# Estimating Hausdorff measure for Denjoy maps

Łukasz Pawelec  Łukasz Pawelec, Instytut Matematyczny Polskiej Akademii Nauk,
ul. Śniadeckich 8, 00-956 Warszawa, Poland<EMAIL_ADDRESS>and  Mariusz
Urbański  Mariusz Urbański, University of North Texas, Department of
Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA
<EMAIL_ADDRESS>
###### Abstract.

By employing the recurrence method worked out in [3], we provide effective
lower estimates of the proper–dimensional Hausdorff measure of minimal sets of
circle homeomorphisms that are not conjugate to any rotation.

###### Key words and phrases:

Hausdorff measure, Denjoy example

###### 1991 Mathematics Subject Classification:

37E10, 37C45

##  1\. Introduction

In this paper we deal with orientation preserving homeomorphisms and, more
specifically, diffeomorphisms of the unit circle $\mathcal{S}^{1}$. A
classical Poincaré’s result states that for every such homeomorphism $f$ there
is a unique number $\alpha\in[0,1)$ such that the map
$f\colon\mathcal{S}^{1}\to\mathcal{S}^{1}$ and the rotation
$R_{\alpha}(t):=t+\alpha\,(\\!\\!\\!\\!\mod\\!1)$ are semi-conjugate, i.e.
$h\circ f=R_{\alpha}\circ h$, for some continuous surjection
$h\colon\mathcal{S}^{1}\to\mathcal{S}^{1}$. The number $\alpha$ is called the
rotation number of $f$. It can be defined more explicitly without bringing up
semi-conjugacies but we do not need such definition in our paper. We do need
semi-conjugacy.

Much later, Denjoy proved that if the derivative of $f$ is of bounded
variation, then there is a (full) conjugacy between $f$ and $R_{\alpha}$, i.e.
the map $h$ is a homeomorphism. However, if the derivative $f^{\prime}$ is
only $\delta$–Hölder continuous for some $0<\delta<1$, then the semi-conjugacy
cannot be made into a full conjugacy. We then call the homeomorphism (or
diffeomorphism) $f$ a _Denjoy map_.

The reason for the non-existence of the full conjugacy is the presence of
wandering intervals. Then the complement of the (forward and backward) orbit
of that interval is the minimal set for the considered system
$f\colon\mathcal{S}^{1}\to\mathcal{S}^{1}$. It is a compact, perfect, nowhere
dense subset of $\mathcal{S}^{1}$, thus a Cantor set. In this paper we study
the Hausdorff dimension and the Hausdorff measure of this unique minimal set
of $f$. We will denote it by $\Omega(f)$ or, even simpler, just by $\Omega$.

Up to our knowledge, the, up to date, best estimates of the Hausdorff and box-
counting dimensions of the minimal set of a Danjoy map were obtained by B. Kra
and J. Schmeling in [2]. They proved a general lower bound of the box-counting
dimension, depending only on the smoothness of the derivative of $f$, namely
that $\dim_{B}(\Omega(f))\geq\delta$. They also showed that the lower bound of
the Hausdorff dimension depends additionally on the Diophantine class $\nu\geq
1$ of the rotation number $\alpha$; see Section 2 for the definition. More
precisely, they proved that $\dim_{H}(\Omega)\geq\delta/\nu$. Moreover, for
the case of so-called _classical_ Denjoy maps (see Example 2), they showed
that both the Hausdorff and box-counting dimensions are in fact equal to their
lower bounds, resp. $\delta/\nu$ and $\delta$, and so they coincide (only) for
$\nu=1$.

In this paper we push these results further. We apply a rather non-standard
method of estimating the Hausdorff measure from below developed by the first
named author in [3]. There it is shown that given a continuous self-map
$T\colon X\to X$ of a compact metric space $X$ preserving a Borel probability
measure $\mu$, a good knowledge of recurrence properties of the map $T$ leads
to lower bounds of the density of $\mu$ with respect to a Hausdorff measure
$H$; in the case when $\mu$ is absolutely continuous with respect to $H$, this
density is the Radon-Nikodym derivative $d\mu/dH$. This bound entails in turn
bounds on $H(X)$.

Applying this method we arrive at an effective bound from below on the
Hausdorff measure, in the relevant Hausdorff dimension, of the minimal set of
a diffeomorphism $f\colon\mathcal{S}^{1}\to\mathcal{S}^{1}$. It depends on the
Diophantine properties of the rotation number and on the lengths of the of the
wandering intervals of $f$, which as we know, are the connected components of
$\mathcal{S}^{1}\setminus\Omega(f)$. See Theorem 10 for a precise statement.
We would like to emphasize that the estimates we obtain are effectively
computable as we show in several examples in the last section of the paper.

Furthermore, our results allow us to recover the lower bounds on the Hausdorff
dimension of $\Omega$ obtained in [2]. In addition, for $\nu=1$ we provide a
simple upper estimate of the Hausdorff dimension of $\Omega$ and it coincides
with the lower bound. In some cases we can even get a meaningful upper
estimate of the Hausdorff measure. For example, our results show that for the
_classical_ Denjoy map the relevant Hausdorff measure is finite and positive
and we provide with concrete bounds.

##  2\. Setting, Definitions and some Tools Used

In this section we introduce notation and bring up the tools which we will use
in the subsequent sections. Throughout the paper we always assume that
$\alpha\in(0,1)$, the rotation number of a considered circle diffeomorphism,
is irrational, and that this diffeomorphism is exactly
$\mathcal{C}^{1+\delta}$ smooth with some $\delta\in(0,1)$. We denote by
$\|x\|$ is the norm (arc-wise distance from the rightmost point) of a point
$x$ on the circle. Lastly, we always assume that the total length of a circle
is equal to $1$.

###  2.1. Construction of Denjoy Maps

We start with recalling briefly the standard construction of Denjoy maps. The
notation we implement here follows [2]. A precise construction may be found
e.g. in [1].

######  Definition 1.

Given a number $\delta\in(0,1)$, a _Denjoy sequence_ of class $\delta$ is any
sequence $\\{\ell_{n}\\}_{n\in{\mathbb{Z}}}$ of non-negative numbers
satisfying the following two conditions.

$\sum_{n\in{\mathbb{Z}}}\ell_{n}=1$ (2.1)

and

$\liminf_{n\to\pm\infty}\frac{\ln|\ell_{n}-\ell_{n+1}|}{\ln\ell_{n}}=1+\delta.$
(2.2)

######  Example 2.

The _classical Denjoy sequence_ of class $\delta$ is given by the following
formula.

$\ell_{n}=c_{\delta}(|n|+1)^{-1/\delta},\ n\geq 1,$ (2.3)

where $c_{\delta}$ is the normalizing constant making the sum
$\sum_{n\in{\mathbb{Z}}}\ell_{n}$ equal to $1$.

Observe that the _classical_ sequence trivially satisfies the requirements of
Definition 1. Such a sequence may be considered as a _best one_ in the class
of Denjoy sequences.

Also, using Riemann’s $\zeta$ function we may state the value of the
normalising constant:

$c_{\delta}=2\zeta\Big{(}\frac{1}{\delta}\Big{)}-1.$ (2.4)

Now, fix a $\delta\in(0,1)$ and take any Denjoy sequence
$(\ell_{n})_{n\in{\mathbb{Z}}}$ of class $\delta$. To obtain a Denjoy map, we
start with a circle of length $1$ with a map that is the rotation by $\alpha$.
Then, we _blow up_ subsequently every point on the orbit of $0$, i.e. point of
the form $\pm n\alpha$, $n\in{\mathbb{N}}_{0}$, to an open interval $J_{n}$ of
length $\ell_{n}$, rescaling at every step the remainder of the circle to have
its length equal equal to $1$.

Figure 1. Denjoy map construction

Then we map $J_{n}$ diffeomorphically onto $J_{n+1}$ with derivative equal to
$1$ at the endpoints. This map has a continuous extension, which we denote by
$f$, to the entire circle $S^{1}$, and, because of (2.2), this extension can
be chosen to be of the class $\mathcal{C}^{1+\delta}$.

The minimal set (whose Hausdorff dimension and Hausdorff measure we want to
estimate) of the map $f$ is then equal to

$\Omega=\Omega_{\alpha}^{\delta}=S^{1}\setminus\bigcup_{n\in{\mathbb{Z}}}J_{n}.$
(2.5)

Observe that $\Omega$ also depends on the choice of the sequence
$(\ell_{n})_{n\in{\mathbb{Z}}}$.

######  Remark 3.

We want to note that in the construction of a Denjoy map $f$, the sum of
lengths of $\ell_{n}$, $n\in{\mathbb{Z}}$, might actually be smaller than $1$,
cf. (2.1). However, in such a situation the minimal set of $f$ would have
Hausdorff dimension trivially equal to $1$ and positive Lebesgue measure. This
is why we have removed such sequences from our considerations.

######  Remark 4.

Observe that every Denjoy map can be obtained by such a construction. The non-
existence of the full conjugation gives rise to the minimal set, whose
_omitted_ intervals must follow the rotation by $\alpha$. Take the longest
such interval, call it $J_{0}$, and then the semi-conjugation with the
rotation about $\alpha$, defines the sequence $(\ell_{n})_{n\in{\mathbb{Z}}}$.
The property (2.2) comes from the smoothness of the diffeomorphism. As we
mentioned in the previous remark, the lengths $\ell_{n}$ need not sum up to
$1$, but – again – we remove such case from our consideration.

