denotes the push-forward of vector fields by $f$:

$f_{\sharp}(X)=(\mathrm{d}f)\circ X\circ f^{-1}~{}~{}\forall
X\in\mathcal{X}(\mathcal{M}_{1})~{}~{}.$

Moreover, the two gradient flows can be identified through $f$ up to
increasing reparameterization (and hence are smoothly topologically
equivalent, see Appendix A) iff
$\frac{1}{\Omega}\,\mathrm{grad}_{\mathcal{G}_{1}}V_{1}$ is $f$-related to
$\mathrm{grad}_{\mathcal{G}_{2}}V_{2}$ for some positive smooth function
$\Omega$ defined on $\mathcal{M}_{1}$ (which induces a Weyl rescaling
$\mathcal{G}_{1}\rightarrow\Omega\mathcal{G}_{1}$ of $\mathcal{G}_{1}$) i.e.
iff we have
$f_{\sharp}(\frac{1}{\Omega}\,\mathrm{grad}_{\mathcal{G}_{1}}V_{1})=\mathrm{grad}_{\mathcal{G}_{2}}V_{2}$.
These observations motivate the following:

######  Definition 3.8.

Consider two scalar triples $(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$. A smooth diffeomorphism
$f:(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$
is called:

  * •

smooth gradient conjugation, if it satisfies the condition:

$f_{\sharp}(\mathrm{grad}_{\mathcal{G}_{1}}V_{1})=\mathrm{grad}_{\mathcal{G}_{2}}V_{2}~{}~{}.$
(63)

  * •

smooth gradient equivalence, if it satisfies the condition:

$f_{\sharp}(\Omega^{-1}\mathrm{grad}_{\mathcal{G}_{1}}V_{1})=\mathrm{grad}_{\mathcal{G}_{2}}V_{2}$
(64)

for some positive smooth function
$\Omega:\mathcal{M}_{1}\rightarrow\mathbb{R}_{>0}$.

The two scalar triples are called smoothly gradient conjugate/equivalent if
there exists a smooth gradient conjugation/equivalence
$f:(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$.

Since the differential pull-back
$f^{\ast}:\mathcal{X}(\mathcal{M}_{2})\rightarrow\mathcal{X}(\mathcal{M}_{1})$
is given by $f^{\ast}=(f_{\sharp})^{-1}$, condition (64) is equivalent with:

$\mathrm{grad}_{\mathcal{G}_{1}}V_{1}=\Omega~{}\mathrm{grad}_{f^{\ast}(\mathcal{G}_{2})}f^{\ast}(V_{2})~{}~{},$
(65)

where $f^{\ast}(V_{2})=V_{2}\circ f$. Gradient conjugation and equivalence are
equivalence relations on scalar triples.

######  Example 3.9.

Any conformal equivalence of scalar triples
$f:(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$
is a gradient equivalence, as can be seen from its defining relations
$f^{\ast}(\mathcal{G}_{2})=\Omega\mathcal{G}_{1}$ and $f^{\ast}(V_{2})=V_{1}$.
However, condition (64) admits solutions $f$ which need not be conformal
equivalences of triples. In local coordinates $(x^{1},\ldots,x^{d})$ on
$\mathcal{M}_{1}$ and $(y^{1},\ldots,y^{d})$ on $\mathcal{M}_{2}$, this
condition takes the form:

$\frac{1}{\Omega(x)}\mathcal{G}_{1}^{jk}(x)\partial_{j}f^{i}(x)\partial_{k}V_{1}(x)=\mathcal{G}_{2}^{ij}(f(x))(\partial_{j}V_{2})(f(x))~{}~{},$
(66)

which is weaker than the defining conditions of a conformal transformation
from $(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ to
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$:

$(\mathcal{G}_{2})_{ij}(f(x))\partial_{k}f^{i}(x)\partial_{l}f^{j}(x)=\Omega(x)(\mathcal{G}_{1})_{kl}(x)~{}~{}\mathrm{and}~{}~{}V_{2}(f(x))=V_{1}(x)~{}~{}.$
(67)

Here $f^{i}(x)=y^{i}(f(x))$. If one chooses coordinates such that
$x^{i}=y^{i}(f(x))$ then $f$ is locally given by $f^{i}(x)=x^{i}$ and (66)
becomes:

$\frac{1}{\Omega(x)}\mathcal{G}_{1}^{ij}(x)\partial_{j}V_{1}(x)=\mathcal{G}_{2}^{ij}(x)\partial_{j}V_{2}(x)$

while (67) reduces to:

$\frac{1}{\Omega(x)}\mathcal{G}_{1}^{ij}(x)=\mathcal{G}_{2}^{ij}(x)~{}\mathrm{and}~{}~{}V_{1}(x)=V_{2}(x)~{}~{}.$

A gradient conjugation of a scalar triple to itself is called a gradient
symmetry, while a gradient equivalence to itself is called a weak gradient
symmetry of that triple. The condition that a diffeomorphism
$f:\mathcal{M}\rightarrow\mathcal{M}$ be a weak gradient symmetry of the
scalar triple $(\mathcal{M},\mathcal{G},V)$ reduces to the requirement that
the gradient vector field $v\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\mathrm{grad}_{\mathcal{G}}V$ (which has components
$v^{i}(x)=\mathcal{G}^{ij}(x)\partial_{j}V(x)$) satisfies
$f_{\ast}(\frac{1}{\Omega}v)=v$, i.e.:

$\partial_{j}f^{i}(x)v^{j}(x)=\Omega(x)v^{i}(f(x))~{}~{}.$

With these preparations, the remarks made above motivate the following:

######  Definition 3.10.

Consider two classical cosmological models parameterized by
$\mathfrak{M}_{1}=(M_{01},\\!\mathcal{M}_{1},\\!\mathcal{G}_{1},\\!\Phi_{1})$
and
$\mathfrak{M}_{2}=(M_{02},\\!\mathcal{M}_{2},\\!\mathcal{G}_{2},\\!\Phi_{2})$
and let $V_{i}\stackrel{{\scriptstyle{\rm def.}}}{{=}}M_{0i}\sqrt{2\Phi_{i}}$
($i=1,2$). We say that the models are

  * •

smoothly IR conjugate and write
$\mathfrak{M}_{1}\equiv_{\mathrm{IR}}\mathfrak{M}_{2}$ if there exists a
smooth gradient conjugation:

$f:(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})~{}~{}.$

  * •

smoothly IR equivalent and write
$\mathfrak{M}_{1}\sim_{\mathrm{IR}}\mathfrak{M}_{2}$ if there exists a smooth
gradient equivalence

$f:(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})~{}~{}.$

In the situations above, $f$ is called respectively a smooth IR conjugation or
IR equivalence between $\mathfrak{M}_{1}$ and $\mathfrak{M}_{2}$.

A (necessarily strict) IR or UV smooth conjugation between a model
parameterized by $(M_{0},\mathcal{M},\mathcal{G},\Phi)$ and itself is called
an IR or UV symmetry of that model.

The binary relations introduced above are equivalence relations on the class
of multifield cosmological models; in fact, they are the isomorphism relations
of associated groupoids. The equivalence classes of $\sim_{\mathrm{UV}}$ and
$\sim_{\mathrm{IR}}$ are called smooth UV and IR cosmological universality
classes.

###  3.5 The late time infrared phase structure

As explained in Section 2, the cosmological curves of a model parameterized by
$(M_{0},\mathcal{M},\mathcal{G},\Phi)$ are approximated by gradient flow
curves of the scalar triple $(\mathcal{M},\mathcal{G},V)$ in the IR limit.
With sufficiently strong assumptions on the behavior of $\mathcal{M}$ and $V$
near the Freudenthal ends of $\mathcal{M}$, the $\omega$\- extended limit
points of such curves coincide with those of certain gradient flow curves of
$(\mathcal{M},\mathcal{G},V)$. In good cases, a cosmological curve $\varphi$
approaches a gradient flow curve $\varphi_{\mathrm{IR}}^{(+)}$ in the distant
future and hence $\varphi$ can be replaced in the infrared by
$\varphi_{\mathrm{IR}}^{(+)}$ for late times.

This statement is familiar when $\mathcal{M}$ is compact and $V$ is a Morse
function. In that case, the gradient flow of $(\mathcal{M},\mathcal{G},V)$ is
complete and hence its maximal flow curves are defined on the entire real
axis. Moreover, any non-constant maximal gradient flow curve $\eta$ satisfies:

$\exists\lim_{t\rightarrow-\infty}\eta(q)=c_{i}~{}~{}\mathrm{and}~{}~{}\exists\lim_{t\rightarrow\infty}\eta(q)=c_{f}~{}~{},$

where $c_{i}$ and $c_{f}$ are critical points of $V$ whose Morse indices obey
$\mathrm{ind}(c_{i})>\mathrm{ind}(c_{f})$. In this situation, the moduli space
of gradient flow orbits has components indexed by the pair $(c_{i},c_{f})$,
while the early and late time behavior of $\eta$ depend respectively on
$c_{i}$ and $c_{f}$. In this case, the late time infrared dynamics of the
cosmological model parameterized by $(M_{0},\mathcal{M},\mathcal{G},\Phi)$ has
a phase structure in which different phases are indexed by the critical points
of $V$; every cosmological curve “settles” at late times in a phase indexed by
a critical point $c$ of Morse index greater that $d\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\dim\mathcal{M}$. The asymptotic behavior of the cosmological flow
for late times in the phase indexed by $c$ depends on the Morse index of $c$
and on the behavior of $\mathcal{G}$ near $c$.

This simple characterization of limit points of the gradient flow of
$(\mathcal{M},\mathcal{G},V)$ no longer holds when $V$ is Morse but
$\mathcal{M}$ is non-compact. With strong enough conditions on the topology of
$\mathcal{M}$ and on the asymptotic behavior of $V$, the gradient flow curves
will still have $\alpha$\- and $\omega$\- extended limits but each of these
can be either a critical point of $V$ or a Freudenthal end of $\mathcal{M}$.
In this case, the IR phases of the model are indexed by critical points of $V$
and by those ends of $\mathcal{M}$ which can arise as $\omega$-limit points of
a cosmological curve.

For general effective potentials $V$ with non-compact $\mathcal{M}$, a maximal
cosmological curve can have more than one extended $\omega$-limit point. In
this situation, IR phases can be classified partially by the allowed
$\omega$-limit sets, but specifying these need not provide a full
classification and more detailed analysis is required. In particular, the
classification of such phases can be quite involved and a simple relation to
the late time behavior of the gradient flow of $(\mathcal{M},\mathcal{G},V)$
may not exist.

##  4 IR universality of two-field models with hyperbolic scalar manifold

The IR conformal invariance property discussed in Subsection 3.3 has a
striking implication for two-field cosmological models when combined with the
uniformization theorem of Poincaré. According to the latter, the Weyl
equivalence class of any Riemannian metric $\mathcal{G}$ defined on a
borderless connected surface $\Sigma$ contains a unique complete metric $G$
(called the uniformizing metric of $\mathcal{G}$) of constant Gaussian
curvature $K$ equal to one of the values $-1$, $0$ or $+1$. The case $K=-1$ is
generic; in this case, the metric $\mathcal{G}$ (and its conformal class) is
called hyperbolizable and $G$ is called the hyperbolization of $\mathcal{G}$.
The cases $K=+1$ and $K=0$ occur only for seven topologies, as follows:

  * •

When $K=+1$, the surface $\Sigma$ must be diffeomorphic with the $2$-sphere
$\mathrm{S}^{2}$ or with the real projective plane
$\mathbb{R}\mathbb{P}^{2}\simeq\mathrm{S}^{2}/\mathbb{Z}_{2}$. Both of these
surfaces admit a unique metric of unit Gaussian curvature.

  * •

When $K=0$, the surface $\Sigma$ must be diffeomorphic with the 2-torus
$\mathrm{T}^{2}$, the Klein bottle
$\mathrm{K}^{2}=\mathbb{R}\mathbb{P}^{2}\times\mathbb{R}\mathbb{P}^{2}\simeq\mathrm{T}^{2}/\mathbb{Z}_{2}$,
the open annulus $\mathrm{A}^{2}$ (which is diffeomorphic with the open
cylinder and with the twice punctured sphere), the open Möbius strip
$\mathrm{M}^{2}\simeq\mathrm{A}^{2}/\mathbb{Z}_{2}$ (which is diffeomorphic
with the once-punctured real projective plane) or with the plane
$\mathbb{R}^{2}$. The 2-torus admits a three-parameter family of flat metrics
while the Klein bottle admits a two-parameter family of such. The open annulus
and open Möbius strip admit a one-parameter family of complete flat metrics
while $\mathbb{R}^{2}$ admits a unique complete flat metric.

The plane $\mathbb{R}^{2}$, the open annulus $\mathrm{A}^{2}$ and the open
Möbius strip $\mathrm{M}^{2}$ are the only three surfaces which admit two
types of uniformizing metrics, namely a Riemannian metric defined on one of
these surfaces is uniformized either by a complete flat metric or by a
hyperbolic metric depending on its conformal class. Namely:

  * •

The plane $\mathbb{R}^{2}$ admits both a complete flat metric and a hyperbolic
metric (the Poincaré metric).

