# Two-colour spectrally multimode integrated SU(1,1) interferometer

Alessandro Ferreri  Institute for Quantum Computing Analytics (PGI-12),
Forschungszentrum Jülich, 52425 Jülich, Germany  Department of Physics,
Paderborn University, Warburger Strasse 100, D-33098 Paderborn, Germany
Polina R. Sharapova  Department of Physics, Paderborn University, Warburger
Strasse 100, D-33098 Paderborn, Germany

###### Abstract

Multimode integrated interferometers have great potential for both spectral
engineering and metrological applications. However, material dispersion of
integrated platforms constitutes an obstacle which limits the performance and
precision of such interferometers. At the same time, two-colour non-linear
interferometers present an important tool for metrological applications, when
measurements in a certain frequency range are difficult. In this manuscript,
we theoretically develop and investigate an integrated multimode two-colour
SU(1,1) interferometer that operates in a supersensitive mode. By ensuring a
proper design of the integrated platform, we suppress dispersion and thereby
significantly increase the visibility of the interference pattern. We
demonstrate that such an interferometer overcomes the classical phase
sensitivity limit for wide parametric gain ranges, when up to $3*10^{4}$
photons are generated.

##  I Introduction

Quantum Metrology is the subject devoted to the fabrication and optimisation
of high-precision interferometers. The interest on both quantum metrology and
interferometry has been growing rapidly, and nowadays it can be acknowledged
as one of the trend topics in quantum optics of the last decades [1, 2, 3, 4].
Surely, a boost in this direction was brought about by the first detection of
gravitational waves, which provided a further precious tool to our
comprehension of the universe [5].

Conveniently, interferometers could be classified in two classes, linear
interferometers and nonlinear or SU(1,1) interferometers, according to the
presence of passive or active components, respectively [6]. In particular, the
linear interferometer is characterized by passive elements, such as beam
splitters, which maintain the total number of photons throughout the entire
interference process; whereas the nonlinear interferometer consists of active
elements, for example, optical parametric amplifiers (OPAs), triggering the
creation of photons. This fundamental difference affects the strategy that has
to be employed to achieve optimal phase sensitivity, i.e. the precision to
trace the interference pattern [7]. Indeed, on the one hand, to beat the
classical shot noise limit (SNL), the linear interferometer requires the
presence of squeezed or quantum-correlated input states [8], such as NOON
state [9], with a specific photon statistics [10, 11]. On the other hand, due
to the generation of the squeezed vacuum state by the OPAs, the SU(1,1)
interferometer can overcome the SNL without any sort of seed, namely with the
quantum vacuum as the input state [12], thereby potentially reaching the
Heisenberg limit (HL), which is the limitation imposed by the time-energy
uncertainty principle [13].

An excellent candidate for the role of photon source in various linear and
nonlinear interferometric schemes is the parametric down-conversion (PDC),
which is one of the most commonly exploited nonlinear processes for the
creation of squeezed and entangled photons [14, 15, 16]. Photons emitted via
PDC, typically called signal and idler photons, are generated in broadband
spatial and spectral modes, whose role in interferometry has already been
proved both theoretically and experimentally [17, 18, 19, 20, 21, 22, 23]. As
an example, it has been observed that the possibility of controlling the
number of spectral modes, and therefore the amount of spectral/ temporal
correlations between signal and idler photons, can determine the behaviour of
the quantum interference pattern in the four-photon Hong-Ou-Mandel scenarios
[24].

It is therefore reasonable to believe that any specific realisation of the
SU(1,1) interferometer based on realistic nonlinear photon sources should take
the multimode structure of the emitted radiation into account. In this
direction, in [25] it has already been demonstrated how a proper state
engineering [26] of the photon source allows one to reduce the dispersion
within the interferometer and therefore improve both the visibility of the
interference pattern and the accuracy in the phase scanning. In addition,
proper spectral engineering of PDC sources allows the creation of correlated
photons of different colours. During the last decade, the realization of such
non-degenerate PDC sources with specific spectral features has become
widespread [27, 28, 29], while their application in interference systems such
as Hong-Ou-Mandel [30] and nonlinear Mach-Zehnder interferometers [19] seems
to be very promising.

Finally, the interest in integrated devices is recently growing due to both
their small footprint and high efficiency [31, 32, 33]. Such technologies are
already utilized in quantum interference scenarios [34, 35], in the
realization of non-degenerate PDC sources suitable for single-photon heralding
[36], as well as in the generation of polarization-entangled photons [37].

In this manuscript, we theoretically develop a spectrally multimode non-
degenerate integrated SU(1,1) interferometer, whose signal and idler photons
are spectrally distinguishable and orthogonally polarized. As an example, we
consider potassium titanyl phosphate (KTP) waveguides, however, the concept of
such an interferometer can be applied to various integrated platforms. We
demonstrate that the presented device works in the supersensitive mode beyond
the classical limit SNL.

The paper is organised as follow: in section II, we discuss the control of the
photon generation process by means of the poling period to ensure the spectral
distinguishability of signal and idler photons and present a theoretical model
for describing integrated two-colour SU(1,1) interferomters based on such non-
degenerate PDC processes. In section III, we show how dispersion within the
two-colour SU(1,1) interferometer can be suppressed by properly designing an
integrated platform, we analyze the phase sensitivity of such a two-colour
interferometer, and compare the obtained results with the performance of its
degenerate counterpart. Finally, we conclude with section IV and comment on
the quantum features of the presented interferometer.

##  II Theoretical model

The integrated intereferometer is sketched in Fig.1, and consists of two type-
II PDC sources, in particular, two periodically poled potassium titanyl
phosphate (ppKTP) waveguides, separated by a non-poled KTP section. The device
is pumped by a continuous wave (CW) laser.

Figure 1: Schematic model of the two-colour SU(1,1) interferometer. To produce
spectrally non-degenerate and orthogonally polarized photons, the pump laser
interacts with two PDC sections consisting of periodically poled KTP
waveguides of length L. Different polarizations are depicted by the red and
blue colours. The PDC sections are separated by the non-poled KTP section of
length l. Two polarization converters, located at $x_{1}$ and $x_{2}$
positions, switch the polarizations of signal and idler photons in order to
ensure a proper compensation of their group velocities when coming to the
second PDC section. An additional phase for the idler photon is provided and
controlled by the phase modulator PM. The signal photon is finally detected.

The Hamiltonian describing the interferometer is given by [20, 25]:

$\hat{H}=i\hbar\Gamma\int
d\omega_{s}d\omega_{i}F(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}\hat{a}_{i}^{\dagger}+h.c.,$
(1)

where $\Gamma$ is the coupling constant containing the second order
susceptibility $\chi^{(2)}$, the pump intensity $I$ and the length of the PDC
sections $L$; $\omega_{s}$ and $\omega_{i}$ are the frequencies of signal and
idler photons, respectively, while $\hat{a}_{s,i}^{\dagger}$ are the creation
operators. The function $F(\omega_{s},\omega_{i})$ is the joint spectral
amplitude (JSA), whose explicit expression will be obtained in the next
section. The JSA contains both the energy and the momentum conservation of the
PDC process, which are expressed by the following relations:

$\displaystyle\omega_{p}-\omega_{s0}-\omega_{i0}=$ $\displaystyle 0,$ (2)
$\displaystyle
k_{o}(\omega_{p})-k_{o}(\omega_{s0})-k_{e}(\omega_{i0})+\frac{2\pi}{\Lambda}=$
$\displaystyle 0,$ (3)

where $\omega_{p}$, $\omega_{s0}$, $\omega_{i0}$ are the central frequencies
of the pump laser, signal and idler photons, respectively, while $k_{x}$ are
their wave vectors. The indices $o$ and $e$ correspond to the ordinary and
extraordinary polarization. The constant $\Lambda$ presents the waveguide
poling period, which is required to close the phasematching condition (fulfill
the momentum conservation), thereby enabling the pair production. In the
following analysis, we neglect the time ordering effect; which is a fair
approximation for SU (1,1) interferometers, taking into account the effective
narrowing of their spectra due to the nonlinear interference [22].

