#  Surfaces of Revolution with Constant Gaussian Curvature in Four-Space

Dang Van Cuong

###### Abstract

In this paper, we show that the constant property of the Gaussian curvature of
surfaces of revolution in both $\mathbb{R}^{4}$ and $\mathbb{R}_{1}^{4}$
depend only on the radius of rotation. We then give necessary and sufficient
conditions for the Gaussian curvature of the general rotational surfaces whose
meridians lie in two dimensional planes in $\mathbb{R}^{4}$ to be constant,
and define the parametrization of the meridians when both the Gaussian
curvature is constant and the rates of rotation are equal.

Mathematics Subject Classification. 53A05, 53C50, 53A35.
Key words and phrases. Surface of revolution, Surfaces with constant Gaussian
curvature.

##  1 Introduction

It is well known that a regular surface in $\mathbb{R}^{3}$ is zero Gaussian
curvature if and only if it is a part of a developable surface. A regular
surface with constant Gaussian curvature, $K,$ is locally isometric to $H^{2}$
provided $K=-1,$ $\mathbb{R}^{2}$ provided $K=0,$ and $S^{2}$ provided $K=1.$
For the surfaces of revolution in $\mathbb{R}^{3},$ it is easy to define the
parametrization of the surfaces with constant Gaussian curvature.
In recent years some mathematicians have taken an interest in the surfaces of
revolution in $\mathbb{R}^{4},$ for example V. Milosheva ([6]), U. Dursun and
N. C. Turgay ([3]), K. Arslan ([1]), …. In [4], V. Milosheva applied
invariance theory of surfaces in the four dimensional Euclidean space to the
class of general rotational surfaces whose meridians lie in two-dimensional
planes in order to find all minimal super-conformal surfaces. These surfaces
were further studied by U. Dursun and N. C. Turgay in [3], which found all
minimal surfaces by solving the differential equation that characterizes
minimal surfaces. They then determined all pseudo-umbilical general rotational
surfaces in $\mathbb{R}^{4}$. K. Arslan et.al in [1] gave the necessary and
sufficient conditions for generalized rotation surfaces to become pseudo-
umbilical, they also shown that each general rotational surface is a Chen
surface in $\mathbb{E}^{4}$ and gave some special classes of generalized
rotational surfaces as examples.
Let $M$ be a spacelike or timelike surface in Lorentz-Minkowski three-space
$\mathbb{R}_{1}^{3}$ generated by a one-parameter family of circular arcs, R.
L${\rm\acute{o}}$pez in [5] shown that if its Gaussian curvature $K$ is a
nonzero constant then $M$ is a surface of revolution, he also described the
parametrizations for $M$ when $K=0.$ In [2], by applying the
$\mathfrak{l}_{r}^{\pm}$-Gauss maps, Cuong defined the parameterizations of
minimal and totally umbilical spacelike surfaces of revolution in
$\mathbb{R}_{1}^{4}$.
In this paper, we introduce the notions of surfaces of revolution in
$\mathbb{R}^{4}$ and in Lorentz-Minkowski $\mathbb{R}_{1}^{4},$ we then give
necessary and sufficient conditions for the Gaussian curvature of these
surfaces to be constant. We then give a differential equation that
characterizes the general rotational surfaces whose meridians lie in two
dimensional planes in $\mathbb{R}^{4}$ with constant Gaussian curvature. In
the case that rates of rotation are equal, we can define the parametrization
of the meridians of these surfaces when its Gaussian curvature is constant.

##  2 Preliminaries

Let $M$ be a semi-Riemannian surface, that is, a semi-Riemannian manifold of
dimension two. For a coordinate system $u,v$ in $M$ the components of the
metric tensor (the coefficients of the first fundamental form) are
traditionally denoted by

$E=g_{11}=\langle\partial_{u},\partial_{u}\rangle,F=g_{12}=\langle\partial_{u},\partial_{v}\rangle,G=g_{22}=\langle\partial_{v},\partial_{v}\rangle.$

Since $M$ is two-dimensional, $T_{p}M$ is the only tangent plane at $p$. Thus
the sectional curvature $K$ becomes a real-valued function on $M$, called
Gaussian curvature of $M.$
Let $u,v$ be an orthogonal coordinate system in a semi-Riemannian surface,
that means $F=\langle\partial_{u},\partial_{v}\rangle=0.$ Then (see
Proposition 4.4, pp. 81, [8])

$K=-\frac{1}{eg}\left[\varepsilon_{1}\left(\frac{g_{u}}{e}\right)_{u}+\varepsilon_{2}\left(\frac{e_{v}}{g}\right)_{v}\right],$

where $e=|E|^{1/2},g=|G|^{1/2}$ and $\varepsilon_{1},\varepsilon_{2}$ are the
sign of $E,G$ , respectively.
If $M$ is a surface immersed in a manifold of constant curvature $C,$ the
normal bundle of $M$ has a orthonormal frame $\\{\nu_{i}\\}_{i=1,\dots,n}$
then by using the equation of Gauss we have

$K=C+\sum_{i=1}^{n}K_{\nu_{i}},$

where $K_{\nu_{i}}$ is $\nu_{i}$-curvature of $M$ associated with $\nu_{i}.$
For more detail, let see [7].
Therefore, if $M$ is a surface immersed in a manifold with constant curvature
we then can define the Gaussian curvature by two ways. In this paper, we only
use the formula of Gaussian curvature in term of the coefficients of the first
fundamental form of $M,$ and apply this formula to define the surfaces of
revolution whose Gaussian curvature are constant.
We now introduce the notion surfaces of revolution in $\mathbb{R}^{4}.$ Let
$C$ be a curve in ${\rm span}\\{e_{1},e_{2},e_{3}\\}$ parametrized by arc-
length

$z(u)=\left(f(u),g(u),\rho(u),0\right),\ u\in I,$ (1)

where $\rho(u)>0$. The orbit of $C$ under the action of the orthogonal
transformations of $\mathbb{R}^{4}$ leaving the plane $Oxy,$

$A=\left[\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\cos v&-\sin v\\\ 0&0&\sin
v&\cos v\end{matrix}\right],\ v\in\mathbb{R},$

is a surface given by

${[SR1]}\qquad{{\rm X}}(u,v)=\left(f(u),g(u),\rho(u)\cos v,\rho(u)\sin
v\right),u\in I,v\in[0,2\pi).$ (2)

Then $[SR_{1}]$ is called surface of revolution in $\mathbb{R}^{4}.$ That
means, $[RS_{1}]$ is orbit of a curve by rotating it around a plane.
We also have the another kind of surface of revolution in $\mathbb{R}^{4},$ it
is the obit of a plane curve rotated around both two planes. This surface is
defined as following. Let $C$ be a regular curve in
$\text{span}\\{e_{1},e_{3}\\}$ parametrized by arc-length

$r(u)=\left(f(u),0,g(u),0\right),\ u\in I,$

and

$B=\left[\begin{matrix}\cos\alpha v&-\sin\alpha v&0&0\\\ \sin\alpha
v&\cos\alpha v&0&0\\\ 0&0&\cos\beta v&-\sin\beta v\\\ 0&0&\sin\beta
v&\cos\beta v\end{matrix}\right],\ v\in\mathbb{R},$

be a subgroup of the orthogonal transformations group on $\mathbb{R}^{4}$,
where $\alpha,\beta$ are positive constants and $(f(u))^{2}+(g(u))^{2}\neq 0$.
The orbit of $C$ under the action of the subgroup $B$ is a surface in
$\mathbb{R}^{4}$ given by

$[SR_{2}]\qquad{\rm X}(u,v)=\left(f(u)\cos\alpha v,f(u)\sin\alpha
v,g(u)\cos\beta v,g(u)\sin\beta v\right),$ (3)

which is called General rotational surface whose meridians lie in two-
dimensional planes. Then $r(u)$ is called meridian and $\alpha,\beta$ are
called the rates of rotation.

Modifying this method, we can introduce the notion surfaces of revolution in
Lorentz-Minkowski space. The Lorentz-Minkowski space $\mathbb{R}^{4}_{1}$ is
the $4$-dimensional vector space
$\mathbb{R}^{4}=\\{(x_{1},\ldots,x_{4}):x_{i}\in\mathbb{R},i=1,\ldots,4\\}$
endowed the pseudo scalar product defined by

$\langle\textbf{x},\textbf{y}\rangle=\sum_{i=1}^{3}x_{i}y_{i}-x_{4}y_{4},$

where $\textbf{x}=(x_{1},\ldots,x_{4}),\textbf{y}=(y_{1},\ldots
y_{4})\in\mathbb{R}^{4}.$
Let $C$ be a spacelike (timelike) curve in
$\text{span}\\{e_{1},e_{2},e_{4}\\}$ parametrized by arc-length,

$z(u)=\left(f(u),g(u),0,\rho(u)\right),\ \rho(u)>0,\ \ u\in I.$

The orbit of $C$ under the action of the orthogonal transformations of
$\mathbb{R}_{1}^{4}$ leaving the spacelike plane $Oxy,$

$A_{S}=\left[\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&\cosh v&\sinh v\\\
0&0&\sinh v&\cosh v\end{matrix}\right],\ v\in\mathbb{R},$

is a surface given by

$[SR_{3}]\qquad{\rm X}(u,v)=\left(f(u),g(u),\rho(u)\sinh v,\rho(u)\cosh
v\right),\ u\in I,\ v\in\mathbb{R}.$ (4)

The surface $[SR_{3}]$ is called the surface of revolution of hyperbolic type
in $\mathbb{R}_{1}^{4}$.
Let $C$ be a spacelike (timelike) curve in
$\text{span}\\{e_{1},e_{3},e_{4}\\}$ parametrized by arc-length,

$z(u)=\left(\rho(u),0,f(u),g(u)\right),\ \rho(u)>0,\ u\in I.$

The orbit of $C$ under the action of the orthogonal transformations of
$\mathbb{R}_{1}^{4}$ leaving the timelike plane $Ozt,$

$A_{T}=\left[\begin{matrix}\cos v&-\sin v&0&0\\\ \sin v&\cos v&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{matrix}\right],\ v\in\mathbb{R},$

is a surface given by

$[SR_{4}]\qquad{\rm X}(u,v)=\left(\rho(u)\cos v,\rho(u)\sin
v,f(u),g(u)\right),\ v\in\mathbb{R}.$ (5)

The surface $[SR_{4}]$ then is called the surface of revolution of elliptic
type in $\mathbb{R}_{1}^{4}.$

##  3 Main Results

The following theorems show that the constant property of Gaussian curvatures
of surfaces of revolution in four-space depends only on the radius of
rotation.