  * •

The open annulus $\mathrm{A}^{2}$ admits a one-parameter family of complete
flat metrics, a hyperbolic metric which makes it into the hyperbolic punctured
disk and a one-parameter family of hyperbolic metrics which produce the
hyperbolic annuli.

  * •

The open Möbius strip $\mathrm{M}^{2}$ admits a one-parameter family of
complete flat metrics and a one-parameter family of hyperbolic metrics which
is obtained by quotienting the hyperbolic annuli through a $\mathbb{Z}_{2}$
group of isometries.

Surfaces on which any metric is hyperbolizable are called of general type;
these include all borderless connected surfaces except
$\mathrm{S}^{2},\mathbb{R}\mathbb{P}^{2},\mathrm{T}^{2},\mathrm{K}^{2},\mathrm{A}^{2},\mathrm{M}^{2}$
and $\mathbb{R}^{2}$. When $\Sigma$ is diffeomorphic with
$\mathrm{A}^{2},\mathrm{M}^{2}$ or $\mathrm{K}^{2}$, only certain conformal
classes are hyperbolizable, while the remaining conformal classes uniformize
to a complete flat metric. When $\Sigma$ is diffeomorphic with
$\mathrm{T}^{2}$ or $\mathrm{K}^{2}$, every conformal class uniformizes to a
complete flat metric, while when it is diffeomorphic with $\mathrm{S}^{2}$ or
$\mathbb{R}\mathbb{P}^{2}$ any conformal class uniformizes to a complete
metric with Gaussian curvature equal to $+1$.

Consider a two-field cosmological model parameterized by
$(M_{0},\Sigma,\mathcal{G},\Phi)$. The observations of Subsection 3.3 imply
that the gradient flow of $V=M_{0}\sqrt{2\Phi}$ computed with respect to
$\mathcal{G}$ has the same oriented orbits as the gradient flow of $V$
computed with respect to the uniformizing metric $G$ and hence differs from
the latter only by an increasing reparameterization of the gradient flow
curves. Thus models whose scalar field metric has constant Gaussian curvature
equal to $-1$, $0$ or $+1$ provide distinguished representatives of the IR
universality classes of all two-field cosmological models. More precisely:

Up to (curve-dependent) increasing reparameterization, the first order IR
approximant of the cosmological flow of a two-field model with scalar triple
$(\Sigma,\mathcal{G},\Phi)$ and rescaled Planck mass $M_{0}$ is described by
the gradient flow of the scalar triple $(\Sigma,G,V)$, where $G$ is the
uniformizing metric of $\mathcal{G}$ and $V\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0}\sqrt{2\Phi}$ is the classical effective scalar potential of
the model.

When $\Sigma$ is of general type, this statement implies that IR universality
classes of two-field models with target $\Sigma$ are classified by gradient
equivalence classes of hyperbolic scalar triples $(\Sigma,G,V)$; more
generally, this holds for two-field models with hyperbolizable scalar manifold
$(\Sigma,\mathcal{G})$. Two-field models whose complete scalar manifold metric
$\mathcal{G}$ has constant negative curvature are called generalized two-field
$\alpha$-attractor models. In this case, one has $\mathcal{G}=3\alpha G$ for
some positive constant $\alpha$ and some hyperbolic metric $G$. Such models
were introduced and studied in genalpha ; elem ; modular  and form a very wide
extension of the two-field cosmological $\alpha$-attractor models previously
introduced in KLR , which correspond to the topologically trivial case when
$(\Sigma,G)$ is the Poincaré disk. It follows that generalized two-field
$\alpha$-attractor models are IR universal among two-field cosmological models
with hyperbolizable target manifold. This gives a conceptual reason to single
out generalized two-field $\alpha$-attractor models for special study.

######  Remark 4.1.

The notion of IR universality employed in this paper differs from the
colloquial use of the same word in the $\alpha$-attractor literature (see for
example AKLWY ), where certain predictions extracted from models whose scalar
manifold is a Poincaré disk were claimed to be ‘universal’ in the sense that
they are insensitive to small enough perturbations of the scalar potential (a
notion which corresponds to structural stability of the cosmological dynamical
system and differs conceptually from our notion of IR universality). Poincaré
disk models cannot be IR universal even in the class of models with
contractible target, since the open disk $\mathrm{D}^{2}\simeq\mathbb{R}^{2}$
admits both metrics which uniformize to the Poincaré metric and metrics which
uniformize to the complete flat metric777While the Poincaré metric is
conformally flat, it is not conformally equivalent to the complete flat metric
on $\mathrm{D}^{2}\simeq\mathbb{R}^{2}$.. As we show in a separate
publication, IR universality classes of two-field models with hyperbolizable
scalar manifold are much more complicated than one might expect from such
works. When $\Sigma$ is not topologically trivial, such models can have
distinct late time IR phases associated to the Freudenthal ends of $\Sigma$,
all of which differ from the single IR phase associated to the plane end of
the Poincaré disk; in fact, the latter is the only hyperbolic surface which
admits a plane end and hence Poincaré disk models are uniquely special among
two-field models with hyperbolizable target manifold. Notice that Freudenthal
end phases arise in addition to those indexed by the critical points of the
scalar potential.

##  5 Conclusions and further directions

We studied the scaling behavior and scaling limits of classical multifield
cosmological models with scalar triple $(\mathcal{M},\mathcal{G},\Phi)$ and
rescaled Planck mass $M_{0}$ and their dynamical renormalization group,
showing that the latter deforms the cosmological flow of such models into a
family of flows defined on $T\mathcal{M}$ which interpolates between a
modification of the geodesic flow of the scalar manifold
$(\mathcal{M},\mathcal{G})$ and a certain lift of the gradient flow of the
classical effective potential $V\stackrel{{\scriptstyle{\rm
def.}}}{{=}}M_{0}\sqrt{2\Phi}$ on this Riemannian manifold. Using the
invariance of oriented gradient flow orbits under Weyl transformations of
$\mathcal{G}$, we found that the first order IR approximants of cosmological
orbits are insensitive to such transformations. This allowed us to give a
mathematical description of dynamical UV and IR universality classes of
classical cosmological models. In the infrared limit, the late time dynamics
of such models is partitioned into “phases” which in good cases are indexed by
critical points of $V$ and by Freudenthal ends of $\mathcal{M}$. These results
provide a realization of ideas familiar from the theory of critical phenomena
within the context of classical cosmology and open up new directions for the
study and classification of such models.

Since the UV and IR approximations are controlled by different conditions from
those governing the slow roll approximation and its slow roll - slow turn
variant, the study of the IR and UV behavior of classical cosmological models
requires new ideas and tools. The UV approximation recovers a modification of
the geodesic flow of $(\mathcal{M},\mathcal{G})$, thus making contact with a
classical subject in Riemannian geometry and dynamical systems theory. The IR
approximation recovers the gradient flow of the classical effective potential
$V$, thus making contact with another classical subject. Since the scalar
manifold $(\mathcal{M},\mathcal{G})$ is generally non-compact, one has to
analyze such flows without the compactness assumptions made in most of the
mathematics literature – and hence the study of cosmological scaling limits
does not reduce to a simple application of known results. For example, one
cannot directly apply classical results of Morse and gradient flow theory to
the study of the IR limit even when a natural smooth compactification of
$\mathcal{M}$ exists and $\Phi$ admits a smooth and Morse extension to that
compactification, because the scalar manifold metric $\mathcal{G}$ does not
generally extend.

For two-field models, the target manifold $\mathcal{M}$ is a connected surface
$\Sigma$. In this case, our results and the uniformization theorem of Poincaré
imply that the IR behavior of the model parameterized by
$(M_{0},\Sigma,\mathcal{G},\Phi)$ coincides with that of the model
parameterized by $(M_{0},\Sigma,G,\Phi)$ to first order in the scale
expansion, where $G$ is the uniformizing metric of $\mathcal{G}$. This implies
that two-field models whose scalar manifold metric has Gaussian curvature
equal to $-1$, $0$ or $1$ are IR universal in the Wilsonian sense that they
provide distinguished representatives of the infrared universality classes of
all two-field models. This gives a powerful conceptual reason to single out
such models for detailed study. In grad , we study the IR behavior of a very
large class of two-field cosmological models using the methods and ideas of
the present paper.

One direction for further research concerns the systematic study of higher
orders of the UV and IR expansions, on which we hope to report shortly. In
this regard, it would be interesting to extract precise asymptotic bounds
which control the error terms at each order. More ambitiously, one can look
for improvements of these expansions which produce uniformly convergent
series; such improvements can be extracted in principle using the method of
multiple scales (see BO ; KC ; Hinch ; Kuehn ). In this context, the
renormalization group action constructed in this paper relates to the work of
Ziane ; DeVille ; Chiba1 ; Chiba2 .

It would also be interesting to study the implications of scale expansions for
cosmological perturbation theory with a view toward developing new
approximation schemes and constructing effective descriptions which take into
account the fact that the underlying classical dynamics of cosmological models
has a dynamical RG flow of its own. In our opinion, this could address the
conceptual criticism of current approaches to effective descriptions in
cosmology made in papers such as Burgess1 ; Burgess2 .

Since the dynamics of multifield cosmological models admits a constrained
Hamiltonian description in the minisuperspace formalism, the RG flow and scale
approximations considered in this paper can be reformulated in that framework,
which allows one to use methods from the theory of Hamiltonian systems. It
would be interesting to see what insight can be gained by pursuing Wilsonian
ideas in that context.

Finally, one can expect that the models with hidden symmetries considered in
Noether1 ; Noether2 ; Hesse  (see Lilia1 ; Lilia2  for some phenomenological
implications) play a special role in the infrared limit, by analogy to similar
situations in the theory of critical phenomena; it would be interesting to
address this conjecture.

When the target manifold $\mathcal{M}$ has dimension greater than two, the
existence of a metric $G$ with special properties in the Weyl-equivalence
class of $\mathcal{G}$ is a classical problem. When $\mathcal{M}$ is non-
compact and one requires $G$ to be complete and of constant scalar curvature,
this is the Yamabe problem for non-compact manifolds, which generally has a
negative answer unless one imposes further restrictions on
$(\mathcal{M},\mathcal{G})$. When $\mathcal{M}$ is three-dimensional, one can
instead use Thurston uniformization, which is different in character. It would
be interesting to see if analogues of our arguments exist that would allow one
to find good representatives of universality classes of models with more than
two scalar fields when appropriate conditions are imposed on the scalar
manifold and potential.

##  Appendix A Topological and smooth equivalence of dynamical systems

Recall that a (smooth) autonomous dynamical system is a pair $(M,X)$, where
$M$ is a manifold and $X$ is a smooth vector field defined on $M$ (see Palis
).

######  Definition A.1.

A flow curve of the autonomous dynamical system $(M,X)$ is a smooth integral
curve $\gamma:I\rightarrow M$ of the vector field $X$, i.e. a solution of the
equation:

$\frac{\mathrm{d}\gamma(t)}{\mathrm{d}t}=X(\gamma(t))~{}~{}\forall t\in
I~{}~{},$

where $I$ is a non-degenerate interval, i.e. a non-empty (open, closed or
semi-closed) interval which is not reduced to a point. The image
$\mathrm{im}\gamma\stackrel{{\scriptstyle{\rm def.}}}{{=}}\gamma(I)\subset M$
of a flow curve is called a flow orbit.

######  Definition A.2.

The flow of the autonomous dynamical system $(M,X)$ is the flow
$\Pi:\mathcal{D}\rightarrow M$ of the vector field $X$ considered on its
maximal domain of definition $\mathcal{D}\subset\mathbb{R}\times M$.

Recall that $\mathcal{D}$ is an open subset of $\mathbb{R}\times M$ which
contains the set ${\widetilde{M}}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\\{0\\}\times M$. It is fibered over $M$ with fiber at $m\in M$
given by the interval of definition $I_{m}$ of the maximal integral curve
$\gamma_{m}:I_{m}\rightarrow M$ of $X$ which satisfies $0\in I_{m}$ and
$\gamma(0)=m$. Notice that the interval $I_{m}$ is open for all $m\in M$. By
definition, we have:

$\Pi(q,m)=\gamma_{m}(q)~{},~{}\forall(q,m)\in\mathcal{D}~{}~{}.$

We denote by $\pi:\mathcal{D}\rightarrow M$ the projection on the second
factor. For each $m\in M$, we have $\Pi(0,m)=m$. Moreover, for all $t_{0}\in
I_{m}$ we have $I_{\gamma(m)(t_{0})}=I_{m}-t_{0}$ and for all $t\in
I_{\gamma_{m}(t_{0})}$ we have:

$\Pi(t,\Pi(t_{0},m))=\Pi(t+t_{0},m)~{}~{}.$

######  Definition A.3.

Let $(M_{1},X_{1})$ and $(M_{2},X_{2})$ be autonomous dynamical systems with
flows $\Pi_{k}:\mathcal{D}_{k}\rightarrow M_{k}$ and domain projections
$\pi_{k}:\mathcal{D}_{k}\rightarrow M_{k}$ ($k=1,2$). Denote by
$I^{(k)}_{m_{k}}\stackrel{{\scriptstyle{\rm def.}}}{{=}}\pi_{k}^{-1}(m_{k})$
the fiber of $\mathcal{D}_{k}$ at $m_{k}\in M_{k}$. Let $h:M_{1}\rightarrow
M_{2}$ be a homeomorphism and $f:\mathcal{D}_{1}\rightarrow\mathcal{D}_{2}$ be
an unbased isomorphism of topological fiber bundles above $h$, thus:

$f(t,m)=(f_{m}(t),h(m))~{}~{}\forall m\in M_{1}~{}~{}\forall t\in
I^{(1)}_{m}~{}~{},$

where $f_{m}:I_{m}^{(1)}\rightarrow I_{h(m)}^{(2)}$ is a homeomorphism for all
$m\in M_{1}$. The pair $(f,h)$ is called:

  1. 1.

topological equivalence, if the following conditions are satisfied:

     * •

We have:

$\Pi_{2}\circ f=h\circ\Pi_{1}~{}~{},$ (68)

i.e. :

$h(\gamma_{m}^{(1)}(t))=\gamma^{(2)}_{h(m)}(f_{m}(t))~{}~{}\forall m\in
M~{}~{}\forall t\in I_{m}^{(1)}~{}~{},$

where $\gamma^{(k)}_{m_{k}}:I_{m_{k}}^{(k)}\rightarrow M_{k}$ is the flow
curve of $\Pi_{k}$ which satisfies $\gamma_{m_{k}}^{(k)}(0)=m_{k}$.