We will denote by $h\colon\Omega\to S^{1}$ the semi-conjugacy given by
Poincaré’s Theorem. By its construction, this semi-conjugacy cannot be a
(full) conjugacy. Nevertheless, $h^{-1}(z)$ is a singleton whenever $z$ does
not belong to the orbit of $0$ under the rotation $R_{\alpha}$, i.e. whenever

$z\notin\mathcal{O}_{R_{\alpha}}:=\\{R^{n}(\alpha):n\in{\mathbb{Z}}\\}.$

Otherwise, $h^{-1}(z)$ is a doubleton. Furthermore, for any two points $x,y\in
S^{1}\setminus O_{R_{\alpha}}$, we also have that

$d(h^{-1}(x),h^{-1}(y))=\sum_{n\in{\mathbb{Z}}:R_{\alpha}^{n}(0)\in(x,y)}|J_{n}|,$
(2.6)

where $|J_{n}|$ denotes the length of interval $J_{n}$ and $(x,y)$ is the
shorter of the two arcs connecting $x$ and $y$; if the intervals have equal
length, take the one in positive orientation.

In general, i.e. for any points $x,y\in S^{1}$, the above formula actually
still holds. More precisely, it holds for those respective preimages of $x$
and $y$ for which the arc connecting them is the shortest.

###  2.2. Hausdorff Measure on Denjoy Minimal Sets

As we are working with subtle measure estimates we deemed it prudent to put
here the precise definitions in use in this paper.

First, recall our circle has length 1. We will use the standard version of the
definition of the Hausdorff measure.

######  Definition 5.

The outer measure is the following

$H_{\beta}(Y)=\lim_{r\to
0}\inf\left\\{\sum_{k=1}^{\infty}(\operatorname{diam}U_{k})^{\beta}:\forall_{k}\operatorname{diam}U_{k}<r\mbox{
and }Y\subset\bigcup_{k=1}^{\infty}U_{k}\right\\},$

where the infimum is take over all countable covers of $Y$ satisfying the
conditions as stated. By Carathéodory’s extension this gives the (typical)
Hausdorff measure.

The next definition is also standard.

######  Definition 6.

The Hausdorff dimension of the set $Y$ is given by the formula

$\dim_{H}(Y)=\inf\\{\beta\geq 0:H_{\beta}(Y)=0\\}.$

###  2.3. Diophantine Properties

We denote the standard continued fraction expansion of an irrational number
$\alpha\in(0,1)$ as $[a_{1},a_{2},\ldots]$. The corresponding convergents are
denoted as $\frac{p_{n}}{q_{n}}$. Recall that the convergents are the
subsequent closest approximations of $\alpha$. Also, the following properties
hold:

$q_{n}=a_{n}q_{n-1}+q_{n-2},$ (2.7)

where $q_{-1}=0$ and $q_{0}=1$, and

$\frac{1}{(a_{n+1}+2)q_{n}}<\frac{1}{q_{n+1}+q_{n}}<\|q_{n}\alpha\|<\frac{1}{q_{n+1}}<\frac{1}{a_{n+1}q_{n}}.$
(2.8)

We now bring up the following classical definition.

######  Definition 7.

An irrational number $\alpha\in(0,1)$ is said to be of _Diophantine class_
$\nu\in[1,+\infty]$, if the inequality

$\|q\alpha\|<\frac{1}{q^{\mu}}.$ (2.9)

has infinitely many positive integral solutions $q$ for every $\mu<\nu$ and at
only finitely many for every $\mu>\nu$. A number of Diophantine class
$\nu=\infty$ is commonly called a Liouville number.

Recall that the set of irrational numbers of Diophantine class $\nu=1$ is of
full Lebesgue measure. However, for any $\nu\in[1,+\infty]$ there exist
$\alpha$ of Diophantine class $\nu$.

######  Remark 8.

Observe that if $\nu>1$, then for infinitely many $n$ we have a very fast
growth of $q_{n}$’s. Combining (2.8) and (2.9) gives $q_{n+1}\approx
q_{n}^{\nu}$ for infinitely many $n$s.

###  2.4. Recurrence Results

To estimate the Hausdorff measure and Hausdorff dimension of the minimal set
$\Omega$, we will apply the method described in a paper by the first named
author, [3].

The result we need is this.

######  Theorem 9.

If $(X,d)$ is a metric space, $\mu$ is a Borel probability measure on $X$, and
$T\colon X\to X$ is a Borel measurable map preserving measure $\mu$, then for
every $\beta>0$ we have that

$\displaystyle\liminf_{n\to\infty}nd^{\beta}(T^{n}(x),x))\leq
g(x):=\limsup\limits_{r\to 0}\frac{H_{\beta}(B(x,r))}{\mu(B(x,r))}$ (2.10)

This result is applied in the following way. For $\mu$-almost, but see the
comment in the next paragraph, every $x\in X$ find a lower bound of the
recurrence speed, i.e. the left hand side of the inequality above. This gives
a lower bound on the above function $g$, giving in turn a lower bound on the
Hausdorff measure of $X$ by using the formula

$H_{\beta}(X)=\int_{X}g(x)d\mu,$

and the fact that $\mu(X)=1$.

Note that we do not have to really care about what the invariant measure
actually is $X$ is compact and the map $T\colon X\to X$ is continuous. Such
measure then exists because of Bogolubov-Krylov Theorem. Indeed, ff we do know
what the invariant measure is, then we may do the calculations for almost
every point with respect to this invariant. If not, then we need to do the
calculations for all points; except countably many of them, if the measure is
non-atomic.

###  2.5. The Main Results

As the general statement of our estimating method is rather complicated, we
will state a few simpler versions. Also, our method can be applied under
weaker assumptions, ex. for some sequences $(\ell_{n})_{n\in{\mathbb{Z}}}$ not
satisfying the requirements of Definition 1; see Example 16.

######  Theorem 10.

Fix $\delta\in(0,1)$ and irrational rotation number $\alpha$. Consider the
corresponding Denjoy map $f\colon S^{1}\to S^{1}$ of the circle $S^{1}$ that
is semi-conjugate to the rotation by $R_{\alpha}$ with gaps $J_{n}$,
$n\in{\mathbb{Z}}$ whose sizes $|J_{n}|=\ell_{n}$ satisfy formulas (2.1) and
(2.2) of Definition 1. For every $n\in{\mathbb{N}}$, set

$N_{n}:=\left[\frac{q_{n}+q_{n+1}}{2}\right].$

Then for any $\beta>0$ we have that

$H_{\beta}(\Omega_{f})\geq\liminf_{n\to\infty}q_{n}\min{}^{\beta}\big{\\{}|J_{i}|:-N_{n}\leq
i\leq N_{n}\big{\\}}.$ (2.11)

In fact, the following more complicated but stronger, estimate holds.

$H_{\beta}(\Omega_{f})\geq\liminf_{n\to\infty}q_{n}\left(\sum_{l=-\infty}^{+\infty}\min\big{\\{}|J_{i}|:(2l-1)N_{n}\leq
i\leq(2l+1)N_{n}\big{\\}}\right)^{\beta}.$ (2.12)

Th strongest estimate we prove is the following. For every $n\in{\mathbb{N}}$,
set $Q_{n}:=q_{n}+q_{n+1}$ and denote by $\mathcal{P}^{n}$ the set of all
sequences $(P^{(n)}_{l})_{l\in{\mathbb{Z}}}$ that satisfy the formula
$P^{(n)}_{l}-P^{(n)}_{l-1}=Q_{n}$. Then

$H_{\beta}(\Omega_{f})\geq\liminf_{n\to\infty}q_{n}\sup{}^{\beta}\left\\{\sum_{l=-\infty}^{+\infty}\min\big{\\{}|J_{i}|:P^{(n)}_{l}\leq
i<P^{(n)}_{l-1}\big{\\}}:P^{(n)}\in\mathcal{P}^{n}\right\\}.$ (2.13)

A direct straightforward application of the first estimate in the above
Theorem 10 gives the following.

######  Corollary 11.

With the notation and hypotheses of Theorem 10, we have that

$\dim_{H}(\Omega_{f})\geq\sup\left\\{\beta\in{\mathbb{R}}^{+}:\liminf_{n\to\infty}q_{n}\bigg{(}\sum_{l=-\infty}^{+\infty}\min\big{\\{}|J_{i}|:(2l-1)N_{n}\leq
i\leq(2l+1)N_{n}\big{\\}}\bigg{)}^{\beta}>0\right\\}$

and

$\dim_{H}(\Omega_{f})\geq\inf\left\\{\beta\in{\mathbb{R}}^{+}:\liminf_{n\to\infty}q_{n}\bigg{(}\sum_{l=-\infty}^{+\infty}\min\big{\\{}|J_{i}|:(2l-1)N_{n}\leq
i\leq(2l+1)N_{n}\big{\\}}\bigg{)}^{\beta}=0\right\\}.$

Lastly, with some additional work, as a consequence of the above corollary, we
will prove (in Section 4) the lower bound on the Hausdorff dimension of
$\Omega_{f}$ which was obtained first by Kra and Schmelling in [2] with an
entirely different method.

######  Proposition 12.

Under assumptions as above

$\dim_{H}(\Omega_{f})\geq\frac{\delta}{\nu}.$ (2.14)

Let us also observe that if the Diophantine class $\nu$ of the rotation number
$\alpha$ is equal to $1$, then we may easily get estimates from above on the
Haudorff dimension (and in some cases the Hausdorff measure).

######  Theorem 13.

For $\nu=1$ and the classic Denjoy sequence (see Example 2) the Hausdorff
dimension equals $\delta$ and the relevant Hausdorff measure is positive and
finite.

##  3\. Proof of Theorem 10

Fix an irrational number $\alpha\in(0,1)$. Take a point $x_{0}$ in
$\Omega_{f}$. For convenience assume that
$x_{0}\notin\mathcal{O}_{R_{\alpha}}$. As we have already remarked, we may
ignore all points in $\mathcal{O}_{R_{\alpha}}$ since this set is countable.
In fact, our estimates also work for such points, but then one needs to be
wary of the somewhat annoying double preimages.