######  Theorem 3.1.

The Gaussian curvature of surface $[SR_{1}]$ is constant if and only if

  1. 1.

$\rho(u)=C_{1}e^{Cu}+C_{2}e^{-Cu},$ provided $K=-C^{2},$

  2. 2.

$\rho(u)=C_{1}\sin(Cu)+C_{2}\cos(Cu),$ provided $K=C^{2},$

  3. 3.

$\rho=C_{1}u+C_{2},$ provided $K=0,$

where $C_{1},C_{2}$ are constant such that for each $u\in I,$ $\rho(u)>0.$

###### Proof..

The coefficients of the first fundamental form of $[SR_{1}]$ are defined

$E=\langle{\rm X}_{u},{\rm X}_{u}\rangle=1,\ F=\langle{\rm X}_{u},{\rm
X}_{v}\rangle=0,\ G=\langle{\rm X}_{v},{\rm X}_{v}\rangle=\rho^{2}.$

Therefore, the Gaussian curvature is defined

$K=-\frac{\rho^{\prime\prime}}{\rho}.$

Solving the equation $K=-C^{2},$ ($C^{2},0$) we have the conclusion of Theorem
3.1. ∎

We have the similar result for surfaces of revolution in $\mathbb{R}_{1}^{4}.$

######  Theorem 3.2.

The Gaussian curvature of Surface $[SR_{3}]$ (or $[SR_{4}]$) is constant,
$K=C,$ if and only if

  1. 1.

$\rho(u)=C_{1}e^{\lambda u}+C_{2}e^{-\lambda u},$ when $\varepsilon C=-2\\\
lambda^{2}<0,$

  2. 2.

$\rho(u)=C_{1}\sin(\lambda u)+C_{2}\cos(\lambda u),$ when $\varepsilon
C=\lambda^{2}>0,$

  3. 3.

$\rho=C_{1}u+C_{2},$ when $C=0,$

where $C_{1},C_{2}$ are constant such that $\rho(u)>0,u\in I$ and
$\varepsilon$ is the sign of $E.$

###### Proof..

The coefficients of the first fundamental form of $[SR_{3}]$ (or $[SR_{4}]$)
are defined

$E=(f^{\prime}(u))^{2}+(g^{\prime}(u))^{2}-(\rho^{\prime}(u))^{2}=\varepsilon,\
F=0,\ G=\left(\rho(u)\right)^{2}>0,$

where $\varepsilon=\pm 1.$ It is similar to the proof of Theorem 3.1, we have
the result of this Theorem.∎

For the general rotational surfaces whose meridians lie in two-dimensional
planes in $\mathbb{R}^{4},$ the following Theorem gives us a differential
equation that characterizes the constant Gaussian curvature surfaces.

######  Theorem 3.3.

The Gaussian curvature of $[SR_{2}]$ is constant if and only if

$f(u)=\frac{\sqrt{G}}{\alpha}\cos\phi(u),\
g(u)=\frac{\sqrt{G}}{\beta}\sin\phi(u),$

where $\phi(u)$ are solutions of the following equation

$G\left(\frac{\sin^{2}\phi}{\alpha^{2}}+\frac{\cos^{2}\phi}{\beta^{2}}\right)(\phi^{\prime})^{2}+2G\sin\phi\cos\phi\left(\frac{1}{\beta^{2}}-\frac{1}{\alpha^{2}}\right)\phi^{\prime}+\frac{(G^{\prime})^{2}}{4G}\left(\frac{\cos^{2}\phi}{\alpha^{2}}+\frac{\sin^{2}\phi}{\beta^{2}}\right)-1=0,$
(6)

and

$G=\left(C_{1}e^{Cu}+C_{2}e^{-Cu}\right)^{2},\ \text{if}\ K=-C^{2},$ (7)
$G=\left(C_{1}\sin(Cu)+C_{2}\cos(Cu)\right)^{2},\ \text{if}\ K=C^{2},$ (8)
$G=\left(C_{1}u+C_{2}\right)^{2},\ \text{if}\ K=0,$ (9)

where $C,C_{1},C_{2}$ are constant such that $G\neq 0$.

###### Proof..

The coefficients of the first fundamental form of $[SR_{2}]$ are defined

$E=(f^{\prime})^{2}+(g^{\prime})^{2}=1,\ F=0,\
G=\alpha^{2}f^{2}+\beta^{2}g^{2}.$ (10)

We have

$K=-\frac{\left(\sqrt{G}\right)_{uu}}{\sqrt{G}}.$

Solving the equation $K=-C^{2}(C^{2},0),$ we have $G.$ Setting

$\alpha f(u)=\sqrt{G}\cos(\phi(u)),\beta g(u)=\sqrt{G}\sin(\phi(u))$

and substituting for (10) we then have (6). ∎

In the case two rates of rotation are equal, $\alpha=\beta,$ we can solve the
equation (6) and find the parametrization of the meridians.

######  Corollary 3.4.

If $\alpha=\beta$ then the Gaussian curvature of $[SR_{2}]$ is constant if and
only if

  1. 1.

In the case $K=C^{2},$

$f(u)=\frac{C_{1}\sin(Cu)+C_{2}\cos(Cu)}{\alpha}\cos\phi(u),g(u)=\frac{C_{1}\sin(Cu)+C_{2}\cos(Cu)}{\alpha}\sin\phi(u)$

where

$\phi(u)=\int\frac{\sqrt{1-C^{2}[C_{1}\cos(Cu)-C_{2}\sin(Cu)]^{2}}}{C_{1}\sin(Cu)+C_{2}\cos(Cu)}du,$

$C_{1},C_{2}$ are constants such that the formula under the integral sign is
defined.

  2. 2.

In the case $K=-C^{2},$

$f(u)=\frac{C_{1}e^{Cu}+C_{2}e^{-Cu}}{\alpha}\cos\phi(u),\
g(u)=\frac{C_{1}e^{Cu}+C_{2}e^{-Cu}}{\alpha}\sin\phi(u),$

where

$\phi(u)=\int\frac{\sqrt{1-C^{2}[C_{1}e^{Cu}-C_{2}e^{-Cu}]^{2}}}{C_{1}e^{Cu}+C_{2}e^{-Cu}}du,$

$C_{1},C_{2}$ are constants such that the formula under the integral sign is
defined.

  1. 3.

In the case $K=0,$

$f(u)=\frac{C_{1}u+C_{2}}{\alpha}\cos\phi(u),g(u)=\frac{C_{1}u+C_{2}}{\alpha}\sin\phi(u),$

where

$\phi(u)=\int{\frac{\sqrt{1-C_{1}^{2}}}{C_{1}u+C_{2}}}du,$

$C_{1},C_{2}$ are constants such that $|C_{1}|\leq 1$ and $C_{1}u+C_{2}\neq
0,\forall u\in I.$

Acknowledgements The author would like to thank Prof. Frank Morgan for his
comment and appended.

## References

  * [1] K. Arslan, B. Bayram, B. Bulca and G. ${\rm\ddot{O}}$zt${\rm\ddot{u}}$rk, General rotation surfaces in $\mathbb{E}^{4}$, Results. Math., (2012), DOI 10.1007/s00025-011-0103-3.
  * [2] D. V. Cuong, $LS_{r}$-valued Gauss maps and spacelike surfaces of revolution in $\mathbb{R}_{1}^{4}$, App. Math. Sci., Vol. 6, (2012), no. 77, 3845 - 3860.
  * [3] U. Dursun, N. C. Turgay, Minimal and Pseudo-Umbilical Rotational Surfaces in Euclidean Space $\mathbb{E}^{4}$, Mediterr. J. Math, (2012), DOI 10.1007/s00009-011-0167-z.
  * [4] G. Ganchev, V. Milousheva, On the Theory of Surfaces in the Four-Dimensional Euclidean Space. Kodai Math. J., 31 (2008), 183-198.
  * [5] R. L${\rm\dot{o}}$pez, Surfaces of constant Gauss curvature in Lorentz-Minkowski three-space, Rocky Mountain Journal of Mathematics, 33 (2003), Number 3, 971-993.
  * [6] V. Milosheva, General rotational surfaces in $\mathbb{R}^{4}$ with meridians lying in two-dimension planes,C. R. Acad. Bulg. Sci., 63, 3, (2010) 339-348.
  * [7] M. Navarro, F. S${\rm\acute{a}}$chez, A theorem of Gauss-Bonnet type in codimension 2 for Riemannian manifolds of even dimension, Abstraction and Application, 1 (2009), 4-17.
  * [8] B. O’Neill, Semi-Riemannian Geometry, Academic Press, Orland 1983.