     * •

$f_{m}:I_{m}^{(1)}\rightarrow I_{h(m)}^{(2)}$ is a strictly increasing
function for all $m\in M$.

  2. 2.

smooth topological equivalence, if it is a topological equivalence and the
maps $f$ and $h$ are smooth diffeomorphisms.

  3. 3.

topological conjugation if it is a topological equivalence and for all $m\in
M$ we have $I_{m}^{(1)}=I_{h(m)}^{(2)}:=I_{m}$ and
$f_{m}=\mathrm{id}_{I_{m}}$.

  4. 4.

smooth topological conjugation if it is a topological conjugation and the maps
$f$ and $h$ are smooth diffeomorphisms.

The two dynamical systems and their flows are called (smoothly) topologically
conjugate/equivalent if there exists a (smooth) topological
conjugation/equivalence $(f,h)$ from $\Pi_{1}$ to $\Pi_{2}$.

(Smooth) topological equivalence and conjugation are equivalence relations. If
$\Pi_{1}$ and $\Pi_{2}$ are topologically equivalent through a pair $(f,h)$,
then $h$ induces a bijection between the sets of orbits of $\Pi_{1}$ and
$\Pi_{2}$ which preserves their orientation.

##  Appendix B Invariance of gradient flow orbits under conformal
transformations

######  Proposition B.1.

Let $\Omega:\mathcal{M}\rightarrow\mathbb{R}_{>0}$ be an everywhere-positive
smooth function defined on $\mathcal{M}$, $\eta:I\rightarrow\mathcal{M}$ be a
gradient flow curve of the scalar triple $(\mathcal{M},\mathcal{G},V)$ and
$t_{0}\in I$ be arbitrary. Consider the increasing reparameterization
$\tau:I\rightarrow J$ defined through:

$\tau(t):=\tau_{\eta}(t)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\Omega(\eta(t^{\prime}))+C~{}~{},$
(69)

where $C$ is an arbitrary constant. Then the reparameterized curve
$\eta_{r}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\eta\circ\tau^{-1}:J\rightarrow\mathcal{M}$ satisfies the gradient
flow equation of the scalar triple $(\mathcal{M},\Omega\mathcal{G},V)$. In
particular, the orbits of the gradient flows of $(\mathcal{M},\mathcal{G},V)$
and $(\mathcal{M},\Omega\mathcal{G},V)$ coincide.

###### Proof.

Consider the Weyl transformation:

$\mathcal{G}\rightarrow\mathcal{G}^{\prime}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Omega\mathcal{G}~{}~{},$

and notice that
$\mathrm{grad}_{\mathcal{G}^{\prime}}V=\frac{1}{\Omega}\mathrm{grad}\,V$. The
reparameterized curve $\eta_{r}$ satisfies:

$\frac{\mathrm{d}\tau}{\mathrm{d}t}\frac{\mathrm{d}\eta_{r}}{\mathrm{d}\tau}+(\mathrm{grad}_{\mathcal{G}}\,V)\circ\eta_{r}=0~{}~{}.$
(70)

Since
$\mathrm{grad}_{\Omega\mathcal{G}}V=\frac{1}{\Omega}\mathrm{grad}_{\mathcal{G}}V$,
this is equivalent to the gradient flow equation of the scalar triple
$(\mathcal{M},\Omega\mathcal{G},V)$ iff
$\frac{\mathrm{d}\tau(t)}{\mathrm{d}t}=\Omega(\eta(t))$, which gives (69). ∎

######  Corollary B.2.

With the notations of the previous proposition, the gradient flows of the
scalar triples $(\mathcal{M},\mathcal{G},V)$ and
$(\mathcal{M},\Omega\mathcal{G},V)$ are smoothly topologically equivalent.

###### Proof.

Let $\mathcal{D}$ and $\mathcal{D}_{r}$ be the maximal domains of definition
of the gradient flows of $(\mathcal{M},\mathcal{G},V)$ and
$(\mathcal{M},\Omega\mathcal{G},V)$. Consider the map
$f:\mathcal{D}\rightarrow\mathcal{D}_{r}$ defined through:

$f(t,m)\stackrel{{\scriptstyle{\rm
def.}}}{{=}}(\tau_{\eta_{m}}(t),m)~{},~{}~{}\forall(t,m)\in\mathcal{D}~{}~{},$

where $\eta_{m}$ is the gradient flow curve of $(\mathcal{M},\mathcal{G},V)$
which satisfies $\eta_{m}(0)=m$. Then $f$ is a diffeomorphism and
$(f,\mathrm{id}_{\mathcal{M}})$ is a smooth topological equivalence from the
gradient flow of $(\mathcal{M},\mathcal{G},V)$ to that of
$(\mathcal{M},\Omega\mathcal{G},V)$. ∎

######  Definition B.3.

A conformal equivalence from the scalar triple
$(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ to the scalar triple
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ is a smooth conformal diffeomorphism
$\Psi:(\mathcal{M}_{1},\mathcal{G}_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2})$
which satisfies the condition $V_{2}\circ\Psi=V_{1}$. The scalar triples
$(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ are called conformally equivalent if
there exists a conformal equivalence from
$(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ to
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$.

It is clear that conformal equivalence is an equivalence relation on the
collection of all scalar triples.

######  Proposition B.4.

Suppose that the scalar triples $(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ are conformally equivalent. Then the
gradient flows of $(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})$ and
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ are smoothly topologically
equivalent.

###### Proof.

Let
$\Psi:(\mathcal{M}_{1},\mathcal{G}_{1},V_{1})\rightarrow(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$
be a conformal diffeomorphism and let
$\Pi_{i}:\mathcal{D}_{i}\rightarrow\mathcal{M}_{i}$ be the gradient flows of
$(\mathcal{M}_{i},\mathcal{G}_{i},V_{i})$ for $i=1,2$. By the definition of
conformal diffeomorphisms, there exists
$\Omega\in\mathcal{C}^{\infty}(\mathcal{M}_{1},\mathbb{R}_{>0})$ such that
$\mathcal{G}^{\prime}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Psi^{\ast}(\mathcal{G}_{2})=\Omega\mathcal{G}_{1}$. Thus $\Psi$
is an isometry from $(\mathcal{M}_{1},\mathcal{G}^{\prime})$ to
$(\mathcal{M}_{2},\mathcal{G}_{2})$ which satisfies
$\Psi^{\ast}(V_{2})=V_{1}$, i.e. an isomorphism of scalar triples from
$(\mathcal{M}_{1},\mathcal{G}^{\prime},V_{1})$ to
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$. Let
$\Pi^{\prime}:\mathcal{D}^{\prime}\rightarrow\mathcal{M}$ be the gradient flow
of $(\mathcal{M}_{1},\mathcal{G}^{\prime},V_{1})$. It is easy to see that a
curve $\eta:I\rightarrow\mathcal{M}_{1}$ is a maximal gradient flow curve of
$(\mathcal{M}_{1},\mathcal{G}^{\prime},V_{1})$ which satisfies $\eta_{1}(0)=m$
(with $m\in\mathcal{M}_{1}$) iff $\eta^{\prime}\stackrel{{\scriptstyle{\rm
def.}}}{{=}}\Psi\circ\eta$ is a maximal gradient flow curve of
$(\mathcal{M}_{2},\mathcal{G}_{2},V_{2})$ which satisfies
$\eta^{\prime}(0)=\Psi(m)$. Thus we have
$\mathcal{D}^{\prime}=\mathcal{D}_{2}$ and
$(\mathrm{id}_{\mathcal{D}_{2}},\Psi)$ is a smooth topological equivalence
from $\Pi^{\prime}$ to $\Pi_{2}$. Since
$\mathcal{G}^{\prime}=\Omega\mathcal{G}_{1}$, Corollary B.2 shows that
$\Pi_{1}$ is smoothly topologically equivalent with $\Pi^{\prime}$. Since
smooth topological equivalence of flows is an equivalence relation, we
conclude. ∎

######  Definition B.5.

A conformal automorphism of the scalar triple $(\mathcal{M},\mathcal{G},V)$ is
a conformal equivalence from this triple to itself.

Conformal automorphisms of $(\mathcal{M},\mathcal{G},V)$ form the stabilizer
of $V$ under the group of conformal transformations of
$(\mathcal{M},\mathcal{G})$.

######  Corollary B.6.

The smooth topological equivalence class of the gradient flow of
$(\mathcal{M},\mathcal{G},V)$ is invariant under conformal automorphisms of
$(\mathcal{M},\mathcal{G},V)$.