We are to estimate from below the number
$\displaystyle\liminf_{n\to\infty}n\left(d(T^{n}(x_{0}),x_{0})\right)^{\beta}$.
Using (2.6), we see that

$\displaystyle d(f^{n}(x_{0}),x_{0})=\sum_{n}|J_{n}|,$ (3.1)

where the sum is taken over all $n$, for which
$n\alpha\in(h(x_{0}),h(f^{n}(x_{0})))$.

To find the lower limit we are only interested in the sequence of subsequent
closest returns. Since the map $h$ semi-conjugates the dynamics on
$\Omega_{f}$ with the rotation by $\alpha$ on the circle, the closest returns
on $\Omega_{f}$ are the closest returns for the rotation. They in turn are
given by the reducts of the continued fraction expansion of $\alpha$. Thus, we
only need to estimate the numbers

$d(f^{q_{n}}(x_{0}),x_{0}).$

Let us denote the $n$-th closest return of $x_{0}$ as
$x_{n}=f^{q_{n}}(x_{0})$. First, we estimate the sum of lengths (3.1) from
below by taking only the biggest element.

$\displaystyle
d(x_{n},x_{0})=\sum\left|J_{k}\right|\geq\left|J_{k_{n}}\right|,$ (3.2)

where $k_{n}\in{\mathbb{Z}}$ is such that
$k_{n}\alpha\in\big{(}h(x_{0}),h(x_{n})\big{)}$ and the number $|J_{k_{n}}|$
is biggest possible (_see Fig. 2_).

Figure 2. Definition of the sequence $k_{n}$

Now, we need to find bounds on $k_{n}$. By the semi-conjugacy
$h(x_{n})=h(x_{0})+q_{n}\alpha$, and the search for $k_{n}$ may by restated as
follows.

_What is the smallest $k\in{\mathbb{N}}$, such that any interval on the circle
of length $q_{n}\alpha$ contains at least one element of the set _

$\big{\\{}-k\alpha,-(k-1)\alpha,\ldots,0,\ldots,(k-1)\alpha,k\alpha\big{\\}}?$

Since we are assuming that $h(x_{0})$ is not in orbit of $0$ under
$R_{\alpha}$, the gap between $h(x_{n})$ and $h(x_{0})$ cannot be equal to any
of the above gaps. The answer to our above question is given in the following
lemma.

######  Lemma 14.

The maximal gap between points in the set $\\{t\alpha:t=-N\ldots 0\ldots N\\}$
is at most of length $\|q_{n}\alpha\|$ if and only if

$N\geq\left[\frac{q_{n}+q_{n+1}}{2}\right].$

###### Proof.

We use the three gaps theorem as stated in [5]. Actually, we only need Lemma
2.1. from this paper, which (with the notation of this paper) states the
following.

> Take a forward orbit of a point under irrational rotation $R_{\alpha}$. For
> $K=q_{n}+q_{n+1}-1$ the set
>
> $\\{\alpha k:k=0,\dots,K\\}$
>
> partitions the circle into $q_{n}$ gaps of length $\|q_{n+1}\alpha\|$ and
> $q_{n+1}$ gaps of length $\|q_{n}\alpha\|$.

The figure below illustrates this phenomenon. We took $\alpha=\sqrt{3}-1$ and
drew the first $25$ iterates. Note that all the gaps are only of two lengths.

Figure 3. Forward orbit of $0$ for $\alpha=\sqrt{3}-1$.

In our setting we are working with $\displaystyle(t\alpha)_{t=-N}^{N}$.
Rotating the circle (and all our points) by $N\alpha$, gives the set
$\displaystyle(t\alpha)_{t=0}^{2N}$ and does not change the gap sizes, leading
straight to the required result. The next figure shows the forward and
backward orbit for the same $\alpha$.

Figure 4. Forward and backward orbit of $0$ for $\alpha=\sqrt{3}-1$.

Note that there is one gap _on the left of the circle_ that is of a different
size. This is the one resulting from the fact that such type of orbit (forward
and backward) always has an odd number of points necessitating the rounding in
our result. Removing $13$ gives the same picture as before but rotated in such
a way that $-13$ becomes $0$. ∎

Recall that we defined $N_{n}=\left[\frac{q_{n}+q_{n+1}}{2}\right]$. By Lemma
14, we know that $|k_{n}|\leq N_{n}$, and we may estimate the size of
$J_{k_{n}}$ from below as

$\left|J_{k_{n}}\right|\geq\min\big{\\{}|J_{i}|:-N_{n}\leq i\leq
N_{n}\big{\\}}.$ (3.3)

Inserting this into our estimate yields

$H_{\beta}(\Omega)\geq\liminf_{n\to\infty}nd^{\beta}(T^{n}(x_{0}),x_{0})\geq\liminf_{n\to\infty}q_{n}\min{}^{\beta}\big{\\{}|J_{i}|:-N_{n}\leq
i\leq N_{n}\Big{\\}}.$ (3.4)

This ends the proof of (2.11).

To get better bounds we need to utilize all the elements of the sum in (3.1).
Recall that the finite sequence $k\alpha$, $k\in[-N_{n},\ldots,N_{n}]$, hits
every gap of size $\|q_{n}\alpha\|$. Similarly, the sequence $k\alpha$,
$k\in[N_{n},\ldots,3N_{n}]$, hits every gap of the same size, and so does
every sequence of type $k\alpha$, $[(2l-1)N_{n},\ldots,(2l+1)N_{n}]$, for
every integer $l$. Summing the lengths of the smallest intervals $J_{i}$ (on
every sequence), leads straightforward to the estimate (2.12).

$H_{\beta}(\Omega)\geq\liminf_{n\to\infty}q_{n}\left(\sum_{l=-\infty}^{+\infty}\min\big{\\{}|J_{i}|:(2l-1)N_{n}\leq
i\leq(2l+1)N_{n}\big{\\}}\right)^{\beta}.$

It is now easy to observe that our finite sequence may be _moved around_ ,
i.e. for any $P\in{\mathbb{Z}}$ the sequence $k\alpha$, for $k\in[P,P+Q_{n})$
hits every gap of size $\|q_{n}\alpha\|$. We used $Q_{n}$ instead of
$N_{n}=[Q_{n}/2]$ since there is no need for the finite sequences to be
_symmetric_ w.r.t. zero. Also, removing the ‘integer part function’ improves
the estimate a little bit.

It remains to divide for every $n\in{\mathbb{N}}$ the set of all integers into
sets of length $Q_{n}$; in the statement of (2.13), we called this ‘dividing’
sequence $\left(P_{l}^{(n)}\right)_{l\in{\mathbb{Z}}}$. Then we find the
minimal length of the intervals $|J_{i}|$ on every finite sequence, i.e.
between $P_{l}^{(n)}$ and $P_{l-1}^{(n)}$, and we sum those values. The proves
the general estimate (2.13).

##  4\. Examples

######  Example 15.

Put $\alpha:=\frac{\sqrt{5}-1}{2}$ and consider the classical Denjoy sequence

$|J_{n}|=c_{\delta}(|n|+1)^{-1/\delta},\ n\in{\mathbb{Z}},$

where, we recall, $c_{\delta}$ is the normalizing constant. Then the minimum
in (2.11) is attained at the largest possible $i$, i.e. at $i=N_{n}$, and,
putting $\alpha_{*}:=\frac{1+\sqrt{5}}{2}$ our simpler estimate gives

$\displaystyle H_{\beta}(\Omega)$
$\displaystyle\geq\liminf_{n\to\infty}c_{\delta}^{\beta}q_{n}\Big{(}\frac{1}{2}(q_{n}+q_{n+1}+1)+1\Big{)}^{-\beta/\delta}\geq\liminf_{n\to\infty}c_{\delta}^{\beta}q_{n}\Big{(}\frac{q_{n+2}}{2}+2\Big{)}^{-\beta/\delta}$
$\displaystyle\geq\liminf_{n\to\infty}\frac{c_{\delta}^{\beta}}{2^{-\beta/\delta}}\left(\frac{\alpha_{*}^{n}-1}{\sqrt{5}}\right)\left(\frac{\alpha_{*}^{n+2}+1}{\sqrt{5}}\right)^{-\beta/\delta}$
$\displaystyle\geq\frac{c_{\delta}^{\beta}\alpha_{*}^{-2\beta/\delta}}{2^{-\beta/\delta}5^{(\delta-\beta)/2\delta}}\lim_{n\to\infty}\alpha_{*}^{n(\delta-\beta)/\delta}.$

Therefore, $H_{\beta}(\Omega)>0$ for $\beta\in(0,\delta]$, thus proving that
$\dim_{H}(\Omega)\geq\delta$, and we get the following estimate on the
Hausdorff measure

$H_{\delta}(\Omega)\geq 8c_{\delta}^{\delta}(1+\sqrt{5})^{-2},$

where $c_{\delta}=\sum_{n\in{\mathbb{Z}}}(|n|+1)^{-1/\delta}$.