Dang Van Cuong
Department of Natural Sciences
Duy Tan University
Danang
Vietnam

E-mail address<EMAIL_ADDRESS>


# Analysis and experimental study on the Jumping Chain

Wenyu Wang, Wu-Long Xu, Yang Xu, Xu-Dong Yang  Faculty of Science, Beijing
University of Technology, Beijing, China

###### Abstract

A freely falling chain from a cup at certain height can jump. The process can
be divided into two parts: a stable suspension and an accelerating procedure.
Variational principle and force analysis demonstrate that the shape of stable
suspension is an inverted catenary. The requirement of the jumping and the
parameters to describe the jumping catenary have been studied in detail, and
experiments have been conducted to verify the theoretical analysis. The
physical picture of the falling chain could be useful in certain falling
systems, providing valuable insight into the dynamical system.

###### pacs:

45.50.Dd, 05.45.-a, 47.54.-r, 05.45.Xt

##  I Introduction

Figure 1: The jumping chain.

There is a very interesting physical phenomenon as shown in FIG. 1: a coiled
but not entangled chain is put in a cup. When the end of the chain is pulled
out of the cup, it will fall freely along the wall of the cup. At a certain
point, usually in a sudden moment, the chain can jump up and remain in a
steady arc shape which can last for a certain duration in the air. This
experiment is often demonstrated in college physics classes as an example of
an intriguing classical mechanical problem. The reader can also observe the
details of the jumping in the supplementary video or on the internet website
vedio . In this paper, we attempt to analyze and study the pattern and related
topics such as the height of the jump and the height of the fall, etc.
Experiments will be conducted and the data will be analyzed to verify our
theoretical analysis wxxy .

In fact, the phenomenon is an application of classical mechanics in string,
rope or fluid physics jordan . The jumping rope is a familiar example of a
slender structure that interacts with the fluid through which it moves
aerodynamics . The chain is usually suspended and swung by a mechanical
apparatus, and the pattern is well studied in the literature hanging . This is
because important class of truly 1D problems involves the motion of lines or
filaments embedded in 3D space, such as vortex lines in a defect lines in
liquid crystals, type-II superconductor, or polymers in imposed flows.
Nonlinear partial differential equations describes the dynamics, and the
twisting, breaking, reconnecting, or the knotting, are the essential
topological parts of the chain dynamics topology . However, the jumping chain
studied in this paper, is not suspended but jump from a higher platform,
falling freely in the space under the attraction of gravity. The damping from
the viscosity of the air seems to be a sub-dominant effect in the whole
process. The gravity and the tension inside the chain play an important role.
The subtle point is that, although the matter is moving from the higher
position to the lower ground, the system is not a fluid or a falling stream. A
detailed study will reveal the different dynamics.

In all, the paper is organized as follows: The phenomenology and dynamics of
the jumping chain will be analyzed in Sec. II; The experimental verification
of our analysis will be presented in Sec. III; The conclusion is given in Sec.
IV.

##  II Phenomenology and Theoretical Analysis

###  II.1 Phenomena

First, let’s describe how the jumping phenomenon takes place. As shown in FIG.
1, the chains are coiled layer by layer inside a cup, and it is kept from
becoming entangled (It is difficult to keep all steel balls untangled in the
preparation, and it is one of the dominant disturbing factors of the
experiment.). Then the cup is placed at a certain height. a small length of
chain is taken out of the cup and allowed to fall freely. In the actual
operation, we also tried to use a soft and non-contractible rope to do the
same experiment. The rope can barely jump up but the phenomenon is rather
obscure, so the thin metal chain composed of steel balls is adopted in our
experiment. The steel balls are loosely connected, and the chain is non-
retractable. It means that the chain do not have flexural capacity. This is
the key difference between the chain and the soft rope. The reason will be
discussed in the following context.

The experiments show that the chain jumps to appropriate higher position from
the cup if the cup is set at some specific heights. Due to the complex way of
chain winding in the cup, the starting point of the jump keeps changing, so
the shape of the jumping from cup to the highest point is very complicated. As
for the pattern between the highest point and the falling off point on the
ground, the chain can maintain an almost stable shape near the highest point
on the top. The shape keeps shaking near the ground, but this does not affect
the stable shape on the top.

Figure 2: The sketch map of our analysis on the jumping chain.

Summarizing our observations, we think this physical phenomenon can be divided
into two sub-processes: the accelerating process and the stable suspension
process. Thus our analysis on the phenomenon is set out as shown in FIG. 2, in
which point A is the beginning of the suspension and the end is point B, which
is the collision point on the ground. The point C is the starting point of the
jumping in the cup where the elements of the chain are static. The part from C
to A is the accelerating process. Note that, as discussed above, point B keeps
shaking, and point C also varies quickly. At the same time, the height of
Point C is decreasing. The definitions of the variables such as
$h_{j},~{}h_{i}$ etc. are given in the following context. One can see that the
stable suspension is obviously of the most concern, and according to following
analysis, we find that this is actually the key point to comprehend the
phenomenon. Thus we analyze the stable suspension in the following subsection.

###  II.2 Analysis on the shape of the stable Suspension

At first glance, the jumping chain appears like a parabolic curve. However,
the truth is not so simple as expect. A free-falling mass point follows a
parabolic trajectory. However, for the non-retractable chain, each element on
the suspended chain falls at the same velocity, only the directions vary.
Based on our observations, the coiled chain in the cup almost moves to the
ground at a constant rate. thus it is natural to assume that the suspended
chain can maintain a constant velocity $v_{0}$ for appropriate interval as an
ideal case. Then, in $\Delta t$ time, there is a part of the chain with mass
$\lambda v_{0}dt$ ($\lambda$ is denoted as the line density) in the cup which
is accelerated from 0 to $v_{0}$, At the point A, the chain feels the dragging
tension from the chain at rest in the cup as follows:

$T_{0}=\lim_{\Delta t\to 0}\frac{\lambda v_{0}^{2}\Delta t}{\Delta t}=\lambda
v_{0}^{2}\,.$ (1)

It should be noted that this is merely the ideal case, and the following study
will show that the actual tension at the end of A will be less than $\lambda
v_{0}^{2}$. At point B, the chain element with velocity $v_{0}$ collides with
the ground. If the collision is elastic, the chain will rebound from the
ground with velocity $v_{0}$. Similar analysis gives that impulsive force to
the ground is $2\lambda v_{0}^{2}$. Of course, the collision is not completely
elastic due to the dragging. We shall provide the specific method for dealing
with the impulsive force at point B later.

We find that, very interestingly, the shape of the suspended chain is actually
an inverted catenary. The exploration of the catenary is an important progress
in the history of classical mechanics and mathematics. Our discovery in fact
compensates a block in the catenary theory. Here we briefly show the
derivation of the catenary and explain the reason for the same shape of the
suspended chain. Suppose the shape of the chain is $y=y(x)$, the solution of
the catenary is to find a $y(x)$ such that the potential or center of gravity
of the chain is at its lowest point. Using the variational principle, the
action of the whole system is the center of gravity in the vertical direction

$S=-\int\lambda g(y-c)\sqrt{1+{y^{\prime}}^{2}}dx=\int\mathcal{L}dx,$ (2)

where $g$ is the acceleration of gravity and $c$ is an arbitrary constant,
$y^{\prime}$ is the derivative of $y$. $\mathcal{L}$ is the Lagrangian of the
system

$\displaystyle\mathcal{L}=-\lambda g(y-c)\sqrt{1+{y^{\prime}}^{2}}.$ (3)

The motion for $\mathcal{L}$ gives the non-linear differential equation of the
catenary

$(y-c)y^{\prime\prime}-{y^{\prime}}^{2}-1=0.$ (4)

The solution is

$y=c+\frac{a}{2}(e^{\frac{x}{a}+b}+e^{-\frac{x}{a}-b}).$ (5)

where $a,~{}b$ are the parameters determined by the inputs of the catenary.
The approach described above is the standard derivation of the catenary. Note
that here suspension chain are hanged at any height, thus an additional
constant $c$ are introduced. In fact, we notice that, although the equation of
motion Eq. (4) is nonlinear, it has a discrete symmetry and $y(x)$ can be
transformed

$y(x)\rightarrow Y(x)=-y(x)+2c.$ (6)

$Y(x)$ is also a solution of the catenary equation. Despite its lack of
practical application, the inverted catenary solution is still a valid
solution of the catenary equation. This solution is an unstable extremum point
of variation, which has been largely overlooked in the literature.
Nevertheless, our research has revealed that this inverted solution is indeed
the exact shape of the suspended chain.

For the jumping chain, the potential energy is the same as that of the
catenary. Therefore only the kinetic term needs to be included in the action

$S=\int\mathcal{L}dx=\int\left(\frac{1}{2}\lambda\sqrt{1+{y^{\prime}}^{2}}v_{0}^{2}-\lambda
gy\sqrt{1+{y^{\prime}}^{2}}\right)dx\,.$ (7)

By comparing Eq. (3) and assuming that $v_{0}^{2}$ is a constant, we can
observe that the Lagrangian of the jumping chain is identical to that of the
catenary, with the only difference being that $c$ is taken as a constant

$c=\frac{v_{0}^{2}}{2g}\,.$ (8)

Due to the given conditions, the solution will be a convexity, specifically,
an inverted catenary

$y=c-\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})\,.$ (9)

Figure 3: The force analysis on the elements of the jumping chain.