###### Proof.

Follows immediately from Proposition B.4 ∎

## References

  * (1) C. Vafa, The string landscape and the swampland, arXiv:hep-th/0509212.
  * (2) H. Ooguri, C. Vafa, On the geometry of the string landscape and the swampland, Nucl. Phys. B 766 (2007) 21-33, arXiv:hep-th/0605264.
  * (3) T. D. Brennan, F. Carta, C. Vafa, The String Landscape, the Swampland, and the Missing Corner, TASI2017 (2017) 015, arXiv:1711.00864 [hep-th].
  * (4) M. van Beest, J. Calderon-Infante, D. Mirfendereski, I. Valenzuela, Lectures on the Swampland Program in String Compactifications, arXiv:2102.01111 [hep-th].
  * (5) A. Achucarro, G.A. Palma, The string swampland constraints require multi-field inflation, JCAP 02 (2019) 041, arXiv:1807.04390 [hep-th].
  * (6) G. Obied, H. Ooguri, L. Spodyneiko, C. Vafa, De Sitter Space and the Swampland, arxiv:1806.08362 [hep-th].
  * (7) S. K. Garg, C. Krishnan, Bounds on Slow Roll and the de Sitter Swampland, JHEP 11 (2019) 075, arXiv:1807.05193 [hep-th].
  * (8) M. Sasaki, E. D. Stewart, A General analytic formula for the spectral index of the density perturbations produced during inflation, Prog. Theor. Phys. 95 (1996) 71, arXiv:astro-ph/9507001.
  * (9) S. Groot Nibbelink, B. J. W. van Tent, Scalar perturbations during multiple field slow-roll inflation, Class. Quant. Grav. 19 (2002) 613 [hep-ph/0107272].
  * (10) J.-O. Gong, T. Tanaka, A covariant approach to general field space metric in multi-field inflation, JCAP 1103 (2011) 015, arXiv:1101.4809 [astro-ph.CO].
  * (11) X. Gao, D. Langlois, S. Mizuno, Influence of heavy modes on perturbations in multiple field inflation, JCAP 10 (2012) 040, arXiv:1205.5275 [hep-th].
  * (12) J. Elliston, D. Seery, R. Tavakol, The inflationary bispectrum with curved field-space, JCAP 1211 (2012) 060, aXiv:1208.6011 [astro-ph.CO].
  * (13) S. Garcia-Saenz, L. Pinol, S. Renaux-Petel, Revisiting non-Gaussianity in multifield inflation with curved field space, JHEP 01 (2020) 073, arXiv:1907.10403 [hep-th].
  * (14) L. Pinol, Multifield inflation beyond $N_{field}=2$: non-Gaussianities and single-field effective theory, JCAP 04 (2021) 002, arXiv:2011.05930 [astro-ph.CO].
  * (15) C. M. Peterson, M. Tegmark, Testing Two-Field Inflation, Phys. Rev. D 83 (2011) 023522, arXiv:1005.4056 [astro-ph.CO].
  * (16) C. M. Peterson, M. Tegmark, Non-Gaussianity in Two-Field Inflation, Phys. Rev. D 84 (2011) 023520, arXiv:1011.6675 [astro-ph.CO].
  * (17) J. Palis Jr., W. De Melo, Geometric theory of dynamical systems: an introduction, Springer, New York, U.S.A. (2012).
  * (18) A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge U.P., 1995.
  * (19) M. Shub, Global stability of dynamical systems, Springer, 1987.
  * (20) C. I. Lazaroiu, C. S. Shahbazi, Generalized two-field $\alpha$-attractor models from geometrically finite hyperbolic surfaces, Nucl. Phys. B 936 (2018) 542-596.
  * (21) S. S. Chern, Sur la géométrie d’un systeme d’équations différentialles du second ordre, Bull. Sci. Math. 63 (1939), 206-212.
  * (22) D. D. Kosambi, Parallelism and path-space, Math. Z. 37 (1933) 608-618.
  * (23) E. Cartan, Observations sur le mémoir précédent, Math. Zeitschrift 37 (1933), 619-622.
  * (24) J. Szilasi, R. L. Lovas, D. C. Kertesz, Connections, sprays and Finsler structures, World Scientific, 2014.
  * (25) P. Antonelli, I. Bucataru, KCC theory of a system of second order differential equations, in Antonelli (ed.), Handbook of Finsler Geometry, vol.1, Kluwer, 2003, pp. 83-174.
  * (26) E. M. Babalic, C. I. Lazaroiu, Cosmological flows on hyperbolic surfaces, Facta Universitatis, Ser. Phys. Chem. Tech. 17 (2019) 1, 1-9, arXiv:1810.00441 [hep-th].
  * (27) H. Freudenthal, Neuaufbau der Endentheorie, Ann. of Math. 43 (1942) 2, 261-279.
  * (28) K. Morita, On bicompactifications of semibicompact spaces, Sci. Rep. Tokyo Bunrika Daigaku A4 (1952), 222-229.
  * (29) J. R. Isbell, Uniform Spaces, Mathematical Surveys and Monographs 12, AMS, 1964.
  * (30) G. Peschke, The theory of ends, Nieuw Archief voor Wiskunde 8 (1990), 1-12.
  * (31) B. Kerekjarto, Vorlesungen über Topologie I, Springer, Berlin, 1923.
  * (32) S. Stoilow, Lecons sur les principes topologiques de la theorie de fonctions analytiques, 2nd ed., Gauthier-Villars, Paris, 1956.
  * (33) I. Richards, On the Classification of Non-Compact Surfaces, Trans. AMS 106 (1963) 2, 259-269.
  * (34) E. M. Babalic, C. I. Lazaroiu, Generalized $\alpha$-attractor models from elementary hyperbolic surfaces, Adv. Math. Phys. 2018 (2018) 7323090, arXiv:1703.01650 [hep-th].
  * (35) E. M. Babalic, C. I. Lazaroiu, Generalized $\alpha$-attractors from the hyperbolic triply-punctured sphere, Nucl. Phys. B 937 (2018) 434-477, arXiv:1703.06033 [hep-th].
  * (36) E. M. Babalic, C. I. Lazaroiu, Two-field cosmological models and the uniformization theorem, Springer Proc. Math. Stat., Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics 2 (2018) 233-241, arXiv:1801.03356 [hep-th].
  * (37) R. Kallosh, A. Linde, D. Roest, Superconformal Inflationary $\alpha$-Attractors, JHEP 11 (2013) 098, arXiv:1311.0472 [hep-th].
  * (38) H. P. de Saint-Gervais, Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem, EMS, 2016.
  * (39) L. Anguelova, E. M. Babalic, C. I. Lazaroiu, Two-field Cosmological $\alpha$-attractors with Noether Symmetry, JHEP 04 (2019) 148, arXiv:1809.10563 [hep-th].
  * (40) L. Anguelova, E. M. Babalic, C. I. Lazaroiu, Hidden symmetries of two-field cosmological models, JHEP 09 (2019) 007, arXiv:1905.01611 [hep-th].
  * (41) E. M. Babalic, C. I. Lazaroiu, Noether Symmetries of Two-Field Cosmological Models, AIP Conf. Proc. 2218 (2020) 050005, arXiv:1910.08441 [hep-th].
  * (42) C. I. Lazaroiu, Hesse manifolds and Hessian symmetries of multifield cosmological models, Rev. Roumaine Math. Pures Appl. 66 (2021) 2, 329-345, arXiv:2009.05117 [hep-th].
  * (43) M. Tsamparlis, A. Palianthanasis, Symmetries of Differential Equations in Cosmology, Symmetry 10 (2018) 7, 233-275, arXiv:1806.05888 [gr-qc].
  * (44) A. Giacomini, E. Gonzalez, G. Leon, A. Paliathanasis, Variational symmetries and superintegrability in multifield cosmology, arXiv:2104.13649 [gr-qc].
  * (45) A. Palianthanasis, New Anisotropic Exact Solution in Multifield Cosmology, Universe 7 (2021) 9, 323-331, arXiv:2108.12154 [gr-qc].
  * (46) S. Mondal, R. Bhaumik, S. Dutta, S. Chakraborty Multi-scalar field cosmological model and possible solutions using Noether symmetry approach, Modern Physics Letters A 36 (2021) 34, 2150246.
  * (47) A. Achucarro, R. Kallosh, A. Linde, D.-G. Wang, Y. Welling, Universality of multi-field $\alpha$-attractors, JCAP 04 (2018) 028, arXiv:1711.09478 [hep-th].
  * (48) S. Kobayashi, A theorem on the affine transformation group of a Riemannian manifold, Nagoya Math. J. 9 (1955) 39-41.
  * (49) S. Ishihara, M. Obata, Affine transformations in a Riemannian manifold, Tohoku Math. J. 7 (1955), 146-150.
  * (50) J. Mikes et. al, Differential Geometry of Special Mappings, Univ. Palackeho v Olomouci, Olomouc 2019, ISBN: 978-80-244-5536-5.
  * (51) M. Crampin, D.J. Saunders, Projective connections, J. Geom. Phys. 57 (2007) 2, 691-727.
  * (52) E. Cartan, Lecons sur la theorie des espaces a connexion projective, Gauthier-Villars, 1937.
  * (53) E. M. Babalic, C. I. Lazaroiu, The infrared behavior of tame two-field cosmological models, arXiv:2203.02297 [gr-qc].
  * (54) C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, 1999.
  * (55) J. Kevorkian, J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, 1996.
  * (56) E. J. Hinch, Perturbation Methods, Cambridge U.P., 1991\.
  * (57) C. Kuehn, Multiple Time Scale Dynamics, Springer, 2015\.
  * (58) P. M. Ziane, On a certain renormalization group method, J. Math. Phys. 41 (2000) 5, 3290-3299.
  * (59) L. DeVille, A. Harkin, M. Holzer, K. Josic, T. J. Kaper, Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Physica D 237 (2008) 1029-1052; Asymptotic normal form theory for nonautonomous equations and its connections with a renormalization group method, SIAM J. Appl. Dyn. Syst (2004) 1-43.
  * (60) H. Chiba, Simplified renormalization group method for ordinary differential equations, J. Diff. Equ. 246 (2009) 1991-2019.
  * (61) H. Chiba Normal Forms of $C^{\infty}$ Vector Fields based on the Renormalization Group, J. Math. Phys. 62 (2021) 062703.
  * (62) C. P. Burgess, S. P. de Alwis, F. Quevedo, Cosmological Trans-Planckian Conjectures are not Effective, JCAP 05 (2021) 037, arXiv:2011.03069 [hep-th].
  * (63) C. P. Burgess, M. Williams, Who You Gonna Call? Runaway Ghosts, Higher Derivatives and Time-Dependence in EFTs, JHEP 08 (2014) 074, arXiv:1404.2236 [gr-qc].
  * (64) L. Anguelova, On Primordial Black Holes from Rapid Turns in Two-field Models, JCAP 06 (2021) 004, arXiv:2012.03705 [hep-th].
  * (65) L. Anguelova, J. Dumancic, R. Gass, L.C.R. Wijewardhana, Dark Energy from Inspiraling in Field Space, arXiv:2111.12136 [hep-th].


# Two-dimensional Control of a Biphoton Joint Spectrum

Anatoly Shukhin  1,* Inbar Hurvitz  2 Sivan Trajtenberg-Mills  2 Ady Arie  2
and Hagai Eisenberg 1  1Racah Institute of Physics, Hebrew University of
Jerusalem, Jerusalem 91904, Israel
2School of Electrical Engineering, Fleischman Faculty of Engineering, Tel Aviv
University, Tel Aviv 69978, Israel<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

Control over the joint spectral amplitude of a photon pair has proved highly
desirable for many quantum applications, since it contains the spectral
quantum correlations, and has crucial effects on the indistinguishability of
photons, as well as promising emerging applications involving complex quantum
functions and frequency encoding of qudits. Until today, this has been
achieved by engineering a single degree of freedom, either by custom poling
nonlinear crystal or by shaping the pump pulse. We present a combined approach
where two degrees of freedom, the phase-matching function, and the pump
spectrum, are controlled. This approach enables the two-dimensional control of
the joint spectral amplitude, generating a variety of spectrally encoded
quantum states - including frequency uncorrelated states, frequency-bin Bell
states, and biphoton qudit states. In addition, the joint spectral amplitude
is controlled by photon bunching and anti-bunching, reflecting the symmetry of
the phase-matching function.

##  1 Introduction

Superposition states of single photons and entangled states of photon pairs
are the core of quantum optics and quantum information technologies. The
ability to efficiently create such quantum states and manipulate their
properties are among the main problems in the way of building long-distance
quantum communications, quantum computing, and quantum sensors [1, 2].
Regarding state generation, one of the most promising approaches is to harness
nonlinear optical effects, e.g., spontaneous parametric down-conversion (SPDC)
[3, 4]. Indeed, for the last decades, SPDC has been serving as a source of
high-purity heralded single-photons, entangled photon pairs, flying qubits,
and other photonic states [5, 6].

In order to build long-distance quantum communication links as well as
scalable quantum networks, in many cases it is preferable to encode
information in the photons’ spectral degree of freedom (DOF)[7, 8, 9] due to
its advantages: i) a quantum state encoded in the frequency domain experiences
virtually no decoherence in comparison with, for example, polarization-encoded
quantum information, and thus information can be transferred through existing
standard optical fiber networks [10]; ii) Unlike polarization, frequency
encoding can carry multilevel quantum information (qudits) [11], which has
benefits when comparing to qubits[12, 13, 14, 15, 16]. In order to take full
advantage of the high-dimensional Hilbert space in the frequency domain, it is
necessary to be able to tailor the spectral properties of the generated
states.

For photon pairs generated via SPDC, this mainly implies controlling their
joint spectral intensity (JSI). There are different ways to control the JSI.
One of them is to use domain-engineered nonlinear crystals [17]. Recently, it
has been shown that shaping the JSI via this approach can produce high-purity
heralded single photons [18, 17, 19, 20, 21, 22], create frequency-bin
entanglement [23], and perform entanglement swapping [24, 25]. Another
approach is active spectral/temporal shaping of the pump field using acousto-
optic or electro-optic modulators, where generation of pure heralded single
photons [26], tailoring the temporal-mode structure of the heralded photons
[27, 28], Schmidt-mode selection [29], and time-domain quantum-optical-
synthesis [30] have been shown. Temporal/spectral shaping can also be applied
to the photons after they are generated [31]. All four frequency-bin Bell
states have been synthesized via pump modulation along with the shaping of
entangled photons [32]. The possibility of the 2D modulation of the JSI was
theoretically shown based on spectrally-resolved two-photon interference [33].
It was theoretically demonstrated that the combination of the domain-
engineered crystals together with pump modulation allows for a tunable
frequency-bin multi-mode squeezed states generation and can be a fruitful path
toward discovering new quantum applications [34].

In this work, we experimentally demonstrate for the first time that utilizing
domain-engineered KTiOPO4 (KTP) crystals along with modulation of a broadband
femtosecond pulsed pump via a simple technique, using only passive optical
elements, can generate a variety of two-photon and single-photon states with
tailored spectral properties (Fig. 1). We also show how by using a single
half-wave plate (HWP) or a linear polarizer after the nonlinear crystal, the
JSI can be manipulated after the photons are generated, by photon bunching in
the frequency domain. We demonstrate the generation of high purity frequency
uncorrelated (separable) states, dual frequency correlated photon pairs as
well as two of the four frequency-bin Bell states, frequency entangled
biphoton 5-mode qudit states and biphoton frequency qutrit states, based on
superposition of 4 pairs of signal/idler modes. The flexibility of generating
different quantum states using the same setup is highly desired for quantum
information applications.

Figure 1: Illustration of the 2-dimensional control over the JSI. Top -
Gaussian-shaped pump spectrum and PMF resulting a round JSI. Bottom - double-
Gaussian-shaped pump spectrum and PMF resulting a 4 lobe JSI.