We may easily improve the estimate by using the more complicated inequality
(2.12). The minima in the sum are attained at the endpoints of the intervals,
i.e. at $(2k\pm 1)N_{n}$ (where the sign depends on whether $k$ is positive or
not). Let us first estimate the sum that occurs in the lower limit

$\displaystyle\left(\sum_{k=-\infty}^{+\infty}|J_{(2k\pm
1)N_{n}}|\right)^{\beta}$
$\displaystyle=c_{\delta}^{\beta}\left((N_{n}+1)^{-1/\delta}+2\sum_{k=1}^{+\infty}\big{(}(2k+1)N_{n}+1\big{)}^{-1/\delta}\right)^{\beta}$
$\displaystyle\geq
c_{\delta}^{\beta}(N_{n}+1)^{-\beta/\delta}\left(1+2\sum_{k=1}^{+\infty}(2k+1)^{-1/\delta}\right)^{\beta}$
$\displaystyle=c_{\delta}^{\beta}(N_{n}+1)^{-\beta/\delta}\left(1+2\Big{(}\zeta\Big{(}\frac{1}{\delta}\Big{)}-1-2^{-1/\delta}\zeta\Big{(}\frac{1}{\delta}\Big{)}\Big{)}\right)^{\beta},$

where we utilised the Riemann $\zeta$ function to simplify the harmonic sums.
We used the following trivial equality:

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\sum_{k=1}^{\infty}\frac{1}{(2k)^{s}}+\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{s}}=2^{-s}\zeta(s)+\sum_{k=1}^{\infty}\frac{1}{(2k+1)^{s}}+1$

Finally, we see that, the only difference in comparison to the previous
estimate is the last (rather complicated) constant. Thus we may repeat the
same calculation, getting at the end $\beta=\delta$. Let us also use the
‘closed form’ of $c_{\delta}$ (2.4) arriving at

$H_{\delta}(\Omega)\geq
8\left(2\zeta\Big{(}\frac{1}{\delta}\Big{)}-1\right)^{\delta}(1+\sqrt{5})^{-2}\left(1+2\Big{(}\zeta\Big{(}\frac{1}{\delta}\Big{)}-1-2^{-1/\delta}\zeta\Big{(}\frac{1}{\delta}\Big{)}\Big{)}\right)^{\delta}.$

######  Example 16.

Keep $\alpha=\frac{\sqrt{5}-1}{2}$ but change the sequence $|J_{n}|$,
$n\in{\mathbb{Z}}$, slightly. Start with $\ell_{n}=(|n|+1)^{-1/\delta}$ but
change the elements $\ell_{\pm 4^{k}}:=7^{-k}$, $k\in{\mathbb{N}}$. Normalize
the sum by $c$ and put $|J_{n}|:=c\ell_{n}$.

_Important note:_ this does not satisfy (2.2)! That is, our construction does
not give a diffeomorphism on the entire circle – only a homeomorphism.
However, the method is still applicable.

Now, using our estimate in the straightforward manner by taking tiny gaps as
the minimum ‘pollutes’ the gap sizes, gives a weak result.

However, recall that the sequences $k\alpha$, $k\in[-N_{n},\ldots,N_{n}]$ and
$k\alpha$, $k\in[N_{n},\ldots,3N_{n}]$, hit every gap of size
$\|q_{n}\alpha\|$. So, the sequence $k\alpha$, $k\in[N_{n},\ldots,5N_{n}]$,
hits every gap at least twice. Now observe that the growth of the sequence
$N_{n}\approx\alpha_{*}^{n}\leq 1.7^{n}$, $n\in{\mathbb{N}}$, is much slower
than the growth of the distances between numbers of gaps with _bad_ sizes,
i.e. $4^{k}$, $k\in{\mathbb{N}}$. Thus our set $[N_{n},\ldots,5N_{n}]$ for $n$
large enough may hit at most one element of the sequence $4^{k}$,
$k\in{\mathbb{N}}$. Since every gap is hit at least twice, we may ignore that
one, and recover the estimate as before, only slightly worsened.

$\displaystyle H_{\beta}(\Omega)$
$\displaystyle\geq\liminf_{n\to\infty}c_{\delta}^{\beta}q_{n}\Big{(}\frac{5}{2}(q_{n}+q_{n+1}+1)+1\Big{)}^{-\beta/\delta}\geq\liminf_{n\to\infty}c_{\delta}^{\beta}q_{n}\Big{(}\frac{5q_{n+2}}{2}+2\Big{)}^{-\beta/\delta}$
$\displaystyle\geq
5^{-\beta/\delta}\frac{5^{-\beta/\delta}c_{\delta}^{\beta}\alpha_{*}^{-2\beta/\delta}}{2^{-\beta/\delta}5^{(\delta-\beta)/2\delta}}\lim_{n\to\infty}\alpha_{*}^{n(\delta-\beta)/\delta}.$

Therefore, $H_{\beta}(\Omega)>0$ for $\beta\in(0,\delta]$, thus proving that
$\dim_{H}(\Omega)\geq\delta$, and we get the following estimate on the
Hausdorff measure

$H_{\delta}(\Omega)\geq\frac{8}{5}c_{\delta}^{\delta}(1+\sqrt{5})^{-2}.$

As above one could improve this estimate by using the whole sum, i.e. applying
(2.13).

######  Example 17.

Take $\alpha\in(0,1)$ whose continued fraction expansion satisfies the
relation

$q_{n+1}=q_{n}^{2}$

for every $n\in{\mathbb{N}}$. Then

$a_{n}\approx c^{2^{n}}$

for some integer $c\geq 2$. $\alpha$ is then of Diophantine class $\nu=2$. Set

$|J_{n}|:=cn^{-3}\ln|n|,$

where $c$ is the corresponding normalizing constant.

We know that for such quickly growing numbers $q_{n}$ we have for any
$\varepsilon>0$ and all integers $n\geq 1$ large enough that

$\displaystyle\frac{q_{n+1}+q_{n}}{2}<\frac{q_{n+1}}{2-\varepsilon}=\frac{q_{n}^{2}}{2-\varepsilon}.$
(4.1)

Using this formula in our estimate, i.e. the computation as in the previous
example, leads in the current case to the following.

$\displaystyle H_{\beta}(\Omega)$
$\displaystyle\geq\liminf_{n\to\infty}q_{n}\frac{c^{\beta}(\ln|q_{n}^{2}|-\ln|2-\varepsilon|)}{q_{n}^{6\beta}(2-\varepsilon)^{-3}}$
$\displaystyle=c^{\beta}(2-\varepsilon)^{3}\liminf_{n\to\infty}q_{n}^{1-6\beta}(\ln|q_{n}^{2}|-\ln|2-\varepsilon|).$

We see from this inequality that if $\beta\leq\frac{1}{6}$, then
$H_{\beta}(\Omega)=+\infty$, giving the estimate

$\dim_{H}(\Omega)\geq 1/6.$

###### Proof of Proposition 12.

To get the desired lower bound on the Hausdorff dimension of $\Omega_{f}$, we
need two ingredients: some kind of a universal bound on the lengths $|J_{n}|$,
$n\in{\mathbb{Z}}$, and some bound on the growth of the denominators of the
continued fraction expansion.

The assumption (2.2) imposed on the sequence $\ell_{n}=|J_{n}|$,
$n\in{\mathbb{Z}}$, easily leads to the following result which was stated and
proved as Lemma 2.3 in [2].

> For all $0<\theta<\delta<1$ there exists $s_{\theta}\geq 1$ such that
>
> $|J_{n}|>n^{-1/\theta}$ (4.2)
>
> for all integers $n>s_{\theta}$.

Now we shall deal with the growth of the $q_{n}$s, $n\in{\mathbb{N}}$.
Combining the continued fraction property (2.8) and the Diophantine class
definition (2.9) we get for every $\varepsilon>0$ and all $n\geq
s_{\varepsilon}$ that

$\frac{1}{q_{n+1}}>\|q_{n}\alpha\|>\frac{1}{q_{n}^{\nu+\varepsilon}}.$

This gives the desired estimate

$q_{n+1}<q_{n}^{\nu+\varepsilon}.$ (4.3)

It remains to estimate below the lower limit

$\liminf_{n\to\infty}q_{n}\min{}^{\beta}\big{\\{}|J_{i}|:-N_{n}\leq i\leq
N_{n}\big{\\}}.$ (4.4)

To do this, first fix $\theta\in(0,\delta)$ and also any $\varepsilon>0$. Then
fix any integer $n>\max(n_{\varepsilon},n_{\theta})$. We will obtain the
desired lower estimate by using first (4.2), then the definition of $N_{n}$
together with the fact $N_{n}<q_{n+1}$, and finally (4.3). With any $\beta>0$,
all of these lead to

$\displaystyle\liminf_{n\to\infty}q_{n}\min{}^{\beta}\big{\\{}|J_{i}|:-N_{n}\leq
i\leq N_{n}\big{\\}}$
$\displaystyle\geq\liminf_{n\to\infty}q_{n}\Big{(}N_{n}^{-1/\theta}\Big{)}^{\beta}\geq\liminf_{n\to\infty}q_{n}\cdot
q_{n+1}^{-\frac{\beta}{\theta}}$
$\displaystyle\geq\liminf_{n\to\infty}q_{n}\cdot
q_{n}^{-(\nu+\varepsilon)\frac{\beta}{\theta}}=\liminf_{n\to\infty}q_{n}^{1-\frac{\nu+\varepsilon}{\theta}\beta}.$

So, this lower limit is positive for $\beta=\frac{\theta}{\nu+\varepsilon}$.
Therefore, by applying Corollary 11, we get that
$\dim_{H}(\Omega)\geq\frac{\theta}{\nu+\varepsilon}$. Hence, by letting first
$\varepsilon$ go to $0$ and then $\theta$ to $\delta$, we obtain that

$\dim_{H}(\Omega)\geq\frac{\delta}{\nu}.$

∎

###### Proof of Theorem 13.

By the previous results, we only need to prove that
$H_{\delta}(\Omega)<+\infty$. Observe the set

$S^{1}\setminus\bigcup_{k=-n}^{n}J_{k}\supset\Omega.$

It consists of $2n+1$ disjoint intervals, let us call them $I_{p}$, whose
total length equals

$1-\sum_{k=-n}^{n}|J_{k}|=\sum_{k<-n}|J_{k}|+\sum_{k>n}|J_{k}|.$

For the classic sequence both of the sums on the RHS are equal and may be
trivially bounded by and integral giving

$\sum_{|k|>n}|J_{k}|\leq
2\frac{\delta}{\delta-1}c_{\delta}(n-1)^{1-1/\delta}.$

As $\delta<1$, function $x^{\delta}$ is concave and application of Jensen’s
inequality leads to the following bound on the covering sum (for clarity, put
$N=2n+1$ until the last equality).