The difference between the deviation of an ordinary catenary and a jumping
chain is that the chain is constantly in motion. The elements of the chain is
continually jumping out of the cup and falling on to the ground, thus changing
the mass of system. This may lead some readers to be skeptical about the
validity of variational principle. However, the same differential equation of
motion can be derived by the force analysis. As shown in FIG. 3, the tension
in the jumping chain is denoted as $T=T(x)$ and the convention of the forward
direction is defined as from point A to point B. Every element of the chain is
moving in a curvilinear motion with the same speed, so the tangential
acceleration is zero, and the infinitesimal tension $dT$ cancels the
tangential component of gravity

$\displaystyle dT=\lambda
g\sqrt{1+{y^{\prime}}^{2}}dx\frac{y^{\prime}}{\sqrt{1+{y^{\prime}}^{2}}}=\lambda
g\frac{dy}{dx}dx=\lambda gdy\,.$ (10)

Thus the tension in the chain can be expressed as

$T=T_{A}+\lambda gy\,.$ (11)

This is a very interesting result that the tension in the stable jumping chain
changes along with the height of the elements, which is the same as the
tension in the ordinary catenary. To analyze the normal component force of the
infinitesimal chain imposed by tension from the curvature of the chain, we can
look to the catenary or hanged rope for comparison. In the case of the
ordinary catenary, the normal component of the tension is canceled by the
gravity, but in the case of the jumping chain, the normal component force
provides the centripetal force together with the gravity. The geometrical
configuration is shown in FIG. 3 where we investigate the centripetal force
acting on the two chain elements $x-dx\to x$ and $x\to x+dx$. From this, we
can easily derive the equation for the centripetal force

$\displaystyle\frac{2\lambda\sqrt{1+y^{\prime
2}}dxv^{2}}{\rho}=2Td\theta+2\lambda g\sqrt{1+y^{\prime
2}}dx\frac{1}{\sqrt{1+y^{\prime 2}}}\,,$

where $\rho$ is the radius of curvature at point $x$. By substituting $\rho$,
$d\theta$, and tension $T$, we can get

$\displaystyle\frac{2\lambda|y^{\prime\prime}|v_{0}^{2}dx}{1+y^{\prime
2}}=\frac{2(T_{A}+\lambda gy)|y^{\prime\prime}|dx}{1+y^{\prime 2}}+2\lambda
gdx\,.$ (12)

Due to the convexity of the jumping chain

$\displaystyle|y^{\prime\prime}|=-y^{\prime\prime}\,.$ (13)

Eq. (12) are reduced to

$\displaystyle-(T_{A}+\lambda gy-\lambda v_{0}^{2})y^{\prime\prime}+\lambda
gy^{\prime 2}+\lambda g=0\,.$ (14)

Given that $T_{A}$ is the variable to be solved, it is convenient to define

$\displaystyle T_{A}-\lambda v_{0}^{2}=-\lambda gc\,.$ (15)

Then the differential equation of $y$ is

$\displaystyle(y-c)y^{\prime\prime}-{y^{\prime}}^{2}-1=0\,.$ (16)

Proved the same catenary equation again.

Although the analysis of force is a bit more complicate than that of the
variational principle of mechanics, the physical picture is easier to
understand because it is an analysis of forces. We can also see that the
forces at the two ends of the catenary and the jumping chain are different. In
case of the ordinary catenary, the tensions at the two ends are dragging the
chain. Whereas, in the case of the jumping chain, the tension at point A is
dragging the chain, while the tension at point B is pushing the chain. Of
course, this is the ideal case; the jumping process will be further analyzed
in the following.

###  II.3 Analysis on the jumping

First, let’s begin with the explanation for why $T_{A}\neq\lambda v_{0}^{2}$.
$T_{A}=\lambda v_{0}^{2}$ corresponds to the ideal case that each chain
element instantaneously accelerates from 0 to $v_{0}$. In contrast, the actual
situation is that the chain is loosely coiled and the chain element needs an
interval for the acceleration, making $T_{A}$ smaller than $\lambda
v_{0}^{2}$. Analytically, if $c=0$, the shape of the suspended chain line is

$\displaystyle y=-\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})\,,$ (17)

then

$\displaystyle y\leq-a\,.$ (18)

It is evident that $y>0$ solution does not exist, as indicated by Eq. (16),
which a non-linear differential equation. Therefore, if the coordinate system
shown in FIG. 2 is used a $y>0$ solution requires a non-zero $c$. From Eq.
(15), it is clear that $T_{A}$ must be less than $\lambda v_{0}^{2}$. As shown
in Fig. 2, the acceleration process for the upper elements in the stable
suspended shape is added. And the height of the acceleration jumping up
process from point C to point A is measured as $h_{i}$. The jumping height of
the stable suspended chain line is $h_{r}$. The total jumping height is thus
the sum of $h_{i}$ and $h_{r}$

$\displaystyle h_{j}=h_{i}+h_{r}\,.$ (19)

Along with the conventions mentioned above, the falling height (the height of
the cup) of the chain is denoted as $H$ and the width of the jump is $2x_{0}$
as shown in Fig. 2. We will further investigate the relationship between these
quantities for jumping chain and attempt to conduct experiment to verify our
analysis.

The tension at point A is

$\displaystyle T_{A}=\lambda v_{0}^{2}-\lambda gc\,.$ (20)

According to the Eq. (11), the tension $T$ from point A to point B is

$\displaystyle T=\lambda v_{0}^{2}+\lambda gy-\lambda gc\,.$ (21)

As discussed previously, the elastic collision implies that the force imposed
from the ground to the chain at point B will be $2\lambda v_{0}^{2}$. This may
appear suspicious to some readers, as a deceleration process is needed at
point B. When the chain finally fall on the ground, the tension should be less
than $2\lambda v_{0}^{2}$. However, we found that the chain elements can
slightly bounce on the ground, thus the actual impulsion to the ground should
be greater than $\lambda v_{0}^{2}$. To account for the force with more
precision, a constant coefficient $\alpha$ can be introduced

$\displaystyle-\alpha\lambda v_{0}^{2}=\lambda v_{0}^{2}-\lambda gH-\lambda
gc\,.$ (22)

The value of $\alpha$ would be greater than 1 and less than 2. It should be
noted that tension at point B should be negative according the direction
defined above. Since falling height $H$ is much larger than the jumping height
$h_{j}$, the tension fluctuation at point B is not sensitive to stable
suspension above. Then

$\displaystyle v_{0}^{2}=\frac{1}{\alpha+1}g(H+c)\,.$ (23)

This is the relation between the falling height and the velocity of the
suspended chain. The jumping height $y(x=0)$ of the stable suspended chain is

$\displaystyle h_{r}=c-a\,.$ (24)

Then all the exploration focuses on the pending parameter $c$ and $\alpha$,
which will be analyzed in the next section when discussing the experimental
verification. Before that, let us analyze the acceleration process.

As the velocity of the chain element changes over time during the acceleration
process, it seems difficult to do the analysis. However, the time parameter
can be eliminated after some substitutions. The velocity of accelerating
elements is denoted as $v$. According to Fig. 2 and Eq. (11), the tangential
acceleration should be

$\displaystyle\lambda vdt\frac{dv}{dt}=\lambda vdv=dT-\lambda gdy\,.$ (25)

Since the chain elements are loosely connected, the normal acceleration
dynamical equation can be disregarded. It can be seen that the time dependence
$dt$ can be eliminated. This is the mathematical formulation of the
transformation from gravity potential to kinetic energy. Carry out the
integration on both sides, we have

$\displaystyle\int^{v_{0}}_{0}\lambda vdv$ $\displaystyle=$
$\displaystyle\int_{0}^{T_{A}}dT-\int_{-h_{i}}^{0}\lambda gdy$ (26)
$\displaystyle=$ $\displaystyle T_{A}-\lambda gh_{i}\,.$

At point A

$\displaystyle\frac{1}{2}\lambda v_{0}^{2}=\lambda v_{0}^{2}-\lambda
gc-\lambda gh_{i}\,,$

namely

$\displaystyle h_{i}=\frac{v_{0}^{2}}{2g}-c\,.$ (27)

Substitute $v_{0}$ from Eq. (23), we can get the relation between $h_{i}$ and
$H$

$\displaystyle h_{i}=\frac{H-(2\alpha+1)c}{2(\alpha+1)}\,.$ (28)

From above equation, the requirement for the jumping can also be got. The jump
can happen only when $h_{i}$ greater than zero, thus the condition for the
jumping is

$\displaystyle H>(2\alpha+1)c\,.$ (29)

The total jumping height is

$\displaystyle h_{j}=h_{i}+h_{r}=\frac{H+c}{2(\alpha+1)}-a\,.$ (30)

It is evident that the jumping height basically is linearly correlated with
the falling height $H$, with additional parameter $c$, $a$ and $\alpha$ to be
determined.

##  III Experimentation and Verification

###  III.1  Analysis and execution of the experiment

Figure 4: The apparatus of the experiment: A, the height calibration on the
wall; B, the cup; C, the chain; D, the chain elements, E, the screenshot.

Through the theoretical analysis in Sec. II, we can verify the relation
through experimentation. Nevertheless, it is important to bear in mind that
all the theoretical analysis are based on ideal conditions, and there are
certain limitations in the actual experiment, such as

  1. 1.

Ideally, the chain in the cup should be loosely connected and freely coiled.
Unfortunately, the elements in chain inevitably become tangled, which prevents
the chain from maintaining homogeneous during the the acceleration and stable
suspension processes. We discovered that this is the primary source of
disturbance in the experiment. Even a small knot can cause a significant
variation of the stable suspension.

  2. 2.

Ideally, the jumping point C and falling point B should remain fixed in order
to obtain a stable suspension. However, these two points tend to shake during
the falling process, making it difficult to accurately measure the shape.
Nevertheless, Since the shape of catenary is an exponential function in which
$y$ increases exponentially with $x$, the shaking chain almost forms a
straight line at the bottom. This makes it challenging to measure the
correlation between the shape and the exponential curve. Furthermore, due to
the upper two points, it is difficult to precisely determine the positions of
points A, B and C. Generally, only the total jumping height $h_{j}$ and
falling height $H$ can be measured.

Though there are these difficulties, it is sure that

$\displaystyle H\gg c,~{}a\,.$ (31)

and the jumping height $h_{j}$, $a$ and $c$ are at the same order. Based on
this, we can assume that the actual shape of the jumping chain resembles and
inverted catenary. The experiment should then test the linear relation between
$h_{j}$ and $H$ (Eq. (30)) and the linear relation between $v_{0}^{2}$ and $H$
(Eq. (23)). Both relations could be used to evaluate the parameter $\alpha$.
which will validate our analysis in Sec. II.