##  2 Theoretical background

SPDC is a nonlinear optical process in which one photon from a pump field is
annihilated in exchange for two photons that are simultaneously created into
fields that are typically called signal and idler [4]. The frequencies and
wave vectors of the three interacting fields obey energy and momentum
conservation laws: $\omega_{p}=\omega_{s}+\omega_{i}$, and
$k_{p}=k_{s}+k_{i}$. Here we focus on the collinear (pump, signal and idler
propagate in the same direction) degenerate (signal and idler share the same
frequency) type-II (signal and idler are orthogonally polarized) SPDC due to
its efficiency and usefulness from the experimental point of view. The state
vector of the two-photon field generated via SPDC in a bulk crystal can be
calculated by first-order perturbation theory [35]:

$\begin{split}|\psi\rangle=&\sqrt{1-\eta^{2}}|0\rangle+\eta\iint
d\omega_{s}d\omega_{i}A(\omega_{s},\omega_{i})a^{\dagger}_{h}(\omega_{s})a^{\dagger}_{v}(\omega_{i})|0\rangle,\end{split}$
(1)

where $|0\rangle$ is the vacuum state, and $a^{\dagger}_{h}(\omega_{s})$ and
$a^{\dagger}_{v}(\omega_{i})$ represent the creation operators into the signal
and idler modes, with horizontal and vertical polarizations, respectively.
$A(\omega_{s},\omega_{i})$ is the joint spectral amplitude (JSA) of the two-
photon field describing the amplitude of generating the signal and idler
photons at the respective frequencies $\omega_{s}$ and $\omega_{i}$. The
conversion efficiency $\eta$ is proportional to the crystal nonlinear
susceptibility $\chi^{(2)}$ and the crystal length $L$. The square of the
absolute value of the JSA is the JSI that describes the probability density of
generating the signal and idler photons at frequencies $\omega_{s}$ and
$\omega_{i}$.

The JSA is given by a product of the phase-matching function (PMF)
$\phi(\Delta k(\omega_{s},\omega_{i}))$ and the spectral amplitude of the pump
field $A_{p}(\omega_{s}+\omega_{i})$ (Fig. 1) [36]. The former, being a
Fourier transform of the crystal’s poling function [36], in the general case
of an arbitrary poling design [37], can be written as:

$\phi(\Delta
k(\omega_{s},\omega_{i}))=\chi^{(2)}\int^{\infty}_{-\infty}d(z)e^{i\Delta
k(\omega_{s},\omega_{i})z}dz,$ (2)

where $d(z)$ is the poling function describing the sign of $\chi^{(2)}$ as a
function of the longitudinal coordinate $z$. $\Delta k(\omega_{s},\omega_{i})$
is the longitudinal wave-vector mismatch which in the case of a constant
poling period is given by [38, 39]:

$\Delta
k(\omega_{s},\omega_{i})=k_{p}(\omega_{s}+\omega_{i})-k_{s}(\omega_{s})-k_{i}(\omega_{i})\pm\frac{2\pi
m}{\Lambda},$ (3)

where $k(\omega_{l})=\omega_{l}n(\omega_{l})/c$, $n(\omega_{l})$ is the
refractive index of the crystal at a frequency $\omega_{l}$ $(l=p,s,i)$, $m$
is the order of quasi-phase-matching and $\Lambda$ is the poling period. Thus,
the PMF is essentially determined by the nonlinearity profile and the phase
mismatch [36].

The nonlinearity profile can be characterized by the poling period and duty
cycle. The latter is the ratio between the lengths of domains of opposite
signs. The poling period affects what wavelengths are phase-matched under a
given pump, while the duty cycle affects generation amplitude at the phase-
matched wavelengths (50% duty cycle for maximum amplitude and 0% for zero
amplitude).

In the simplest case of a crystal without poling or with a constant poling
period and duty cycle, the PMF can be expressed in the following way:

$\phi(\Delta k(\omega_{s},\omega_{i}))=\int^{L}_{0}e^{i\Delta
k(\omega_{s},\omega_{i})z}dz=L\operatorname{sinc}\left(\frac{\Delta
k(\omega_{s},\omega_{i})L}{2}\right)\exp\left({i\frac{\Delta
k(\omega_{s},\omega_{i})L}{2}}\right).$ (4)

The poling period and duty cycle that vary along the crystal, locally
determine phase-matched wavelengths and their generation amplitude. By using a
domain-engineered nonlinear crystal, it is possible to obtain the required PMF
and therefore spectral properties of the generated photon pairs along the
difference-frequency axis on a JSI graph.

Obtaining the desired spectral shape of the PMF $\phi(\Delta
k(\omega_{s},\omega_{i}))$, was previously done by numerical optimization
[17]. However, this shape can be obtained analytically: by taking the Fourier
transform of the PMF, which we denote $u(z)$, we can design the nonlinear
crystal as follows [37, 40]:

$\begin{split}d_{j}(z)=\text{sign}\left[\cos\left(\frac{2\pi}{\Lambda}z+\varphi_{j}(z)\right)-\cos(\pi
q_{j}(z))\right],\end{split}$ (5)

where $q(z)=\arcsin(|u(z)|/\pi)$, and $\varphi$ is the phase of $u(z)$. In
this work, we used three different crystal designs, $d_{j}(z)$, where
$j=0,1,2$ is for a sinc-, a Gaussian-, and a first-order Hermite-Gaussian-
shaped PMF, respectively (Table 1).

Table 1: A detailed table of the periodically poled, Gaussian and Hermite-
Gaussian designs, providing crystal lengths, $\phi(\Delta
k(\omega_{s},\omega_{i}))$, and figures that show the phase-matching function
vs. $\Delta k$ (which is given in Eq.3). For the Gaussian and Hermite-Gaussian
crystals $\sigma\approx 3.7\cdot 10^{-4}[1/\mu m]$. For the Gaussian and HG1
designs we ignore the finite length of the crystal, since the efficiency at
the edges of the crystal is negligible.

The spectral amplitude of the pump, which is a bandwidth-limited sech2 wave
packet of duration $\tau_{p}$ (FWHM), can be approximated for simplicity by a
Gaussian of the same FWHM as

$A_{p}(\omega_{s}+\omega_{i})=A_{0}\exp\left({-\frac{(\omega_{p}-\omega_{p_{0}})^{2}}{2\sigma^{2}_{\omega}}}\right),$
(6)

where $A_{0}$ and $\omega_{p_{0}}$ are the amplitude and the central frequency
of the pump field. The pump bandwidth (FWHM) is then defined as
$\Delta\omega_{p}=2\sqrt{2\ln 2}~{}\sigma_{\omega}=0.315\cdot 2\pi/\tau_{p}$,
where 0.315 is the time-bandwidth product for a sech2 pulse.

The JSI reveals spectral properties, including frequency correlations of the
two-photon field and therefore it is one of the main functions characterizing
SPDC sources. Shaping the JSI enables one to generate quantum states ranging
from pure heralded single-photons and entangled two-photon states [41] to
frequency-entangled qudits [42, 43] and grid states [44]. As shown above, the
JSI can be manipulated by modifying both the PMF and the pump spectrum. In
most cases, the PMF has a sinc-shaped profile due to the constant-period
poling or homogeneous poling in the nonlinear crystals commonly used in
experiments. On the other hand, the pump spectrum is usually the outcome of
laser radiation, approximated by the Gaussian spectrum, as described earlier.
As a result, the spectrum of a generated state, as well as the state itself
(since in some cases changing a spectrum leads to a different quantum state
(see the following sections) are unlikely to have the required
characteristics, such as purity and fidelity, approaching 1.

Nonetheless, as it will be shown in Secs. 4.2$-$4.3, a simple way of pump
modulation along with non-uniform crystal poling provides a powerful tool for
building an adaptable source that is able to generate a variety of quantum
states while keeping the changes in the experimental setup minimal.

Figure 2: Experimental arrangement for observing SPDC in domain-engineered KTP
crystals under the modulated pump. The radiation of the pulsed laser is split
into two arms via a beam splitter (BS). The pump is modulated (shown in the
yellow box), focused into the domain-engineered (DE) KTP crystal (shown in the
pink box) where SPDC occurs, and then filtered out. Photon pairs, generated
inside the crystal, are split into two arms by polarization, are coupled into
single-mode fibers, and are detected by single-photon detectors. Arrival times
of electrical signals from the detectors are compared using a time tagger. A
small fraction of the pump power is detected by a photodiode, electrical
signals from which are used as a reference clock allowing to measure the
relative arrival time of the photon pairs. A HWP3 or a POL2 are used only for
Hong-Ou-Mandel interference (shown in the purple box), described in sec. 4.4.
See text for more details.

##  3 Setup description

The experimental setup, illustrated in Fig. 2, involved the utilization of a
tunable pulsed Titanium-Sapphire laser as the pump source with a pulse
duration of $\approx$ 90 fs, a pulse repetition rate of 76 MHz, and a spectral
width of approximately 7 nm (FWHM). The central wavelength $\lambda_{p_{0}}$
was adjusted to 780 nm.

The pump spectral modulation was acheived as follows (see the yellow box in
Fig. 2): a vertically polarized pump pulse passes through a half-wave plate
(HWP1) rotated at $22.5^{\circ}$, followed by two movable quartz wedges (QW)
with fast and slow axes in the {H,V} basis and a variable overall thickness in
the range of $(2-18)\pm 0.1$ mm. The wedges split each diagonally polarized
pump pulse into two pulses with orthogonal polarizations. The delay $\tau$
between them depends on the properties of the wedges:
$\tau=|L_{\textrm{QW}}(n^{\textrm{gr}}_{\textrm{H}}-n^{\textrm{gr}}_{\textrm{V}})/c|$,
where $L_{\textrm{QW}}$ is the overall thickness of the wedges,
$n^{\textrm{gr}}_{\textrm{H}}$ and $n^{\textrm{gr}}_{\textrm{V}}$ are the
group indices of quartz for the horizontally and vertically polarized light,
respectively, and $c$ is the speed of light in vacuum. The mentioned thickness
range allows a relative delay from $64$ to $572~{}(\pm 2)$ fs. Next, the two
pulses are polarized using a linear polarizer (POL) positioned at an angle of
$45^{\circ}$ relative to the reference frame of the wedges. The pump passes
through another half-wave plate (HWP2) to match its polarization to the KTP
axes. Afterward, the pump is focused by the L1 lens into the domain-engineered
KTP crystal (Raicol Crystals Ltd.), where collinear type-II SPDC occurs.

After the crystal, the pump is filtered out using three dichroic mirrors (DM)
with 40 dB extinction for each mirror between the down-converted photons
around 1560 nm and the pump. The signal and idler photons are then collimated
by the L2 lens, split into two arms via a polarizing beam splitter (PBS), and
coupled into single-mode fibers using coupling stages (CS1, CS2). The signal
and idler photons are then detected by two InGaAs infrared single-photon
avalanche detectors D1, D2 (Micro Photon Devices Srl), that are triggered by
the laser pulses. The detection efficiency is $25\%$ at 1550 nm, and the dark
count rate in gated mode is $\sim 0.8$ kHz, at $6$ V of excess bias voltage,
gate duration of 1 ns, and $30\text{\,}\mathrm{\SIUnitSymbolMicro s}$ hold-off
time. The detected electrical pulses arrival times were recorded using a time
tagger (HRMTime, SensL) with a temporal resolution of 27 ps. Based on the
rates of detected coincidences ($0.7\pm 0.1$ kHz/mW) and single counts ($14\pm
1$ kHz/mW), and correcting for the hold-off time, as well as the losses in the
setup and the detectors’ efficiencies, the internal pair generation rate was
$300\pm 30$ kHz/mW.

To measure the JSI of the generated photons, we used a time-of-flight
spectroscopy [45, 46, 47]. This technique leverages the natural capability of
sufficiently long and dispersive fibers to perform a frequency-time Fourier
transform on incoming light [48]. The benefits of using this method are that
it does not require alignment and offers a faster measurement process compared
to alternative methods. We used 10 km of SMF-28 optical fiber in both arms of
the signal and idler photons. The fiber dispersion value is 18 ps$/$nm$/$km at
1550 nm, hence the spectral delay is 180 ps/nm for a 10-km fiber. By comparing
the arrival times of the correlated photons and the reference clock from the
photodiode, we reconstruct the JSI.

The temporal resolution of our setup is limited by the total timing jitter,
which consists of the jitters of the detectors, the time-tagger, and the clock
photodiode. The combined effect of the latter two components was determined by
analyzing the width of the distribution of the period between the times
originating from the clock, which was found to be $\approx 130$ ps. Quadrature
summation of the mentioned jitter values results in a temporal resolution of
150 ps (FWHM).

In addition to timing jitter, the temporal resolution is also constrained by
the marginal (unconditional) duration of the single photons being analyzed.
However, in our case, the single photon duration is shorter than 1 ps, which
is significantly smaller than the timing jitter and has a negligible impact on
the overall resolution.

Taking into account the dispersive fiber length tolerance (5% of the fiber
length), the resolution in wavelength $\Delta\lambda$ was calculated to be
$0.83\pm 0.04$ nm, using the relation $\Delta\lambda=\Delta
t/DL_{\textrm{fiber}}$, where $L_{\textrm{fiber}}$ is the fiber length and $D$
is the fiber dispersion. The resolution can be further improved by using
longer fibers and higher dispersion, usually at the expense of higher photon
loss.

##  4 Experimental results

###  4.1 Domain-engineered nonlinear crystals

Three KTP crystals, each having a constant poling period of $46\,\mu m$, were
used in the experiments (see Table 1 for details). The first crystal is 5 mm
long and has a constant duty cycle of $50\%$. The second crystal has a duty
cycle that gradually varies from the crystal’s center (50%) towards the edges
(3% due to the finite domain width), describing a Gaussian function (zeroth-
order Hermite-Gaussian HG0) which results in an apodized nonlinearity along
the crystal [49, 50, 21]. The first reason for apodization is to eliminate the
side lobes that appear in an ordinary sinc-shaped phase-matching function.
These side lobes, being arranged at an angle to the signal/idler wavelength
axes (see, for instance, Fig. 4a), reduce the purity of heralded single-photon
states, generated in non-apodized crystals [4, 18, 17, 19, 20, 21, 22].