$\sum_{p=1}^{N}|I_{p}|^{\delta}\leq
N\left(\sum_{p=1}^{N}\frac{1}{N}|I_{p}|\right)^{\delta}\leq
N^{1-\delta}\left(2\frac{\delta}{\delta-1}c_{\delta}(n-1)^{1-1/\delta}\right)^{\delta}=\left(2\frac{\delta}{\delta-1}c_{\delta}\right)^{\delta}\left(\frac{2n+1}{n-1}\right)^{1-\delta}.$

Clearly, the lengths (diameters) of the intervals $|I_{p}|$ shrink to 0 as
$n\to+\infty$. Thus, we arrive at

$H_{\delta}(\Omega)\leq
2\left(\frac{\delta}{\delta-1}c_{\delta}\right)^{\delta}.\qed$

######  Remark 18.

Note that the calculation above works for all irrational $\alpha$, but
whenever $\nu>1$ it does not give the proper upper estimate on the dimension.
It should not come as a surprise as in the proof above we essentially treat
the cover as its elements were of comparable sizes.

On the other hand, if $\nu>1$, then looking at the rotation by $\alpha$
sometimes the orbit returns very close to the starting point. So in the proof
above some of the sets $I_{p}$ will have very small length in comparison with
others. Thus, a cover were the sizes are similar may not be optimal.

Following the proof of the previous result we get the following
generalisation.

######  Corollary 19.

Assume $\nu=1$. If for every $\varepsilon>0$, there exists a subsequence
$(n_{s})$ and constant $C<\infty$ such that

$\sum_{|k|>n_{s}}|J_{k}|\leq Cn_{s}^{1-\frac{1}{\delta+\varepsilon}},$

then $\dim_{H}(\Omega)=\delta$.

## Acknowledgements

The research of Łukasz Pawelec was supported in part by National Science
Centre, Poland, Grant OPUS21 “Holomorphic dynamics, fractals, thermodynamic
formalism”, 2021/41/B/ST1/00461.

The research of Mariusz Urbanski was supported in part by the Simons Grant:
581668.

## References

  * [1] A. Katok and B. Hasselblatt.  Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, 1995).
  * [2] B. Kra and J. Schmelling.  Diophantine classes, dimension and Denjoy maps.  Acta Arithmetica 105.4 (2002), 323–340.
  * [3] Ł. Pawelec.  Estimating the Hausdorff measure by recurrence.  ArXiv e-prints.
  * [4] F. Przytycki and M. Urbański.  Conformal Fractals: Ergodic Theory Methods (Cambridge University Press, 2010).
  * [5] T. van Ravenstein.  The three gap theorem (Steinhaus conjecture).  J. Austral. Math. Soc. 45 (1988), 360–370.


# Designing High-Power, Octave Spanning Entangled Photon Sources for Quantum
Spectroscopy

S. Szoke  Division of Engineering and Applied Sciences, California Institute
of Technology, Pasadena, CA 91125, USA  M. He  B.P. Hickam  S.K. Cushing
<EMAIL_ADDRESS>Division of Chemistry and Chemical Engineering,
California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

Entangled photon spectroscopy is a nascent field that has important
implications for measurement and imaging across chemical, biology, and
materials fields. Entangled photon spectroscopy potentially offers improved
spatial and temporal-frequency resolutions, increased cross sections for
multiphoton and nonlinear measurements, and new abilities in inducing or
measuring quantum correlations. A critical step in enabling entangled photon
spectroscopies is the creation of high-flux entangled sources that can use
conventional detectors, as well as provide redundancy for the losses in
realistic samples. Here, we report a periodically poled, chirped, lithium
tantalate platform that generates entangled photon pairs with a $\sim 10^{-7}$
efficiency. For a near watt level diode laser, this results in a near
$\mu$W-level flux. The single photon per mode limit that is necessary to
maintain non-classical photon behavior is still satisfied by distributing this
power over up to an octave-spanning bandwidth. The spectral-temporal photon
correlations are observed via a Michelson-type interferometer that measures
the broadband Hong-Ou-Mandel two-photon interference. A coherence time of 245
fs for a 10 nm bandwidth in the collinear case and a 62 fs for a 125 nm
bandwidth in the non-collinear case is measured using a CW pump laser, and,
essentially, collecting the full photon cone. We outline in detail the
numerical methods used for designing and tailoring the entangled photons
source, such as changing center wavelength or bandwidth, with the ultimate aim
of increasing the availability of high-flux UV-Vis entangled photon sources in
the optical spectroscopy community.

##  I Introduction

An entangled photon pair is generated by splitting a single photon into two
photons via spontaneous parametric down conversion (SPDC) Burnham and Weinberg
(1970). The quantum correlations between the two photons have several effects.
First, when coherently recombined in time and space, the two entangled photons
behave as the initial pump photon. In other words, the two entangled photons
can be thought of as acting like a single photon in light-matter interactions
Szoke  _et al._ (2020). As a result, two entangled photons leave the same side
of a beamsplitter Hong, Ou, and Mandel (1987) and diffract from a grating at
the wavelength of the pump photon Mitev  _et al._ (2020); Shimizu, Edamatsu,
and Itoh (2003); Abouraddy  _et al._ (2001). Entangled photon pairs also
linearize multiphoton and nonlinear spectroscopy, as has been well-studied in
two photon absorption and fluorescence Javanainen and Gould (1990); Dayan  _et
al._ (2005); Villabona-Monsalve, Burdick, and Goodson (2020); Raymer  _et al._
(2013); Tabakaev  _et al._ (2021); Li  _et al._ (2020); Kang  _et al._ (2020).
Entangled photons also have non-Fourier reciprocal spectral-temporal
resolutions and are able to yield spatial resolutions which scale inversely
with the number of correlated entangled photons N Boto  _et al._ (2000);
Steuernagel (2004); Oka (2011, 2018); Schlawin, Dorfman, and Mukamel (2018).

These properties, plus the more well known sub-shot noise and classical light
rejection behaviors, make entangled photons tempting for use in ultrafast
laser spectroscopy Dorfman, Schlawin, and Mukamel (2016); MacLean, Schwarz,
and Resch (2019). The fragility of the entangled states, and the difficulty
and expense associated with single photon counting, necessitates a need for
high flux (nW-$\mu$W) entangled photon sources that would easily interface
with existing ultrafast optics setups Nasr  _et al._ (2005). However, commonly
used nonlinear crystals (BBO, KDP), usually have entangled photon creation
efficiencies in the $10^{-10}$ to $10^{-12}$ range Kwiat  _et al._ (1995);
Friberg, Hong, and Mandel (1985); Kwiat  _et al._ (1999). Reliance on
’birefringent phase matching’ limits the highest usable nonlinear coefficients
of the crystal as well as the bandwidth of entangled photons that can be
created Karan  _et al._ (2020); Nikogosyan (1991); Midwinter and Warner
(1965). The possible phase-matching conditions result in bandwidths in the
tens of nanometers range, which corresponds to 100s of femtoseconds to
picoseconds in correlation times in experiments. Given the current assumption
that the entangled photon correlation time must be less than a molecule or
material’s dephasing time to allow a nonlinear or multiphoton enhancement, the
narrow bandwidth of birefringent phase matching presents critical limits on
entangled spectroscopy’s feasibility Dorfman, Schlawin, and Mukamel (2016);
Roslyak and Mukamel (2009).

In contrast, the ferroelectric properties of materials such as KTP and lithium
niobate allow the crystals to be periodically poled and utilize a ’quasi
phase-matching’ (QPM) technique Wang  _et al._ (1999); Yu  _et al._ (2002);
Lin  _et al._ (2019). Quasi-phase-matching allows the use of Type-0 collinear
phase matching Chen  _et al._ (2009), SPDC efficiencies of $10^{-6}$ to
$10^{-10}$ Bock  _et al._ (2016); Shi and Tomita (2004), a broadly tunable
center wavelength (set by the grating period instead of crystal angle)
Armstrong  _et al._ (1962), and easy implementation of waveguides that can
saturate single photon detectors with $\sim$mW of CW diode laser input power
Tanzilli  _et al._ (2001). Broad bandwidth phase matching is achieved by
introducing a chirp into the poling period of the structure. The nonlinear
poling allows for more wavelengths to be phase matched, while also retaining
the benefits associated with longer crystals, namely a larger downconverted
photon flux. The increased bandwidth directly affects the effective temporal
resolution of experiments Nasr  _et al._ (2008); Pe’er  _et al._ (2005); Ou
and Lu (1999).The broad bandwidth also allows increased power densities, as
the the single photon per wavelength mode occupation limit necessary for non-
classical effects is not exceeded (calculated as the flux in photons/s divided
by the bandwidth in Hz) Hadfield (2009). While KTP and lithium niobate have
been the primary workhorse for entangled photon generation in quantum
information system settings, they have limitations with regards to their
damage thresholds and UV cut-offs. In contrast, lithium tantalate is more
ideal for physical chemistry and materials research because it has higher
power handling capabilities (240 MW/cm2) Zverev  _et al._ (1972) and a lower
UV absorption edge ($\approx$280 nm) Antonov  _et al._ (1975); Meyn and Fejer
(1997).