Figure 5: Left: The recorded results between the jumping height and falling
height; Right: the velocity $v_{0}^{2}$ of the stable shape and the falling
height

The apparatus used in our experiment is illustrated in Fig. 4. We drawn a
height calibration on the wall, which can be seen in penal A of Fig. 4. The
chain and the cup are also depicted in the figure, and the details of chain
are provided in the Tab. 1.

Diameter of the steal ball |  $5\pm 0.5$mm
---|---
Length of the steal ball |  $3\pm 0.5$mm
Length of the chain | 8.4$\pm$0.1m
Density $\lambda$ | 0.036$\pm$0.03kg/m
Height of the cup |  $95\pm 0.5$mm
Diameter of the cup |  $66\pm 0.5$mm
Table 1: The setup of the exepriment.

The experiment was conducted as follows: The chain was pulled out of the cup
from a height 200cm, then gradually lowered by 5cm each time. When the height
reached 100cm, it became increasingly difficult to lower the chain further.
Therefore, we chose to lower the chain by 10cm for heights lower than 100cm,
until the height reached 40cm. At each specific height, a cellphone was used
to record the jumping chain, while the camera were placed at the same height
at the top of jumping chain (see penal E of Fig. 4). Due to the presence of
various disturbances in the experiment, multiple vedios were taken in order to
select the best one for the data collection.

In Principle, catenaries with different $c$ and $a$ can exist when the
requirements discussed in Sec. II are satisfied. However, since the chain
needs to jump from the cup, thus the shape of the jumping chain depends on
specific initial conditions, namely, the diameter of the cup, the depth of the
cup at the beginning of the jump, etc. Thus though the shape of the jumping at
a specific height is not in a single parameters set, We found that $c$, $a$
are at the same order of the diameter of the cup. thus the linear relation
could be found in the following.

The data collection was carried out as follows. each frame of the chosen vedio
was examined, and the best moment with the highest jumping height and stable
shape was identified. The falling height and jumping height were then recorded
base on the frame at the best moment. To measure $v_{0}$, the time interval
between a specific height of the chain left in the cup and the final moment
was recorded. With these measured raw data, we can analyze on the relation
between the physical quantities talked in the Sec. II

###  III.2 Analysis on the data

As discussed in the previous subseciton, linear relation between $h_{j}$ and
$H$ (Eq. (30)) and the linear relation between $v_{0}^{2}$ and $H$ (Eq. (23))
can be used to verify our analysis. The corresponding results are shown in
Fig. 5, with the left panel displaying the recorded jumping heights and
falling height, and the right panel shows the calculated velocity $v_{0}$ of
the stable suspension shape and falling height. Note that due to presence of
various disturbances in the experiment, only data with falling higher than
100cm (black points) were chosen for the analysis. As can be seen in Fig. 5,
the linear relations are realized and the slopes give the values of the
$\alpha_{1}$ (left panel) and $\alpha_{2}$ (right panel)

$\displaystyle\alpha_{1}=2.6^{+5}_{-1.3},~{}~{}~{}\alpha_{2}=2.5^{+7.0}_{-0.7}\,.$
(32)

We can see that the errors are quite large and the central values exceed 2.
However, given the difficult setup of the experiment, these two measurements
on $\alpha$ are acceptable. Note that although $\alpha_{1}$ and $\alpha_{2}$
come from the same series of derivations, Eq. (30) and Eq. (23) are assumed to
independent with each other. Thus, we can conclude that $\alpha_{1}$ and
$\alpha_{2}$ are in agreement with each other. Another interest point is that
the extension line in the left panel intersects the $H$ coordinate axis at a
point greater than 0. This point should correspond to the required height of
the jumping condition in Eq. (29), Although this is only a rough estimation.

Finally, we can explain why the soft rope cannot effectively jump. the
elements of chain are connected loosely, and the rope cannot achieve this
condition. This is also the reason why the catenary cannot be realized by the
soft rope. Specifically, the soft rope lacks the non-flexural capacity and the
absence of disturbance from normal force, making the jumping impossible.

##  IV Conclusion

The jumping of a free-falling chain is an intriguing phenomenon, and this
paper provides a detailed analysis of the physics behind it. The analysis are
divided into two parts: the stable suspension and the jumping. Both
variational principle and force analysis demonstrate that the stable
suspension is an inverted catenary. The parameters that describe the
phenomenon are studied and tested through experiments, The requirement for the
jumping are discussed. The measurements of the parameter $\alpha$ meet our
expectations. The physical picture of the falling chain could be useful in
some falling systems. and the inverted catenary may be a complement to
classical mechanics.

## References

  * (1) https://v.qq.com/x/page/o0363pisyik.html?
  * (2) Wenyu Wang, Wu-Long Xu, Yang Xu, Xu-Dong Yang, Physics and Engineering, 2018, v.28; No.185(03) 38-43+50. In Chinese.
  * (3) D. T. Almeida Jordan, Chain-oscillations and polygons in physics, master thesis of university of Amsterdam.
  * (4) J. M. Aristoff and H. A. stone, Proc. R. Soc. A (2012) 468, 720–730.
  * (5) Andrew Belmonte, Michael J. Shelley, Shaden T. Eldakar, and Chris H. Wiggins, Phys. Rev. Lett. 87, 114301.
  * (6) A. Thess, O. Zikanov, and A. Nepomnyashchy, Phys. Rev. E 59, 3637 (1999).


# Resonant squeezed light from photonic Cooper pairs

Sanker Timsina  Department of Physics and Astronomy, University of Victoria,
Victoria, British Columbia V8W 2Y2, Canada  Centre for Advanced Materials and
Related Technology, University of Victoria, Victoria, British Columbia V8W
2Y2, Canada  Taha Hammadia  Department of Physics and Astronomy, University of
Victoria, Victoria, British Columbia V8W 2Y2, Canada  Centre for Advanced
Materials and Related Technology, University of Victoria, Victoria, British
Columbia V8W 2Y2, Canada  École Polytechnique, Institut Polytechnique de
Paris, 91120 Palaiseau, France  Sahar Gholami Milani  Department of Physics
and Astronomy, University of Victoria, Victoria, British Columbia V8W 2Y2,
Canada  Centre for Advanced Materials and Related Technology, University of
Victoria, Victoria, British Columbia V8W 2Y2, Canada  Filomeno S. de Aguiar
Júnior  Department of Chemistry, University of Victoria, Victoria, British
Columbia V8W 2Y2, Canada  Centre for Advanced Materials and Related
Technology, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
Alexandre Brolo  Department of Chemistry, University of Victoria, Victoria,
British Columbia V8W 2Y2, Canada  Centre for Advanced Materials and Related
Technology, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
Rogério de Sousa<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Victoria, Victoria, British Columbia V8W 2Y2, Canada  Centre for
Advanced Materials and Related Technology, University of Victoria, Victoria,
British Columbia V8W 2Y2, Canada

###### Abstract

Raman scattering of photons into phonons gives rise to entangled photon pairs
when the phonon emitted in a Stokes process is coherently absorbed in
antiStokes scattering, forming the photonic analog of Cooper pairs. We present
a nonperturbative theory for the time evolution of photonic Cooper pairs that
treats interacting photons and phonons as a hybrid excitation, the Ramaniton.
As the Ramaniton propagates in a wave guide it displays quantum oscillations
between photon and phonon occupation, leading to resonant squeezed Stokes-
antiStokes light when the phonon occupation becomes equal to zero. This
phenomenon is predicted to generate up to 28 dB of squeezed light even in
standard silicon on insulator waveguides.

Figure 1: Two and four-photon Raman scattering processes. (a) Stokes: An
incident laser photon with frequency $\omega_{L}$ and momentum $\bm{k}_{L}$
scatters into a red shifted Stokes photon with frequency $\omega_{L}-\Omega$
and momentum $\bm{k}_{L}-\bm{Q}$, and emits a phonon with frequency $\Omega$
and momentum $\bm{Q}$. (b) anti-Stokes process: The laser photon absorbs a
phonon and emits a blue shifted photon with frequency $\omega_{L}+\Omega$ and
momentum $\bm{k}_{L}+\bm{Q}$. (c) Stokes-antiStokes pair: The phonon emitted
by a Stokes event is coherently absorbed by another incident laser photon,
leading to an entangled Stokes-antiStokes (SaS) pair. The frequency of the
Stokes/antiStokes photons can be $\omega_{L}\pm\delta$ with $\delta\neq\Omega$
for a virtual phonon excitation. When the phonon resonance condition
$\delta=\Omega$ is satisfied, a “real” phonon is coherently emitted and
absorbed. This phonon resonance happens when the normalized Raman shift
$q=\delta/\Omega=c^{\prime}Q/\Omega=1$, where $c^{\prime}$ is the speed of
light in the material.

##  I Introduction

A key challenge for the development of quantum photonic technology is the
efficient generation of photon entanglement in solid-state chips [1]. Low
noise generation of entangled photon pairs is a necessary condition for
several applications, ranging from quantum sensing [2] and quantum computing
[3] with squeezed states to distributed quantum computing in a future quantum
internet [4]. Realizing these applications requires maximization of photon
entanglement [5], together with minimization of noise from photon loss and
thermal emission in the solid-state environment. To achieve this, it is
desirable to develop microscopic models that account for light-matter
interaction beyond the usual free-space quantum optic phenomenology.

