59)

describes the two-mode squeezing, namely the creation of two photons in two
different modes of the same TL.

Finally, we present the two external coupling terms, which have important
implications in the interaction between the LHTL and the RHTL, and for this
reason are largely discussed in the main text. As first, we introduce the
hopping effect, associated with the Hamiltonian contribution

$\displaystyle\mathcal{\hat{H}}_{\textrm{HP}}=$
$\displaystyle\frac{\chi(t)}{\sqrt{C_{\textrm{l}}C_{\textrm{r}}}}\sum_{i,j}^{N/2}\frac{1}{\sqrt{\omega_{i}\upsilon_{j}}}\left(\hat{a}_{i}\hat{b}_{j}^{\dagger}e^{i(k_{i}-p_{j})\Delta
x}+\textrm{h.c.}\right).$ (60)

This term allows the propagation of excitations form one TL to the other one.
Thank to the time-dependence of the Josephson energy, the hopping effect can
occur also between modes having different energies. In this case, this effect
corresponds to an external Raman scattering.

To conclude, the last contribution to the Hamiltonian is

$\displaystyle\mathcal{\hat{H}}_{\textrm{IS}}=-\frac{\hbar
E(t)}{N\sqrt{C_{\textrm{l}}C_{\textrm{r}}}}\left(\frac{2\pi}{\phi_{0}}\right)^{2}\sum_{i,j}^{N/2}\frac{1}{\sqrt{\omega_{i}\upsilon_{j}}}\left(a_{i}^{\dagger}\hat{b}_{j}^{\dagger}+\hat{a}_{i}\hat{b}_{j}\right).$
(61)

This term implies the possibility to squeeze two modes stemming from two
different TL.