Figure 3: SPDC in the domain-engineered 15.9 mm long crystal, that is
described by the double-lobe PMF (see Table1, bottom line). (a) poling design,
where the domains in the two halves of the crystal are arranged in the
opposite order (b) calculated PMF, (c) and (d) calculated and experimentally
measured JSI without pump filtering, respectively, (e) and (f) calculated and
experimentally measured JSI under the filtered pump of 1.5 nm bandwidth,
respectively. The generated state is the frequency-bin $|\Psi^{-}\rangle$ Bell
state. The lines in (c) and (e) correspond to the $1/e^{2}$ of the PMF and
pump intensity distributions.

The poling pattern of the third crystal (Fig. 3a) results in a first-order
Hermite-Gaussian $\text{HG}_{1}$ PMF, consisting of two main lobes (having
positive and negative nonlinear response) with no side-lobes (Fig. 3b).
Multiplying this PMF by the pump spectrum and taking the absolute squared
value, results with the JSI shown in Fig. 3c. The experimentally measured JSI
is shown in Fig. 3d. An interesting feature of this JSI is that while each
lobe, being diagonally elongated, corresponds to the frequency-correlated
state, the two lobes as a whole are anti-diagonally arranged and demonstrate
the frequency anti-correlated state. As opposed to the other methods that rely
on interference of the generated biphotons (such as in Ref. [30]), the
resulting modulation from the domain-engineered crystals does not require
precise stabilization and alignment of interfering beams, since the
interference occurs inside the nonlinear crystal.

If the pump is filtered such that its bandwidth is equal to that of the single
lobe of the PMF, the JSI results in two round lobes, positioned along the
difference-frequency axis and equidistant from the exact degenerate regime.
This corresponds to a frequency-bin $|\Psi^{-}\rangle$ Bell state [32]:
$|\omega^{s}_{1}\omega^{i}_{2}\rangle-|\omega^{s}_{2}\omega^{i}_{1}\rangle$,
where $\omega^{s}_{1,2}$ and $\omega^{i}_{1,2}$ are the central frequencies of
each lobe for the signal and idler field, respectively. The sign between the
two terms in the above expression reflects the relative phase between the two
lobes of the PMF, which in this case is $\pi$. The simulated and measured JSIs
for this case are shown in Fig. 3e and Fig. 3f, respectively. Notice, that all
the graphs in this paper are normalized to a maximum value of 1. A convenient
way to quantify the degree of entanglement of a given state is the
cooperativity parameter $\mathcal{K}$ (also known as the Schmidt number) [41].
In the case of the ideal Bell state, the cooperativity is equal to 2, whereas
for the JSI in Fig. 3f it is reduced to $1.706\pm 0.003$ due to the presence
of background noise.

Using a crystal with two in-phase phase-matching lobes allows for creation of
the frequency-bin $|\Psi^{+}\rangle$ Bell as well.

Generally, the domain engineering technique gives the ability to tailor the
generated state, affecting the generation probability amplitude within a
nonlinear crystal. The following sections will discuss other options to
control the JSI before and after the nonlinear crystal.

###  4.2 Pump modulation

Table 2: A detailed table of the modulation of the pump spectra. Three pump
spectra used in the experiments are shown here in both the time and the
frequency domains.

Splitting each pump pulse into two pulses with a delay (determined by the
thickness and refractive indices of the quartz wedges) allows for a modulation
of the pump spectrum (Table 2). This translates into a modulation of the JSI
along the sum-frequency axis. Specifically, a Fourier transform (denoted below
as $\mathcal{F}$) of the two Gaussian wave packets with a carrier frequency
$\omega_{p_{0}}=2\pi c/\lambda_{p_{0}}$ and with a relative delay $\tau$ can
be written as:

$\displaystyle\mathcal{F}\left\\{e^{i\omega_{p}t}\left(e^{\frac{-\left(t+\frac{\tau}{2}\right)^{2}}{2\sigma^{2}_{t}}}+e^{\frac{-\left(t-\frac{\tau}{2}\right)^{2}}{2\sigma^{2}_{t}}}e^{i\phi}\right)+\textrm{c.c.}\right\\}=e^{\frac{-\left(\omega_{p}-\omega_{p_{0}}\right)^{2}}{2\sigma^{2}_{\omega}}}\cos\left((\omega_{p}-\omega_{p_{0}})\tau+\frac{\phi}{2}\right),$
(7)

where $\phi$ is the phase difference between the two pulses accumulated during
their propagation through the quartz wedges of thickness $L_{\textrm{QW}}$. By
employing quartz wedges with variable thicknesses, we can adjust the pump
spectrum modulation frequency and the position of its maxima and minima. The
different modulations of the pump’s temporal shape are shown in Table 2,
together with their spectral shape.

Figure 4: JSI graphs, obtained from the 5 mm long crystal (top row) and the
15.9 mm long crystal with the single-lobe PMF (bottom row), (see Table1 for
details). (a) and (d) The non-modulated Gaussian pump of 7 nm bandwidth. (b)
and (e) The modulated pump; $\tau=226\pm 2$ fs, $L_{\textrm{QW}}=7.0\pm
0.1~{}$mm and $\tau=382\pm 2~{}$fs, $L_{\textrm{QW}}=12.0\pm 0.1~{}$mm,
respectively. (c) and (f) The non-modulated pump of 3 nm and 1 nm bandwidth,
respectively. The graphs (a),(d),(c), and (f) are obtained without HWP1, QW,
and POL in the setup (see Fig. 2). Fig. (b) represents the frequency-bin
$|\Phi^{-}\rangle$ Bell-state. The pump in (a) and (d) corresponds to the
first column of Table 2, while the pump in (b) and (e) corresponds to the
second and third columns, respectively.

####  4.2.1 5 mm periodically poled KTP crystal

The effect of pump modulation on the JSI for the 5 mm long crystal can be seen
in Fig. 4 (top row). Without modulation, the JSI is an ellipse elongated along
the sum-frequency axis (Fig. 4a). Such a state corresponds to the frequency-
correlated photon pairs. Once the pump is modulated, the initial single-lobe
ellipse turns into multiple lobes. Adjusting the delay $\tau$, it is possible
to attain two round lobes (Fig. 4b). The state in this case corresponds to the
frequency-bin $|\Phi^{-}\rangle$ Bell-state [32]:
$|\omega^{s}_{1}\omega^{i}_{1}\rangle-|\omega^{s}_{2}\omega^{i}_{2}\rangle$.
The minus sign reflects the fact that the phase difference between the two
adjacent lobes is $\pi$ since the modulating cosine function of the pump
changes its sign on every zero crossing. $\omega_{1}$ and $\omega_{2}$ are the
central frequencies of the two lobes, which are correlated for the signal and
idler field. The cooperativity parameter in this case is $1.933\pm 0.003$. It
is also possible to use two pairs of quartz wedges of equal thicknesses along
with a half-wave plate at $22.5^{\circ}$ placed in between, to obtain two in-
phase lobes in the pump spectrum. Such spectrum generates the
$|\Phi^{+}\rangle$ Bell state without the need for active electro-optical
components. Spectral filtering of the pump selects one of the lobes, resulting
in a round-shaped JSI. In this case, the generated photon pairs are frequency-
uncorrelated, and the heralded single photons are in a pure state [4]. The
condition for a round-shaped JSI (assuming non-modulated pump and the single-
lobe PMF) is the equality between the widths of the JSI along the pump and PMF
directions. This is achievable by the pump filtering whenever the PMF’s
bandwidth is narrower than that of the pump. For the 5 mm long crystal, this
condition corresponds to a pump filter of 3 nm bandwidth. The measured JSI
under such filtering is demonstrated in Fig. 4c. The Schmidt decomposition
analysis [51] of this JSI resulted in heralded single-photon purity
($1/\mathcal{K}$) as high as $0.996\pm 0.003$. This is based on the assumption
that the shape of the JSI mirrors the shape of the JSA, disregarding any joint
phase effects, such as $\exp(i\omega_{s}\omega_{i}t^{2})$ in the case of a
chirped pump, which render the JSA non-factorable in terms of phase. Thus, it
is possible to efficiently switch between the generation of frequency-
correlated and frequency-uncorrelated states, which increases the versatility
of this source.

####  4.2.2 15.9 mm single-lobe phase matched KTP crystal

Using longer crystals leads to a narrower bandwidth of the PMF. Spectrally
uncorrelated two-photon states may still be generated by pump filtering. Fig.
4d and Fig. 4f show the JSIs obtained from a 15.9 mm long crystal without and
with a spectral bandpass filter of 1.5 nm bandwidth placed in the pump beam.
As seen in the graphs, such a filter bandwidth matches the PMF’s bandwidth,
generating a round-shaped JSI (without side lobes due to apodization). The
heralded single-photon purity is $0.988\pm 0.003$.

Frequency-bin entanglement between two photons does not necessarily equate to
entanglement between two qubits. The JSI shown in Fig. 4b indeed describes the
frequency-entangled state of two qubits, since each field is in the
superposition of two frequency bins (lobes): $\omega_{1}$ or $\omega_{2}$.
Nevertheless, a longer delay $\tau$ increases the modulation frequency of the
pump and hence also the number of lobes in the JSI (Fig. 4e). As a result,
each biphoton is in a superposition of more than two spectral bins, leading to
the generation of a frequency-entangled two-qudit state

$|\psi\rangle=\sum_{n}(-1)^{n}c_{n}|\omega^{s}_{n}\omega^{i}_{n}\rangle,$ (8)

where $n$ and $c_{n}$ are the indices and the weights of the frequency bins.
The cooperativity parameter in this case is $2.745\pm 0.003$. Notice that the
described two-qubit and two-qudit frequency-entangled states are pure only if
each lobe has a round shape and the projections of the lobes onto the
frequency axes are not overlapping. By choosing the delay $\tau$
appropriately, the first requirement may be satisfied for different crystal
designs and lengths, as well as for different pump bandwidths. The second
requirement is satisfied by further controlling the shape of the pump pulse.
We should note again that using multiple quartz wedges with different
thicknesses can enable more intricate modulation in the JSI, providing the
ability to control the relative phase between different lobes in the pump and
photon pairs’ spectrum. Frequency-bin qudit states were recently reported in
[23], by using structured crystals. In contrary to what is shown here, in that
case the states are aligned along the orthogonal line of the joint spectrum,
with different frequencies of the signal and idler in each of the separate
lobes.

###  4.3 Two-dimensional control of the JSI

The combination of the modulated pump and the domain-engineered nonlinear
crystal allows for two-dimensional control of the JSI. The control directions
are along the sum- and difference-frequency axes. The 15.9 mm long KTP
crystal, characterized by the double-lobe PMF, enables the generation of
unusual quantum states such as a frequency-entangled two-qutrit state. The
calculated and measured JSIs, consisting of four lobes are presented in Fig.
5a and Fig. 5b, respectively.

Figure 5: A JSI from the 15.9 mm long crystal with the double-lobe PMF (see
Table1 bottom line) and modulated pump spectrum: (a) simulated, (b)
experimentally measured. The lines in (a) correspond to the $1/e^{2}$ of the
PMF and pump intensity distributions.

In order to create this JSI, the pump spectrum was modulated so that it
consists of two lobes, whose bandwidths are equal to that of the PMF lobes.
Under this condition, the four lobes are round. In the basis of the marginal
spectra, each photon is projected onto three spectral lobes that correspond to
the basis states of a qutrit. The generated two-qutrit state is neither
correlated nor anti-correlated, in contrast to the regular frequency-entangled
states. This two-photon state can be written as

$|\psi\rangle=|\omega^{s}_{2}\rangle\otimes\left(|\omega^{i}_{1}\rangle+|\omega^{i}_{3}\rangle\right)-\left(|\omega^{s}_{1}\rangle+|\omega^{s}_{3}\rangle\right)\otimes|\omega^{i}_{2}\rangle,$
(9)

where the lower indices $1,2$, and $3$ correspond to the left, central, and
the right maxima of each field, respectively. The state, however, can be
turned into either correlated or anti-correlated by post-selecting
corresponding pairs of lobes. The cooperativity parameter for the calculated
JSI (Fig. 5a) is 2, corresponding to the two-mode entanglement: . The same
parameter for the measured JSI (Fig. 5b) is reduced to $1.506\pm 0.003$ due to
the presence of background noise. More recently, this state has been
theoretically investigated in the context of the time-frequency quantum
metrology [52].

An interesting aspect of this state is revealed when the signal and idler
photons are sent to Alice and Bob, respectively. Whenever Alice detects her
photons in the state $|\omega_{2}\rangle$, she gets no information about the
corresponding outcome of Bob, which can be either $|\omega_{1}\rangle$ or
$|\omega_{3}\rangle$. On the other hand, whichever Bob’s result in this case
is, he knows unambiguously what is the outcome of Alice. Conversely, taking
into account the symmetry of the state, if Alice measures either
$|\omega_{1}\rangle$ or $|\omega_{3}\rangle$, the roles are switched.