A recent development was the realization that vibrational modes of molecules
and crystals (phonons) can act as mediators for photon-photon interaction in a
wide variety of substances [6, 7]. The origin of this phenomenon is a
correlated Raman process, whereby a phonon emitted by a Stokes photon
scattering event is coherently absorbed by another incident photon, generating
a Stokes-antiStokes (SaS) photon pair, see Fig. 1. This phenomenon is the
photon analogue of the attractive interaction that forms Cooper pairs in
superconductors, leading to the idea that the SaS state is a “photonic Cooper
pair” [8].

So far experimental and theoretical studies of photonic Cooper pairs have
focused solely on the regime of short time evolution or scattering [9], and
the question of what photonic state emerges for longer time evolution in a
waveguide is open. As suggested in Fig. 1(c), photonic Cooper pair formation
provides an additional mechanism for spontaneous four-wave mixing; however, it
is not known how such a mechanism, when mediated by a lower energy excitation
such as a phonon, distinguishes itself from the usual photon-photon
interaction mediated by higher-energy virtual transitions to electronic states
of the material [10]. The modelling of photonic devices traditionally assumes
phenomenological quartic-in-electric-field interactions to describe
spontaneous four-wave mixing [11]; it is not known whether the dominant
mechanism is of phononic or electronic origin.

Here we propose a nonperturbative theory for phonon-mediated generation of
entangled photon pairs in Raman scattering. The theory introduces the concept
of the Ramaniton, a hybrid photon-phonon excitation that is different from the
polariton [12] in at least two ways: First, Ramanitons have zero electric
dipole moment and do not cause photon loss [13]; and second, the excited
states of Ramanitons are made of superpositions of photons/phonons as well as
their corresponding antiparticles (holes), a direct consequence of the
presence of Cooper-pair correlations. Interpreting the photonic excitations
inside the material as Ramanitons allows the prediction of the quantum state
of light that arises in a gas of photonic Cooper pairs. Our theory
demonstrates _resonant squeezing_ for certain photon propagation lengths,
leading to applications in the optimization of quantum photonic devices.

##  II Model for light-matter interaction mediated by a Raman phonon

We consider a model for Stokes (S) and antiStokes (aS) photons interacting
with an optical phonon, the THz lattice vibration of a crystal. Their
noninteracting Lagrangian density is given by [12, 14]

$\displaystyle{\cal L}_{0}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\left(\left|\bm{E}_{S}\right|^{2}+\left|\bm{E}_{aS}\right|^{2}\right)-\frac{1}{2\mu}\left(\left|\bm{B}_{S}\right|^{2}+\left|\bm{B}_{aS}\right|^{2}\right)$
(1)
$\displaystyle+\frac{\rho}{2}\left|\dot{u}\right|^{2}-\frac{\rho}{2}\Omega^{2}\left|u\right|^{2},$

where vectors $\bm{E}_{j}=-\dot{\bm{A}}_{j}$ and
$\bm{B}_{j}=\nabla\times\bm{A}_{j}$, $j=S,aS$, are the electric and magnetic
fields for the Stokes (S) and antiStokes (aS) photons, which are written in
terms of vector potentials $\bm{A}_{j}$. The scalar $u$ models a phonon
displacement with mass density $\rho$ and frequency $\Omega$ independent of
wave vector $\bm{Q}$ (a dispersionless mode, describing the flat band of
optical phonons near $Q=0$).

We assume $u$ in Eq. (1) is a _pure_ Raman mode, i.e. one that is invariant
under the inversion symmetry operation that takes $\bm{E}_{j}$ into
$-\bm{E}_{j}$. This necessarily implies that the lowest order interaction
between $u$ and photons is given by

${\cal L}_{{\rm
int}}=\frac{\xi}{2}u\bm{E}_{L}\cdot\left(\bm{E}_{S}+\bm{E}_{aS}\right)+{\rm
c.c.},$ (2)

where $\xi$ is a coupling constant and $\bm{E}_{L}$ is the electric field for
the pump laser. Such an interaction is present in all materials, including the
ones that have an inversion center, such as crystalline Si and associated
amorphous dielectrics SiOx and SiNx [1], where phonons are either pure
infrared ($u$ changes sign under inversion) or pure Raman ($u$ does not change
sign under inversion). While materials without an inversion center have
phonons with mixed symmetry, interaction (2) is still present in them provided
we interpret $u$ as the component of a phonon displacement that does not
change sign under inversion. That is the part of the phonon that does not have
electric dipole moment and as a consequence does not cause photon loss [13].

Assuming the pump power is much larger than the S, aS output allows us to take
the “parametric approximation” and replace $\bm{E}_{L}$ by a complex classical
field, $\bm{E}_{L}=E_{L}e^{i(k_{L}z-\omega_{L}t)}\bm{\hat{x}}$ where $E_{L}$
is the pump amplitude, and $\omega_{L}=c^{\prime}k_{L}$ is the frequency of
the laser, with $c^{\prime}=1/\sqrt{\epsilon\mu}=c/n_{0}$ the speed of light
in the material.

The quantum Hamiltonian is obtained from the total Lagrangian density ${\cal
L}={\cal L}_{0}+{\cal L}_{{\rm int}}$ by assuming the dynamical variables
$\bm{A}_{S}$ and $\bm{A}_{aS}$ are quantum operators with usual commutation
relations with their canonical momenta $\partial{\cal
L}/\partial\dot{\bm{A}}_{j}$. After going to $k$ space we get the following
Hamiltonian,

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\hbar\sum_{\bm{Q}}\left[\omega_{\bm{k}_{L}+\bm{Q}}\left(b^{{\dagger}}_{\bm{Q}}b_{\bm{Q}}+\frac{1}{2}\right)+\omega_{\bm{k}_{L}-\bm{Q}}\left(b^{{\dagger}}_{-\bm{Q}}b_{-\bm{Q}}+\frac{1}{2}\right)+\Omega\left(c^{{\dagger}}_{\bm{Q}}c_{\bm{Q}}+\frac{1}{2}\right)\right.$
(3)
$\displaystyle\left.+i\frac{\eta}{2}\cos(\omega_{L}t)\sqrt{\Omega\omega_{\bm{k}_{L}+\bm{Q}}}\left(c^{\dagger}_{\bm{Q}}+c_{\bm{Q}}\right)b^{\dagger}_{\bm{Q}}+i\frac{\eta}{2}\cos(\omega_{L}t)\sqrt{\Omega\omega_{\bm{k}_{L}-\bm{Q}}}\left(c^{\dagger}_{\bm{Q}}+c_{\bm{Q}}\right)b^{\dagger}_{-\bm{Q}}+{\rm
H.c.}\right],$

where $b^{{\dagger}}_{\pm\bm{Q}},b_{\pm\bm{Q}}$ are creation and annihilation
operators for photons with momentum $\bm{k}_{L}\pm\bm{Q}$ and frequency
$\omega_{\bm{k}_{L}\pm\bm{Q}}=c^{\prime}|\bm{k}_{L}\pm\bm{Q}|$;
$c^{{\dagger}}_{\bm{Q}},c_{\bm{Q}}$ are creation and annihilation operators
for the phonon mode. The dimensionless parameter $\eta=\xi
E_{L}/(\Omega\sqrt{\rho\epsilon})$ is the figure of merit for the strength of
the photon-phonon coupling; it plays a key role in our theory below.

We move to a rotating frame by applying the unitary transformation
$e^{i\omega_{L}t\left(b^{{\dagger}}_{\bm{Q}}b_{\bm{Q}}+b^{{\dagger}}_{-\bm{Q}}b_{-\bm{Q}}\right)}$,
and take the rotating wave approximation by dropping the terms in Eq. (3) that
do not conserve energy, $c_{\bm{Q}}b^{{\dagger}}_{-\bm{Q}}$ and
$c^{{\dagger}}_{\bm{Q}}b^{{\dagger}}_{\bm{Q}}$. These terms lead to time-
oscillatory contributions with high frequency and small amplitude for all
observables of interest. Also note that phase matching is trivial in the
absence of dispersion in $c^{\prime}$. Without loss of generality we focus on
the case of forward scattering where $\bm{Q}\parallel\bm{k}_{L}$; this implies
the mode $b_{-\bm{Q}}\equiv b_{S}$ is red shifted in frequency, and
$b_{\bm{Q}}\equiv b_{aS}$ is blue shifted. Altogether the approximations lead
to a time-independent Hamiltonian that we call the Ramaniton model:

$\displaystyle{\cal H}_{{\rm R}}$ $\displaystyle=$
$\displaystyle\hbar\Omega\left[-qb^{{\dagger}}_{S}b_{S}+qb^{{\dagger}}_{aS}b_{aS}+c^{\dagger}c+\frac{1}{2}\right.$
(4)
$\displaystyle+\left.i\eta_{-}\left(c^{\dagger}b^{\dagger}_{S}-cb_{S}\right)+i\eta_{+}\left(cb^{\dagger}_{aS}-c^{\dagger}b_{aS}\right)\right],$

where we wrote the dimensionless Raman shift as
$q=\frac{c^{\prime}Q}{\Omega}$, and the dimensionless coupling constants as
$\eta_{\pm}=\frac{\eta}{4}\sqrt{\frac{\omega_{L}}{\Omega}\pm q}$ (note $0\leq
q\leq\omega_{L}/\Omega$). Hamiltonian (4) provides the microscopic realization
of the phenomenological model proposed in [8]. This shows that the degree of
light-matter interaction set by $\eta_{\pm}$ can be tuned by the pump laser
frequency $\omega_{L}$ and amplitude $E_{L}$ via $\eta$.

As a first attempt to tackle the phonon-mediated interaction we follow Saraiva
et al. [8] and apply the Schrieffer-Wolff canonical transformation to Eq. (4).
This converts the terms linear in $\eta_{\pm}$ to a Cooper-pair like
interaction,

${\cal H}_{{\rm
S-W}}=\frac{\hbar\Omega\eta_{+}\eta_{-}}{1-q}\left(b^{{\dagger}}_{S}b^{{\dagger}}_{aS}+b_{aS}b_{S}\right),$
(5)

plus a series of other terms such as
$\eta_{-}^{3}/(1-q)^{2}b^{{\dagger}}_{S}c^{{\dagger}}$.