To accelerate the adoption of entangled photon techniques, we outline how to
create an entangled photon source that can achieve near $\mu$W powers of
entangled photons down to UV wavelengths with temporal resolutions of tens of
femtoseconds. These entangled photon fluxes are high enough to be visible by
eye when starting from a 1W CW pump laser, greatly facilitating the alignment
of entangled experiments. Here, the generated photon pairs are centered around
a degenerate wavelength of 812 nm, resulting from a 406 nm pump source, but
this wavelength can be tuned as outlined by the formulas in the paper. The 8%
MgO doped congruent lithium tantalate (CLT) gratings use a Type-0, collinear
quasi-phase matching configuration. A Michelson type interferometric scheme
Lopez-Mago and Novotny (2012) measures the fourth-order interference of the
entangled photons and gives coherence times of 245 fs for a bandwidth of 10 nm
and 62 fs for a bandwidth of 125 nm. Of course, simply producing a broad
bandwidth of entangled photons is not the same as carrying out experiments
using them. We discuss our attempts to utilize the full SPDC cone and
bandwidth in subsequent experiments, emphasizing where optics development is
still needed to achieve maximum temporal resolution and flux. Using the full
cone is in contrast to most experiments where irises are used to select two
points from the SPDC cone, thereby severely reducing the total photon flux at
the gain of state purity. The paper aims to give quick access to the sources
needed for the emerging field of nonlinear entangled photon optics as well as
outlining the next steps needed in the field.

##  II Theory

###  II.1 Designing Broadband SPDC in chirped nonlinear materials

Prior to quantifying the experimental behavior of the SPDC sources, we lay out
the theoretical and numerical details used in their design.

In SPDC, the creation of two entangled photons arises from the quantum
mechanical process of a single pump photon mixing with the underlying vacuum
state. The two down-converted daughter photons display strong correlations in
time, energy, and momentum due to the parametric mixing process Burnham and
Weinberg (1970); Klyshko, Penin, and Polkovnikov (1970). The time and energy
correlations originate from the individual photons being generated
simultaneously from the pump photon:

$E_{p}\left(\omega_{p}\right)=E_{s}\left(\omega_{s}\right)+E_{i}\left(\omega_{i}\right)$
(1) $\therefore\;\omega_{i}=\omega_{p}-\omega_{s}$ (2)

where $\omega_{p}$, $\omega_{s}$, and $\omega_{i}$ are the frequencies of the
pump, signal, and idler photon respectively. The momentum correlation is
dictated by the phase matching condition, which is a consequence of the wave-
like nature of the interacting fields. Each field has a corresponding wave
vector defined via its respective phase velocity
$\textit{v}_{p}=c/n\left(\omega,T\right)$ in the medium:

$k\left(\omega,T\right)=\frac{\omega n\left(\omega,T\right)}{c}$ (3)

with $n\left(\omega,T\right)$ being the frequency and temperature dependent
refractive index of the material. The following vector sum, the longitudinal
component of which is depicted in Fig.1, should be satisfied in order to
ensure that a proper phase relationship between the interacting waves is
maintained throughout the nonlinear crystal.

$\displaystyle\Delta\textbf{{k}}\left(\omega_{p},\omega_{s},\omega_{i}\right)=\textbf{{k}}_{p}\left(\omega_{p},n_{p}\left(\omega_{p},T\right)\right)-\textbf{{k}}_{s}\left(\omega_{s},n_{s}\left(\omega_{s},T\right)\right)$
(4)
$\displaystyle-\textbf{{k}}_{i}\left(\omega_{i},n_{i}\left(\omega_{i},T\right)\right)$

It is only when this relationship holds true $\left(\Delta k=0\right)$ that
constructive interference between the propagating waves can occur and energy
is transferred dominantly in one direction, rather than oscillating back and
forth between the two optical modes Boyd and Prato (2008).

Figure 1: Phase-matching diagram for a two-photon downconversion process,
showing the longitudinal phase-mismatch component $\Delta k_{z}$. The
entangled photons (signal and idler) are each shown to have their distinct
wave vectors k and emission angles $\theta$. The phase-mismatch component must
be compensated to create entangled photons by spontaneous parametric down
conversion (SPDC).

A natural extension of this phase-matching principle, termed quasi-phase-
matching, is apparent if one considers the duality between position and
momentum space. A periodic lattice in position space directly corresponds to a
well-defined wave vector k in momentum space via a Fourier transform.
Therefore, if the nonlinear dielectric tensor domains of ferroelectric
nonlinear crystal are modulated with a specific periodicity $\Lambda$, a
quasi-momentum term $\Gamma\left(\Lambda\right)$ arises, which can be used to
satisfy a non-phase-matched nonlinear interaction.

$\Delta\tilde{\textbf{{k}}}\left(\omega_{p},\omega_{s},\omega_{i}\right)=\Delta\textbf{{k}}\left(\omega_{p},\omega_{s},\omega_{i}\right)-\Gamma\left(\Lambda\right)$
(5)

The quasi-phase-matching term $\Gamma\left(\Lambda\right)$ also has an
additional advantage, in that it can be chosen to be higher-order. Here, the
order of the configuration is defined by m.

$\Gamma\left(\Lambda\right)=\frac{2\pi m}{\Lambda},\;m\in
2\mathbb{Z}+1\left(\neg 0\right)$ (6)

This serves a particular experimental benefit as it allows for nonlinear
interactions to be phase-matched (albeit with lower efficiency), even if the
required $1^{st}$ order periodicity is too small to be fabricated Fejer  _et
al._ (1992).

We now describe the design of a grating in congruent lithium tantalate with
the design goal of a 406 nm pump wavelength that creates a >400 nm bandwidth
around the degenerate 812 nm SPDC point. The source must also be capable of
creating near $\mu$W powers given a 1 W input power. The first step is to
obtain the Sellmeier equations that describe the frequency and temperature
dependent refractive indices of the material Moutzouris  _et al._ (2011). The
numeric values and a representative plot are found in the supplementary
information. Using the Sellmeier equations, the phase mismatch (Eq.5) is then
evaluated for a wide range of pump-signal/idler wavelength configurations
while keeping one entangled photon frequency a constant. External conditions
such as the desired temperature range and minimum achievable poling
periodicity are also enforced such that the parameter space remains within
experimentally realistic boundaries. The root of Eq.5 can then be found by a
root-finding algorithm. For the congruent lithium tantalate and a 406-812 SPDC
process, and imposing a realistic < 200°C temperature condition, solving for
the root of Eq.5 yields a poling period of 9.5 $\mu$m at 133°C (see figure in
SI for the graphical solution).

Quasi-phase-matching is a strongly temperature dependent technique. By solving
Eq.5 for a range of temperatures near the calculated degeneracy point (133°C),
the splitting and tuning of the SPDC spectrum can be calculated. Tuning the
SPDC spectrum via the crystal temperature is useful for controlling what
intermediate states are involved in, the magnitude of, and the temporal
resolution of an entangled two photon process. The temperature dependent
behavior is shown in Fig.2. Here, the transition between the degenerate and
non-degenerate domains is clearly observable, with lower temperatures
corresponding to a weaker degenerate emission profile. The lower plot of Fig.2
also clearly shows a very strong narrowing of the signal and idler bandwidths
as one temperature tunes further away from degeneracy.

Figure 2: SPDC emission spectrum for ppCLT at various phase-matching
temperatures showing transition from degenerate to non-degenerate
downconversion. Bottom plot showing spectra at three temperatures
corresponding to cross sections of the 2D plot. The phase matching temperature
results in the broadest bandwidth.

Additionally, both signal and idler have an inherent angular emission property
determined by the momentum conservation condition in Eq.5. The strong
frequency dependence of the outgoing angles can be numerically calculated from
the momentum conservation equation if one decomposes the wave vectors into
their longitudinal and transverse components, and solves the trigonometric
equations for transverse phase-matching. Subsequently computing the emission
angle for each frequency of interest, in conjunction with the frequency
dependent refractive index of the nonlinear crystal, the SPDC cone’s spectral
characteristic for various crystal temperatures can be plotted (Fig.3). As
predicted, at the degenerate emission temperature of 133°C, the plot shows
that the emission angle of the 812 nm signal and idler photons is 0°. More
importantly however, comparing the results to those in Fig.2, the shaded non-
degenerate region also corresponds to a collinear emission profile. Vice
versa, temperatures below 133°C display a (albeit weak) degenerate, non-
collinear character.

Figure 3: Broadband signal/idler emission angles in ppCLT with 9.5 $\mu$m
poling period, showing how the SPDC source does not act as an ideal dipole
emitter when broadband entangled photon pairs are desired. Points where curves
reach a value of zero correspond to collinear, non-degenerate emission. Note
that at certain temperatures, there need not be any SPDC emission in certain
directions, governed by the spectral emission characteristics in Fig.2. Arrow
indicating direction of increasing temperature.

In a chirped crystal, the bandwidth of the down-converted entangled photons is
increased by changing the poling period throughout the crystal. The increasing
width elements correspond to different phase-matched wavelength configurations
of Eq.5 (Fig.4). The photons still add up to the narrow linewidth of the pump
laser but the temporal resolution is given by the temporal alignment and down-
converted bandwidth of the entangled photons. This is why entangled photon
experiments are said to have independent spectral and temporal resolutions.
The importance of the pump laser’s linewidth in determining the entangled
photon state’s purity, and thus the access to non-classical spectroscopic
properties, is discussed in the SI.

Figure 4: Depiction of a chirped periodically poled crystal with total length
$L_{T}$. Each crystal segment of length $L_{n}$ is associated with a sinc-
shaped downconversion amplitude around a different center wavelength.
Equivalently to a normal periodically poled crystal, the domain orientation is
reversed in adjacent elements.

Following a similar treatment to Di Giuseppe  _et al._ (2002), the crystal can
be decomposed into its individual domains, in each of which the downconversion
amplitude follows the usual sinc function-like phase-matching dependence
Burlakov  _et al._ (1997). This is attributed to the Fourier transform of a
rectangular function in real-space representing the probability of a
downconversion event occurring inside the crystal element.