The lack of degeneracy characterizing two-colour spectra, namely
$\omega_{s0}\neq\omega_{i0}$, is achieved by properly setting the poling
period $\Lambda$ of the PDC section. This concretely means that the degree of
freedom given by the choice of the periodic pole allows us to determine the
central frequencies of the signal and idler photons, and consequently the
separation of their spectra. In order to realise the fully spectral
distinguishability of signal and idler photons, we firstly estimate the full
width at half maximum (FWHM) of their spectra $\Delta\omega$ in a case of the
degenerate PDC source with the poling period $\Lambda_{deg}$ [25], note that
the spectra of signal and idler photons are identical in this case due to the
use of the CW laser. Changing the poling period allows signal and idler
photons to be generated at different central frequencies with a proper
spectral separation. To characterize the splitting of the spectra, we impose a
spectral detuning $\delta\omega$, namely a spectral distance between the
degenerate frequency $\omega_{p}/2$ and the central frequency of the signal
(idler) photon, $\omega_{s0}=\omega_{p}/2+\delta\omega$ and
$\omega_{i0}=\omega_{p}/2-\delta\omega$, such that
$\delta\omega\gg\Delta\omega$. This condition ensures the complete spectral
splitting of signal and idler photons. The spectral detuning $\delta\omega$ is
identical for both photons in order to fulfill the energy conservation of the
process. Eventually we consider the spectral detunings which are much smaller
than the degenerate frequency $\omega_{p}/2$, so that
$\Delta\omega\ll\delta\omega\ll\omega_{p}/2$. Then, the new value of the
poling period $\Lambda$, which allows the photon pair creation at frequencies
$\omega_{s0}$ and $\omega_{i0}$, is calculated by means of the momentum
conservation in Eq (3). In contrast to the single-colour SU(1,1)
interferometer described in [25], the current device is characterized by two
polarization converters (PC) which are required for the proper compensation of
the group velocities of signal and idler photons. The presence of two
polarization conversions along the non-poled section enables the preparation
of only one periodic pole. Indeed, if we use only one PC, the lack of
degeneracy would push us to utilize a second waveguide with a poling period
$\Lambda_{2}$ for the second PDC section, different with respect to the poling
period $\Lambda$ of the first PDC section, because of different frequencies of
the signal and idler photons. The presence of the second PC solves the
problem, however, this PC must be properly arranged with respect to the first
PC, whose exact position will be calculated in the next section. For the sake
of simplicity, we can assume the second PC to be located in front of the
second PDC section. The function describing the periodic pole is given by:

$g(z)=\begin{cases}square_{\Lambda}(z)&0<z<L\\\ 1&L<z<l+L\\\
square_{\Lambda}(z-(L+l))&L+l<z<2L+l\end{cases},$ (4)

where the periodic pole of the second PDC is not in phase with the periodic
pole of the first one. This fact results in the generation of a global phase
in the joint spectral amplitude (JSA).

In a general form, the phasematching function of the whole nonlinear
interferometer of length $2L+l$ including several poled regions can be given
by the expression [38, 39, 40, 41]:

$\Phi(\omega_{s},\omega_{i})=\int_{0}^{2L+l}dz\,g(z)e^{i\int_{0}^{z}d\xi\Delta
k(\xi)},$ (5)

where $\Delta k(\xi)$ is the phasematching at the position $(\xi)$. Using
Eq.(4) and taking into account only the first-order terms in the Fourier
decomposition of $g(z)$, the phasematching function describing the whole
interferometer presented in Fig.1 with two periodically poled regions given by
Eq.(4) becomes:

$\displaystyle\Phi(\omega_{s},\omega_{i})=$
$\displaystyle\int_{0}^{L}dz\,e^{i\frac{2\pi
z}{\Lambda}}e^{i\int_{0}^{z}d\xi\Delta k(\xi)}$
$\displaystyle+\int_{L+l}^{2L+l}dz\,e^{i\frac{2\pi(z-L-l)}{\Lambda}}e^{i\int_{0}^{z}d\xi\Delta
k(\xi)},$ (6)

where the phasematching $\Delta k(\xi)$ takes the following expressions in
different regions of the interferometer:

$\Delta k(\xi)=\begin{cases}\Delta
k=k_{o}(\omega_{p})-k_{o}(\omega_{s})-k_{e}(\omega_{i})&0<z<L\\\ \Delta
k^{\prime}=k_{o}(\omega_{p})-k_{o}(\omega_{s})-k_{e}^{\prime}(\omega_{i})&L<z<L+x_{1}\\\
\mkern 1.5mu\overline{\mkern-1.5mu\Delta k\mkern-1.5mu}\mkern
1.5mu=k_{o}(\omega_{p})-k_{e}(\omega_{s})-k_{o}(\omega_{i})&L+x_{1}<z<L+x_{1}+x_{2}=L+l\\\
\Delta
k=k_{o}(\omega_{p})-k_{o}(\omega_{s})-k_{e}(\omega_{i})&L+l<z<2L+l,\end{cases}$
(7)

where $k_{e}^{\prime}(\omega_{i})=k_{e}(\omega_{i})+\delta k_{e}(\omega_{i})$
is the modified wavevector of the idler photon in the phase modulation region.
Such a change of the wavevector can be performed by applying external voltage.
Hence, introducing an average phase imparted by external voltage
$\varphi=\delta k_{e}(\omega_{i0})x_{1}$ with respect to the central frequency
of the idler photon $\omega_{i0}$, one can define the difference of optical
paths in the phase modulation region as $\Delta k^{\prime}x_{1}=\Delta
kx_{1}+\delta k_{e}(\omega_{i})x_{1}=\Delta kx_{1}+\varphi\frac{\delta
k_{e}(\omega_{i})}{\delta k_{e}(\omega_{i0})}\approx\Delta kx_{1}+\varphi$,
since the relation $\frac{\delta k_{e}(\omega_{i})}{\delta
k_{e}(\omega_{i0})}\approx 1$ holds in the vicinity of $\omega_{i0}$.

We now evaluate two integrals in Eq.(6). The first integral provides the JSA
of the first PDC section:

$\displaystyle\int_{0}^{L}dz\,e^{i\frac{2\pi
z}{\Lambda}}e^{i\int_{0}^{z}d\xi\Delta k(\xi)}=$
$\displaystyle\int_{0}^{L}dz\,e^{i\frac{2\pi z}{\Lambda}}e^{\Delta kz}$
$\displaystyle=$ $\displaystyle L\operatorname{sinc}\left[\frac{\Delta\beta
L}{2}\right]e^{i\frac{\Delta\beta L}{2}},$ (8)

where $\Delta\beta=\Delta k+\frac{2\pi}{\Lambda}$.

The second integral can be evaluated as follows:

$\displaystyle\int_{L+l}^{2L+l}dz\,e^{i\frac{2\pi(z-L-l)}{\Lambda}}e^{i\int_{0}^{z}d\xi\Delta
k(\xi)}=$
$\displaystyle\int_{L+l}^{2L+l}dz\,e^{i\frac{2\pi(z-L-l)}{\Lambda}}e^{i\Delta
kL}e^{i\Delta k^{\prime}x_{1}}e^{i\mkern 1.5mu\overline{\mkern-1.5mu\Delta
k\mkern-1.5mu}\mkern 1.5mux_{2}}e^{i\Delta k(z-x_{1}-x_{2}-L)}$
$\displaystyle=$ $\displaystyle L\operatorname{sinc}\left[\frac{\Delta\beta
L}{2}\right]e^{i\frac{\Delta\beta
L}{2}}e^{i\frac{2\pi(-L-l)}{\Lambda}}e^{i(\Delta k^{\prime}-\Delta
k)x_{1}}e^{i(\mkern 1.5mu\overline{\mkern-1.5mu\Delta k\mkern-1.5mu}\mkern
1.5mu-\Delta k)x_{2}}e^{i\Delta\beta(L+l)}.$ (9)

Using the definition of the averaged phase introduced above and collecting
Eqs.(6), (8) and (9), one can write the expression for the joint spectral
amplitude, which under the condition of the CW pump has the form:

$\displaystyle F(\omega_{s},\omega_{i})=$ $\displaystyle
C\delta(\omega_{p}-\omega_{s}-\omega_{i})\operatorname{sinc}\bigg{[}\frac{\Delta\beta
L}{2}\bigg{]}$ $\displaystyle\times e^{i\frac{\Delta\beta
L}{2}}\bigg{[}1+e^{i\varphi}e^{i\mkern 1.5mu\overline{\mkern-1.5mu\Delta
k\mkern-1.5mu}\mkern 1.5mux_{2}}e^{i\Delta k(L+x_{1})}\bigg{]},$ (10)

where the $\delta$-function indicates the CW pump, while $C$ is the
normalization constant. We notice that the length of the PDC source within the
sinc-function fashions the width of the joint spectral intensity (JSI) and
therefore the FWHM of the spectra of both signal and idler photons. A pondered
choice of L therefore plays a crucial role in the realisation of the spectral
distinguishability of signal and idler photons. The spectra of the signal
(idler) photons can be calculated as follows

$\displaystyle I(\omega_{s(i)})=\int
d\omega_{i(s)}|F(\omega_{s},\omega_{i})|^{2},$ (11)

and are presented in Fig.2 for the length L=8mm and the poling period
$\Lambda=133\mu m$.

Figure 2: Normalized spectra of (a) signal and (b) idler photons. The FWHM of
both is $\Delta\omega$= 2*$10^{12}$ rad/s and the spectral detuning of both is
$\delta\omega$=10*$10^{12}$ rad/s. The following parameters were used for
calculations: L = 8 mm, $x_{1}\simeq 1.038$ mm, $x_{2}\simeq 8.962$ mm,
$\Lambda=133\mu m$, pump wavelength $\lambda_{p}$ = 766 nm,
$\phi=\varphi+\Phi=0$ (see Eq. (LABEL:Phi) for more details).

##  III Dispersion suppression and phase sensitivity

The regime of destructive interference of SU(1,1) interferometers is very
important for reducing noise and improving phase sensitivity [7]. Analyzing a
degenerate multimode SU(1,1) interferometer, we have already found that the
shape of the JSI and the number of generated photons at different imparted
phase $\varphi$ are strongly conditioned by the arrangement of various
components within the integrated device [25]. In this section, to improve the
phase sensitivity of the non-degenerate two-colour SU(1,1) interferometer, we
elaborate a strategy for reducing the number of generated photons in the
destructive interference regime by properly placing polarization converters.