Interactions similar to (5) are known to generate two-mode squeezed states of
light that play a key role in quantum sensing and computing [15]. However, the
Schrieffer-Wolff method leads to a perturbative series in powers of
$\eta/(1-q)$. The series blows up when $q\rightarrow 1$, the phonon resonance
condition for “real” emission/absorption of a phonon. Below we propose a
nonperturbative solution to this problem.

##  III Nonperturbative theory

Here we propose instead to diagonalize the Ramaniton model exactly using the
Nambu method, required to find quantum excitations of superconducting [16] and
antiferromagnetic materials [17]. In the Nambu representation the Hamiltonian
(4) is written as

${\cal H}_{R}=\frac{1}{2}\bm{v}^{{\dagger}}\cdot\mathbb{L}\cdot\bm{v},$ (6)

where
$\bm{v}=\left(b_{S},c,b_{aS},b^{{\dagger}}_{S},c^{{\dagger}},b^{{\dagger}}_{aS}\right)^{T}$
is a particle-antiparticle column vector, and $\mathbb{L}$ is a $6\times 6$
Hermitian matrix. The goal of the Nambu method is to find the set of operators
$\bm{\alpha}=\mathbb{U}^{-1}\cdot\bm{v}$ that leads to

${\cal
H}_{R}=\sum_{j=1}^{3}\hbar\omega_{j}\left(\alpha^{{\dagger}}_{j}\alpha_{j}+\frac{1}{2}\right).$
(7)

To see how this is done, first note that in order to preserve the Bosonic
commutation relations for the new operators $\bm{\alpha}$ we need to use a
nonunitary canonical transformation $\mathbb{U}$ that satisfies
$\mathbb{U}^{-1}=\mathbb{Z}\cdot\mathbb{U}^{{\dagger}}\cdot\mathbb{Z}$, where
$\mathbb{Z}={\rm diag}\\{1,1,1,-1,-1,-1\\}$ is the plus (minus) identity in
the particle (antiparticle or hole) subspace. From Eq. (7) we get
$\left[\bm{\alpha},{\cal
H}_{R}\right]=\mathbb{Z}\cdot\mathbb{W}\cdot\bm{\alpha}$, where
$\mathbb{W}={\rm
diag}\\{\omega_{1},\omega_{2},\omega_{3},\omega_{1},\omega_{2},\omega_{3}\\}$.
Compare to $\left[\mathbb{U}^{-1}\cdot\bm{v},{\cal
H}_{R}\right]=\mathbb{U}^{-1}\cdot\mathbb{Z}\cdot\mathbb{L}\cdot\bm{v}$ to get
$(\mathbb{Z}\cdot\mathbb{L})\cdot\mathbb{U}=\mathbb{U}\cdot(\mathbb{Z}\cdot\mathbb{W})$,
showing that the columns of $\mathbb{U}$ are the right eigenvectors of
$\mathbb{Z}\cdot\mathbb{L}$, with eigenvalues $\pm\omega_{j}$. Thus the Nambu
method is to diagonalize the non-Hermitian matrix $\mathbb{Z}\cdot\mathbb{L}$,
use its eigenvectors to form the columns of $\mathbb{U}$, and make sure the
eigenvector normalization satisfies
$\mathbb{U}\cdot\mathbb{Z}\cdot\mathbb{U}^{{\dagger}}\cdot\mathbb{Z}=\mathbb{I}$,
where $\mathbb{I}$ is the identity matrix.

Figure 2: Ramaniton dispersions for $\omega_{L}/\Omega=12.4$, $\eta=0.1$
(dashed lines), and $\eta=1$ (solid lines). The $\omega_{1}$ Ramaniton is
decoupled from the phonon and is independent of $\eta$. When $\eta\ll 1$,
photon-phonon hybridization is significant only in the neighbourhood of $q=1$
(phonon resonance), where there is avoided crossing between $\omega_{2}$ and
$\omega_{3}$. In contrast, for larger $\eta$ the modes $\omega_{2},\omega_{3}$
are hybridized in a wider range of $q$.

The operators $\alpha^{{\dagger}}_{j}$ create the hybrid phonon-photon
excitations that we call Ramanitons; exact diagonalization of
$\mathbb{Z}\cdot\mathbb{L}$ leads to 6 eigenvalues, but the canonical
transformation constraint
$\mathbb{U}\cdot\mathbb{Z}\cdot\mathbb{U}^{{\dagger}}\cdot\mathbb{Z}=\mathbb{I}$
shows that only 3 are physical modes. These are the Ramaniton dispersion
relations,

$\displaystyle\omega_{1}$ $\displaystyle=$ $\displaystyle-q\Omega,$ (8a)
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{q+1}{2}-\frac{1}{2}\sqrt{(q-1)^{2}+\frac{\eta^{2}q}{2}}\right)\Omega,$
(8b) $\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\left(\frac{q+1}{2}+\frac{1}{2}\sqrt{(q-1)^{2}+\frac{\eta^{2}q}{2}}\right)\Omega.$
(8c)

Some of the Ramaniton frequencies are negative in the rotating frame; as a
check, note that taking the limit $\eta\rightarrow 0$ leads to frequencies
$\pm q\Omega=\pm c^{\prime}Q$ and $\Omega$, consistent with Eq. (4) when
$\eta=0$.

The mode with frequency $\omega_{1}$ is a quantum superposition of a
“particle-like” Stokes and a “hole-like” antiStokes photon; this is a pure
photon mode that is _decoupled_ from the phonon. Antiparticle or hole-like
states appear because the vacuum of the Ramaniton is “dressed” with
excitations of photons and phonons. The other excitations
$\omega_{2},\omega_{3}$ are hybrid in that they mix (anti)photons and
(anti)phonons. They lead to avoided-crossing of photon and phonon dispersions,
with maximum hybridization happening close to the phonon resonance condition
$q=1$. The laser amplitude $E_{L}$ is able to modify these modes by increasing
$\eta$. The dispersions are shown in Fig. 2.

For $q\ll 1$, Eq. (8b) becomes
$\omega_{2}\approx(1-\frac{\eta^{2}}{8})c^{\prime}Q$, showing that the index
of refraction depends on the laser intensity $I=c^{\prime}\epsilon
E_{L}^{2}/2$ according to $n\approx n_{0}(1+\frac{\eta^{2}}{8})\equiv
n_{0}+n_{2}I$ in the low $I$ regime. This shows that $\eta$ contributes to the
Kerr effect (intensity dependence of the index of refraction), giving rise to
a nonlinear index of refraction $n_{2}$ via the relation
$\eta=\sqrt{8n_{2}I/n_{0}}$. This observation can be used to estimate $\eta$
in photonic materials/devices.

Raman spectra of bulk silicon shows a single strong sharp resonance at
$\Omega/2\pi=15.6$ THz [18], indicating that the single Raman phonon model (4)
is a good approximation. In addition, for the $1550$ nm telecom wavelength
($\omega_{L}/2\pi=193$ THz) photon loss is so small it can hardly be detected
[19], further supporting a model that neglects photon loss into channels other
than the Raman phonon. For this regime $\omega_{L}/\Omega=12.4$, with
measurements of $n_{0}=3.42$, and $n_{2}=4.5\times 10^{-18}$ m${}^{2}/$W [20].
Using intensities achievable on chip ($I=10^{11}$ W/m2 [21]) we estimate
$\eta\approx 10^{-3}$ for common silicon waveguides.

Figure 3: Time dependence of phonon occupation $\langle N_{c}(t)\rangle$, and
photon Stokes $\langle N_{S}(t)\rangle$ and antiStokes modes $\langle
N_{aS}(t)\rangle$ at the phonon resonance ($q=1$). The parameters are
realistic for silicon chips, $\omega_{L}/\Omega=12.4$ and $\eta=10^{-3}$. When
$\langle N_{c}(t)\rangle=0$ (at e.g. $\Omega t=8.89\times 10^{3}$), the SaS
state is “paired up” giving rise to maximum squeezing.

##  IV Photon vacuum squeezing and correlation

The Cooper pair-like interaction (5) obtained by perturbation theory suggests
the time evolution under the Ramaniton Hamiltonian gives rise to a two-mode
squeezed state. We now use our nonperturbative theory to predict the dynamics
of the gas of photonic Cooper pairs. This can be done by writing bare
operators in terms of Ramaniton ones, and obtaining their Heisenberg
representation from $\alpha_{j}(t)=e^{i{\cal
H}_{R}t/\hbar}\alpha_{j}(0)e^{-i{\cal
H}_{R}t/\hbar}=e^{-i\omega_{j}t}\alpha_{j}(0)$.

One important constraint is that the Ramaniton model (4) has the symmetry
$[{\cal H}_{R},N_{S}-N_{aS}-N_{c}]=0$, where $N_{c}=c^{{\dagger}}c$ and
$N_{j}=b^{{\dagger}}_{j}b_{j}$ with $j=S,aS$ are bare phonon and photon number
operators. As a result the quantity $N_{S}-N_{aS}-N_{c}$ is a constant
independent of time. If the initial state is the vacuum,
$|\psi(t=0)\rangle=|0\rangle_{S}|0\rangle_{c}|0\rangle_{aS}$, their average
occupation changes in time according to $\langle N_{S}(t)\rangle-\langle
N_{aS}(t)\rangle=\langle N_{c}(t)\rangle$. Therefore, $\langle
N_{S}(t)\rangle-\langle N_{aS}(t)\rangle$ oscillates as time evolves due to
coherent population transfer between photons and phonons. This is shown in
Fig. 3.