Increasing the number of peaks in the pump and/or in the PMF allows for the
generation of frequency-bin cluster states [53, 54] and paves the way for the
generation of more intricate states, such as Gottesman, Kitaev and Preskill
(GKP) states, that can be used for quantum error correction [44].

###  4.4 Shaping the JSI via Hong-Ou-Mandel interference

Generation of states with multiple lobes along the sum-frequency direction,
such as those presented in Fig. 3 and Fig. 5 is possible even if the crystal
is not domain-engineered. In this method, after the photons are already
created, the JSA interferes with its own spectrally-flipped (around
$\omega_{s}=\omega_{i}$ axis) replica. This interference can be achieved in
two ways. The first method is to rotate the polarization of the signal and
idler photons by $45^{\circ}$ by placing a half-wave plate (HWP3 in Fig. 2) at
$22.5^{\circ}$. The biphoton state from the source is described by the Fock
representation in Eq. 1. After the wave-plate rotation, the state is
transformed into [55]

$\begin{split}|\psi\rangle=&\frac{1}{2}\iint
d\omega_{1}d\omega_{2}A(\omega_{1},\omega_{2})\times\\\
&[a^{\dagger}_{h}(\omega_{1})a^{\dagger}_{h}(\omega_{2})+a^{\dagger}_{h}(\omega_{1})a^{\dagger}_{v}(\omega_{2})-a^{\dagger}_{v}(\omega_{1})a^{\dagger}_{h}(\omega_{2})+a^{\dagger}_{v}(\omega_{1})a^{\dagger}_{v}(\omega_{2})]|0\rangle.\end{split}$
(10)

At the following PBS (see Fig. 2), the first and last terms do not yield to
eventual coincidences, while the two middle terms do. By re-labeling the
second term that is left after the post-selection, the detected normalized
state is

$\begin{split}|\psi\rangle=&\iint
d\omega_{1}d\omega_{2}[A(\omega_{1},\omega_{2})-A(\omega_{2},\omega_{1})]a^{\dagger}_{h}(\omega_{1})a^{\dagger}_{v}(\omega_{2})|0\rangle.\end{split}$
(11)

As long as the JSA is symmetric (even) for the exchange between
$\omega_{1}\Leftrightarrow\omega_{2}$,
$A(\omega_{1},\omega_{2})=A(\omega_{2},\omega_{1})$ and these two terms cancel
each other, a manifestation of the Hong-Ou-Mandel bunching effect. In the case
of an anti-symmetric (odd) JSA
$A(\omega_{1},\omega_{2})=-A(\omega_{2},\omega_{1})$ these two terms interfere
constructively and the photons anti-bunch. The polarization walk-off
$\tau_{wo}$ between the two photons in the generating crystal at the reference
frame of the $\omega_{1}$ photon, can also break the symmetry of the JSA,
which becomes $A(\omega_{1},\omega_{2})e^{-i\omega_{2}\tau_{wo}}$.
Substituting a JSA from a symmetric PMF (such as from the $\text{HG}_{0}$
crystal) and walk-off into Eq. 11, the observed JSI is [56, 57, 58, 55, 59,
57, 60, 61, 43, 58]

$\begin{split}I_{\text{HWP}}=&|A(\omega_{1},\omega_{2})e^{-i\omega_{2}\tau_{wo}}-A(\omega_{2},\omega_{1})e^{-i\omega_{1}\tau_{wo}}|^{2}=|A(\omega_{1},\omega_{2})|^{2}\sin^{2}\left(\frac{\omega_{2}-\omega_{1}}{2}\tau_{wo}\right).\end{split}$
(12)

The biphotons of the $\omega_{1}=\omega_{2}$ degenerate line are
indistinguishable, and thus result in no coincidences. The JSI oscillates
along the sum-frequency direction. If the JSA without walk-off is anti-
symmetric (such as from the $\text{HG}_{1}$ crystal) the sine function in the
JSI is replaced by a cosine function.

A second method for interfering the biphoton spectrum with itself is by
placing a linear polarizer (POL2 in Fig. 2) at $45^{\circ}$ before the PBS.
The original state of Eq. 1 can be written in the $\pm 45^{\circ}$ diagonal
polarization basis as

$\begin{split}|\psi\rangle=&\frac{1}{2}\iint
d\omega_{1}d\omega_{2}A(\omega_{1},\omega_{2})\times\\\
&[a^{\dagger}_{p}(\omega_{1})a^{\dagger}_{p}(\omega_{2})+a^{\dagger}_{p}(\omega_{1})a^{\dagger}_{m}(\omega_{2})-a^{\dagger}_{m}(\omega_{1})a^{\dagger}_{p}(\omega_{2})+a^{\dagger}_{m}(\omega_{1})a^{\dagger}_{m}(\omega_{2})]|0\rangle,\end{split}$
(13)

where $a^{\dagger}_{p}$ and $a^{\dagger}_{m}$ represent the creation operators
of a $+45^{\circ}$ and a $-45^{\circ}$ polarized photons, respectively. The
polarizer transmits only the $p$-polarized photons, where only the first term
can contribute two photons for the eventual coincidence measurement. As the
PBS projects on the {H,V} basis, the normalized transmitted state in this
basis is

$\begin{split}|\psi\rangle=&\frac{1}{2}\iint
d\omega_{1}d\omega_{2}A(\omega_{1},\omega_{2})\times\\\
&[a^{\dagger}_{h}(\omega_{1})a^{\dagger}_{h}(\omega_{2})+a^{\dagger}_{h}(\omega_{1})a^{\dagger}_{v}(\omega_{2})+a^{\dagger}_{v}(\omega_{1})a^{\dagger}_{h}(\omega_{2})+a^{\dagger}_{v}(\omega_{1})a^{\dagger}_{v}(\omega_{2})]|0\rangle.\end{split}$
(14)

Again, only the two middle terms can result in coincidence detection.
Following a similar derivation as for the half-wave plate case, the opposite
sign of the third term in Eq. 14 results in the following JSI

$\begin{split}I_{\text{POL}}=&|A(\omega_{1},\omega_{2})e^{-i\omega_{2}\tau_{wo}}+A(\omega_{2},\omega_{1})e^{-i\omega_{1}\tau_{wo}}|^{2}=|A(\omega_{1},\omega_{2})|^{2}\cos^{2}\left(\frac{\omega_{2}-\omega_{1}}{2}\tau_{wo}\right).\end{split}$
(15)

As the polarizer erases the polarization difference between the two photons,
in this case the biphoton spectrum and its flipped version interfere
constructively along the degenerate line where the photons anti-bunch. The
oscillation is along the perpendicular direction as in the previous case. The
PMF, in this case, is symmetric (e.g. $\text{HG}_{0}$), but an anti-symmetric
function will result in bunching along the degenerate line, and the cosine
will be replaced by a sine function.

Figure 6: Frequency-resolved Hong-Ou-Mandel interference from a 5 mm long
crystal. Top row - calculated JSIs, bottom row - experimentally measured JSIs.
(a) and (d) The original JSI, measured without the HWP3 and POL2 in the setup
(see Fig. 2), (b) and (e) the JSI, measured by adding the HWP3 at
$22.5^{\circ}$ before the PBS, and (c) and (f) the JSI when the HWP3 is
replaced by the POL2 at $45^{\circ}$. See Table1 for crystal details

We demonstrated the above two methods using a 5 mm long crystal with
$\text{HG}_{0}$ PMF and a 15.9 mm long crystal with $\text{HG}_{1}$ double-
lobe PMF. The average walk-off delay values $\tau_{wo}$ of these crystals are
$0.74~{}$ps (at 1560 nm) and $2.36~{}$ps (at 1582 nm), respectively.

Theoretically simulated and experimentally obtained results for the 5 mm long
crystal are shown at the top and bottom row in Fig. 6, respectively. After
placing the HWP3 at $22.5^{\circ}$, the original JSI (Fig. 6a, d), consisting
of one lobe, splits into two lobes separated by the zero-intensity bunching
(Fig. 6b, e), along the degenerate line. Measuring the JSI and its modulation
frequency experimentally determines the temporal walk-off value or any other
unknown delay between signal and idler photons.

The results when the HWP3 is replaced by the POL2 at $45^{\circ}$ are
presented in Fig. 6c, f. As was derived above, in this case, anti-bunching is
observed along the degenerate line.

Similar results for both methods were also obtained for a 15.9 mm long
$\text{HG}_{0}$ crystal (not presented). The longer crystal results in a
proportionally longer walk-off delay, as well as in a narrower PMF’s spectral
width. As the periodicity of the spectral oscillations is inversely
proportional to the walk-off delay, the total number of observed oscillations
is preserved, regardless of the crystal length.

The JSI was measured for the other 15.9 mm long crystal, which is
characterized by the $\text{HG}_{1}$ PMF. Anti-symmetry of the PMF was
experimentally revealed by Hong-Ou-Mandel interference. The JSIs, measured
with the HWP3 and POL2 are the original JSI, modulated according to the
symmetry of the PMF (see Eq. 12 and Eq. 15). This unambiguously reveals the
symmetry of the original JSA, which is anti-symmetric in our case. If the PMF
was symmetric, the graphs for HWP and POL would be swapped. The original JSI
for this crystal is presented in Fig. 7a (simulated) and Fig. 7d (measured).
As the walk-off is larger, but the PMF is as wide as for the shorter crystal,
more oscillations are observed for the HWP and for the polarizer cases (Fig.
7b,c). Fitting the oscillations (using the walk-off value of 2.54 ps instead
of the expected 2.36 ps due to the not perfectly collinear alignment of the
setup or slightly different Sellmeier equations) for both measurements well
agrees with the anti-symmetric $\text{HG}_{1}$ PMF. Thus, the presented
methods can distinguish between symmetric and anti-symmetric PMF by bunching
or anti-bunching the produced biphotons.

Figure 7: Frequency-resolved Hong-Ou-Mandel interference from 15.9 mm long
crystal with the double-lobe PMF (see Table1, bottom line). Top row -
calculated JSIs, bottom row - experimentally measured JSIs. (a) and (d) The
original JSI, measured without the HWP3 and POL2 in the setup (see Fig. 2),
(b) and (e) the JSI, measured by adding the HWP3 at $22.5^{\circ}$ before the
PBS, and (c) and (f) the JSI when the HWP3 is replaced by the POL2 at
$45^{\circ}$. The pump wavelength is 791 nm.

The walk-off $\tau_{wo}$ can be controlled by adding a birefringent crystal
before the HWP or the polarizer. Different walk-off values lead to different
modulation frequencies, thereby allowing one to further control the JSI along
the difference-frequency axis in a similar way as pump modulation enables one
to control the JSI along the sum-frequency axis.

##  5 Conclusion

We demonstrated type-II SPDC using both domain-engineered aperiodically poled
nonlinear crystals, along with pump spectral modulation. This enables
versatile control of the joint spectral intensity of photon pairs along the
sum- and difference-frequency axes. Such control can generate novel quantum
states in the spectral domain and stimulate the development of new protocols
in quantum optics and quantum information science.

Using those methods we have generated a variety of quantum states.
Specifically, we have shown two of the four frequency-bin Bell states, but
with minor modification the two additional Bell states can be generated.
Moreover, one can attain two lobes like those shown in Fig. 3f but that do not
overlap in wavelength, which increases the fidelity of the frequency-bin Bell
state. Moreover, a dichroic mirror placed in the path of the photons described
by such a JSI will create the near-degenerate polarization-entangled state
$|\textrm{H}_{\omega_{1}}\textrm{V}_{\omega_{2}}\rangle-|\textrm{V}_{\omega_{1}}\textrm{H}_{\omega_{2}}\rangle$.
Remarkably, a JSA consisting of non-overlapping lobes can be decomposed into
separable Schmidt modes that can be easily filtered by regular spectral
filters [62]. This results in a measurable Schmidt decomposition that enables
one to tailor the degree of entanglement through Schmidt modes synthesis.

The 2D shaping methods of the JSI that we demonstrated here was used to
generate states with interesting correlations between 4 pairs of signal and
idler frequencies. Moreover, it can enable the creation of time-frequency GKP
states, that are useful for quantum error correction [44].

Leveraging the Hong-Ou-Mandel interference enables further manipulation of the
JSI along the difference-frequency axis after the photons’ generation. Even if
the photons do not overlap in time and time-domain Hong-Ou-Mandel experiments
do not show any interference pattern, the two-photon interference still
manifests in the multi-lobe JSI. This extends the dynamic range of
interferometers and, along with a broadband pump, can be of a great advantage
to quantum remote synchronization [43], quantum-enhanced interferometric
spectroscopy, metrology, and spectral domain quantum optical coherence
tomography that are unattainable with classical states of light [52, 63, 64,
63, 43, 58, 65, 66, 33, 67, 68, 69].

We also discussed that frequency-resolved Hong-Ou-Mandel interference can
serve as a valuable tool for distinguishing between even and odd PMFs and
measuring relative phases between the PMF’s lobes. Experiments were done at
the telecom wavelength of $1.5\,\mu$m, making the presented source compatible
with optical fiber networks. When combined, our methods open a wide range of
possibilities for preparing and manipulating quantum states of light for
various tasks of quantum optics and quantum information technologies. The
methods discussed in this paper can be easily generalized to crystals other
than KTP and to nonlinear waveguides and various types of integrated photonic
circuits where periodic poling is possible. Furthermore, with stronger pump
power, the same setup can be used to shape bright squeezed vacuum [54, 34].