We start by noticing that according to the last line of Eq.(10), the JSA is
modulated by a cosine function, whose argument depends on $\varphi$. However,
the two exponential terms in the end of the second line explicitly depend on
the frequencies of both signal and idler photons; this causes by dispersion
that prevents the suppression of photon generation in the destructive
interference regime. Nevertheless, the use of proper positions of polarization
converters $x_{1}$ and $x_{2}$ allows us to compensate for the first-order
dispersion terms and thereby reduce the arguments of such exponential factors
to a mere additional global phase.

To find such positions, we apply the Taylor expansion to the wavevectors of
the signal and idler photons around their central frequencies $\omega_{s0}$
and $\omega_{i0}$, respectively. In the zeroth order we have

$\Delta k^{(0)}(L+x_{1})+\mkern 1.5mu\overline{\mkern-1.5mu\Delta
k\mkern-1.5mu}\mkern
1.5mu^{(0)}x_{2}=-\frac{2\pi}{\Lambda}(L+x_{1})+Kx_{2}\equiv\Phi,$ (12)

where $K=k_{o}(\omega_{p})-k_{e}(\omega_{s0})-k_{o}(\omega_{i0})$. One can
observe that the zero-order term results in a global phase $\Phi$. In the
following consideration, we will shift the value of the phase $\varphi$ by the
global phase $\Phi$, in order to have the maximum of JSA Eq.(10) for the phase
$\phi=\varphi+\Phi=0$.

In the first order, one can write the following system of equations:

$\displaystyle\Delta k^{(1)}(L+x_{1})+\mkern 1.5mu\overline{\mkern-1.5mu\Delta
k\mkern-1.5mu}\mkern 1.5mu^{(1)}x_{2}=0,$ (13) $\displaystyle x_{1}+x_{2}=l.$
(14)

By exploiting the presence of strong frequency correlations between signal and
idler photons caused by the use of CW pump laser,
$\omega_{i}=\omega_{p}-\omega_{s}$, Eq.(13) becomes:

$\left(-\frac{\Omega}{v_{o,b}}+\frac{\Omega}{v_{e,r}}\right)(L+x_{1})+\left(-\frac{\Omega}{v_{e,b}}+\frac{\Omega}{v_{o,r}}\right)x_{2}=0,$
(15)

where $\Omega=\omega_{s}-\omega_{p}/2-\delta\omega$ is the frequency detuning,
and $v=(\partial k/\partial\omega)^{-1}$ are the group velocities calculated
with respect to the central frequencies of photons. Here we determine the
central frequencies of photons by the indices $r$ and $b$, namely, $r$
corresponds to the ”red” idler photon
$\omega_{r}\equiv\omega_{i0}=\omega_{p}/2-\delta\omega$, while $b$ \- to the
”blue” signal photon $\omega_{b}\equiv\omega_{s0}=\omega_{p}/2+\delta\omega$.
The solution of the coupled equations (14),(15) allows us to find the proper
positions of both polarization converters in order to suppress the first-order
dispersion terms and obtain quasi-perfect interference at the output of the
two-colour SU(1,1) interferometer:

$\displaystyle x_{1}=$
$\displaystyle\frac{v_{e,r}v_{o,b}(v_{e,b}-v_{o,r})l+v_{e,b}v_{o,r}(v_{o,b}-v_{e,r})L}{v_{e,b}v_{e,r}v_{o,b}+v_{e,b}v_{e,r}v_{o,r}-v_{e,b}v_{o,b}v_{o,r}-v_{e,r}v_{o,b}v_{o,r}},$
(16) $\displaystyle x_{2}=$
$\displaystyle\frac{v_{o,r}v_{e,b}(v_{e,r}-v_{o,b})(l+L)}{v_{e,b}v_{e,r}v_{o,b}+v_{e,b}v_{e,r}v_{o,r}-v_{e,b}v_{o,b}v_{o,r}-v_{e,r}v_{o,b}v_{o,r}}.$
(17)

Figure 3: Normalized spectra of signal photons at (a) $\phi=0$, (b)
$\phi=\pi/2$, and (c) $\phi=\pi$. The normalization is performed with respect
to the maximum intensity in the constructive interference case, when $\phi=0$.
The following parameters are chosen and fixed for all further calculations:
$\lambda_{p}$ = 766 nm, L = 8 mm, $x_{1}\simeq 1.038$ mm, $x_{2}\simeq 8.962$
mm, $\Lambda=133\mu m$. The choice of $\delta\omega=10^{13}$rad/s ensures the
fully spectral distinguishability of signal and idler photons.

Fig.3 presents the signal spectra normalized to the maximum intensity for
three different phases corresponding to the regimes of constructive $\phi=0$
(Fig.3 a) and destructive $\phi=\pi$ (Fig.3 c) interference, as well as to the
intermediate regime with the phase $\phi=\pi/2$ (Fig.3 b). As it can be
observed in Fig.3c, the intensity of the signal photon in the destructive
interference case is drastically reduced with respect to the constructive
interference regime, although a noise at large frequency detunings
$\lvert\Omega\rvert>0$ is present. The presence of this noise is caused by the
fact that our technique for compensating the group velocities of signal and
idler photons succeeds in the proximity of the central frequencies (namely
where the main peaks of the intensity spectra are expected), whereas it fails
for frequencies far from $\omega_{s0}$ and $\omega_{i0}$, resulting in the
emergence of residual photons. The same issue has been already observed in the
degenerate SU(1,1) interferometer [25]. In both degenerate and two-colour
frameworks, the presence of this residual radiation hinders the perfect
interference, and therefore affects the precision of the interferometer, which
can be estimated by means of the phase sensitivity:

$\lvert\Delta\phi\rvert=\bigg{\lvert}\frac{\Delta N}{\partial\langle
N\rangle/\partial\phi}\bigg{\rvert},$ (18)

where $\Delta N$ is the standard deviation of the number of photons, while
$\langle N\rangle$ is the mean number of photons.

Fig.4 shows the trend of the phase sensitivity calculated with respect to the
signal photons and normalized to the shot noise limit (SNL) when varying the
internal phase of the interferometer. The shot noise limit, that determines
the classical phase sensitivity bound, is calculated relative to the number of
photons inside the interferometer $\langle N_{in}\rangle$:

$\lvert\Delta\phi_{SNL}\rvert=\frac{1}{\sqrt{\langle N_{in}\rangle}}.$ (19)

Since the phase sensitivity is a symmetrical function with respect to the
point $\phi=\pi$, we only report the range $[0,\pi]$. As already experienced
in [25], the normalized phase sensitivity diverges in the proximity of both
$\phi=0$ and $\phi=\pi$. The presence of these divergences can be
mathematically clarified by looking at Eq.(18): At $\phi=0$ both the number of
photons and its variance reach the maximum value, whereas the derivative is
identically equal to zero; at $\phi=\pi$, the presence of residual photons
prevents the variance of the photon number dropping to zero, whereas the
derivative of $\langle N\rangle$ is identically equal to zero, being at the
point of minimum intensity.

Figure 4: The phase sensitivity normalized to SNL versus phase at different
gains. The SNL is shown by the black line.

As can be seen from Fig.4, the presented interferometer can perform the phase
scan in a supersensitive mode, where the phase sensitivity overcomes the
classical limit set by SNL. One can observe that the normalized phase
sensitivity dramatically depends on the parametric gain, which is defined by
$\gamma=G(\phi=0)\sqrt{\lambda_{1}(\phi=0)}$, where $G(\phi=0)$ includes both
the coupling constant of the PDC process and the normalization factor of the
JSA, and $\lambda_{1}(\phi=0)$ is the first eigenvalue of the spectral
Schmidt-mode decomposition of JSA [42]. Both the parameter G and the first
Schmidt eigenvalue $\lambda_{1}$ are evaluated at the maximum of intensity
when $\phi=0$ [25]. We stress that by varying the gain opportunely, we can let
the number of photons to vary from $\langle N(\phi=0)\rangle\approx 0.12$ for
$\gamma\simeq 0.04$, to $\langle N(\phi=0)\rangle\approx 10^{9}$ for
$\gamma\simeq 10.0$.

11footnotetext: Since the primary interest of this work is to develop a
multimode description of two-colour integrated SU(1,1) interferometers, we
decided to avoid the consideration of losses and focus our investigation on
the role of dispersion in the phase estimation problem. Accounting for losses,
whose influence on the phase sensitivity is already well-known in literature,
would require a considerable extension of the existing theory.

The dependence of the normalized phase sensitivity on the gain is underlined
in Fig.5. It can be seen that the two-colour SU (1,1) interferometer operates
in the supersensitive regime for the same gain region as the degenerate
(single-colour) SU (1,1) interferometer [25]. Moreover, the phase sensitivity
of both interferometers behaves in the same way, and its degradation is
observed with an increase in the gain. We attribute the worsening of the
performance at higher gains to the high-order dispersion terms, which cause
the emergence of the central peak at $\phi=\pi$ in Fig. 4, and prevent the
improvement of the phase sensitivity with increasing gain, in contrast to the
single plane-wave mode scenarios [43, 44, 45, 46], where the worsening of the
phase sensitivity is associated only with the presence of internal losses [47,
48, 49, 50, 51]. However, despite deterioration in phase sensitivity at high
gains, we notice that the presence of a supersensitivity range is maintained
at gain values of about 4, namely, when the interferometer produces about
3*$10^{4}$ photons.

Figure 5: The minimum values of the normalized phase sensitivity presented in
Fig.4 versus gain $\gamma$ in comparison with the degenerate case reported in
[25]. The higher the gain is, the faster the phase sensitivity grows. The SNL
is plotted by the black line.