When $\langle N_{c}(t)\rangle=0$ we have $\langle N_{S}(t)\rangle=\langle
N_{aS}(t)\rangle$, a special situation where the photonic state is “paired
up”. As we now show this gives rise to maximum squeezing.

Define the generalised quadrature

$X(t)=\frac{1}{2^{3/2}}\left\\{e^{-i\phi}\left[b_{S}(t)+b_{aS}(t)\right]+{\rm
H.c.}\right\\},$ (9)

where $b_{j}(t)=e^{i{\cal H}_{R}t/\hbar}b(0)e^{-i{\cal H}_{R}t/\hbar}$ are
operators in the Heisenberg representation, and $\phi$ is a phase. The amount
of vacuum noise squeezing as a function of time in Decibel units (dB) is given
by

$S(t,\phi)=-10\log_{10}\left\\{\frac{\left\langle\left[\Delta
X(t)\right]^{2}\right\rangle}{\left\langle\left[\Delta
X(0)\right]^{2}\right\rangle}\right\\},$ (10)

where $\Delta X(t)=X(t)-\langle X\rangle$. The averages are taken by assuming
the initial state is the vacuum of the bare modes in the rotating frame,
$|\psi(t=0)\rangle=|0\rangle_{S}|0\rangle_{c}|0\rangle_{aS}$. The denominator
in Eq. (10) is the variance of $X$ in the vacuum, $\langle\left[\Delta
X(0)\right]^{2}\rangle=1/4$. This corresponds to the quantum noise limit (QNL)
of $0$ dB. The generalized quadrature is said to be squeezed (anti-squeezed)
whenever the quantum noise of $X$ is below (above) the QNL.

Our calculations show that the amount of vacuum squeezing $S(t,\phi)$ is
oscillatory as a function of the propagation time $t$ or length
$L=c^{\prime}t$ of a waveguide; this is shown in Fig. 4, where $\Omega t$ is
now replaced by $L/(c^{\prime}/\Omega)$, with $L$ the length of a photonic
waveguide. The quantum oscillations as a function of $L$ arise from the
population exchange between bare phonon and photons in the time evolution of
the Ramaniton. As a consequence, for each Raman shift $q$ there is a value of
nonzero $t$ or $L$ that gives rise to $\langle N_{c}\rangle=0$ and maximum
squeezing. For real $\xi$ the optimal $\phi$ is equal to $\frac{\pi}{2}$.
Figure 5 shows vacuum squeezing as a function of Raman shift $q$ for three
different waveguide lengths $L$. The global maximum for squeezing is obtained
for $L\Omega/c^{\prime}=8.89\times 10^{3}$ and $q=1$, when both the condition
for resonant squeezing ($\langle N_{c}\rangle=0$) and the condition for phonon
resonance ($q=1$, emission/absorption of a real phonon) are _simultaneously_
satisfied. For other values of $L$, the condition for resonant squeezing
$\langle N_{c}\rangle=0$ is satisfied away from $q=1$; as a result squeezing
shows local maxima at these points.

Figure 4: Vacuum noise squeezing $S(t,\phi)$ for $\phi=\pi/2$ as a function of
propagation length $L$ using parameters realistic for silicon on insulator
waveguides. The parameters are $\omega_{L}/\Omega=12.4$, $\eta=10^{-3}$, and
Raman shifts $q=0.9995$ (blue) and $q=1$ (red). The black line represents the
quantum noise limit of 0 dB. Figure 5: Vacuum noise squeezing for realistic
silicon waveguides as a function of Raman shift $q$, for three different
waveguide lengths $L$. Global maximum squeezing of 28 dB is achieved for
$L=8.89\times 10^{3}c^{\prime}/\Omega$ and $q=1$, the phonon resonance
condition when a real phonon is emitted/absorbed. For other propagation
lengths squeezing has local maxima away from $q=1$. The parameters used are
$\omega_{L}/\Omega=12.4$ and $\eta=10^{-3}$.

For additional insight, we turn to calculations of the second-order cross-
correlation between photons in S and aS modes. The zero-time delay two-mode
intensity correlation function is defined as [15]

$g^{(2)}_{SaS}(0)=\frac{\langle
b_{S}^{\dagger}(t)b_{aS}^{\dagger}(t)b_{aS}(t)b_{S}(t)\rangle}{\langle
b_{S}^{\dagger}(t)b_{S}(t)\rangle\langle
b_{aS}^{\dagger}(t)b_{aS}(t)\rangle}.$ (11)

When the initial state is the vacuum, we can use the Nambu method to obtain
the exact result $g^{(2)}_{SaS}(0)=2+1/\langle N_{S}(t)\rangle$. This shows
that in the Ramaniton model $g^{(2)}_{SaS}(0)$ is a proxy for the number of
photons in the S mode. Figure 6 shows $g^{(2)}_{SaS}(0)$ as a function of $q$
for $L=8.89\times 10^{3}c^{\prime}/\Omega$ and other parameters the same as in
Fig. 5. We see that $g^{(2)}_{SaS}(0)$ is minimum at the global maximum
squeezing point at $q=1$. For this case $g^{(2)}_{SaS}(0)$ _anticorrelates_
with the amount of squeezing.

The inset of Fig. 6 shows the number of S and aS photons as a function of $q$
for the same parameters. Away from $q=1$, $\langle
N_{S}(t)\rangle\approx\langle N_{aS}(t)\rangle$ decreases significantly and
$g^{(2)}_{SaS}(0)$ increases; when $q\ll 1$ or $q\gg 1$, both $\langle
N_{S}(t)\rangle$ and $\langle N_{aS}(t)\rangle$ are quite small, leading to
$g^{(2)}_{SaS}(0)\gg 1$ (not shown). This occurs because away from the phonon
resonance S and aS photons are only created as pairs, maximizing bunching
correlations (Fig. 1(c)). However, this bunched state is not bright enough to
generate high levels of squeezing.

Figure 6: Two-mode intensity cross-correlation function $g^{(2)}_{SaS}(0)$ as
a function of Raman shift $q$ for the parameters leading to a global maximum
in squeezing. This is waveguide length $L\Omega/c^{\prime}=8.89\times 10^{3}$
and other parameters as in Fig. 5. The inset is the plot of average number of
Stokes and anti-Stokes photons for the same set of parameters.

Once again, for the particular case of vacuum for all bare modes at the
beginning of the waveguide ($L=0$ or $t=0$), the Nambu method can be used to
obtain an exact analytical result relating the amount of squeezing to the
number of photons in the bare modes:

$\frac{\left\langle\left[\Delta
X(t)\right]^{2}\right\rangle}{\left\langle\left[\Delta
X(0)\right]^{2}\right\rangle}=\left|\sqrt{1+\langle
N_{S}(t)\rangle}-\sqrt{\langle N_{S}(t)\rangle-\langle
N_{c}(t)\rangle}\right|^{2}.$ (12)

This equation is minimized (maximum squeezing) when $\langle
N_{c}(t)\rangle=0$ simultaneously with $\langle
N_{S}(t)\rangle\rightarrow\infty$, i.e. when the brightness of the Stokes
channel is maximized. The latter takes place at the phonon resonance.

##  V Conclusions

We presented a nonperturbative theory for phonon-mediated photon-photon
interaction based on the concept of the Ramaniton, an excitation that is
qualitatively different than the polariton in that it mixes particle and
antiparticle (hole-like) phonon/photon modes. This distinctive feature arises
from Cooper-pair-like correlations that are _universal_ to all Raman
scattering processes. The Ramaniton removes the distinction between real and
virtual phonon processes in correlated Raman scattering, regularizing the
singularity that appears in the perturbative treatment for this problem [8].

The usefulness of the Ramaniton is evident from Fig. 3. As time evolves, the
number of photons/phonons goes from $0$ to $\sim 10^{2}$; capturing this range
of modes with exact numerical diagonalization is challenging. The Ramaniton
solves this problem and enables analytical determination of the regimes where
optimal squeezing takes place.

We show that the amount of vacuum noise squeezing, a property that is well-
known to arise from entanglement [5] shows local maxima (resonances) as a
function of photon time of flight $t$ or waveguide propagation length $L$.
“Resonant squeezing” occurs when the average number of excited phonons
$\langle N_{c}\rangle$ becomes equal to zero for nonzero $t$ or $L$. When this
happens the number of Stokes and antiStokes photons become equal, leading to
“photonic pairing”, with squeezing measured in dB increasing logarithmically
with the number of photons.

The global maximum for squeezing happens when both the condition for resonant
squeezing $\langle N_{c}\rangle=0$ and the condition for phonon resonance
$q=1$ are simultaneously satisfied. In this case the “paired state” has
maximum brightness. As seen in the $q=1$ curve of Fig. 4, the global maximum
peak is relatively broad as a function of $L$, so maximum squeezing can be
achieved without the need of high precision on the value of $L$. This leads to
our prediction that up to 28 dB of squeezing may be reached in silicon on
insulator waveguides, provided that light can be extracted near the optimal
waveguide length $L=8.89\times 10^{3}c^{\prime}/\Omega=7.95$ mm. Achieving
this would be a significant improvement on the 15 dB of squeezing measured in
an optical table [22] and also on the 11 dB of squeezing inferred before
extraction on a photonic chip [23].

In conclusion, we described a theory for phonon-mediated generation of
squeezed light. The theory opens up avenues for material and device
optimization in integrated quantum photonics for quantum information and
sensing applications.

## Acknowledgements

We thank S. Barzanjeh, B. King, and A. MacRae for encouragement and insightful
discussions in this topic. This work was supported by the Natural Sciences and
Engineering Research Council of Canada (NSERC) through its Discovery
(RGPIN-2020-04328), CREATE (543245-2020), and USRA programs. We acknowledge
the support of the Government of Canada’s New Frontiers in Research Fund
(NFRF), [NFRFE-2019-01104].

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