$A_{n}\left(\omega_{s},\omega_{i}\right)=L_{n}sinc\left(\frac{L_{n}\Delta
k}{2}\right)$ (7)

With $L_{n}$ being the length of the $n^{th}$ crystal element. Similarly, a
cumulative-phase term can be defined, which is related to a forward
propagating wave with phase-mismatch $\Delta k$ traversing a crystal length
$L_{prop}$ after it has been generated in the SPDC process at crystal element
n.

$\varphi=\Delta kL_{prop}=\Delta
k\left(\frac{L_{n}}{2}+\sum_{m=n+1}^{N}L_{m}\right)$ (8)

Carrying out a sum over all individual crystal elements then allows for the
resulting SPDC spectrum to be calculated (Eq.9) and thereby observe the
effects of different chirping parameters. For lithium tantalate, this is shown
in Fig.5. In the case of CLT, a $\pm$10% chirp around the degenerate poling
periodicity calculated above yields a full octave spanning SPDC bandwidth.
Trivially, in the limit of no chirp, where all crystal elements are equal in
size, the solution for a periodically poled crystal of length L is recovered.

$\displaystyle
A\left(\omega_{s},\omega_{i}\right)=\chi_{0}\sum_{n=1}^{N}\left(-1\right)^{n}L_{n}sinc\left(\frac{L_{n}\Delta
k}{2}\right)$ (9) $\displaystyle\times e^{-i\Delta
k\left(\frac{L_{n}}{2}+\sum_{m=n+1}^{N}L_{m}\right)}$

Figure 5: a) SPDC emission bandwidth broadening as a function of chirp
parameter, defined as total percentage deviation from the degenerate poling
period. b) Numerically calculated FWHM bandwidth trend as a function of poling
period chirp.

By plotting the solution of Eq.9 as a function of the position inside the
crystal along the propagation direction, we can gain some intuition for how
the constructive interference results in a broadened spectrum. In the case of
CLT, a $\pm$10% chirp around the degenerate poling periodicity calculated
above yields a full octave spanning SPDC bandwidth as shown in Fig.6. Here we
used a total grating length of 18 mm equal to the chip used in subsequent
experiments.

Figure 6: a) Build up of the collective SPDC spectrum as a function of the
position inside a chirped ppCLT grating in the direction of propagation. b)
Line cuts across the 2D spectrum plot in a) showing the normalized spectrum
(in frequency space) at the positions designated as L1, L2, L3, and L4
respectively.

The 2D plot in Fig.6 provides some additional insight into how modulation of
the poling period affects the build-up of the emitted spectrum. In particular,
it’s important to note that while the first 8mm length of the crystal appears
to show no downconversion taking place, this is indeed not the case. While all
crystal elements in this region satisfy a particular phase-matching condition,
it is only after a certain propagation length that the phases of each spectral
mode coherently sum to a strong enough amplitude that would be perceivable on
the plot. Furthermore, the broadening of the downconverted bandwidth does have
a particular drawback, in that the frequencies close to degeneracy become
weaker in intensity. This is best observed in the line cuts for positions L3
and L4, where instead of a constant intensity across all frequencies, the
emission spectrum peaks at the lower and higher cut-offs but begins to droop
more severely as the bandwidth increases. Thus, increasing the chirping of the
poling period indefinitely does not necessarily yield a flat power spectral
density. It should be pointed out that this issue can be addressed via chirp
profiles which follow not a linear but some higher order functional form
Brańczyk  _et al._ (2011).

In contrast to narrowband SPDC where the emission angles for the photon pairs
are well confined within the two narrow arcs satisfying momentum conservation,
such extremely broadband downconversion poses inherent difficulties in the
spatial degree of freedom due angular (chromatic) dispersion of the resulting
beam. As Fig.3 shows, the emission angles increase quickly away from
degeneracy. This presents several experimental issues, as the beam no longer
represents an ideal point source, which will be discussed in the experimental
section below. Taking into account the physical dimensions of the grating
cross section (detailed in the experimental section below), it is easy to see
that the output aperture of the grating can at most only accommodate
$arctan(0.15mm/9mm)\approx 1\degree$ of emission in the horizontal axis, and
$arctan(0.5mm/9mm)\approx 3.2\degree$ along the vertical axis relative to the
center point of the crystal where we focus. Beyond this, large parts of the
SPDC emission cone will begin to interact with the domain boundaries which
form the grating channel. While in principle the refractive index is the same
for both un-poled and poled crystal regions, the periodic poling process
causes some stress at the domain walls. Therefore, the edge of the each
pattern in the lateral direction acts as a kind of scattering center, possibly
resulting in unwanted interference patterns. Therefore special attention
should be paid to accommodate the spatial distribution of the cone as it
diverges inside the channel.

A final point to consider is the total length of the grating.is to be chosen
by considering two interlinked criteria. Firstly, the expected power spectral
density of the entangled photons at the output scales as $L^{2}$. Emission
intensity can therefore be significantly increased by utilizing longer
structures. Conversely, the expected power spectral density also depends on a
$sinc\left(\frac{L_{n}\Delta k}{2}\right)$ multiplicative factor. Hence,
longer gratings result in narrower emission bandwidths. This ‘dual’-argument
also applies to poling patterns which have a chirp, as while increasing the
amount of variation in the poling periodicity serves to broaden the emission
bandwidth, each spectral component will be less populated with photons and
thus less brilliant.

##  III Experiment

###  III.1 SPDC Characterization

The periodically poled congruent lithium tantalate gratings were manufactured
by HC Photonics. The unchirped chip consists of 8 gratings with different
poling periods (8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12 $\mu$m). It is 20 mm long
and each channel has a 0.9 mm by 0.5 mm cross section. The chirped chip
consists of 6 gratings with varying chirp parameters and entrance poling
periods (Table 1). Each grating is 18 mm long and has a 1 mm by 0.3 mm cross
section. The gratings are AR-coated on both input and output faces at
406nm(R<0.5%)/812nm(R<0.5%).

Grating |  $\Lambda_{1}$[$\mu$m] |  $\Lambda_{c}$[$\mu$m] |  $\Lambda_{2}$[$\mu$m] |  Chirp
---|---|---|---|---
1 |  9.000 |  9.45 |  9.900 |  10%
2 |  9.048 |  9.50 |  9.952 |  10%
3 |  9.095 |  9.55 |  10.005 |  10%
4 |  9.143 |  9.60 |  10.057 |  10%
5 |  9.190 |  9.65 |  10.110 |  10%
6 |  9.238 |  9.70 |  10.162 |  10%
Table 1: Chip parameters

The experimental setup, as illustrated in Fig.8, consisted of a computer-
tunable, CW Ti:Sapphire laser (M2 SolsTiS) and an external SHG cavity to act
as the pump source. The maximal output power of the SHG at 406 nm was 1.15 W
with a 0.89 nm linewidth. The pump beam is focused through the ppCLT grating
using an aspheric lens with a 40 cm focal length. This focal length is chosen
such that the size of the focused beam was smaller than the grating’s cross
sectional dimension and further optimized for peak SPDC brightness. The
position of the crystal can be finely adjusted in the XYZ directions using a
piezoelectric stage, which is critical for maximizing the entangled photon
flux. The angular deviation of the grating’s long axis relative to the input
beam was not as crucial as the X-Y centering of the chip around the focused
beam and the z-control over exact focal point positioning. The temperature of
the crystal is controlled by a ceramic heater and PID loop (Covesion) to an
accuracy of 10 mK.

To measure the SPDC spectrum as well as the power of the downconverted flux,
the output of the grating is passed through a total of three OD-4 500 nm
longpass filters (Edmund Optics) to eliminate the residual 406 nm pump. The
SPDC cone is collimated with an off-axis parabolic mirror and then focused
into a USB spectrometer (Thorlabs). For more detailed characterization, the
SPDC cone is collimated with a telephoto lens and then focused into a
spectrometer (Princeton Instruments IsoPlane) with a 15cm focal length lens.
In this configuration, spectral and power measurements are performed with an
electron multiplying intensified CCD camera (emICCD, Princeton Instruments
MAX4). For the broadband spectral measurements, spectra are collected and
stitched together as a 800 nm blazed grating with 150 grooves/mm is scanned
across several center wavelengths. These spectra are then background
corrected, quantum efficiency corrected according to the detection efficiency
of the emICCD camera, and normalized.

Fig.7 (a) depicts the SPDC spectra for the 9.50 µm unchirped grating as the
temperature of the crystal is varied from 60°C to 160°C. Above 150°C, the
shift from degenerate to non-degenerate emission conditions is apparent from
the decrease in intensity around the degenerate wavelength region of 812 nm
(gray dashed line). In comparison, as shown in Fig.7 (b), at 145 °C, the
chirped chip (blue curve) exhibits much broader and flatter emission than the
unchirped counterpart (red curve).

Figure 7: a) SPDC spectra for a 9.50 µm unchirped grating, gray reference line
marks the degenerate wavelength of 812 nm. Arrow a guide in the direction of
increasing temperature. b) SPDC spectra for a chirped grating with a 9.50 µm
center poling period and 10% chirp rate. Both spectra taken at a temperature
of 145°C. The chirped emission is broader and has a flatter top than the
unchirped emission. Note the detector cut-off at 900 nm.