##  IV Conclusion

In this paper we modelled a high-performance integrated SU(1,1) interfermeter
whose output photons are characterized by frequency distinguishability. The
spectral separation of signal and idler photons was achieved by the correct
choice of the periodic pole of both PDC sources composing the interferometer.
At the same time, a compensation of the group velocities of photons was
preformed by using two polarization converters placed at the appropriate
positions inside the interferometer, that results in a suppression of
dispersion and, consequently, leads to an improvement in the accuracy of the
interferometer. We demonstrated that the phase sensitivity of the presented
multimode non-degenerate interferometer overcomes the classical limit and
reaches the same magnitudes as in the case of a degenerate SU(1,1)
interferometer.

Our results show that the possibility of fully controlling the intensity of
the output radiation, the spectral characteristics and distinguishability of
signal and idler photons offers a high degree of manipulation, and makes this
device a useful photon source for a large variety of optical scenarios.
Moreover, the discussed interferometer benefits from its reduced footprint due
to the integrated design as well as from its outstanding performance in the
phase scanning. The latter is confirmed by the fact that the phase sensitivity
of the interferometer can overcome the shot noise limit, while the
interferometer can still generate a considerable amount of photons. This
specific feature of the presented interferometer is the result of a
combination of two factors: the use of a CW laser and the presence of two
polarization converters. By properly locating these polarization converters,
we were able to reduce dispersion and drastically decrease the number of
photons in the destructive interference regime and thereby maximize the
visibility of the interference pattern.

The presented two-colour SU(1,1) interferometer can be of wide interest for
metrological applications where the measurement over one of the photon is
challenging. The efficiency of this interferometer can be further tested by
making use of different seeding strategies. Furthermore, we leave the
realisation of the high-performance two-colour SU(1,1) interferometer, whose
emitted photons preserve both frequency and polarization entanglement, for
future work. Finally, this model will soon be upgrated by accounting for
internal losses, which typically further erodes the efficiency of
interferometers.

##  V Acknowledgements

We acknowledge the financial support of the Deutsche Forschungsgemeinschaft
(DFG) via TRR 142/2, project C02 and via project SH 1228/3-1. We also thank
the PC2 (Paderborn Center for Parallel Computing) for providing computing
time.

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# Wave manipulation using a bistable chain with reversible impurities

Hiromi Yasuda  Department of Mechanical Engineering and Applied Mechanics,
University of Pennsylvania, Philadelphia, PA 19104, USA  Efstathios G.
Charalampidis  Mathematics Department, California Polytechnic State
University, San Luis Obispo, CA 93407-0403, USA  Prashant K. Purohit
Department of Mechanical Engineering and Applied Mechanics, University of
Pennsylvania, Philadelphia, PA 19104, USA  Panayotis G. Kevrekidis  Department
of Mathematics and Statistics, University of Massachusetts, Amherst, MA
01003-4515, USA  Jordan R. Raney 111Corresponding author Department of
Mechanical Engineering and Applied Mechanics, University of Pennsylvania,
Philadelphia, PA 19104, USA

###### Abstract

We systematically study linear and nonlinear wave propagation in a chain
composed of piecewise-linear bistable springs. Such bistable systems are ideal
testbeds for supporting nonlinear wave dynamical features including transition
and (supersonic) solitary waves. We show that bistable chains can support the
propagation of subsonic wavepackets which in turn can be trapped by a low-
energy phase to induce energy localization. The spatial distribution of these
energy foci strongly affects the propagation of linear waves, typically
causing scattering, but, in special cases, leading to a reflectionless mode
analogous to the Ramsauer-Townsend (RT) effect. Further, we show that the
propagation of nonlinear waves can spontaneously generate or remove additional
foci, which act as effective “impurities”. This behavior serves as a novel
mechanism for reversibly programming the dynamic response of bistable chains.

###### pacs:

##  I Introduction

Multistable mechanical metamaterials have attracted increasing attention from
various research communities in physics, engineering, and materials science,
both because of their unique static and dynamic behavior Shan et al. (2015);
Rafsanjani et al. (2015); Bertoldi et al. (2017). Indeed, multistability has
been leveraged to achieve controlled state changes in engineering applications
such as reconfigurable robots Hawkes et al. (2010); Chen et al. (2018),
deployable space structures Seffen and Pellegrino (1999), medical devices
Kuribayashi et al. (2006), and many others.

Even simple discrete systems composed of bistable elements (i.e., bistable
lattices) are capable of rich and varied nonlinear dynamics. For example, by
tailoring the energy landscape of a bistable element, a bistable lattice can
allow the propagation of transition fronts Nadkarni et al. (2014, 2016); Raney
et al. (2016); Jin et al. (2020); Yasuda et al. (2020) where the propagating
directions and speed can be manipulated. In addition, recent studies have
shown that a bistable chain can support the propagation of solitary waves Katz
and Givli (2018, 2019); Vainchtein (2020).

Here, we study the in situ manipulation of the dynamic response of a bistable
chain in the broader context of linear/nonlinear wave dynamics. In particular,
we showcase a mechanism for controllably inserting reversible effective
“impurities” in a bistable chain, arising from the trapping of propagating
wavepackets. We numerically investigate the amplitude-dependent behavior of a
bistable chain that can support not only supersonic solitary waves but,
depending on the impact speed, also subsonically propagating wavepackets. We
find that the propagation of the latter is brought to a halt in the bulk of
the chain, leading to the formation of effective impurities (albeit within a
homogeneous medium) that may play a central role in subsequent dynamics. More
specifically, we demonstrate that the presence of such energy foci affects the
scattering of linear waves. Interestingly, if two such effective impurities
are placed next to each other and the system is driven at a specific frequency
that leads to the propagation of a linear wave of a particular wavenumber,
complete transmission without reflection can occur in a bistable lattice
Martínez et al. (2016). This is reminiscent of the Ramsauer-Townsend (RT)
effect, which is well-known in quantum mechanics Sakurai (1994), the simplest
instance of which involves the perfect transmission at select energies in one-
dimensional scattering from a square well. Multiple impurities can be created
via collisions of a propagating wavepacket with an impurity. Note that
hereafter we will use the term impurity in this loose sense, implying the
presence of a standing wave, even though no spatial inhomogeneity is present
in the lattice.

Our presentation is organized as follows: Section II discusses the model
setup, explains the dynamic response of a 1D bistable chain under impact
excitation, and identifies the formation of impurities due to the trapping of
a subsonic wavepacket. In Section III, we investigate the scattering behavior
between plane waves and single impurities. Section IV explores scattering for
the case with multiple impurities. Finally, Section V summarizes our findings
and presents directions for future studies.

Figure 1: (a) Schematic illustration of the system composed of bistable
springs. The impact is applied to the left-most unit ($n=1$) by setting an
initial velocity. The right-most mass ($n=N$) is connected to a dashpot. (b)
The relationship between force and strain of a bistable spring is expressed by
a trilinear function [cf. Eq. (1)]. The stiffness parameters are
$(K_{I},K_{S},K_{II})=(1,-1,1.5)$ and the strain parameters are
$(r_{S},r_{II})=(0.4,1)$. (c) The bistable energy landscape obtained from the
trilinear force-strain relationship of panel (b).

##  II Bistable chains

Figure 2:  Amplitude-dependent behavior for a bistable chain composed of
strain-hardening springs ($K_{II}/K_{I}=1.5$). All unit cells are initially in
Phase I. We apply an impact velocity to the first unit cell ($n=1$) by setting
$v_{1}(t=0)$. We consider four different values of the impact velocities: (a)
$v_{1}(t=0)=0.1$, (b) $0.552$, (c) $2$, and (d) $15$. The color indicates the
strain $r_{n}$ as a function of the unit index $n$ and time $t$.

We begin our discussion by examining the dynamic response of a 1D lattice
under an elastic impact excitation caused by a striker (see Fig. 1(a)). We
define $u_{n}$ as the relative displacement of the $n$th particle measured
from its equilibrium point. We define the strain of the $n$th particle as
$r_{n}\coloneqq u_{n}-u_{n+1}$. Motivated by the reconfigurability of the
bistable lattice we consider the trilinear spring element (see Fig. 1(b))
defined by

$F(r)=\left\\{\begin{array}[]{lcr}{{K}_{I}}r&r\leq r_{a}\\\
K_{S}\left(r-r_{S}\right)&r_{a}\leq r\leq r_{b}\\\
K_{II}\left(r-r_{II}\right)&r_{b}\leq r\end{array}\right.,$ (1)

where $K_{I}$, $K_{S}$, and $K_{II}$ are the stiffness parameters. Therefore,
Eq. (1) is a double-well potential function as shown in Fig. 1(c). In this
study, we use the following set of numerical values for the parameters:
$(K_{I},K_{S},K_{II})=(1,-1,1.5)$, and $(r_{S},r_{II})=(0.4,1)$. By using
these values, we can determine the other strain parameters $(r_{a},r_{b})$.
Here, we define the “Phase I” and “Phase II” regimes corresponding to the
stable states around $r_{I}=0$ and $r_{II}=1$, respectively (see also Fig.
1(c)). These two regimes are connected by the spinodal region corresponding to
the negative stiffness range (see the shaded gray area in Fig. 1(b)).