Funding This work is supported by the Israel Science Foundation (Grants No.
2085/18, 969/22), Israel Ministry of Science.

Acknowledgments We thank Ziv Gefen from Raicol Crystals for assistance in
preparing the nonlinear crystals. I.H. acknowledges support by the Weinstein
Research Institute for signal processing. I.H. acknowledges support by the
Israeli planning and budgeting committee for quantum science and technology
research.

Disclosures The authors declare no conflicts of interest.

Data availability The data that support the findings of this study are
available from the corresponding author upon reasonable request.

## References

  * [1] J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics 3, 687–695 (2009).
  * [2] F. Flamini, N. Spagnolo, and F. Sciarrino, “Photonic quantum information processing: a review,” Reports on Progress in Physics 82, 016001 (2018).
  * [3] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: Single-photon sources and detectors,” Review of scientific instruments 82, 071101 (2011).
  * [4] A. Migdall, S. V. Polyakov, J. Fan, and J. C. Bienfang, _Single-photon generation and detection: physics and applications_ (Academic Press, 2013).
  * [5] H.-S. Zhong, H. Wang, Y.-H. Deng, _et al._ , “Quantum computational advantage using photons,” Science 370, 1460–1463 (2020).
  * [6] C. Couteau, “Spontaneous parametric down-conversion,” Contemporary Physics 59, 291–304 (2018).
  * [7] M. Kues, C. Reimer, J. M. Lukens, _et al._ , “Quantum optical microcombs,” Nature Photonics 13, 170–179 (2019).
  * [8] M. C. Ponce, A. L. M. Muniz, M. Huber, and F. Steinlechner, “Unlocking the frequency domain for high-dimensional quantum information processing,” arXiv preprint arXiv:2206.00969 (2022).
  * [9] H.-H. Lu, J. M. Lukens, N. A. Peters, _et al._ , “Quantum interference and correlation control of frequency-bin qubits,” Optica 5, 1455–1460 (2018).
  * [10] J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scalable quantum information processing,” Optica 4, 8–16 (2017).
  * [11] H.-H. Lu, P. Imany, N. B. Lingaraju, _et al._ , “Quantum information processing in the frequency domain,” Optics and Photonics News 30 (2019).
  * [12] H.-H. Lu, K. V. Myilswamy, R. S. Bennink, _et al._ , “Bayesian tomography of high-dimensional on-chip biphoton frequency combs with randomized measurements,” Nature Communications 13, 1–12 (2022).
  * [13] Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, “Qudits and high-dimensional quantum computing,” Frontiers in Physics 8, 589504 (2020).
  * [14] D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, “High-dimensional quantum communication: benefits, progress, and future challenges,” Advanced Quantum Technologies 2, 1900038 (2019).
  * [15] G. Lima, L. Neves, I. F. Santos, _et al._ , “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
  * [16] L. Bulla, M. Pivoluska, K. Hjorth, _et al._ , “Nonlocal temporal interferometry for highly resilient free-space quantum communication,” Physical Review X 13, 021001 (2023).
  * [17] A. Dosseva, Ł. Cincio, and A. M. Brańczyk, “Shaping the joint spectrum of down-converted photons through optimized custom poling,” Physical Review A 93, 013801 (2016).
  * [18] B. Baghdasaryan, F. Steinlechner, and S. Fritzsche, “Enhancing the purity of single photons in parametric down-conversion through simultaneous pump-beam and crystal-domain engineering,” Physical Review A 108, 023718 (2023).
  * [19] F. Graffitti, P. Barrow, M. Proietti, _et al._ , “Independent high-purity photons created in domain-engineered crystals,” Optica 5, 514–517 (2018).
  * [20] F. Graffitti, D. Kundys, D. T. Reid, _et al._ , “Pure down-conversion photons through sub-coherence-length domain engineering,” Quantum Science and Technology 2, 035001 (2017).
  * [21] A. Pickston, F. Graffitti, P. Barrow, _et al._ , “Optimised domain-engineered crystals for pure telecom photon sources,” Optics Express 29, 6991–7002 (2021).
  * [22] A. M. Brańczyk, A. Fedrizzi, T. M. Stace, _et al._ , “Engineered optical nonlinearity for quantum light sources,” Optics Express 19, 55–65 (2011).
  * [23] C. L. Morrison, F. Graffitti, P. Barrow, _et al._ , “Frequency-bin entanglement from domain-engineered down-conversion,” APL Photonics 7, 066102 (2022).
  * [24] F. Graffitti, P. Barrow, A. Pickston, _et al._ , “Direct generation of tailored pulse-mode entanglement,” Physical Review Letters 124, 053603 (2020).
  * [25] S. Merkouche, V. Thiel, A. O. C. Davis, and B. J. Smith, “Heralding multiple photonic pulsed bell pairs via frequency-resolved entanglement swapping,” Phys. Rev. Lett. 128, 063602 (2022).
  * [26] J. Li, J. Su, L. Cui, _et al._ , “Generation of pure-state single photons with high heralding efficiency by using a three-stage nonlinear interferometer,” Applied Physics Letters 116, 204002 (2020).
  * [27] V. Ansari, J. M. Donohue, M. Allgaier, _et al._ , “Tomography and purification of the temporal-mode structure of quantum light,” Physical Review Letters 120, 213601 (2018).
  * [28] B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Physical Review X 5, 041017 (2015).
  * [29] B. Brecht, A. Eckstein, R. Ricken, _et al._ , “Demonstration of coherent time-frequency schmidt mode selection using dispersion-engineered frequency conversion,” Phys. Rev. A 90, 030302(R) (2014).
  * [30] R.-B. Jin, H. Oshima, T. Yagisawa, _et al._ , “Two-photon spectral modulation via temporal manipulation: Quantum optical synthesis of spectral modes from temporal square waves,” Applied Physics Letters 121, 244002 (2022).
  * [31] A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, “Temporal shaping of entangled photons,” Phys. Rev. Lett. 94, 073601 (2005).
  * [32] S. Seshadri, H.-H. Lu, D. E. Leaird, _et al._ , “Complete frequency-bin bell basis synthesizer,” Phys. Rev. Lett. 129, 230505 (2022).
  * [33] B. Li, B. Yuan, C. Chen, _et al._ , “Spectrally resolved two-photon interference in a modified hong–ou–mandel interferometer,” Optics & Laser Technology 159, 109039 (2023).
  * [34] C. Drago and A. M. Brańczyk, “Tunable frequency-bin multimode squeezed vacuum states of light,” Physical Review A 106, 043714 (2022).
  * [35] T. E. Keller and M. H. Rubin, “Theory of two-photon entanglement for spontaneous parametric down-conversion driven by a narrow pump pulse,” Physical Review A 56, 1534 (1997).
  * [36] F. Kaneda, J. Oikawa, M. Yabuno, _et al._ , “Generation of spectrally factorable photon pairs via multi-order quasi-phase-matched spontaneous parametric downconversion,” arXiv preprint arXiv:2111.10981 (2021).
  * [37] R. Shiloh and A. Arie, “Spectral and temporal holograms with nonlinear optics,” Optics Letters 37, 3591–3593 (2012).
  * [38] M. Fiorentino, S. M. Spillane, R. G. Beausoleil, _et al._ , “Spontaneous parametric down-conversion in periodically poled ktp waveguides and bulk crystals,” Optics Express 15, 7479–7488 (2007).
  * [39] D. S. Hum and M. M. Fejer, “Quasi-phasematching,” Comptes Rendus Physique 8, 180–198 (2007).
  * [40] A. Leshem, R. Shiloh, and A. Arie, “Experimental realization of spectral shaping using nonlinear optical holograms,” Optics Letters 39, 5370–5373 (2014).
  * [41] A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” arXiv preprint quant-ph/0305192 (2003).
  * [42] C. Bernhard, B. Bessire, T. Feurer, and A. Stefanov, “Shaping frequency-entangled qudits,” Physical Review A 88, 032322 (2013).
  * [43] R.-B. Jin, R. Shimizu, M. Fujiwara, _et al._ , “Simple method of generating and distributing frequency-entangled qudits,” Quantum Science and Technology 1, 015004 (2016).
  * [44] N. Fabre, G. Maltese, F. Appas, _et al._ , “Generation of a time-frequency grid state with integrated biphoton frequency combs,” Phys. Rev. A 102, 012607 (2020).
  * [45] M. Avenhaus, A. Eckstein, P. J. Mosley, and C. Silberhorn, “Fiber-assisted single-photon spectrograph,” Optics letters 34, 2873–2875 (2009).
  * [46] T. Gerrits, M. J. Stevens, B. Baek, _et al._ , “Generation of degenerate, factorizable, pulsed squeezed light at telecom wavelengths,” Optics Express 19, 24434–24447 (2011).
  * [47] K. Zielnicki, K. Garay-Palmett, D. Cruz-Delgado, _et al._ , “Joint spectral characterization of photon-pair sources,” Journal of Modern Optics 65, 1141–1160 (2018).
  * [48] R. Salem, M. A. Foster, and A. L. Gaeta, “Application of space-time duality to ultrahigh-speed optical signal processing,” Advances in Optics and Photonics 5, 274–317 (2013).
  * [49] C. Phillips, C. Langrock, D. Chang, _et al._ , “Apodization of chirped quasi-phasematching devices,” JOSA B 30, 1551–1568 (2013).
  * [50] H. Suchowski, G. Porat, and A. Arie, “Adiabatic processes in frequency conversion,” Laser & Photonics Reviews 8, 333–367 (2014).
  * [51] C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
  * [52] E. Descamps, N. Fabre, A. Keller, and P. Milman, “Quantum metrology using time-frequency as quantum continuous variables: Resources, sub-shot-noise precision and phase space representation,” Physical Review Letters 131, 030801 (2023).
  * [53] C. Reimer, S. Sciara, P. Roztocki, _et al._ , “High-dimensional one-way quantum processing implemented on d-level cluster states,” Nature Physics 15, 148–153 (2019).
  * [54] I. Hurvitz, A. Karnieli, and A. Arie, “Frequency-domain engineering of bright squeezed vacuum for continuous-variable quantum information,” Optics Express 31, 20387–20397 (2023).
  * [55] K. Wang, “Quantum theory of two-photon wavepacket interference in a beamsplitter,” Journal of Physics B: Atomic, Molecular and Optical Physics 39, R293 (2006).
  * [56] P. S. Kuo, T. Gerrits, V. B. Verma, and S. W. Nam, “Spectral correlation and interference in non-degenerate photon pairs at telecom wavelengths,” Optics Letters 41, 5074–5077 (2016).
  * [57] R.-B. Jin, T. Gerrits, M. Fujiwara, _et al._ , “Spectrally resolved hong-ou-mandel interference between independent photon sources,” Optics Express 23, 28836–28848 (2015).
  * [58] Y. Chen, S. Ecker, L. Chen, _et al._ , “Temporal distinguishability in hong-ou-mandel interference for harnessing high-dimensional frequency entanglement,” npj Quantum Information 7, 1–7 (2021).
  * [59] C. Chen, Y. Chen, and L. Chen, “Spectrally resolved hong-ou-mandel interferometry with discrete color entanglement,” Physical Review Applied 19, 054092 (2023).
  * [60] T. Gerrits, F. Marsili, V. B. Verma, _et al._ , “Spectral correlation measurements at the hong-ou-mandel interference dip,” Phys. Rev. A 91, 013830 (2015).
  * [61] V. V. Orre, E. A. Goldschmidt, A. Deshpande, _et al._ , “Interference of temporally distinguishable photons using frequency-resolved detection,” Physical Review Letters 123, 123603 (2019).
  * [62] A. Avella, M. Gramegna, A. Shurupov, _et al._ , “Separable schmidt modes of a nonseparable state,” Physical Review A 89, 023808 (2014).
  * [63] S. M. Kolenderska, F. Vanholsbeeck, and P. Kolenderski, “Fourier domain quantum optical coherence tomography,” Optics Express 28, 29576–29589 (2020).
  * [64] P. Yepiz-Graciano, A. M. A. Martínez, D. Lopez-Mago, _et al._ , “Spectrally resolved hong–ou–mandel interferometry for quantum-optical coherence tomography,” Photon. Res. 8, 1023–1034 (2020).
  * [65] J. P. Dowling, “Quantum optical metrology–the lowdown on high-n00n states,” Contemporary physics 49, 125–143 (2008).
  * [66] R.-B. Jin and R. Shimizu, “Extended wiener–khinchin theorem for quantum spectral analysis,” Optica 5, 93–98 (2018).
  * [67] D. Triggiani, G. Psaroudis, and V. Tamma, “Quantum enhancement in time-delays estimation through spectrally resolved two-photon interference,” arXiv preprint arXiv:2112.12102 (2021).
  * [68] Y. Chen, L. Hong, and L. Chen, “Quantum interferometric metrology with entangled photons,” Frontiers in Physics p. 818 (2022).
  * [69] L. A. Rozema, J. D. Bateman, D. H. Mahler, _et al._ , “Scalable spatial superresolution using entangled photons,” Physical Review Letters 112, 223602 (2014).