###  III.2 Two-Photon Interference

Figure 8: Entangled photon pair Michelson interferometer used to measure the
fourth order interference. SHG: second harmonic generation unit, L1: focusing
lens, ppLT: periodically poled lithium tantalate chip, LP: longpass filter,
OAP: off-axis parabolic mirror, C1: cylindrical lens, BS1 and BS2: 50:50
beamsplitters, M1 and M2: mirrors, QWP: quarter-wave plate, D1 and D2:
multimode fiber coupled single photon avalanche diodes connected to
coincidence counting unit. The flip mirrors allow re-configuring such that the
Ti:Sapph output is picked off and used for aid in alignment of downstream
optics. A 406 nm diode laser is used as an alternate pump source. (Inset) Chip
diagram showing the grating layout and their corresponding entrance
($\Lambda_{1}$), center ($\Lambda_{c}$), and end ($\Lambda_{2}$) poling
periodicity locations in reference to Table.1.

The fourth-order interference between entangled photon pairs was quantified
with a two-photon Michelson interferometer, depicted in Fig.8. The Michelson
interferometer is chosen because the configuration allows a broader bandwidth
of the SPDC cone to be easily utilized, in contrast to a Mach-Zehnder
configuration. Following the end of the ppLT chip, the entangled photons are
collimated with an off-axis parabolic mirror and a cylindrical lens. These
photons are directed through a broadband nonpolarizing 50:50 beamsplitter
(BS1, Layertec) which separates the incident photons into two paths. In the
reflected path, the photons pass through an achromatic quarter-waveplate
(Thorlabs) twice to rotate their polarizations. The optical path length
difference between the two arms is adjusted by scanning a mirror mounted to a
linear stage in the transmitted arm. Coincidence counts vs. optical path
length difference are then collected at one of the output arms of BS1. The
coincidence counting setup consists of a second broadband 50:50 beamsplitter
(BS2, Layertec), and two free-space-to-fiber coupling setups connecting to
single-photon avalanche diodes (SPADs, Laser Component COUNT) and time-tagging
electronics (PicoHarp 300). Because of the high SPDC flux, the pump beam must
first be dimmed by six orders of magnitude to avoid SPAD saturation (counting
rate $10^{6}$ photons/s). Full counting rate experiments can be completed
using the CCD alone Reichert, Defienne, and Fleischer (2018); Zhang  _et al._
(2020) but were not implemented at the time of writing this paper.

The free-space-to-fiber coupling setups each consist of 2 mirrors to align the
beamsplitter outputs through a 4.51 mm focal length asphere, which then
focuses the beam into a multimode fiber (105 $\mu m$ core diameter, 0.22 NA)
connected to the SPADs. For these measurements, the coincidence time-bin
resolution is set to 8 ps and coincidences are summed within a 10 ns window.
Fourth-order interferograms were measured for different bandwidths of
entangled photon pairs. For the first experiment, bandpass filters centered at
810 nm with a 10 nm FWHM are mounted to the front of each of the fiber
coupling units. Note that this filter is not at the 812nm degenerate
wavelength and reduces the degree of entanglement. For the second experiment,
the 810 nm bandpass filters are removed to couple the broadband, collinear
SPDC into the SPADs. Finally, the oven is allowed to cool to the non-collinear
but degenerate temperature of 150°C, and broadband two-photon interference was
measured again. Note that due to the large bandwidth of the SPDC, the fiber
coupling efficiency of the non-degenerate photons is also poor at less than
10$\%$ of total power. As discussed at the end of this section, optics
development is still needed similar to using classical broadband ultrafast
lasers.

Figure 9: Simulated (top row) and measured (bottom row) fourth-order
interference for narrowband collinear (a,d), broadband collinear (b,e), and
broadband non-collinear (c,f) entangled photons. Coincidence counts are
measured with 15 s integration time. The insets in each figure depict the SPDC
spectra used to simulate or measure the corresponding interference in each
detector arm. For simulations, Gaussian functions are used to approximate SPDC
spectral envelopes. As shown in Table 2, the theroretical and measured
coherence times are 200 fs and 245 fs for collinear narrowband, 45 fs and 57.4
fs for collinear broadband, 18 fs and 62 fs for non-collinear broadband. The
growing difference between expected time resolution from the SPDC bandwidth
and the measured interference is due to the difficulty with fiber coupling the
full cone emission profile.

Fig.9 shows the simulated fourth-order interferences using the produced
bandwidth of the SPDC source and then the measured fourth-order interferences
for two collinear bandwidths and a third non-collinear, full spectrum
measurement. When the quarter-wave plate is set to 0°, the photons arriving at
BS1 have the same polarization, and interfere coherently as correlated pairs.
The interference pattern is robust against pump power fluctuations. When the
quarter-wave plate is rotated to 45°, the reflected arm becomes
perpendicularly polarized to the transmitted arm, the interference pattern
disappears, and the coincidence counts follow the average pump power reading
collected before and after the coincidence detection (SI Fig.8-9). The
resulting coherence length of the total flux can be determined from the width
of the interference peak. Table 2 summarizes the theoretical and measured
coherence lengths.

|  Collinear narrowband |  Collinear broadband |  Non-collinear broadband
---|---|---|---
Theoretical coherence time [fs] |  200 |  45 |  18
Measured coherence time [fs] |  245 |  57.4 |  62
Table 2: Coherence times of different configurations and bandwidths of SPDC

For the measurement with 810-(10) nm bandpass filters mounted to the front of
the fiber coupling units, shown in Fig.9(d), the coherence length is
approximately 73 µm (or 245 fs). The simulated interference pattern as well as
the coherence length of 60µm (200 fs), shown in Fig.9(a), agree fairly with
the measurement given that the 10nm bandpass filter is offset from the
degenerate wavelength of 812 nm. Note that the measured bandwidth after the
fiber is roughly double what it actually is because the spectrometer slit had
to be run wide to get sufficient flux for spectral analysis after the fiber
coupling. The theoretical plot uses the correct bandwidth. Fig.9(e) shows the
interference from collinear broadband SPDC flux without bandpass filters. The
bandwidth is reduced and the SPDC spectrum is split (nondegenerate) because
the phase matching temperature is maintained at the elevated collinear
emission (Fig.2). From the width of the center peak, the coherence length is
approximately 17.2 µm (or 57.4 fs). For reference, the simulated coherence
length is approximately 13 µm (45 fs). Similarly, Fig.9(f) shows the lower
temperature non-collinear broadband interference. Here, the full 200 nm
bandwidth of the SPDC source is attempted to be used (expected temporal width
8fs). However the spectrum reaching the detector is limited to 125 nm because
of collimation issues, upstream optics, and fiber coupling. The coherence
length calculated from the measured width of the peak is 18.9 µm (or 62 fs).
This indicates that the degree of entanglement has degraded due to poor fiber
coupling at non-collinear temperatures and other upstream optics. For example,
the spectrum after the multiple beamsplitters is not balanced (Fig.9(f),
inset) as compared to the other cases. The increased temporal width is also
indicative of the need for dispersion management upstream, just as is similar
to a broadband ultrafast laser pulse of an expected 20 fs pulse. The same
would be true for the chirped (>200 nm) SPDC source so an interferometer is
not shown here.

The lack of fiber coupling efficiency in the broadband, non-collinear case is
not simply one of input optics. Immediately following the chip, the spatial
profile of the entangled photons is cone-like with a wavelength-dependent
distribution (Fig.10 left). The measured angular deviation with wavelength
matches the simulated entangled photon emission. Within a non-degenerate pair,
the idler photon has a larger emission angle than the signal photon. The
wavelength-dependent angular emission cone present a challenge in the
implementation of free-space broadband entangled photon setups because the
emission does not act like single point source. We attempted to collimate the
emission cone using a variety of free-space optical configurations and found
that a telephoto lens or an off-axis parabolic mirror (sometimes with an
additional cylindrical lens for beam shaping) provides the best long range
spatial profiles. After a propagation length of 45 cm into the far field, the
entangled photon cone collapses (Fig.10 middle). This spatial profile collapse
is most likely due to the interference of photon pairs created along the
length of the crystal and scattering effects near the domain boundaries of the
grating. The end result shows characteristics of somewhere between a Laguerre
and a Hermite mode which would match the circular emission in a square profile
created by the source. The beam profile will likely be improved by using
adaptive optics. The far field effects could also be reduced by using a
waveguide instead of a grating; however, nanophotonic implementations cannot
handle the high input powers used in this paper and fiber coupling of the
broadband emission spectrum from the waveguide is not well developed. Another
obvious option is, instead of using fiber coupled SPADs for the detection, is
to move to emICCD photon counting techniques that would not require focusing
of the collapsed beam profile. However, this approach still needs exploration
and requires an emICCD even more costly than SPADs. Further investigation into
the creation and collimation of broadband entangled photons is critical for
the implementation of short temporal length, high flux entangled photon
spectroscopy.

Figure 10: Images of collimated SPDC emission cone at 145 °C in the near field
(left), far field (middle) after collimation with an off axis parabolic
mirror, and far field after collimation and focusing by a 15.45cm focal length
lens (right). The left and middle images are collected with a standard cell-
phone camera, the right image is collected with an emICCD.

##  IV Conclusion

Quasi-phase-matching in periodically poled crystals offers significant
benefits in the experimental implementation of broadband entangled photon
sources for use in spectroscopy. Simulated parameters for unchirped,
periodically poled lithium tantalate gratings were utilized to design and
construct a broadband entangled photon source spanning nearly an octave in
frequency. This broadband source was characterized by measuring the output
spectra of the grating and the fourth order interference by a two-photon
Michelson interferometer. The two-photon Michelson interferometer was used to
emphasize where further optics and detector development is still need to
utilize the broadband and short temporal width sources. As in, the SPDC
spectrum can be reliably created from the designed chip, however, the down-
stream optics and detectors still need development to fully utilize this
source. Given the theoretical and experimentally demonstrated advantages of
entangled photons in studies of chemical, biological, and material systems, we
anticipate the widespread applicability of this particular energy-time
entangled photon source.

## Acknowledgment

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, under Award Number DE-
SC0020151.

## Data Availability

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

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