Based on this trilinear spring element, the dynamic response of our bistable
chain is analyzed by using the following equation of motion:

${m}{{\ddot{u}}_{n}}=F\left({{u}_{n-1}}-{{u}_{n}}\right)-F\left({{u}_{n}}-{{u}_{n+1}}\right),$
(2)

where $m$ is the mass of the particle whose value is fixed to $m=1$, and
$n=1,\dots,N$ with $N$ being the number of particles in the chain. It should
be noted that we can calculate two sound speeds: $c_{I}=\sqrt{K_{I}/m}=1$ and
$c_{II}=\sqrt{K_{II}/m}=\sqrt{1.5}$ corresponding to Phase I and Phase II,
respectively.

###  II.1 Amplitude-dependent behavior

We analyze strain profiles of propagating waves in a chain consisting of
$N=400$ particles under impact. We numerically simulate the effect of impact
by setting the velocity of the first particle to a non-zero value, that is, we
set $v_{1}(t=0)\neq 0$. All other initial conditions are set to zero (i.e.,
positions $u_{n}(t=0)$ and velocities $v_{n}(t=0)$). Then, we advance Eq. (2)
forward in time by using a standard fourth-order Runge-Kutta method. To
minimize the effect of reflected waves at the other end of the chain
($n=400$), the last particle is connected to a dashpot in which the damping
force is linearly proportional to the velocity of the last particle
($F_{\nu}=\nu v_{N}$ with $\nu=1$). The right end of the chain is attached to
a rigid wall, i.e., $u_{401}(t)\equiv v_{401}(t)=0\,\,\forall t\geq 0$.

We now discuss Fig. 2 which shows the space-time contour plots of the strain
variable for four different values of impact velocities: $v_{1}(t=0)=0.1$,
$0.552$, $2$, and $15$. In particular, if $v_{1}(t=0)=0.1$, we only observe
small-amplitude wave propagation and the chain configuration remains in Phase
I (see Fig. 2(a)). If we increase the impact velocity to $v_{1}(t=0)=0.552$,
we observe the formation of a propagating localized wavepacket with amplitude
(in the strain variable) greater than $r_{S}=0.4$ (see Fig. 2(b)). This
finding indicates that spring elements undergo a transition from Phase I to
Phase II and then back to Phase I as the wavepacket propagates in the chain.
It is noteworthy that the speed of this excitation ($V_{tw}=0.358$) is slower
than that of the small-amplitude wave front (see Fig. 2(a-b)). That is, this
is a subsonic propagating wavepacket. Interestingly, this structure is brought
to a halt before it reaches the end of the chain, and two adjacent spatial
nodes are trapped in Phase II (see the red arrow in Fig. 2(b)), forming the
effective impurity discussed above. Note that we observe this trapping
behavior for the hardening case (i.e., $K_{II}/K_{I}=1.5$), however, the chain
with softening springs (e.g., $K_{II}/K_{I}=0.5$) does not show the formation
of impurities or of propagating excitations (see Appendix A for details).
Although propagation of supersonic solitary waves in a nonintegrable Fermi-
Pasta-Ulam chain Friesecke and Pego (1999); Truskinovsky and Vainchtein
(2014), specifically a bistable chain Katz and Givli (2018, 2019); Vainchtein
(2020), has been reported previously, the formation of impurities arising from
the spontaneous trapping behavior we observe in the present work has been
unexplored, to the best of our knowledge.

If $v_{1}(t=0)$ is further increased to $2$, the chain exhibits phase
transformation of multiple units which propagate from the left end of the
chain to the other end as shown in Fig. 2(c). Finally, when we apply an
extremely large amplitude input to the system ($v_{1}(t=0)=15$), we find both
propagation of a solitary wave ($V_{tw}=1.076$) and a phase transformation
front (see Fig. 2(d)). This solitary wave propagates faster than the (linear)
sonic wave speed (i.e., it is a supersonic solitary wave). Therefore, we find
that this bistable chain can support coherent structures of both subsonic and
supersonic nature, depending on the magnitude of the impact velocity. We will
discuss the trapping of subsonically propagating wavepackets in the next
section.

###  II.2 Trapping behavior

Figure 3: Trapping behavior. (a) The spatial distribution of the subsonic
wavepacket generated by the impact velocity $v_{1}(t=0)=0.552$ is shown at
different instants of time $t$. The waveforms for $t=585$ and $660$ correspond
to the black vertical lines in Fig. 2(b), respectively. Each strain profile is
shifted in the space domain so that the peak of the strain profile is located
at $n=0$ to ease visualization. (b) The strain change of $n=70$ is shown as a
function of time $t$. The shaded gray area indicates the spinodal regime (see
also Fig. 1(c)). To quantify the wave shape, we measure time widths in which
the strain is in the spinodal region, $t_{SpL}$ and $t_{SpR}$, bounded by the
red vertical lines. (c) Surface plot of the energy
$E_{n}(t)=\frac{1}{2}m\dot{u}_{n}^{2}(t)+U_{n}(t)$ where $U_{n}$ is the
bistable potential energy function of Fig. 1(c). (d) The peak energy of the
propagating wavepacket is shown as a function of $t$. $\Delta U$ is the energy
barrier measured from the energy minimum in Phase II (see Fig. 1(c)). The red
vertical line indicates the trapping point at which the peak energy becomes
lower than that of the energy barrier $\Delta U$. (e) We calculate the time
difference $t_{SpR}-t_{SpL}$, which indicates the symmetric (if
$t_{SpR}-t_{SpL}=0$) or asymmetric waveform (otherwise).

To understand this trapping behavior thoroughly, we examine the time evolution
of the subsonic propagating wavepacket by extracting the strain waveforms at
different time frames, especially right before the trapping occurs. In
particular, Fig. 3(a) shows the strain profiles for four different time
frames: $t=450$, $520$, $585$, and $660$, (the shapes at $t=585$ and $660$
correspond to the black vertical lines in Fig. 2(b)). Note that the wave
profiles are shifted in the spatial domain so that the peak of the wave is
located at the center of the domain to ease visualization. The subsonic
localized pattern propagates without any noticeable distortions initially (see
also Fig. 3(b) for the strain change of $n=70$ as a function of time $t$),
although as time passes, the profile features an oscillatory tail as can be
discerned at $t=585$ (just before the wave stops) which is denoted by the red
dashed ellipse in Fig. 3(a). This tail pattern of the wavepacket stems from
its resonance with the linear modes associated with the (zero) background.

Indeed, as the wave propagates the profile loses energy in the form of emitted
radiation. This is investigated in panels (c) and (d) of Fig. 3, which depict
the spatio-temporal evolution of the total energy of the system and the peak
of the energy as a function of time, respectively. As the wave propagates
within the chain, the peak energy remains nearly constant, until an abrupt
drop in the form of radiation immediately prior to trapping (i.e., when the
total energy becomes lower than $\Delta U$, the energy barrier needed to be
overcome to return to Phase I). In addition to these energy considerations, we
analyze the change of the wave shape by considering the strain in the spinodal
region, which is associated with negative stiffness (Fig. 1(b)). To
characterize the wave shape, we measure two time widths, $t_{SpL}$ and
$t_{SpR}$, in which the strain is in the spinodal region (Fig. 3(b)). We plot
the time width difference $t_{SpR}-t_{SpL}$ as shown in Fig. 3(e) in which a
positive value of $t_{SpR}-t_{SpL}$ indicates that the strain stays longer in
the spinodal region when the unit goes back to the initial state from Phase
II. The value of $t_{SpR}-t_{SpL}$ is nearly zero initially (i.e., symmetric
wave shape). However, this value increases significantly, most notably right
before trapping. This is indicative of rapid growth of distortion. There are
two competing processes: 1) potential energy release (and conversion to
kinetic) due to the transition from Phase I to Phase II, and 2) potential
energy re-balancing from the wavepacket due to its jumping over the energy
barrier (and also the spinodal region). The propagation of the subsonic
wavepacket shows that these two competing processes are not in the perfect
balance needed for genuine traveling. Rather, the energy lost in the form of
leakage of radiation wavepackets dominates the energy release, as is evidenced
by monitoring the spinodal region. As a result, the wavepacket eventually gets
stuck in Phase II, nucleating a new stable phase due to the metastability (cf.
Refs. Balk et al. (2001); Ngan and Truskinovsky (2002)). These act effectively
as impurities with respect to wave propagation, as described in the next
section.

##  III Scattering between plane waves and impurities

Based on the impurity formation described above, we now turn our focus to the
scattering of a plane wave by such impurities in the system. We analyze the
propagation of linear waves by considering the following linearized equation
of motion posed on an infinite lattice:

$m{{\ddot{u}}_{n}}={{K}_{n-1}}{{u}_{n-1}}+{{K}_{n}}{{u}_{n+1}}-\left({{K}_{n-1}}+{{K}_{n}}\right){{u}_{n}},$
(3)

where $K_{n}$ is $K_{II}$ ($K_{I}$) for impurities (rest of the particles). We
can thus determine the normal modes of the system by introducing the plane
wave ansatz $u_{n}=e^{i\left(kn-\omega t\right)}$ where $k$ and $\omega$ are
the wave number and frequency, respectively (with $i=\sqrt{-1}$). Without
impurities, we can obtain the customary dispersion relation:

$\omega=2\pi f=\sqrt{\frac{2K_{I}}{m}\left[1-\cos(k)\right]}.$ (4)

We systematically study scattering of plane waves by introducing a single
impurity (i.e., only a single spring element is in Phase II) or double
impurity (two adjacent elements in Phase II) into a host chain in which all
elements are initially in Phase I. Figures 4(a) and 4(b) show the schematic
illustrations for a chain with a single impurity ($K_{n=0}=K_{II}$) and double
impurity ($K_{n=0}=K_{n=1}=K_{II}$).

###  III.1 Theoretical analysis

Figure 4: (a-b) Schematic illustrations for a chain with a (a) single and (b)
double impurity. Analytical transmission and reflection coefficients for (c)
single- and (d) double-impurity cases. The two different stiffness ratio cases
(left) $\gamma=0.5$ and (right) $\gamma=2.5$, are shown. The reflection and
transmission coefficients for (e) single- and (f) double-impurity cases are
plotted as a function of stiffness ratio ($\gamma$) and wavenumber ($k$). The
gray dashed lines indicate $|R|=0$ (or $|T|=1$. (e-f)

To examine the effect of impurities on the scattering problem, we employ the
following ansatz:

$u_{n}=\left\\{\begin{array}[]{lcr}e^{i(kn-\omega t)}+Re^{-i(kn+\omega
t)}\,,&\quad\text{if}&n\leq 0\\\ Te^{i(kn-\omega
t)}\,,&\quad\text{if}&n>0\end{array}\right.\,,$ (5)

composed of incident (denoted by $I$ in Fig. 4(a)), reflected ($R$), and
transmitted waves ($T$). This way, $|R|^{2}$ and $|T|^{2}$ are the reflection
and transmission coefficients, respectively. By following the same procedure
used for the granular chain with Hertzian interactions Martínez et al. (2016),
we plug Eq. (5) into Eq. (3), and obtain the linear system of equations:

$\mathbf{Ax}=\mathbf{b},$ (6)

with

$\displaystyle\mathbf{A}$
$\displaystyle=\left[\begin{matrix}\gamma{{e}^{ik}}&\frac{{{\omega}^{2}}m}{K_{I}}+{{e}^{ik}}-1-\gamma\\\
\left(1+\gamma\right){{e}^{ik}}-{{e}^{2ik}}-\frac{{{\omega}^{2}}m}{K_{I}}{{e}^{ik}}&-\gamma\\\
\end{matrix}\right],$ (7a) $\displaystyle\mathbf{x}$
$\displaystyle={{\left[\begin{matrix}T,&R\\\ \end{matrix}\right]}^{T}},$ (7b)
$\displaystyle\mathbf{b}$
$\displaystyle={{\left[\begin{matrix}-\frac{{{\omega}^{2}}m}{K_{I}}+1+\gamma-{{e}^{-ik}},&\gamma\\\
\end{matrix}\right]}^{T}},$ (7c)

for the single-impurity case. Here, we define the stiffness ratio as
$\gamma=K_{II}/K_{I}$. Solving Eq. (6) yields respectively the reflection and
transmission coefficients:

$\displaystyle{{\left|R\right|}^{2}}$
$\displaystyle={{\left|\frac{\left(1-\gamma\right)\left(-1+{{e}^{ik}}\right){{e}^{ik}}}{\left(-1+{{e}^{ik}}\right)-2\gamma{{e}^{ik}}}\right|}^{2}},$
(8a) $\displaystyle{{\left|T\right|}^{2}}$
$\displaystyle={{\left|\frac{\gamma\left(1+{{e}^{ik}}\right)}{\left(1-{{e}^{ik}}\right)+2\gamma{{e}^{ik}}}\right|}^{2}}.$
(8b)

For a chain with a double impurity, we use the following ansatz instead:

$u_{n}=\left\\{\begin{array}[]{lcr}e^{i(kn-\omega t)}+Re^{-i(kn+\omega
t)}&\quad\text{if}&n<0\\\ {{T}_{1}}{{e}^{i\left(kn-\omega
t\right)}}+{{R}_{1}}{{e}^{-i\left(kn+\omega t\right)}}&\quad\text{if}&n=0,1\\\
Te^{i(kn-\omega t)}&\quad\text{if}&n>1\end{array}\right..$ (9)

Here, $|R_{1}|^{2}$ and $|T_{1}|^{2}$ correspond to the reflection and
transmission coefficients inside the double-impurity region (see Fig. 4(b)).
Then, following the same procedure as above, we arrive at

$\displaystyle\mathbf{A}$
$\displaystyle=\left[\begin{matrix}0&{{e}^{2ik}}-\left(1+\gamma-\frac{{{\omega}^{2}}m}{K_{I}}\right){{e}^{ik}}&\gamma&\gamma\\\
0&\gamma{{e}^{ik}}&\gamma{{e}^{ik}}-2\gamma+\frac{{{\omega}^{2}}m}{K_{I}}&\gamma{{e}^{-ik}}-2\gamma+\frac{{{\omega}^{2}}m}{K_{I}}\\\
{{e}^{2ik}}&0&\gamma-\left(\gamma+1-\frac{{{\omega}^{2}}m}{K_{I}}\right){{e}^{ik}}&\gamma-\left(\gamma+1-\frac{{{\omega}^{2}}m}{K_{I}}\right){{e}^{-ik}}\\\
\left(\frac{{{\omega}^{2}}m}{K_{I}}-2\right){{e}^{2ik}}+{{e}^{3ik}}&0&{{e}^{ik}}&{{e}^{-ik}}\\\
\end{matrix}\right],$ (10a) $\displaystyle\mathbf{x}$
$\displaystyle={{\left[\begin{matrix}T,&R,&{{T}_{1}},&{{R}_{1}}\\\
\end{matrix}\right]}^{T}},$ (10b) $\displaystyle\mathbf{b}$
$\displaystyle={{\left[\begin{matrix}\left(-\frac{{{\omega}^{2}}m}{K_{I}}+1+\gamma\right){{e}^{-ik}}-{{e}^{-2ik}},&-\gamma{{e}^{-ik}},&0,&0\\\
\end{matrix}\right]}^{T}}.$ (10c)

In this way, we have

$\displaystyle{{\left|R\right|}^{2}}$
$\displaystyle={{\left|\frac{\left(1-\gamma\right)\left(-1+{{e}^{ik}}\right)\left\\{{{\left(-1+{{e}^{ik}}\right)}^{2}}+2\gamma{{e}^{ik}}\right\\}}{-{{e}^{ik}}+\left(3-4\gamma\right){{e}^{2ik}}-3{{\left(1-\gamma\right)}^{2}}{{e}^{3ik}}+{{\left(1-\gamma\right)}^{2}}{{e}^{4ik}}}\right|}^{2}},$
(11a) $\displaystyle{{\left|T\right|}^{2}}$
$\displaystyle={{\left|\frac{{{\gamma}^{2}}\left(1+{{e}^{ik}}\right)}{{{\left(-1+{{e}^{ik}}\right)}^{3}}-2\gamma\left(2-3{{e}^{ik}}+{{e}^{2ik}}\right){{e}^{ik}}+{{\gamma}^{2}}\left(-3+{{e}^{ik}}\right){{e}^{2ik}}}\right|}^{2}}.$
(11b)

For a single-impurity chain, Fig. 4(c) depicts the reflection (blue) and
transmission (red) efficiency for $\gamma=0.5$ (left panel) and $\gamma=2.5$
(right), as a function of the wavenumber $k$. Both stiffness ratio cases show
similar monotonic increase (decrease) of the reflection (transmission). On the
other hand, the double-impurity case with $\gamma=0.5$ exhibits a non-
monotonic change of reflection and transmission. In particular, we find that
there exists a reflectionless mode (i.e., the reflection coefficient is
identically equal to zero) at a non-zero wavenumber, and complete transmission
takes place at that wavenumber, in a way reminiscent of the Ramsauer-Townsend
effect. However, if we increase the stiffness value from $\gamma=0.5$ to 2.5,
these unique features disappear.

To thoroughly analyze the change of reflection and transmission coefficients,
we plot the latter two as a function of the stiffness ratio $\gamma$ and
wavenumber $k$. These are shown in Figs. 4(e)-(f) for single- and double-
impurity chains, respectively. In these figures, the dashed lines indicate
regions with $|R|=0$ or $|T|=1$. The single-impurity chains show monotonic
changes of reflection and transmission for any stiffness ratios as we increase
or decrease the wavenumber from $k=0$. As far as the double-impurity chains
are concerned, our analysis shows an additional minimum reflection valley and
transmission peak at non-zero wavenumbers if $\gamma\leq 2$.

###  III.2 Numerical simulations

Figure 5: Analysis of transmission. (a) Comparison between analytics (black
line) and numerics (red markers) for a stiffness ratio of $\gamma=0.5$. (b)
Spatio-temporal plots of velocity profiles for excitation frequencies of
(left) $\omega=0.6$ and (right) $\omega=1.0$, respectively. The inset shows
the temporal dependence of the velocity $v_{n}$ for (gray) $n=-269$ and (red)
$n=31$. (c-d) Same as the top panels but for the case corresponding to a
double-impurity chain.

We now verify the above theoretical considerations by solving Eq. (2)
directly. To draw comparisons between the theoretical analysis and numerical
simulations we measure the transmission coefficient numerically by analyzing
velocity profiles of incident and transmitted waves under harmonic excitation
Martínez et al. (2016). For our numerical computations, we consider $N=601$
particles ($n\in[-300,300]$) and embed an effective impurity. The latter is in
the form of a compactly supported nodes at the center of the chain
corresponding to $n=0$ for a single-impurity, and $n=0,1$ for a double-
impurity. We apply harmonic excitation to the left end of the chain ($n=-300$)
in the form of a force input $F_{ex}=F_{0}\sin(\omega t)$ with $F_{0}=10^{-3}$
N. We then calculate the transmission coefficient from numerical simulations,
and analyze the velocity profile of the $n=-269$ particle for incident waves
and that of the $n=31$ particle for transmitted waves (see Appendix B for
details about the calculation of the transmission from direct numerical
simulations).

In Fig. 5, we compare theoretical results with numerical simulations for the
stiffness ratio $\gamma=0.5$. Figure 5(a) shows the transmission coefficients
from numerical simulations (red markers) for the single-impurity case, which
demonstrates excellent agreement with the analytical prediction of Eq. (8b)
(solid black line). For a single-impurity chain, the reflection coefficient
increases with frequency (i.e., wavenumber). In particular, we observe
reflected waves with larger amplitude for a value of the excitation frequency
of $\omega=1.0$, compared with the case with $\omega=0.6$, as shown in Fig.
5(b). In the case of a double-impurity chain, numerical computations capture a
transmission peak at non-zero excitation frequency of about $\omega=1.0$,
denoted by the gray arrow in Fig. 5(c). In addition, we observe reflected
waves for the $\omega=0.6$ case (see the left panel of Fig. 5(d)), however,
such reflected waves are unnoticeable if $\omega=1.0$ is applied to a double-
impurity chain (see the right panel of Fig. 5(d)), which corresponds to the
reflectionless mode of the RT resonance.

##  IV Chain with multiple impurities

Figure 6: (a) Schematic illustration of a collision of a propagating
wavepacket with an impurity. (b-c) Spatio-temporal plots of strain wave
profiles for the (b) single and (c) double impurity cases. We apply the
initial velocity ($v_{1}(t=0)=0.55$) to the first particle to generate the
subsonic propagating wavepacket.

Figure 7: Collision of a (subsonic) propagating wavepacket with a single
impurity. (a) Surface plot of the energy
$E_{n}(t)=\frac{1}{2}m\dot{u}_{n}^{2}(t)+U_{n}(t)$ where $U_{n}$ is the
bistable potential energy function as shown in Fig. 1(c). The red vertical
line indicates the collision point. (b) The peak energy of the propagating
wavepacket is shown as a function of time.

Figure 8: Scattering by multiple double impurities. (a) The schematic
illustration highlights the application of a harmonic excitation (with
excitation frequency $\omega_{ex}$) to the first unit ($n=1$) of the chain
with multiple double impurities. The stiffness parameters for the host and
impurities are $K_{host}=1$ and $K_{imp}=1.5$, respectively. (b) The
transmission and reflection coefficients for the double impurity with
$\gamma=1.5$ are shown as a function of frequency $\omega$. Surface plots of
strain wave propagation are presented for: (c) $\omega_{ex}=1.10$ and (d)
$\omega_{ex}=1.73$.

With a firm understanding of scattering by single and double effective
impurities, we consider now the formation of multiple impurities in the chain
and demonstrate an interesting application of the reflectionless mode,
specifically for reconfigurable manipulation of linear wave propagation. In
Sec. II, we observed the formation of a (double) impurity due to the trapped
energy transported by a propagating wavepacket. In this section, we show that
the collision of such a wavepacket with an impurity can introduce additional
impurities.

To demonstrate this, we perform numerical simulations by generating a
propagating wavepacket in a chain with a single or double impurity under
impact (see Fig. 6(a)). The stiffness parameters used in this section are
$K_{I}=1$ and $K_{II}=1.5$ with an initial velocity of $v_{1}=0.55$ applied to
the first particle. This generates a wavepacket as shown in Fig. 6(b) which
corresponds to the single-impurity case. We find that a wavepacket penetrates
through a single-impurity but its propagation is stopped before reaching the
end of the chain, leading to an additional double impurity besides the initial
single impurity. Note that the initial impurity is shifted backward by one
spatial node. If a wavepacket collides with a double impurity, then an
additional double impurity is formed once the solitary wave passes through the
impurity region (Fig. 6(c)).

We examine such interactions between a subsonic wavepacket and an impurity by
considering the energy change before and after the collisions. In Fig. 7,
panel (a) shows the spatio-temporal plot of the total energy $E_{n}$ for the
single-impurity case (corresponding to Fig. 6(a)), and we also track the peak
energy of the subsonic wavepacket as shown in panel (b). After the collision,
some of the energy is reflected (see also the energy decrease of the
transmitted wavepacket as shown in panel (b)). However, this reflected energy
is not sufficient to generate another propagating wavepacket, and it partly
gets trapped causing the backward shift of the impurity, which is followed by
the trapping of the transmitted wavepacket. Therefore, our numerical results
demonstrate the feasibility of creating multiple impurities via collisions of
the propagating wavepacket with impurities, instead of directly manipulating
the phase of the individual units.

Having confirmed the formation of multiple impurities in our system, a natural
question to ask is whether the reflectionless mode can be observed in a chain
with multiple impurities (see Fig. 8(a) for a schematic). Figure 8(b) shows
the analytical reflection and transmission coefficients for a chain with
multiple double impurities ($\gamma=1.5$) where we identify the reflectionless
mode at $\omega=1.73$. To examine the scattering between plane waves and
multiple impurities, we consider a chain in which $4$ double impurities are
embedded. Then, we apply two different harmonic excitation inputs to the chain
for comparison: $\omega_{ex}=1.10$ and $1.73$. Here, the amplitude of
excitation force is $F_{0}=0.14$. Figure 8(c) shows the space-time contour
plot of strain wave propagation for an excitation frequency of
$\omega_{ex}=1.10$. As predicted, we observe the reflected waves. Once these
waves reach the left end of the chain, the amplitude of the strain becomes
large enough to trigger a phase transition toward Phase II propagating in the
chain due to the amplitude-dependent nature of our bistable system. On the
other hand, if $\omega_{ex}=1.73$, the wave passes through the impurity region
without noticeable reflected waves and no particle overcomes the energy
barrier necessary for a phase transition. Once again, our detailed
understanding of the scattering problem enables a characterization of the
associated reflectionless dynamics within the chain.

##  V Conclusions & Future Directions

In the present work, we investigated the formation of effective impurities in
a bistable chain arising from the trapping of propagating wavepackets formed
upon impact. We numerically studied the amplitude dependence of the impact
speed on the behavior of linear and nonlinear wave propagation in the bistable
chain whose spring elements exhibit an asymmetric energy landscape. In
particular, we used numerical simulations to investigate the propagation of
wavepackets formed upon impact. Interestingly, our numerical analysis revealed
that due to the formation of an oscillatory tail during time propagation, the
resulting moving wavepacket loses its energy and eventually gets trapped into
a lower energy stationary state.

Based on this emergent localization, we systematically studied the interaction
of impurities with linear waves and moving wavepackets. For the linear wave
case, we examined the scattering between plane waves and localized structures.
In the system with a single impurity, our analytical and numerical results
showed that the transmission coefficient monotonically decreases as the
harmonic input excitation increases. On the other hand, the bistable chain
with two adjacent units in the lower energy phase exhibits a reflectionless
mode due to the analog of the well-known Ramsauer-Townsend effect.

Also, we explored the interaction between a single/double impurity and a
propagating wavepacket. The latter can pass through the impurity, instead of
being reflected or merged into the existing impurity, however, the wavepacket
eventually gets trapped in a low-energy state, which creates an additional
impurity in the chain. Based on this feasibility of adding multiple
impurities, together with the reflectionless mode, we further demonstrated
that a phase transition front, or a reflectionless propagation can be
“engineered” in the chain with multiple double impurities, depending on the
input harmonic frequency. Given that the impurities created by moving
wavepackets in a bistable chain can change the linear/nonlinear wave dynamics,
such bistable systems can be useful for controlling a reconfigurable structure
in a flexible manner. It should be noted in passing that such impurities can
be removed from the system by generating the propagation of phase transition
fronts Yasuda et al. (2020) (see also Appendix C for numerical
demonstrations). This in principle allows reprogramming of the system’s linear
dynamic response.

This work paves the path for future efforts. From the theoretical perspective,
the analytical derivation of exact traveling solutions in the present setup
would be an interesting problem to pursue by using techniques such as those
discussed in Refs. Vainchtein (2010); Vainchtein and Kevrekidis (2012) for a
diatomic lattice (see also Truskinovsky and Vainchtein (2014)). On the other
hand, a systematic existence/bifurcation and stability analysis of traveling
waves will be of particular relevance to the present setting. To that end,
spectral collocation techniques such as those presented in Ref. Eilbeck and
Flesch (1990) could be employed. An especially important aspect of such
considerations concerns the nature of the propagating subsonic wavepackets.
While these spontaneously emerge as a result of the initial impact, they also
shed energy away only to eventually (and spontaneously) transform themselves
to compactly supported standing excitations. A further understanding of
potential traveling solutions associated with these wavepackets and of the
energetics of the above process would be especially useful for the further
unveiling of the dynamics of such bistable lattices. Of course, here we have
only considered a piecewise linear problem, while realistic systems could be
smooth, nonlinear variants thereof. It will be especially interesting to
explore which of the ramifications of the present setting (including potential
subsonic traveling patterns, compactly supported standing waves and their
respective stability traits and scattering implications) are particular to the
present setting and which generalize, including, potentially, in higher-
dimensional settings. These directions are currently under consideration.

## Acknowledgements

The authors thank Professor Anna Vainchtein (University of Pittsburgh) for
numerous useful discussions. HY and JRR gratefully acknowledge the support of
ARO grant number W911NF-17–1–0147, DARPA Young Faculty Award number
W911NF2010278, and NSF grant number CMMI-2041410. PKP and JRR acknowledge
support from Penn’s Materials Research Science and Engineering Center (MRSEC)
DMR-1720530. This material is based upon work supported by the US National
Science Foundation under Grant No. DMS-1809074 (PGK).

## Appendix A: Amplitude-dependent phase transition

Figure A1: Amplitude-dependent phase transition behavior. (a) Schematic
illustration for examining the steady-state configuration after an impact is
applied to the chain. We extract the final configurations of the chain at
$t=3000$ and present them as a function of the impact velocity $v_{1}$ for (b)
the softening ($K_{II}/K_{I}=0.5$) and hardening ($K_{II}/K_{I}=1.5$) chains.

To investigate the impact amplitude-dependent phase transition, we conduct
numerical simulations on a chain with $100$ particles by applying various
amplitude impacts as shown in Fig. A1(a). All of the spring elements are
initially in Phase I, and an initial impact is applied to the first particle.
Numerical simulations are performed for $t\in[0,3000]$ to allow sufficient
time to achieve a steady-state configuration. We demonstrate the final
configuration at $t=3000$, representing the steady-state configuration after
impact, as a function of the amplitude of impact velocity. The corresponding
numerical simulation results for the softening case ($K_{II}/K_{I}=0.5$) are
shown in Fig. A1(b), whereas the those for the hardening case
($K_{II}/K_{I}=1.5$) in Fig. A1(c). In the figures, the white and red colors
indicate the final states in Phase I and Phase II, respectively. For the
softening case, the phase transition tends to be localized around the left end
of the chain for smaller amplitude impact, and as we increase the amplitude of
the impact input, the phase transition wave propagates in the chain and most
of the spring elements move to Phase II states.

For the hardening chain, we obtain more interesting behavior, specifically the
formation of a propagating wavepacket. In Fig. A1(c), we find that phase
transition occurs only near the left end of the chain for smaller amplitude
impact, which is similar to the softening case. However, and in the specific
amplitude regime (denoted by the black arrow in the figure), no particle is in
Phase II in the final configuration due to the formation of a (subsonic)
wavepacket propagating from one end to the other one of the chain. If we
increase the impact amplitude from that regime, a (subsonic) wavepacket stops
propagating in the middle of the chain, which forms an impurity, as we
discussed in Sec. II. Note that when the propagation of a wavepacket is
brought to halt, waves propagating toward the left end of the chain are
generated (see the red dashed circle in the inset plot of Fig. A1(c)), which
sometimes induces phase transition behavior.

Figure A2: Numerical simulations to obtain the transmission coefficient. (a)
Spatio-temporal plot of velocity. We embed a single impurity (stiffness ratio
$\gamma=0.5$) in a chain composed of 601 units, and apply harmonic excitation
($\omega=0.4$) to the chain. (b) The temporal distribution of the velocity
profiles $v_{n}$ for: (gray) $n=-269$ and (red) $n=31$.

## Appendix B: Transmission for various stiffness ratios

Figure A3: Analytical transmission compared with numerical simulations for
(a,b) single impurity and (c,d) double impurity cases. We calculate
transmission coefficients for various stiffness ratios: (left) $\gamma=1.5$
and (right) $\gamma=2.5$.

Figure A2(a) shows a spatio-temporal plot of particle velocities for the case
corresponding to $\omega=0.4$. To calculate the transmission coefficient from
numerical simulations, we analyze the velocity profile of the $n=-269$
particle for incident waves and that of $n=31$ particle for transmitted waves.
In Fig. A2(b), we present the temporal distribution of the velocity profiles
$v_{n}$ for the above mentioned values of $n$ where we calculate the average
amplitudes from the steady-state region bounded by the vertical dashed lines.
Let $A_{I}$ and $A_{T}$ be the average amplitudes of the incident waves
measured at $n=-269$ and transmitted waves at $n=31$. Then, we calculate
numerically the transmission as $T_{sim}=A_{T}/A_{I}$.

Besides the case with stiffness ratio $\gamma=0.5$ discussed in Sec. III, we
additionally conduct numerical simulations for $\gamma=1.5$ and $2.5$. In Fig.
A3, we show numerical results for a chain with (a,b) a single-impurity, and
(c,d) a double-impurity. The left (right) two plots are obtained for stiffness
ratio of $\gamma=1.5$ ($\gamma=2.5$). Similarly to the comparison for
$\gamma=0.5$ in the main text, we confirm excellent agreement between
numerical simulation results and analytical transmission coefficients for all
four cases.

## Appendix C: Impurity disappearance via propagation of phase boundary

Figure A4: Effect of multiple double impurities on phase transitions. Spatio-
temporal plots for (a) a homogeneous chain (Phase I) and (b) a chain with
double impurities ($K_{imp}=K_{II}=1.5$). We control the displacement of the
first unit at constant speed $v_{1}=1.2$. The gray (black) dashed lines
indicate the phase boundary (linear wave front). (c) We plot the speed of the
phase boundary ($V_{PB}$) as a function of the applied speed $v_{1}$. The
inset plot shows the difference between the phase boundary speeds for the
homogeneous chain without impurities and the chain with multiple impurities,
which is defined as $\Delta V_{PB}=V_{PB}^{i}-V_{PB}^{h}$.

To further reveal the nature of a bistable lattice, we present in this section
a way to bring a chain with pre-existing effective impurities to a homogeneous
configuration by utilizing the propagation of phase boundaries. To achieve
this, we control the displacement of the first particle at constant speed
$v_{1}$. The numerical parameters used in our analysis are the same as those
of Sec. IV. Figure A4(a) shows the strain profiles for a homogeneous chain in
which all spring elements are initially in Phase I. We apply a constant
velocity of $v_{1}=1.2$ to the first particle in order to control its
displacement. As one can observe in other bistable lattices (e.g., Refs.
Slepyan et al. (2005); Truskinovsky and Vainchtein (2006); Deng et al.
(2020)), phase transitions from Phase I to Phase II take place and the phase
boundary separating the phases (denoted by the gray dashed line) moves at
constant speed in the chain. Note that the linear (sonic) waves propagate
faster than the phase boundary (see the black dashed line). By embedding
multiple double impurities in a host homogeneous chain, we perform the same
analysis. In Fig. A4(b), our numerical results show the propagation of phase
boundaries in a manner similar to the homogeneous case. These phase
transitions bring all spring elements to Phase II. In this way, the bistable
chain can achieve a homogeneous configuration (Phase II) without impurities.
Although the overall strain profiles for the multiple impurities case is
similar to that for the homogeneous case, the propagating phase boundaries
experience phase shifts at the locations of the impurities.

To quantify this phase shifting behavior we calculate the speed of the phase
boundary propagating in the system by fitting a straight line to the phase
boundary (see the gray dashed line in Fig. A4(b)). In Fig. A4(c) the phase
boundary speed $V_{PB}$ is presented as a function of the applied velocity for
displacement control $v_{1}$. The gray and red markers indicate the phase
boundary speed for a homogeneous chain and a chain with multiple impurities,
respectively. The solid black line is the analytical phase boundary speed for
a homogeneous chain Abeyaratne and Knowles (2006); Zhao and Purohit (2014);
Khajehtourian and Kochmann (2020). If the applied velocity is small, we find
that the phase boundary speed $v_{PB}$ (obtained from a chain with impurities)
is higher than that of the homogeneous case because the linear waves propagate
faster in Phase II (impurity locations) due to $K_{II}>K_{I}$. The inset panel
in Fig. A4(c) shows the difference between the homogeneous and multiple
impurities cases, defined as $\Delta V_{PB}=(V_{PB}^{i}-V_{PB}^{h})/c_{I}$,
where $V_{PB}^{h}$ and $V_{PB}^{i}$ are the phase boundary speeds calculated
from the homogeneous and multiple impurities cases, respectively. As was
already mentioned above, the gap between these two cases is larger for smaller
applied velocity and smaller for larger applied velocity, which means that the
propagation of phase boundary is insensitive to multiple impurities in a
chain. This is due to the fact that the phase boundary speed at high impact
velocities approaches the sonic wave speed.

Figure A5: Initialize a chain with multiple impurities under tension. To apply
the tension, we control the displacement of the first particle at constant
speed $v_{1}$. Spatio-temporal plots of the strain wave profiles for (a)
$v_{1}=-0.1$ and (b) $v_{1}=-0.5$.

If we control the displacement of the first particle in the opposite direction
it is also possible to bring all spring elements to Phase I. Figure A5 shows
the numerical simulation results for two different applied velocities: (a)
$v_{1}=-0.1$ and (b) $v_{1}=-0.5$. Interestingly, if the applied velocity is
not sufficiently large, the embedded impurities remain in Phase II as shown in
Fig. A5(a). However, if $v_{1}=-0.5$, the large-amplitude waves are generated
and the impurities are brought back to Phase I. Here, we also observe that the
waves experience phase shifts due to impurities.

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